FractionalBirkhoffianMechanicsBasedonQuasi-Fractional

Research ArticleFractional Birkhoffian Mechanics Based on Quasi-FractionalDynamics Models and Its Noether Symmetry

Yun-Die Jia1 and Yi Zhang 2

1College of Mathematical Sciences Suzhou University of Science and Technology Suzhou 215009 China2College of Civil Engineering Suzhou University of Science and Technology Suzhou 215011 China

Correspondence should be addressed to Yi Zhang weidiezhgmailcom

Received 14 December 2020 Revised 18 February 2021 Accepted 23 February 2021 Published 27 April 2021

Academic Editor Gilberto Espinosa-Paredes

Copyright copy 2021 Yun-Die Jia and Yi Zhang -is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

-is paper focuses on the exploration of fractional Birkhoffian mechanics and its fractional Noether theorems under quasi-fractional dynamics models -e quasi-fractional dynamics models under study are nonconservative dynamics models proposedby El-Nabulsi including three cases extended by RiemannndashLiouville fractional integral (abbreviated as ERLFI) extended byexponential fractional integral (abbreviated as EEFI) and extended by periodic fractional integral (abbreviated as EPFI) First thefractional PfaffndashBirkhoff principles based on quasi-fractional dynamics models are proposed in which the Pfaff action containsthe fractional-order derivative terms and the corresponding fractional Birkhoffrsquos equations are obtained Second the Noethersymmetries and conservation laws of the systems are studied Finally three concrete examples are given to demonstrate thevalidity of the results

1 Introduction

Symmetry theory plays an important role in mathematicsphysics and mechanics and the study of symmetry prop-erties of dynamic systems has become a very effectivemethod to solve some practical problems -e most im-portant and common symmetries are mainly of two kindsnamely Noether symmetry and Lie symmetry Noetherrsquossymmetry theory originated in 1918 and was first put for-ward by the famous mathematician Emmy Noether [1] Inthis method the relationship between symmetry and con-served quantity was established by using the invariance ofHamilton action under the infinitesimal group transfor-mation of time and generalized coordinates Candotti [2]and Desloge [3] applied Noetherrsquos theorem to classicalmechanics Djukic [4] established Noetherrsquos theorem fornonconservative systems Liu [5] generalized Noetherrsquostheorem to nonholonomic mechanical systems In 1979Lutzky [6] applied the Lie method [7] of invariance ofdifferential equations under infinitesimal group transfor-mations to differential equations of motion for dynamical

systems and started the study of Lie symmetry and con-served quantity of mechanical systems Ibragimov [8] andBluman [9] elaborated the role of Lie algebra and Lie groupin studying the invariance of differential equations Zhao[10] extended Lie symmetry theory to nonconservativemechanical systems Mei [11 12] systematically studiedNoether symmetry Lie symmetry of constrainedmechanicalsystems and corresponding conserved quantities Recentlysome new progress has been made in the study of these twosymmetries (cf [13ndash24] and references therein)

Fractional calculus is an important mathematical tool inscience and engineering [25ndash28] In recent decades theresearch of fractional calculus has developed greatly and itsapplication fields have expanded to automatic controlquantum mechanics and mechanical systems [29ndash35]Riewe [36 37] introduced the fractional variational problemfor the first time in the study of nonconservative mechanicsIn 2005 El-Nabulsi established a dynamical model ofnonconservative systems under the framework of fractionalcalculus [38] based on the definition of RiemannndashLiouvillefractional integral (ERLFI) El-Nabulsi expanded the idea of

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 6694709 17 pageshttpsdoiorg10115520216694709

dynamics modeling and successively put forward the dy-namical models of nonconservative systems which areextended by exponentially fractional integral (EEFI) andextended by periodic laws fractional integral (EPFI) [39 40]respectively -e equations obtained from quasi-fractionaldynamics models are similar to dynamical equations ofclassical conservative systems which contain the gener-alized fractional external forces corresponding to dissi-pative forces but the term with the fractional derivativedoes not show up Different from other models the frac-tional time integration of quasi-fractional dynamics modelsonly needs one parameter In this way it simplifies thecalculation of complex fractional calculus and provides amodeling method for nonconservative systems -ereforethe quasi-fractional dynamics models can be used to studycomplex dynamical systems more conveniently Fredericoand Torres [41] first presented fractional Noetherrsquos theo-rems Since then studies on fractional Noether symmetryand conservation laws have been extensively developed[42ndash49] In addition Torres and Frederico studiedNoetherrsquos theorems of fractional action-like variationproblems [50 51] In recent years nonconservative dy-namical systems based on quasi-fractional dynamicalmodels have been studied deeply and the correspondingdynamical equations and Noether conservation laws havebeen obtained [52ndash55] However most of the previousstudies on the variational problems of quasi-fractionaldynamics models are confined to Lagrangian frameworkand Hamiltonian framework

It is well known that Birkhoffian mechanics is a newstage in the development of Hamiltonian mechanics[56ndash58] Under canonical transformation Hamilton ca-nonical equation remains unchanged but under generalnoncanonical transformation it becomes Birkhoffrsquos equa-tion Santilli [57] and Mei [58] both pointed out that Bir-khoffian mechanics is the most general possible mechanicswhich can be applied to hadron physics space mechanicsstatistical mechanics biophysics engineering and otherfields Zhang and Zhai [59] has proposed the fractionalPfaffndashBirkhoff principle and fractional Birkhoffrsquos equationsand proved that the fractional Hamilton principle is thespecial case of the fractional PfaffndashBirkhoff principle and thefractional Hamilton equations and the fractional Lagrangeequations are the special cases of the fractional Birkhoffrsquosequations Zhang and Zhou [52] proposed the quasi-frac-tional Pfaff-Birkhoff principle and derived correspondingquasi-fractional Birkhoffrsquos equations which is based on thequasi-fractional model given by [38] Up to now someresults have been obtained on Noether symmetry of frac-tional or quasi-fractional Birkhoffian systems such as[52 59ndash66] However the results of these quasi-fractionalBirkhoffian systems are limited to the Pfaff action containingonly integral-order derivative terms Here we will furtherextend fractional Birkhoffian mechanics on the basis of threequasi-fractional dynamical models given in [38ndash40] wherefor the Pfaff actions we consider contain fractional-orderderivative terms -e quasi-fractional Lagrangian systemand quasi-fractional Hamiltonian system are special cases ofthe results presented in this paper

-e text is organized as follows In Section 2 the frac-tional PfaffndashBirkhoff principles under quasi-fractional dy-namics models are presented and Birkhoffrsquos equations aregiven and nonisochronous variational formulae of the Pfaffaction are driven In Section 3 fractional Noether sym-metries are well defined and their criteria are established InSection 4 fractional Noether theorems are proved For il-lustrating the application of the methods and results in thistext three examples are given in Section 5 In Section 6 wecome to the conclusions

2 Fractional Birkhoffrsquos Equations andVariation of Fractional Pfaff Action underQuasi-Fractional Dynamics Models

For an introduction to fractional calculus and its basictheory please refer to the monographs [27 28]

21 Fractional Birkhoffian System Based on ERLFI Weconsider a fractional Birkhoffian system determined byBirkhoffrsquos variables aμ(μ 1 2 2n) whose Birkhoffrsquosfunctions are Rμ Rμ(τ a]) the Birkhoffian is B B(τ a])β is the order of fractional derivative and 0le βlt 1

Under the model of ERLFI we define the Pfaff action as

SR 1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113960 1113961(t minus τ)

αminus 1dτ

(1)

where aDβτaμ (μ 1 2 2n) is the fractional derivative

term-e variational principle

δSR 0 (2)

with commutative relation

δaDβτa

μ aD

βτδa

μ (3)

and boundary conditions

aμ|τa a

μ1

aμ|τb a

μ2

(4)

is called the fractional PfaffndashBirkhoff principle based onERLFI

According to principle (2) we drive

zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)

αminus 11113960 1113961 0

(μ 1 2 2n)

(5)

Equation (5) is the fractional Birkhoffrsquos equations basedon ERLFI

If β⟶ 1 equation (5) becomes Birkhoffrsquos equationsbased on ERLFI If β⟶ 1 and α⟶ 1 equation (5) be-comes classical Birkhoffrsquos equations [58]

Take the infinitesimal transformations

2 Mathematical Problems in Engineering

τ τ + Δτ

aμ(τ) a

μ(τ) + Δaμ

(μ 1 2 2n)(6)

and their first-order extensions

τ τ + εσξσ0 τ a

]( 1113857

aμ(τ) a

μ(τ) + εσξ

σμ τ a

]( 1113857 (μ 1 2 2n)

(7)

where εσ is the infinitesimal parameter and ξσ0 and ξσμ are thegenerating functions

Under transformation (6) the Pfaff action (1) is trans-formed into

SR(c) 1Γ(α)

1113946b

aRμ τ a

]( 1113857 aD

βτa

μminus B τ a

]( 11138571113876 1113877(t minus τ)

αminus 1dτ

(8)

And we have

SR(c) minus SR(c)

1Γ(α)

1113946b

aRμ τ a

]( 1113857 aD

βτa

μminus B τ a

]( 11138571113876 1113877

(t minus τ)αminus 1dτ

minus1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113960 1113961


1Γ(α)

1113946b

aRμ τ + Δτ a

]+ Δa]

( 11138571113960 11139611113966 1113967

aDβτa

μ+ aD

βτΔa

μminus aD

βτ _a

μΔτ( 1113857 + ΔτaDβτ _a

μ1113872 1113873

minus B τ + Δτ a]

+ Δa]( 11138571113859

(t minus (τ + Δτ))αminus 1 1 +

ddτΔτ1113888 1113889

minus Rμ τ a]

( 1113857aDβτa

μminus B τ a

]( 11138571113960 1113961(t minus τ)

αminus 11113967dτ

(9)

Let ΔSR be nonisochronous variation of SR which isthe main line part of SR(c) minus SR(c) relative to ε and weobtain

ΔSR 1Γ(α)

1113946b

a

zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889(t minus τ)

αminus 1Δa]1113896 1113897

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889(t minus τ)

αminus 1Δτ

+ RμaDβτa

μminus B1113872 1113873(t minus τ)

αminus 1 ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a


μ1113872 1113873(t minus τ)

αminus 1

minus RμaDβτa

μminus B1113872 1113873(α minus 1)(t minus τ)

αminus 2Δτ1113967dτ

(10)

Since

δaμ

Δaμminus _a

μΔτ

Δ _aμ

ddτΔaμ

( 1113857 minus _aμ ddτ

(Δτ)

(11)

then we obtain

ΔSR 1Γ(α)

1113946b

a

ddτ

RμaDβτa


αminus 1Δτ11139601113896

+ 1113946τ

aRμaD

βτδa

μ(t minus s)

αminus 1minus δa

μsD

βb Rμ(t minus s)

αminus 11113872 11138731113872 1113873ds1113877

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)

αminus 11113872 11138731113890 1113891δa

μ1113897dτ

(12)

By using formula (7) we obtain

ΔSR 1Γ(α)

1113946b

aεσ

ddτ

RμaDβτa


αminus 1ξσ011139601113896

+ 1113946τ

a

RμaDβsξσμ minus _a

μξσ01113872 1113873(t minus s)αminus 1

minus ξσμ minus _aμξσ01113872 1113873sD

βb Rμ(t minus s)

αminus 11113872 11138731113874 1113875ds1113877

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)

αminus 11113872 11138731113890 1113891 ξσμ minus _a

μξσ01113872 1113873dτ

(13)

Mathematical Problems in Engineering 3

Equations (10) and (13) are two mutually equivalentformulas derived from Pfaff action (1)

22 Fractional Birkhoffian System Based on EEFI Under themodel of EEFI we define the Pfaff action as

SE 1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113960 1113961(cosh t minus cosh τ)

αminus 1dτ

(14)

-e fractional PfaffndashBirkhoff principle is

δSE 0 (15)

under commutative relation

δaDβτa

μ aD

βτδa

μ (16)


aμ|τa a

μ1

aμ|τb a

μ2

(17)

-e fractional Birkhoffrsquos equations are

zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(cosh t minus cosh τ)

αminus 1

+ τDβb Rμ(cosh t minus cosh τ)

αminus 11113960 1113961 0 (μ 1 2 2n)

(18)

If β⟶ 1 equation (18) becomes Birkhoffrsquos equationsbased on EEFI If β⟶ 1 and α⟶ 1 equation (18) be-comes classical Birkhoffrsquos equations [58]

According to formula (6) action (14) is transformed into

SE(c) 1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a


αminus 1dτ

(19)

and we have

SE(c) minus SE(c)

1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113876 1113877

(cosht minus coshτ)αminus 1dτ

minus1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113960 1113961


1Γ(α)

1113946b

aRμ τ + Δτ a

]+ Δa]

( 11138571113960 11139611113966 1113967

aDβτa

μ+ aD

βτΔa

μminus aD

βτ _a


μ1113872 1113873


+ Δa]( 11138571113859

(cosht minus cosh(τ + Δτ))αminus 1 1 +

ddτΔτ1113888 1113889

minus Rμ τ a]

( 1113857aDβτa

μminus B τ a

]( 11138571113960 1113961(cosht minus coshτ)

αminus 11113967dτ

(20)

So the nonisochronous variation ΔSE of action SE is

ΔSE 1Γα

middot 1113946b

a

zRμ

za] aD

βτa

μminus

zB

za]cos htminus cos hταminus 1Δa]

1113896

+zRμ

zτ aDβτa

μminus

zB

zτcos htminus cos hταminus 1Δτ

+ RμaDβτa

μminus Bcos htminus cos hταminus 1 d

dτΔτ

+ RμaDβτΔa

μminus aD

βτ _a

μΔτ+ΔτaDβτ _a

μcos htminus cos hταminus 1

+ αminus 1sinhτcos htminus cos hταminus 2Δτ⎫⎬

⎭dτ

(21)

Equation (21) can also be written as

ΔSE 1Γα

middot 1113946b

a

d

dτRμaD

βτa

μminus Bcosh tminus coshταminus 1Δτ11138761113896

+ 1113946τ

aRμaD

βτδa

μcosh tminus cosh sαminus 1

minus δaμ

sDβbRμcosh tminus cosh s

αminus 1ds1113877

+zR]za

μaDβτa

]minus

zB

zaμcos h tminus cos hταminus 1

1113890

+τDβbRμcosh tminus cosh ταminus 1⎤⎦δa

μ⎫⎬

⎭dτ

(22)

By using formula (7) we have

ΔSE 1Γα

middot 1113946b

aεσ

d

dτRμaD

βτa

μminus Bcoshtminus coshταminus 1ξσ011138761113896

+ 1113946τ

aRμaD

βs ξ

σμminus _a

μξσ0cosh tminus cosh sαminus 1

minus ξσμminus _aμξσ0 sD

βbRμcosh tminus cosh s

αminus 1ds1113877

+zR]za

μaDβτa

]minus

zB

zaμcosh tminus cosh ταminus 1

1113890

+τDβbRμcosh tminus cosh ταminus 1⎤⎦ξσμminus _a

μξσ0⎫⎬

⎭dτ

(23)



23 Fractional Birkhoffian System Based on EPFI Under themodel of EPFI we define the Pfaff action as

SP 1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113960 1113961sin1113966

middot (α minus 1)(t minus τ) +π2

1113874 11138751113883dτ

(24)


δSP 0 (25)


δaDβτa

μ aD

βτδa

μ (26)


aμ|τa a

μ1

aμ|τb a

μ2

(27)


zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889sin (α minus 1)(t minus τ) +

π2

1113874 1113875

+ τDβb Rμsin (α minus 1)(t minus τ) +

π2

1113874 11138751113876 1113877 0 (μ 1 2 2n)

(28)

If β⟶ 1 equation (28) becomes Birkhoffrsquos equationsbased on EPFI If β⟶ 1 and α⟶ 1 equation (28) be-comes the classical Birkhoffrsquos equations [58]


SP(c) 1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113876 1113877sin


1113874 1113875dτ

(29)

and we have

SP(c) minus SP(c)

1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113876 1113877

sin (α minus 1)(t minus τ) +π2

1113874 1113875dτ

minus1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113960 1113961


1113874 1113875dτ

1Γ(α)

1113946b

aRμ τ + Δτ a

]+ Δa]

( 11138571113960 11139611113966 1113967

aDβτa

μ+ aD

βτΔa

μ1113872 1113873

minus aDβτ _a


μ1113872 1113873


+ Δa]( 11138571113859

sin (α minus 1)(t minus (τ + Δτ)) +π2

1113874 1113875

1 +ddτΔτ1113888 1113889

minus Rμ τ a]

( 1113857aDβτa

μminus B τ a

]( 11138571113960 1113961


1113874 11138751113883dτ

(30)

So we have

ΔSP 1Γ(α)

1113946b

a

zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889sin (α minus 1)(t minus τ) +

π2

1113874 1113875Δa]+

zRμ

zτ aD

βτa

μminus

zB

zτ1113888 1113889sin (α minus 1)(t minus τ) +

π2

1113874 1113875Δτ1113896

+ RμaDβτa

μminus B1113872 1113873sin (α minus 1)(t minus τ) +

π2

1113874 1113875ddτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a


μ1113872 1113873sin (α minus 1)(t minus τ) +

π2

1113874 1113875

minus (α minus 1)cos (α minus 1)(t minus τ) +π2

1113874 11138751113883dτ

(31)



ΔSP 1Γ(α)

1113946b

a

ddτ

RμaDβτa


π2

1113874 1113875Δτ + 1113946τ

aRμaD

βs δa

μsin (α minus 1)(t minus s) +π2

1113874 1113875111387411138761113896

minus δaμ

sDβb Rμsin (α minus 1)(t minus s) +

π2

1113874 11138751113874 11138751113875ds1113877 +zR]za

μaDβτa

]minus

zB


π2

1113874 11138751113890


π2

1113874 11138751113874 11138751113877δaμ1113883dτ

(32)


ΔSP 1Γ(α)

1113946b

aεσ

ddτ

RμaDβτa


π2

1113874 1113875ξσ011138761113896

+ 1113946τ

aRμaD

βs ξσμ minus _a

μξσ01113872 1113873sin (α minus 1)(t minus s) +π2

1113874 1113875 minus ξσμ minus _aμξσ01113872 1113873sD

βb Rμsin (α minus 1)(t minus s) +

π2

1113874 11138751113874 11138751113874 11138751113877ds

+zR]za

μaDβτa

]minus

zB


π2

1113874 1113875 + τDβb Rμsin (α minus 1)(t minus τ) +

π2

1113874 11138751113874 11138751113890 1113891 ξσμ minus _aμξσ01113872 11138731113897dτ

(33)


3 Fractional Noether Symmetries under Quasi-Fractional Dynamics Models

Next we will define the Noether symmetries of the systemunder three quasi-fractional dynamics models and establishtheir criteria

31 Fractional Noether Symmetries Based on ERLFI

Definition 1 If the Pfaff action (1) satisfies the equality

ΔSR 0 (34)

then transformation (6) is said to be Noether symmetric forsystem (5)

According to Definition 1 using formulas (10) and (13)we have the following

Criterion 1 If transformation (6) is Noether symmetricthen the equation

zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113960 1113961 minus RμaDβτa

μminus B1113872 1113873

α minus 1t minus τΔτ 0

(35)

needs to be satisfied Equation (35) can be written as r equations

zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873 minus RμaDβτa

μminus B1113872 1113873

α minus 1t minus τ

ξσ0 0 (σ 1 2 r)

(36)


If r 1 equation (36) gives the fractional Noetheridentity based on ERLFI

Criterion 2 If transformation (7) is Noether symmetricthen the following r equations

ddτ

RμaDβτa


αminus 1ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds1113882 1113883

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873 0 (σ 1 2 r)

(37)

need to be satisfied


ΔSR minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (38)

where ΔG εσGσ andGσ Gσ(τ a]) is the gauge functionthen transformation (6) is said to be Noether quasi-symmetricfor system (5)


Criterion 3 If transformation (6) is Noether quasi-sym-metric then the equation

zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ

+ RμaDβτa

μminus B1113872 1113873

ddτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113960 1113961

minus RμaDβτa

μminus B1113872 1113873

α minus 1t minus τΔτ minus

ddτ

(ΔG)(t minus τ)1minus α

(39)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0

+ RμaDβτa

μminus B1113872 1113873 _ξ

σ0 + Rμ aD

βτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873

minus RμaDβτa

μminus B1113872 1113873


ξσ0 minus _Gσ(t minus τ)

1minus α (σ 1 2 r)

(40)

If r 1 equation (40) also gives the fractional Noetheridentity based on ERLFI

Criterion 4 If transformation (7) is Noether quasi-sym-metric then the following r equations

ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds1113882 1113883

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873 minus _Gσ (σ 1 2 r)

(41)



32 Fractional Noether Symmetries Based on EEFI


ΔSE 0 (42)


According to Definition 3 using formulas (21) and (23)we have


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113872 1113873 + RμaDβτa

μminus B1113872 1113873

(α minus 1)sinh τcosh t minus cosh τ

Δτ 0

(43)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873 + RμaDβτa

μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(44)

If r 1 equation (44) gives the fractional Noetheridentity based on EEFI


ddτ

RμaDβτa

μminus B1113872 1113873(cosh t minus cosh τ)


aRμaD

βs ξσμ minus _a

μξσ01113872 1113873(cosh t minus cosh s)αminus 111138761113882


βb Rμ(cosh t minus cosh s)αminus 11113960 11139611113877ds1113883

+zR]za

μaDβτa

]minus

zB


αminus 1+ τD

βb Rμ(cosh t minus cosh τ)


μξσ01113872 1113873 0 (σ 1 2 r)

(45)



ΔSE minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (46)

where ΔG εσGσ andGσ Gσ(τ a]) is the gauge functionthen transformation (6) is said to beNoether quasi-symmetricfor system (18)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ

+ RμaDβτa

μminus B1113872 1113873

d

dτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113872 1113873

+ RμaDβτa

μminus B1113872 1113873


Δτ minusd

dτ(ΔG)(cosh t minus cosh τ)

1minus α

(47)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0

+ RμaDβτa

μminus B1113872 1113873 _ξ

σ0 + Rμ aD

βτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873

+ RμaDβτa

μminus B1113872 1113873


ξσ0 minus _Gσ(cosh t minus cosh τ)


(48)

If r 1 equation (48) also gives the fractional Noetheridentity based on EEFI


ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a




+zR]za

μaDβτa

]minus

zB


αminus 1+ τD



μξσ01113872 1113873 minus _Gσ (σ 1 2 r)

(49)


33 Fractional Noether Symmetries Based on EPFI

Definition 5 If the Pfaff action 24 satisfies the equality

ΔSP 0 (50)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873

α minus 1tan((α minus 1)(t minus τ) +(π2))

Δτ 0

(51)

needs to be satisfied


Equation (51) can be written as r equations

zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(52)

If r 1 equation (52) gives the fractional Noetheridentity based on EPFI


ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a

μξσ01113872 1113873sin (α minus 1)(t minus s) +

π2

1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891 ξσμ minus _aμξσ01113872 1113873 0 (σ 1 2 r)

(53)



ΔSP minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (54)

where ΔG εσGσ andGσ Gσ(τ a]) is the gauge functionthen transformation (6) is said to be Noether quasi-sym-metric for system (28)



zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873


Δτ minus(ddτ)(ΔG)

sin((α minus 1)(t minus τ) +(π2))

(55)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 minus_Gσ


(56)



ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891

ξσμ minus _aμξσ01113872 1113873 minus _G

σ (σ 1 2 r)

(57)


4 Fractional Noetherrsquos Theorems under Quasi-Fractional Dynamics Models

Now we prove Noetherrsquos theorems for fractional Birkhoffiansystems under three quasi-fractional dynamics models

41 Fractional Noetherrsquos lteorems Based on ERLFI

Theorem 1 If transformation (7) is Noether symmetric ofsystem (5) based on ERLFI then

Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds c

σ (σ 1 2 r)

(58)

are r linearly independent conserved quantities Proof From Definition 1 we get ΔSR 0 namely

1Γ(α)

1113946b

aεσ

ddτ

RμaDβτa



a




βb Rμ(t minus s)

αminus 11113872 11138731113874 1113875ds1113876 11138771113896

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873dτ1113897 0

(59)


By substituting (5) into the above formula and con-sidering the arbitrariness of the integral interval and theindependence of εσ we obtain

ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113872 11138731113872 1113873ds1113876 1113877 0 (60)

So -eorem 1 is proved Theorem 2 If transformation (7) is Noether quasi-sym-metric of system (5) based on ERLFI then

Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds

+ Gσ

cσ (σ 1 2 r)

(61)

are r linearly independent conserved quantities

Proof Combining Definition 2 and formula (13) usingequation (5) and considering the arbitrariness of the integralinterval and the independence of εσ the conclusion isobtained

42 Fractional Noetherrsquos lteorems Based on EEFI

Theorem 3 If transformation (7) is Noether symmetric ofsystem (18) based on EEFI then

Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876


βb Rμ(cosh t minus cos hs)

αminus 11113960 11139611113877ds c

σ (σ 1 2 r)

(62)

are r linearly independent conserved quantities Theorem 4 If transformation (7) is Noether quasi-sym-metric of system (18) based on EEFI then

Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876


βb Rμ(cosh t minus cosh s)

αminus 11113960 11139611113877ds + G

σ c

σ (σ 1 2 r)

(63)


43 Fractional Noetherrsquos lteorems Based on EPFI

Theorem 5 If transformation (7) is Noether symmetric ofsystem (28) based on EPFI then

Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 11138751113876



π2

1113874 11138751113876 1113877]ds cσ (σ 1 2 r)

(64)



Theorem 6 If transformation (7) is Noether quasi-sym-metric of system (28) based on EPFI then

Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βτ ξσμ minus _a


1113874 11138751113876



π2

1113874 11138751113876 11138771113877ds + Gσ

cσ (σ 1 2 r)

(65)


Obviously if β⟶ 1 then-eorems 1ndash6 give Noetherrsquostheorems for quasi-fractional Birkhoffian systems If α⟶ 1and β⟶ 1 -eorems 1ndash6 give Noetherrsquos theorems forclassical Birkhoffian systems [58]

5 Examples

51 Example 1 Consider a fractional Birkhoffian systembased on ERLFI -e Pfaff action is

SR 1Γ(α)

1113946b

aa2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113966 1113967(t minus τ)αminus 1

dτ

(66)

where the Birkhoffian is B a2a3 and Birkhoffrsquos functionsare R1 a2 R2 0 R3 a4 andR4 0

From equation (5) Birkhoffrsquos equations are

τDβb a

2(t minus τ)

αminus 11113960 1113961 0

aDβτa

1minus a

3 0

minus a2(t minus τ)

αminus 1+ τD

βb a

4(t minus τ)

αminus 11113960 1113961 0

aDβτa

3 0

(67)

According to (40) the Noether identity gives

aDβτa

1minus a

31113872 1113873ξσ2 minus a

2ξσ3 + aDβτa

3ξσ4 + a2

aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113960 1113961 _ξσ0 minus


middot a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113872 1113873ξσ0 minus _Gσ(t minus τ)

1minus α

(68)

Let

ξσ0 1

ξσ1 a1

ξσ2 1

ξσ3 a3

ξσ4 1

Gσ

0

(69)

By -eorem 2 we obtain

I a2

aDβτa

1+ a

4aD

βτa

3minus a

3a2

1113872 1113873(t minus τ)αminus 1

const (70)

-e conserved quantity (70) corresponds the Noethersymmetry (69)

When β⟶ 1 formula (70) is reduced to

I a2a1

+ a4a3

minus a2a3

1113872 1113873(t minus τ)αminus 1

const (71)

Formula (71) is the conserved quantity of Birkhoffiansystem based on ERLFI

When β⟶ 1 and α⟶ 1 formula (70) is reduced to

I a2a1

+ a4a3

minus a2a3

const (72)

Formula (72) is the classical conserved quantity

52 Example 2 Consider a fractional Birkhoffian systembased on EEFI -e Pfaff action is


SE 1Γ(α)

1113946b

aa2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

+ a3

1113872 11138732

1113874 11138751113882 1113883 middot (cosh t minus cosh τ)αminus 1dτ (73)

where B a2a3 + (a3)2 R1 a2 + a3 R2 0 R3

a4 andR4 0From equation (18) Birkhoffrsquos equations are

τDβb a

2+ a

31113872 1113873(cosh t minus cosh τ)

αminus 11113960 1113961 0

aDβτa

1minus a

3 0

aDβτa

1minus a

2minus 2a


αminus 1+ τD

βb a

4(cosh t minus cosh τ)

αminus 11113960 1113961 0

aDβτa

3 0

(74)

According to (48) the Noether identity is

aDβτa

1minus a

31113872 1113873ξσ2 + aD

βτa

1minus a

2minus 2a

31113872 1113873ξσ3 + aD

βτa

3ξσ4 + a2

+ a3

1113872 1113873 aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877 _ξσ0

+(α minus 1)sinh τcosh t minus cosh τ

a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113874 1113875ξσ0 minus _Gσ(cosh t minus cosh τ)

1minus α

(75)

Let

ξ0 1

ξ1 a1

ξ2 0

ξ3 a3

ξ4 0

Gσ

0

(76)


I a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877

middot (cosh t minus cosh τ)αminus 1

const(77)

-e conserved quantity (77) corresponds to the Noethersymmetry (76)

If β⟶ 1 then we obtain

I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

1113876 1113877


const(78)

Formula (78) is the conserved quantity of Birkhoffiansystem based on EEFI

If β⟶ 1 and α⟶ 1 then we obtain

I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

const (79)


53 Example 3 Consider a fractional Birkhoffian systembased on EPFI -e Pfaff action is

SP 1Γ(α)

1113946b

aa3

aDβτa

1+ a

4aD

βτa

2minus12

a3

1113872 11138732

minus12

a4

1113872 11138732

1113876 1113877 middot sin (α minus 1)(t minus τ) +π2

1113874 11138751113882 1113883dτ (80)

where B (12)(a3)2 + (12)(a4)2 R1 a3 R2

a4 andR3 R4 0



τDβb a

3sin (α minus 1)(t minus τ) +π2

1113874 11138751113876 1113877 0

τDβb a


1113874 11138751113876 1113877 0

aDβτa

1minus a

3 0

aDβτa

2minus a

4 0

(81)


aDβτa

1minus a

31113872 1113873ξ3 + aD

βτa

2minus a

41113872 1113873ξ4 + a

3aD

βτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113872 1113873

+ a4

aDβτξ

σ2 minus aD

βτ _a

2ξσ01113872 1113873 + aDβτ _a

2ξσ01113872 1113873 + a3

aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875 _ξσ0

minusα minus 1

tan((α minus 1)(t minus τ) +(π2))a3

aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875ξσ0

minus _Gσ 1sin((α minus 1)(t minus τ) +(π2))

(82)

Let

ξ0 0

ξ1 a2

ξ2 minus a1

ξ3 ξ4 0

Gσ

0

(83)


I 1113946τ

aa3

aDβτa

2sin (α minus 1)(t minus s) +π2

1113874 11138751113882 1113883

minus a4

aDβτa


1113874 1113875

minus a2

sDβb a


1113874 11138751113876 1113877

+a1

sDβb a


1113874 11138751113876 11138771113883ds

const

(84)


When β⟶ 1 formula (84) becomes

I a3a2

minus a4a1

1113872 1113873sin (α minus 1)(t minus τ) +π2

1113874 1113875 const (85)

Formula (85) is the conserved quantity of Birkhoffiansystem based on EPFI

When β⟶ 1 and α⟶ 1 formula (84) becomes

I a3a2

minus a4a1

const (86)


6 Conclusions

By introducing fractional calculus into the dynamic mod-eling of nonconservative systems the dynamic behavior andphysical process of complex systems can be described moreaccurately which provides the possibility for the quanti-zation of nonconservative problems Compared with frac-tional models the quasi-fractional model greatly simplifiesthe calculation of complex fractional-order calculus so it canbe used to study complex nonconservative dynamic systemsmore conveniently-e dynamics of Birkhoffian system is anextension of Hamiltonian mechanics and the fractionalBirkhoffian system is an extension of integer Birkhoffiansystem -erefore fractional Birkhoffian dynamics is a re-search field worthy of further study and full of vitality

-e main contributions of this paper are as follows Firstlybased on three quasi-fractional dynamicsmodels the fractionalPfaffndashBirkhoff principles and fractional Birkhoffrsquos equationsare established in which the Pfaff action contains fractional-order derivative terms Secondly the fractional Noethersymmetry is explored and its definitions and criteria are


established -irdly Noetherrsquos theorems for fractional Bir-khoffian systems under three quasi-fractional dynamicsmodelsare proved and fractional conservation laws are obtained

Obviously the results of the following two systems arespecial cases of this paper (1) the quasi-fractional Bir-khoffian systems based on quasi-fractional dynamicsmodels in which the Pfaff action contains only integer-orderderivative terms (2) the classical Birkhoffian systems underinteger-order models -erefore our study is of greatsignificance

Data Availability

-e data used to support the findings of this study are in-cluded within the article

Conflicts of Interest

-e author declares that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

-is work was supported by the National Natural ScienceFoundation of China (nos 11972241 11572212 and11272227) and Natural Science Foundation of JiangsuProvince of China (no BK20191454)

References

[1] A E Noether ldquoInvariante variationsproblemerdquo Nachr AkadWiss Gottingen Math-Phys vol 2 pp 235ndash257 1918

[2] E Candotti C Palmieri and B Vitale ldquoOn the inversion ofnoetherrsquos theorem in classical dynamical systemsrdquo AmericanJournal of Physics vol 40 no 3 pp 424ndash429 1972

[3] E A Desloge and R I Karch ldquoNoetherrsquos theorem in classicalmechanicsrdquo American Journal of Physics vol 45 no 4pp 336ndash339 1977

[4] D S Vujanovic and B D Vujanovic ldquoNoetherrsquos theory inclassical nonconservative mechanicsrdquo Acta Mechanicavol 23 no 1-2 pp 17ndash27 1975

[5] D Liu ldquoNoether theorem and its inverse for nonholonomicconservative dynamical systemsrdquo Science in China Series Avol 20 no 11 pp 1189ndash1197 1991

[6] M Lutzky ldquoDynamical symmetries and conserved quanti-tiesrdquo Journal of Physics A Mathematical and General vol 12no 7 pp 973ndash981 1979

[7] P J Olver Applications of Lie Groups to Differential Equa-tions Springer-Verlag New York NY USA 1986

[8] N H Ibragimov CRC Handbook of Lie Group Analysis ofDifferential Equations CRC Press Boca Raton FL USA 1994

[9] G W Bluman and S Kumei Symmetries and DifferentialEquations Springer-Verlag New York NY USA 1989

[10] Y Y Zhao ldquoConservative quantities and Lie symmetries ofnonconservative dynamical systemsrdquo Acta Mechanica Sinicavol 26 pp 380ndash384 1994

[11] F X Mei Applications of Lie Groups and Lie Algebras toConstrained Mechanical Systems Science Press BeijingChina 1999

[12] F X Mei Symmetries and Conserved Quantities of Con-strained Mechanical Systems Beijing Institute of TechnologyBeijing China 2004

[13] Y Zhang ldquoNoetherrsquos theorem for a time-delayed Birkhoffiansystem of Herglotz typerdquo International Journal of Non-linearMechanics vol 101 pp 36ndash43 2018

[14] L J Zhang and Y Zhang ldquoNon-standard Birkhoffian dy-namics and its Noetherrsquos theoremsrdquo Communications inNonlinear Science and Numerical Simulation vol 91 p 11Article ID 105435 2020

[15] C J Song and Y Chen ldquoNoetherrsquos theorems for nonshifteddynamic systems on time scalesrdquo Applied Mathematics andComputation vol 374 p 12 Article ID 125086 2020

[16] H-B Zhang and H-B Chen ldquoNoetherrsquos theorem of Ham-iltonian systems with generalized fractional derivative oper-atorsrdquo International Journal of Non-linear Mechanicsvol 107 pp 34ndash41 2018

[17] Y Zhang ldquoHerglotzrsquos variational problem for nonconserva-tive system with delayed arguments under Lagrangianframework and its Noetherrsquos theoremrdquo Symmetry vol 12no 5 p 13 Article ID 845 2020

[18] S X Jin and Y Zhang ldquoNoether theorem for nonholonomicsystems with time delayrdquo Mathematical Problems in Engi-neering vol 2015 p 9 Article ID 539276 2015

[19] P-P Cai J-L Fu and Y-X Guo ldquoLie symmetries andconserved quantities of the constraint mechanical systems ontime scalesrdquo Reports on Mathematical Physics vol 79 no 3pp 279ndash298 2017

[20] M J Lazo J Paiva and G S F Frederico ldquoNoether theoremfor action-dependent Lagrangian functions conservation lawsfor non-conservative systemsrdquo Nonlinear Dynamics vol 97no 2 pp 1125ndash1136 2019

[21] Y Zhang ldquoLie symmetry and invariants for a generalizedBirkhoffian system on time scalesrdquo Chaos Solitons amp Fractalsvol 128 pp 306ndash312 2019

[22] F X Mei H B Wu and Y F Zhang ldquoSymmetries andconserved quantities of constrained mechanical systemsrdquoInternational Journal of Dynamics and Control vol 2 no 3pp 285ndash303 2014

[23] M Inc A Yusuf A I Aliyu and D Baleanu ldquoLie symmetryanalysis explicit solutions and conservation laws for thespace-time fractional nonlinear evolution equationsrdquo PhysicaA Statistical Mechanics and Its Applications vol 496pp 371ndash383 2018

[24] D Baleanu M Inc A Yusuf and A I Aliyu ldquoLie symmetryanalysis exact solutions and conservation laws for the timefractional Caudrey-Dodd-Gibbon-Sawada-Kotera equationrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 59 pp 222ndash234 2018

[25] K B Oldham and J Spanier lte Fractional Calculus Aca-demic Press San Diego CL USA 1974

[26] K S Miller and B Ross An Introduction to the FractionalIntegrals and Derivatives-lteory and Applications JohnWileyamp Sons New York NY USA 1993

[27] I Podlubny Fractional Differential Equations AcademicPress San Diego CL USA 1999

[28] A A Kilbas H M Srivastava and J J Trujillo lteory andApplications of Fractional Differential Equations Elsevier B VAmsterdam -e Netherlands 2006

[29] R HilferApplications of Fractional Calculus in Physics WorldScientific Singapore 2000

[30] R Herrmann Fractional Calculus An Introduction forPhysicists World Scientific Publishing Singapore 2014

[31] V E Tarasov ldquoFractional diffusion equations for openquantum systemrdquo Nonlinear Dynamics vol 71 no 4pp 663ndash670 2013


[32] R E Gutierrez J M Rosario and J T Machado ldquoFractionalorder calculus basic concepts and engineering applicationsrdquoMathematical Problems in Engineering vol 2010 Article ID375858 19 pages 2010

[33] J A T Machado M F Silva R S Barbosa et al ldquoSomeapplications of fractional calculus in engineeringrdquo Mathe-matical Problems in Engineering vol 2010 Article ID 63980134 pages 2010

[34] D Baleanu J H Asad and I Petras ldquoFractional bateman-feshbach tikochinsky oscillatorrdquo Communications in lteo-retical Physics vol 61 no 2 pp 221ndash225 2014

[35] Y Zhang ldquoFractional differential equations of motion interms of combined Riemann-Liouville derivativerdquo ChinesePhysics B vol 21 no 8 5 pages Article ID 084502 2012

[36] F Riewe ldquoNonconservative Lagrangian and Hamiltonianmechanicsrdquo Physical Review E vol 53 no 2 pp 1890ndash18991996

[37] F Riewe ldquoMechanics with fractional derivativesrdquo PhysicalReview E vol 55 no 3 pp 3581ndash3592 1997

[38] R A El-Nabulsi ldquoA fractional approach to nonconservativeLagrangian dynamical systemsrdquo Fizika A vol 14 no 4pp 289ndash298 2005

[39] R A El-Nabulsi ldquoFractional variational problems from ex-tended exponentially fractional integralrdquo Computers ampMathematics with Applications vol 217 no 22 pp 9492ndash9496 2011

[40] R A El-Nabulsi ldquoA periodic functional approach to thecalculus of variations and the problem of time-dependentdamped harmonic oscillatorsrdquo Appled Mathematics Lettersvol 24 no 10 pp 1647ndash1653 2011

[41] G S F Frederico and D F M Torres ldquoA formulation ofNoetherrsquos theorem for fractional problems of the calculus ofvariationsrdquo Journal of Mathematical Analysis and Applica-tions vol 334 no 2 pp 834ndash846 2007

[42] T M Atanackovic S Konjik S Pilipovic and S SimicldquoVariational problems with fractional derivatives invarianceconditions and Noetherrsquos theoremrdquo Nonlinear Analysisvol 71 no 5-6 pp 1504ndash1517 2009

[43] A B Malinowska and D F M Torres Introduction to theFractional Calculus of Variations Imperial College PressLondon UK 2012

[44] R Almeida S Pooseh and D F M Torres ComputationalMethods in the Fractional Calculus of Variations ImperialCollege Press Singapore 2015

[45] J Cresson and A Szafranska ldquoAbout the Noetherrsquos theoremfor fractional Lagrangian systems and a generalization of theclassical Jost method of proofrdquo Fractional Calculus andApplied Analysis vol 22 no 4 pp 871ndash898 2019

[46] X Tian and Y Zhang ldquoNoetherrsquos theorem for fractionalHerglotz variational principle in phase spacerdquo Chaos Solitonsamp Fractals vol 119 pp 50ndash54 2019

[47] X Tian and Y Zhang ldquoFractional time-scales Noether the-orem with Caputo Δ derivatives for Hamiltonian systemsrdquoApplied Mathematics and Computation vol 393 Article ID125753 15 pages 2021

[48] S Zhou H Fu and J Fu ldquoSymmetry theories of Hamiltoniansystems with fractional derivativesrdquo Science China PhysicsMechanics and Astronomy vol 54 no 10 pp 1847ndash18532011

[49] C J Song ldquoNoether symmetry for fractional Hamiltoniansystemrdquo Physics Letters A vol 383 Article ID 125914 2019

[50] G S F Frederico and D F M Torres ldquoConstants of motionfor fractional action-like variational problemsrdquo International

Journal of Applied Mathematics vol 19 no 1 pp 97ndash1042006

[51] G S F Frederico and D F M Torres ldquoNonconservativeNoetherrsquos theorem for fractional action-like variationalproblems with intrinsic and observer timesrdquo InternationalJournal of Ecological Economics and Statistics vol 91pp 74ndash82 2007

[52] Y Zhang and Y Zhou ldquoSymmetries and conserved quantitiesfor fractional action-like Pfaffian variational problemsrdquoNonlinear Dynamics vol 73 no 1-2 pp 783ndash793 2013

[53] Z-X Long and Y Zhang ldquoFractional Noether theorembased on extended exponentially fractional integralrdquo In-ternational Journal of lteoretical Physics vol 53 no 3pp 841ndash855 2014

[54] Z-X Long and Y Zhang ldquoNoetherrsquos theorem for fractionalvariational problem from El-Nabulsi extended exponentiallyfractional integral in phase spacerdquo Acta Mechanica vol 225no 1 pp 77ndash90 2014

[55] Z X Long and Y Zhang ldquoNoetherrsquos theorem for noncon-servative Hamilton system based on El-Nabulsi dynamicalmodel extended by periodic lawsrdquo Chinese Physics B vol 23no 11 9 pages Article ID 114501 2014

[56] G D Birkhoff Dynamical Systems AMS College PublisherProvidence RI USA 1927

[57] R M Santilli Foundations of lteoretical Mechanics IISpringer New York NY USA 1983

[58] F X Mei R C Shi Y Zhang and H B Wu Dynamics ofBirkhoffian Systems Beijing Institute of Technology BeijingChina 1996

[59] Y Zhang and X-H Zhai ldquoNoether symmetries and conservedquantities for fractional Birkhoffian systemsrdquo NonlinearDynamics vol 81 no 1-2 pp 469ndash480 2015

[60] X-H Zhai and Y Zhang ldquoNoether symmetries and conservedquantities for fractional Birkhoffian systems with time delayrdquoCommunications in Nonlinear Science and Numerical Simu-lation vol 36 no 1ndash2 pp 81ndash97 2016

[61] Y Zhou and Y Zhang ldquoNoether theorems of a fractionalBirkhoffian system within Riemann-Liouville derivativesrdquoChinese Physics B vol 23 no 12 Article ID 124502 2014

[62] C-J Song and Y Zhang ldquoNoether symmetry and conservedquantity for fractional Birkhoffian mechanics and its appli-cationsrdquo Fractional Calculus and Applied Analysis vol 21no 2 pp 509ndash526 2018

[63] Q Jia H Wu and F Mei ldquoNoether symmetries and con-served quantities for fractional forced Birkhoffian systemsrdquoJournal of Mathematical Analysis and Applications vol 442no 2 pp 782ndash795 2016

[64] H-B Zhang andH-B Chen ldquoNoetherrsquos theorem of fractionalBirkhoffian systemsrdquo Journal of Mathematical Analysis andApplications vol 456 no 2 pp 1442ndash1456 2017

[65] X Tian and Y Zhang ldquoNoether symmetry and conservedquantities of fractional Birkhoffian system in terms of Her-glotz variational problemrdquo Communications in lteoreticalPhysics vol 70 no 3 pp 280ndash288 2018

[66] C J Song ldquoAdiabatic invariants for generalized fractionalBirkhoffian mechanics and their applicationsrdquo MathematicalProblems in Engineering vol 2018 Article ID 641496018 pages 2018


dynamics modeling and successively put forward the dy-namical models of nonconservative systems which areextended by exponentially fractional integral (EEFI) andextended by periodic laws fractional integral (EPFI) [39 40]respectively -e equations obtained from quasi-fractionaldynamics models are similar to dynamical equations ofclassical conservative systems which contain the gener-alized fractional external forces corresponding to dissi-pative forces but the term with the fractional derivativedoes not show up Different from other models the frac-tional time integration of quasi-fractional dynamics modelsonly needs one parameter In this way it simplifies thecalculation of complex fractional calculus and provides amodeling method for nonconservative systems -ereforethe quasi-fractional dynamics models can be used to studycomplex dynamical systems more conveniently Fredericoand Torres [41] first presented fractional Noetherrsquos theo-rems Since then studies on fractional Noether symmetryand conservation laws have been extensively developed[42ndash49] In addition Torres and Frederico studiedNoetherrsquos theorems of fractional action-like variationproblems [50 51] In recent years nonconservative dy-namical systems based on quasi-fractional dynamicalmodels have been studied deeply and the correspondingdynamical equations and Noether conservation laws havebeen obtained [52ndash55] However most of the previousstudies on the variational problems of quasi-fractionaldynamics models are confined to Lagrangian frameworkand Hamiltonian framework

It is well known that Birkhoffian mechanics is a newstage in the development of Hamiltonian mechanics[56ndash58] Under canonical transformation Hamilton ca-nonical equation remains unchanged but under generalnoncanonical transformation it becomes Birkhoffrsquos equa-tion Santilli [57] and Mei [58] both pointed out that Bir-khoffian mechanics is the most general possible mechanicswhich can be applied to hadron physics space mechanicsstatistical mechanics biophysics engineering and otherfields Zhang and Zhai [59] has proposed the fractionalPfaffndashBirkhoff principle and fractional Birkhoffrsquos equationsand proved that the fractional Hamilton principle is thespecial case of the fractional PfaffndashBirkhoff principle and thefractional Hamilton equations and the fractional Lagrangeequations are the special cases of the fractional Birkhoffrsquosequations Zhang and Zhou [52] proposed the quasi-frac-tional Pfaff-Birkhoff principle and derived correspondingquasi-fractional Birkhoffrsquos equations which is based on thequasi-fractional model given by [38] Up to now someresults have been obtained on Noether symmetry of frac-tional or quasi-fractional Birkhoffian systems such as[52 59ndash66] However the results of these quasi-fractionalBirkhoffian systems are limited to the Pfaff action containingonly integral-order derivative terms Here we will furtherextend fractional Birkhoffian mechanics on the basis of threequasi-fractional dynamical models given in [38ndash40] wherefor the Pfaff actions we consider contain fractional-orderderivative terms -e quasi-fractional Lagrangian systemand quasi-fractional Hamiltonian system are special cases ofthe results presented in this paper

-e text is organized as follows In Section 2 the frac-tional PfaffndashBirkhoff principles under quasi-fractional dy-namics models are presented and Birkhoffrsquos equations aregiven and nonisochronous variational formulae of the Pfaffaction are driven In Section 3 fractional Noether sym-metries are well defined and their criteria are established InSection 4 fractional Noether theorems are proved For il-lustrating the application of the methods and results in thistext three examples are given in Section 5 In Section 6 wecome to the conclusions

2 Fractional Birkhoffrsquos Equations andVariation of Fractional Pfaff Action underQuasi-Fractional Dynamics Models

For an introduction to fractional calculus and its basictheory please refer to the monographs [27 28]

21 Fractional Birkhoffian System Based on ERLFI Weconsider a fractional Birkhoffian system determined byBirkhoffrsquos variables aμ(μ 1 2 2n) whose Birkhoffrsquosfunctions are Rμ Rμ(τ a]) the Birkhoffian is B B(τ a])β is the order of fractional derivative and 0le βlt 1

Under the model of ERLFI we define the Pfaff action as

SR 1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113960 1113961(t minus τ)

αminus 1dτ

(1)

where aDβτaμ (μ 1 2 2n) is the fractional derivative

term-e variational principle

δSR 0 (2)

with commutative relation

δaDβτa

μ aD

βτδa

μ (3)


aμ|τa a

μ1

aμ|τb a

μ2

(4)

is called the fractional PfaffndashBirkhoff principle based onERLFI

According to principle (2) we drive

zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)

αminus 11113960 1113961 0

(μ 1 2 2n)

(5)

Equation (5) is the fractional Birkhoffrsquos equations basedon ERLFI

If β⟶ 1 equation (5) becomes Birkhoffrsquos equationsbased on ERLFI If β⟶ 1 and α⟶ 1 equation (5) be-comes classical Birkhoffrsquos equations [58]

Take the infinitesimal transformations


τ τ + Δτ

aμ(τ) a

μ(τ) + Δaμ

(μ 1 2 2n)(6)



]( 1113857

aμ(τ) a

μ(τ) + εσξ

σμ τ a

]( 1113857 (μ 1 2 2n)

(7)



SR(c) 1Γ(α)

1113946b

aRμ τ a

]( 1113857 aD

βτa

μminus B τ a

]( 11138571113876 1113877(t minus τ)

αminus 1dτ

(8)

And we have

SR(c) minus SR(c)

1Γ(α)

1113946b

aRμ τ a

]( 1113857 aD

βτa

μminus B τ a

]( 11138571113876 1113877


minus1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113960 1113961


1Γ(α)

1113946b

aRμ τ + Δτ a

]+ Δa]

( 11138571113960 11139611113966 1113967

aDβτa

μ+ aD

βτΔa

μminus aD

βτ _a


μ1113872 1113873


+ Δa]( 11138571113859


ddτΔτ1113888 1113889

minus Rμ τ a]

( 1113857aDβτa

μminus B τ a

]( 11138571113960 1113961(t minus τ)

αminus 11113967dτ

(9)


ΔSR 1Γ(α)

1113946b

a

zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889(t minus τ)

αminus 1Δa]1113896 1113897

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889(t minus τ)

αminus 1Δτ

+ RμaDβτa


αminus 1 ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a


μ1113872 1113873(t minus τ)

αminus 1

minus RμaDβτa



(10)

Since

δaμ

Δaμminus _a

μΔτ

Δ _aμ

ddτΔaμ


(Δτ)

(11)

then we obtain

ΔSR 1Γ(α)

1113946b

a

ddτ

RμaDβτa


αminus 1Δτ11139601113896

+ 1113946τ

aRμaD

βτδa

μ(t minus s)

αminus 1minus δa

μsD

βb Rμ(t minus s)

αminus 11113872 11138731113872 1113873ds1113877

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)

αminus 11113872 11138731113890 1113891δa

μ1113897dτ

(12)


ΔSR 1Γ(α)

1113946b

aεσ

ddτ

RμaDβτa


αminus 1ξσ011139601113896

+ 1113946τ

a




βb Rμ(t minus s)

αminus 11113872 11138731113874 1113875ds1113877

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873dτ

(13)




SE 1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a


αminus 1dτ

(14)


δSE 0 (15)


δaDβτa

μ aD

βτδa

μ (16)


aμ|τa a

μ1

aμ|τb a

μ2

(17)


zR]za

μaDβτa

]minus

zB


αminus 1


αminus 11113960 1113961 0 (μ 1 2 2n)

(18)



SE(c) 1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a


αminus 1dτ

(19)

and we have

SE(c) minus SE(c)

1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113876 1113877


minus1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113960 1113961


1Γ(α)

1113946b

aRμ τ + Δτ a

]+ Δa]

( 11138571113960 11139611113966 1113967

aDβτa

μ+ aD

βτΔa

μminus aD

βτ _a


μ1113872 1113873


+ Δa]( 11138571113859


ddτΔτ1113888 1113889

minus Rμ τ a]

( 1113857aDβτa

μminus B τ a


αminus 11113967dτ

(20)


ΔSE 1Γα

middot 1113946b

a

zRμ

za] aD

βτa

μminus

zB


1113896

+zRμ

zτ aDβτa

μminus

zB


+ RμaDβτa


dτΔτ

+ RμaDβτΔa

μminus aD

βτ _a




⎭dτ

(21)


ΔSE 1Γα

middot 1113946b

a

d

dτRμaD

βτa


+ 1113946τ

aRμaD

βτδa


minus δaμ


αminus 1ds1113877

+zR]za

μaDβτa

]minus

zB


1113890


μ⎫⎬

⎭dτ

(22)


ΔSE 1Γα

middot 1113946b

aεσ

d

dτRμaD

βτa


+ 1113946τ

aRμaD

βs ξ

σμminus _a




αminus 1ds1113877

+zR]za

μaDβτa

]minus

zB


1113890


μξσ0⎫⎬

⎭dτ

(23)




SP 1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113960 1113961sin1113966


1113874 11138751113883dτ

(24)


δSP 0 (25)


δaDβτa

μ aD

βτδa

μ (26)


aμ|τa a

μ1

aμ|τb a

μ2

(27)


zR]za

μaDβτa

]minus

zB


π2

1113874 1113875


π2

1113874 11138751113876 1113877 0 (μ 1 2 2n)

(28)



SP(c) 1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113876 1113877sin


1113874 1113875dτ

(29)

and we have

SP(c) minus SP(c)

1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113876 1113877


1113874 1113875dτ

minus1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113960 1113961


1113874 1113875dτ

1Γ(α)

1113946b

aRμ τ + Δτ a

]+ Δa]

( 11138571113960 11139611113966 1113967

aDβτa

μ+ aD

βτΔa

μ1113872 1113873

minus aDβτ _a


μ1113872 1113873


+ Δa]( 11138571113859


1113874 1113875

1 +ddτΔτ1113888 1113889

minus Rμ τ a]

( 1113857aDβτa

μminus B τ a

]( 11138571113960 1113961


1113874 11138751113883dτ

(30)

So we have

ΔSP 1Γ(α)

1113946b

a

zRμ

za] aD

βτa

μminus

zB


π2

1113874 1113875Δa]+

zRμ

zτ aD

βτa

μminus

zB


π2

1113874 1113875Δτ1113896

+ RμaDβτa


π2

1113874 1113875ddτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a



π2

1113874 1113875


1113874 11138751113883dτ

(31)



ΔSP 1Γ(α)

1113946b

a

ddτ

RμaDβτa


π2

1113874 1113875Δτ + 1113946τ

aRμaD

βs δa


1113874 1113875111387411138761113896

minus δaμ


π2

1113874 11138751113874 11138751113875ds1113877 +zR]za

μaDβτa

]minus

zB


π2

1113874 11138751113890


π2

1113874 11138751113874 11138751113877δaμ1113883dτ

(32)


ΔSP 1Γ(α)

1113946b

aεσ

ddτ

RμaDβτa


π2

1113874 1113875ξσ011138761113896

+ 1113946τ

aRμaD

βs ξσμ minus _a




π2

1113874 11138751113874 11138751113874 11138751113877ds

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891 ξσμ minus _aμξσ01113872 11138731113897dτ

(33)






ΔSR 0 (34)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873


(35)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(36)




ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds1113882 1113883

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873 0 (σ 1 2 r)

(37)



ΔSR minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (38)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ

+ RμaDβτa

μminus B1113872 1113873

ddτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113960 1113961

minus RμaDβτa

μminus B1113872 1113873


ddτ


(39)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0

+ RμaDβτa

μminus B1113872 1113873 _ξ

σ0 + Rμ aD

βτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873

minus RμaDβτa

μminus B1113872 1113873




(40)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds1113882 1113883

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873 minus _Gσ (σ 1 2 r)

(41)





ΔSE 0 (42)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113872 1113873 + RμaDβτa

μminus B1113872 1113873


Δτ 0

(43)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873 + RμaDβτa

μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(44)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a




+zR]za

μaDβτa

]minus

zB


αminus 1+ τD



μξσ01113872 1113873 0 (σ 1 2 r)

(45)



ΔSE minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (46)





zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ

+ RμaDβτa

μminus B1113872 1113873

d

dτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113872 1113873

+ RμaDβτa

μminus B1113872 1113873


Δτ minusd


1minus α

(47)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0

+ RμaDβτa

μminus B1113872 1113873 _ξ

σ0 + Rμ aD

βτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873

+ RμaDβτa

μminus B1113872 1113873




(48)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a




+zR]za

μaDβτa

]minus

zB


αminus 1+ τD



μξσ01113872 1113873 minus _Gσ (σ 1 2 r)

(49)




ΔSP 0 (50)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873


Δτ 0

(51)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(52)



ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


π2

1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891 ξσμ minus _aμξσ01113872 1113873 0 (σ 1 2 r)

(53)



ΔSP minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (54)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873




(55)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 minus_Gσ


(56)



ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891


σ (σ 1 2 r)

(57)






Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds c

σ (σ 1 2 r)

(58)


1Γ(α)

1113946b

aεσ

ddτ

RμaDβτa



a




βb Rμ(t minus s)

αminus 11113872 11138731113874 1113875ds1113876 11138771113896

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873dτ1113897 0

(59)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113872 11138731113872 1113873ds1113876 1113877 0 (60)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds

+ Gσ

cσ (σ 1 2 r)

(61)





Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds c

σ (σ 1 2 r)

(62)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds + G

σ c

σ (σ 1 2 r)

(63)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 11138751113876



π2

1113874 11138751113876 1113877]ds cσ (σ 1 2 r)

(64)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD



1113874 11138751113876



π2

1113874 11138751113876 11138771113877ds + Gσ

cσ (σ 1 2 r)

(65)



5 Examples


SR 1Γ(α)

1113946b

aa2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113966 1113967(t minus τ)αminus 1

dτ

(66)



τDβb a

2(t minus τ)

αminus 11113960 1113961 0

aDβτa

1minus a

3 0


αminus 1+ τD

βb a

4(t minus τ)

αminus 11113960 1113961 0

aDβτa

3 0

(67)


aDβτa

1minus a

31113872 1113873ξσ2 minus a

2ξσ3 + aDβτa

3ξσ4 + a2

aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113960 1113961 _ξσ0 minus


middot a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3


1minus α

(68)

Let

ξσ0 1

ξσ1 a1

ξσ2 1

ξσ3 a3

ξσ4 1

Gσ

0

(69)


I a2

aDβτa

1+ a

4aD

βτa

3minus a

3a2

1113872 1113873(t minus τ)αminus 1

const (70)



I a2a1

+ a4a3

minus a2a3

1113872 1113873(t minus τ)αminus 1

const (71)



I a2a1

+ a4a3

minus a2a3

const (72)




SE 1Γ(α)

1113946b

aa2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

+ a3

1113872 11138732


where B a2a3 + (a3)2 R1 a2 + a3 R2 0 R3


τDβb a

2+ a


αminus 11113960 1113961 0

aDβτa

1minus a

3 0

aDβτa

1minus a

2minus 2a


αminus 1+ τD

βb a


αminus 11113960 1113961 0

aDβτa

3 0

(74)


aDβτa

1minus a

31113872 1113873ξσ2 + aD

βτa

1minus a

2minus 2a

31113872 1113873ξσ3 + aD

βτa

3ξσ4 + a2

+ a3

1113872 1113873 aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877 _ξσ0


a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732


1minus α

(75)

Let

ξ0 1

ξ1 a1

ξ2 0

ξ3 a3

ξ4 0

Gσ

0

(76)


I a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877


const(77)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

1113876 1113877


const(78)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

const (79)



SP 1Γ(α)

1113946b

aa3

aDβτa

1+ a

4aD

βτa

2minus12

a3

1113872 11138732

minus12

a4

1113872 11138732


1113874 11138751113882 1113883dτ (80)

where B (12)(a3)2 + (12)(a4)2 R1 a3 R2

a4 andR3 R4 0



τDβb a


1113874 11138751113876 1113877 0

τDβb a


1113874 11138751113876 1113877 0

aDβτa

1minus a

3 0

aDβτa

2minus a

4 0

(81)


aDβτa

1minus a

31113872 1113873ξ3 + aD

βτa

2minus a

41113872 1113873ξ4 + a

3aD

βτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113872 1113873

+ a4

aDβτξ

σ2 minus aD

βτ _a

2ξσ01113872 1113873 + aDβτ _a

2ξσ01113872 1113873 + a3

aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875 _ξσ0

minusα minus 1


aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875ξσ0


(82)

Let

ξ0 0

ξ1 a2

ξ2 minus a1

ξ3 ξ4 0

Gσ

0

(83)


I 1113946τ

aa3

aDβτa


1113874 11138751113882 1113883

minus a4

aDβτa


1113874 1113875

minus a2

sDβb a


1113874 11138751113876 1113877

+a1

sDβb a


1113874 11138751113876 11138771113883ds

const

(84)



I a3a2

minus a4a1


1113874 1113875 const (85)



I a3a2

minus a4a1

const (86)


6 Conclusions






Data Availability




Acknowledgments


References






































































τ τ + Δτ

aμ(τ) a

μ(τ) + Δaμ

(μ 1 2 2n)(6)



]( 1113857

aμ(τ) a

μ(τ) + εσξ

σμ τ a

]( 1113857 (μ 1 2 2n)

(7)



SR(c) 1Γ(α)

1113946b

aRμ τ a

]( 1113857 aD

βτa

μminus B τ a

]( 11138571113876 1113877(t minus τ)

αminus 1dτ

(8)

And we have

SR(c) minus SR(c)

1Γ(α)

1113946b

aRμ τ a

]( 1113857 aD

βτa

μminus B τ a

]( 11138571113876 1113877


minus1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113960 1113961


1Γ(α)

1113946b

aRμ τ + Δτ a

]+ Δa]

( 11138571113960 11139611113966 1113967

aDβτa

μ+ aD

βτΔa

μminus aD

βτ _a


μ1113872 1113873


+ Δa]( 11138571113859


ddτΔτ1113888 1113889

minus Rμ τ a]

( 1113857aDβτa

μminus B τ a

]( 11138571113960 1113961(t minus τ)

αminus 11113967dτ

(9)


ΔSR 1Γ(α)

1113946b

a

zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889(t minus τ)

αminus 1Δa]1113896 1113897

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889(t minus τ)

αminus 1Δτ

+ RμaDβτa


αminus 1 ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a


μ1113872 1113873(t minus τ)

αminus 1

minus RμaDβτa



(10)

Since

δaμ

Δaμminus _a

μΔτ

Δ _aμ

ddτΔaμ


(Δτ)

(11)

then we obtain

ΔSR 1Γ(α)

1113946b

a

ddτ

RμaDβτa


αminus 1Δτ11139601113896

+ 1113946τ

aRμaD

βτδa

μ(t minus s)

αminus 1minus δa

μsD

βb Rμ(t minus s)

αminus 11113872 11138731113872 1113873ds1113877

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)

αminus 11113872 11138731113890 1113891δa

μ1113897dτ

(12)


ΔSR 1Γ(α)

1113946b

aεσ

ddτ

RμaDβτa


αminus 1ξσ011139601113896

+ 1113946τ

a




βb Rμ(t minus s)

αminus 11113872 11138731113874 1113875ds1113877

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873dτ

(13)




SE 1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a


αminus 1dτ

(14)


δSE 0 (15)


δaDβτa

μ aD

βτδa

μ (16)


aμ|τa a

μ1

aμ|τb a

μ2

(17)


zR]za

μaDβτa

]minus

zB


αminus 1


αminus 11113960 1113961 0 (μ 1 2 2n)

(18)



SE(c) 1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a


αminus 1dτ

(19)

and we have

SE(c) minus SE(c)

1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113876 1113877


minus1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113960 1113961


1Γ(α)

1113946b

aRμ τ + Δτ a

]+ Δa]

( 11138571113960 11139611113966 1113967

aDβτa

μ+ aD

βτΔa

μminus aD

βτ _a


μ1113872 1113873


+ Δa]( 11138571113859


ddτΔτ1113888 1113889

minus Rμ τ a]

( 1113857aDβτa

μminus B τ a


αminus 11113967dτ

(20)


ΔSE 1Γα

middot 1113946b

a

zRμ

za] aD

βτa

μminus

zB


1113896

+zRμ

zτ aDβτa

μminus

zB


+ RμaDβτa


dτΔτ

+ RμaDβτΔa

μminus aD

βτ _a




⎭dτ

(21)


ΔSE 1Γα

middot 1113946b

a

d

dτRμaD

βτa


+ 1113946τ

aRμaD

βτδa


minus δaμ


αminus 1ds1113877

+zR]za

μaDβτa

]minus

zB


1113890


μ⎫⎬

⎭dτ

(22)


ΔSE 1Γα

middot 1113946b

aεσ

d

dτRμaD

βτa


+ 1113946τ

aRμaD

βs ξ

σμminus _a




αminus 1ds1113877

+zR]za

μaDβτa

]minus

zB


1113890


μξσ0⎫⎬

⎭dτ

(23)




SP 1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113960 1113961sin1113966


1113874 11138751113883dτ

(24)


δSP 0 (25)


δaDβτa

μ aD

βτδa

μ (26)


aμ|τa a

μ1

aμ|τb a

μ2

(27)


zR]za

μaDβτa

]minus

zB


π2

1113874 1113875


π2

1113874 11138751113876 1113877 0 (μ 1 2 2n)

(28)



SP(c) 1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113876 1113877sin


1113874 1113875dτ

(29)

and we have

SP(c) minus SP(c)

1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113876 1113877


1113874 1113875dτ

minus1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113960 1113961


1113874 1113875dτ

1Γ(α)

1113946b

aRμ τ + Δτ a

]+ Δa]

( 11138571113960 11139611113966 1113967

aDβτa

μ+ aD

βτΔa

μ1113872 1113873

minus aDβτ _a


μ1113872 1113873


+ Δa]( 11138571113859


1113874 1113875

1 +ddτΔτ1113888 1113889

minus Rμ τ a]

( 1113857aDβτa

μminus B τ a

]( 11138571113960 1113961


1113874 11138751113883dτ

(30)

So we have

ΔSP 1Γ(α)

1113946b

a

zRμ

za] aD

βτa

μminus

zB


π2

1113874 1113875Δa]+

zRμ

zτ aD

βτa

μminus

zB


π2

1113874 1113875Δτ1113896

+ RμaDβτa


π2

1113874 1113875ddτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a



π2

1113874 1113875


1113874 11138751113883dτ

(31)



ΔSP 1Γ(α)

1113946b

a

ddτ

RμaDβτa


π2

1113874 1113875Δτ + 1113946τ

aRμaD

βs δa


1113874 1113875111387411138761113896

minus δaμ


π2

1113874 11138751113874 11138751113875ds1113877 +zR]za

μaDβτa

]minus

zB


π2

1113874 11138751113890


π2

1113874 11138751113874 11138751113877δaμ1113883dτ

(32)


ΔSP 1Γ(α)

1113946b

aεσ

ddτ

RμaDβτa


π2

1113874 1113875ξσ011138761113896

+ 1113946τ

aRμaD

βs ξσμ minus _a




π2

1113874 11138751113874 11138751113874 11138751113877ds

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891 ξσμ minus _aμξσ01113872 11138731113897dτ

(33)






ΔSR 0 (34)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873


(35)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(36)




ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds1113882 1113883

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873 0 (σ 1 2 r)

(37)



ΔSR minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (38)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ

+ RμaDβτa

μminus B1113872 1113873

ddτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113960 1113961

minus RμaDβτa

μminus B1113872 1113873


ddτ


(39)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0

+ RμaDβτa

μminus B1113872 1113873 _ξ

σ0 + Rμ aD

βτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873

minus RμaDβτa

μminus B1113872 1113873




(40)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds1113882 1113883

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873 minus _Gσ (σ 1 2 r)

(41)





ΔSE 0 (42)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113872 1113873 + RμaDβτa

μminus B1113872 1113873


Δτ 0

(43)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873 + RμaDβτa

μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(44)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a




+zR]za

μaDβτa

]minus

zB


αminus 1+ τD



μξσ01113872 1113873 0 (σ 1 2 r)

(45)



ΔSE minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (46)





zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ

+ RμaDβτa

μminus B1113872 1113873

d

dτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113872 1113873

+ RμaDβτa

μminus B1113872 1113873


Δτ minusd


1minus α

(47)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0

+ RμaDβτa

μminus B1113872 1113873 _ξ

σ0 + Rμ aD

βτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873

+ RμaDβτa

μminus B1113872 1113873




(48)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a




+zR]za

μaDβτa

]minus

zB


αminus 1+ τD



μξσ01113872 1113873 minus _Gσ (σ 1 2 r)

(49)




ΔSP 0 (50)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873


Δτ 0

(51)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(52)



ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


π2

1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891 ξσμ minus _aμξσ01113872 1113873 0 (σ 1 2 r)

(53)



ΔSP minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (54)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873




(55)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 minus_Gσ


(56)



ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891


σ (σ 1 2 r)

(57)






Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds c

σ (σ 1 2 r)

(58)


1Γ(α)

1113946b

aεσ

ddτ

RμaDβτa



a




βb Rμ(t minus s)

αminus 11113872 11138731113874 1113875ds1113876 11138771113896

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873dτ1113897 0

(59)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113872 11138731113872 1113873ds1113876 1113877 0 (60)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds

+ Gσ

cσ (σ 1 2 r)

(61)





Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds c

σ (σ 1 2 r)

(62)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds + G

σ c

σ (σ 1 2 r)

(63)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 11138751113876



π2

1113874 11138751113876 1113877]ds cσ (σ 1 2 r)

(64)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD



1113874 11138751113876



π2

1113874 11138751113876 11138771113877ds + Gσ

cσ (σ 1 2 r)

(65)



5 Examples


SR 1Γ(α)

1113946b

aa2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113966 1113967(t minus τ)αminus 1

dτ

(66)



τDβb a

2(t minus τ)

αminus 11113960 1113961 0

aDβτa

1minus a

3 0


αminus 1+ τD

βb a

4(t minus τ)

αminus 11113960 1113961 0

aDβτa

3 0

(67)


aDβτa

1minus a

31113872 1113873ξσ2 minus a

2ξσ3 + aDβτa

3ξσ4 + a2

aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113960 1113961 _ξσ0 minus


middot a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3


1minus α

(68)

Let

ξσ0 1

ξσ1 a1

ξσ2 1

ξσ3 a3

ξσ4 1

Gσ

0

(69)


I a2

aDβτa

1+ a

4aD

βτa

3minus a

3a2

1113872 1113873(t minus τ)αminus 1

const (70)



I a2a1

+ a4a3

minus a2a3

1113872 1113873(t minus τ)αminus 1

const (71)



I a2a1

+ a4a3

minus a2a3

const (72)




SE 1Γ(α)

1113946b

aa2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

+ a3

1113872 11138732


where B a2a3 + (a3)2 R1 a2 + a3 R2 0 R3


τDβb a

2+ a


αminus 11113960 1113961 0

aDβτa

1minus a

3 0

aDβτa

1minus a

2minus 2a


αminus 1+ τD

βb a


αminus 11113960 1113961 0

aDβτa

3 0

(74)


aDβτa

1minus a

31113872 1113873ξσ2 + aD

βτa

1minus a

2minus 2a

31113872 1113873ξσ3 + aD

βτa

3ξσ4 + a2

+ a3

1113872 1113873 aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877 _ξσ0


a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732


1minus α

(75)

Let

ξ0 1

ξ1 a1

ξ2 0

ξ3 a3

ξ4 0

Gσ

0

(76)


I a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877


const(77)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

1113876 1113877


const(78)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

const (79)



SP 1Γ(α)

1113946b

aa3

aDβτa

1+ a

4aD

βτa

2minus12

a3

1113872 11138732

minus12

a4

1113872 11138732


1113874 11138751113882 1113883dτ (80)

where B (12)(a3)2 + (12)(a4)2 R1 a3 R2

a4 andR3 R4 0



τDβb a


1113874 11138751113876 1113877 0

τDβb a


1113874 11138751113876 1113877 0

aDβτa

1minus a

3 0

aDβτa

2minus a

4 0

(81)


aDβτa

1minus a

31113872 1113873ξ3 + aD

βτa

2minus a

41113872 1113873ξ4 + a

3aD

βτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113872 1113873

+ a4

aDβτξ

σ2 minus aD

βτ _a

2ξσ01113872 1113873 + aDβτ _a

2ξσ01113872 1113873 + a3

aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875 _ξσ0

minusα minus 1


aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875ξσ0


(82)

Let

ξ0 0

ξ1 a2

ξ2 minus a1

ξ3 ξ4 0

Gσ

0

(83)


I 1113946τ

aa3

aDβτa


1113874 11138751113882 1113883

minus a4

aDβτa


1113874 1113875

minus a2

sDβb a


1113874 11138751113876 1113877

+a1

sDβb a


1113874 11138751113876 11138771113883ds

const

(84)



I a3a2

minus a4a1


1113874 1113875 const (85)



I a3a2

minus a4a1

const (86)


6 Conclusions






Data Availability




Acknowledgments


References








































































SE 1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a


αminus 1dτ

(14)


δSE 0 (15)


δaDβτa

μ aD

βτδa

μ (16)


aμ|τa a

μ1

aμ|τb a

μ2

(17)


zR]za

μaDβτa

]minus

zB


αminus 1


αminus 11113960 1113961 0 (μ 1 2 2n)

(18)



SE(c) 1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a


αminus 1dτ

(19)

and we have

SE(c) minus SE(c)

1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113876 1113877


minus1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113960 1113961


1Γ(α)

1113946b

aRμ τ + Δτ a

]+ Δa]

( 11138571113960 11139611113966 1113967

aDβτa

μ+ aD

βτΔa

μminus aD

βτ _a


μ1113872 1113873


+ Δa]( 11138571113859


ddτΔτ1113888 1113889

minus Rμ τ a]

( 1113857aDβτa

μminus B τ a


αminus 11113967dτ

(20)


ΔSE 1Γα

middot 1113946b

a

zRμ

za] aD

βτa

μminus

zB


1113896

+zRμ

zτ aDβτa

μminus

zB


+ RμaDβτa


dτΔτ

+ RμaDβτΔa

μminus aD

βτ _a




⎭dτ

(21)


ΔSE 1Γα

middot 1113946b

a

d

dτRμaD

βτa


+ 1113946τ

aRμaD

βτδa


minus δaμ


αminus 1ds1113877

+zR]za

μaDβτa

]minus

zB


1113890


μ⎫⎬

⎭dτ

(22)


ΔSE 1Γα

middot 1113946b

aεσ

d

dτRμaD

βτa


+ 1113946τ

aRμaD

βs ξ

σμminus _a




αminus 1ds1113877

+zR]za

μaDβτa

]minus

zB


1113890


μξσ0⎫⎬

⎭dτ

(23)




SP 1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113960 1113961sin1113966


1113874 11138751113883dτ

(24)


δSP 0 (25)


δaDβτa

μ aD

βτδa

μ (26)


aμ|τa a

μ1

aμ|τb a

μ2

(27)


zR]za

μaDβτa

]minus

zB


π2

1113874 1113875


π2

1113874 11138751113876 1113877 0 (μ 1 2 2n)

(28)



SP(c) 1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113876 1113877sin


1113874 1113875dτ

(29)

and we have

SP(c) minus SP(c)

1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113876 1113877


1113874 1113875dτ

minus1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113960 1113961


1113874 1113875dτ

1Γ(α)

1113946b

aRμ τ + Δτ a

]+ Δa]

( 11138571113960 11139611113966 1113967

aDβτa

μ+ aD

βτΔa

μ1113872 1113873

minus aDβτ _a


μ1113872 1113873


+ Δa]( 11138571113859


1113874 1113875

1 +ddτΔτ1113888 1113889

minus Rμ τ a]

( 1113857aDβτa

μminus B τ a

]( 11138571113960 1113961


1113874 11138751113883dτ

(30)

So we have

ΔSP 1Γ(α)

1113946b

a

zRμ

za] aD

βτa

μminus

zB


π2

1113874 1113875Δa]+

zRμ

zτ aD

βτa

μminus

zB


π2

1113874 1113875Δτ1113896

+ RμaDβτa


π2

1113874 1113875ddτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a



π2

1113874 1113875


1113874 11138751113883dτ

(31)



ΔSP 1Γ(α)

1113946b

a

ddτ

RμaDβτa


π2

1113874 1113875Δτ + 1113946τ

aRμaD

βs δa


1113874 1113875111387411138761113896

minus δaμ


π2

1113874 11138751113874 11138751113875ds1113877 +zR]za

μaDβτa

]minus

zB


π2

1113874 11138751113890


π2

1113874 11138751113874 11138751113877δaμ1113883dτ

(32)


ΔSP 1Γ(α)

1113946b

aεσ

ddτ

RμaDβτa


π2

1113874 1113875ξσ011138761113896

+ 1113946τ

aRμaD

βs ξσμ minus _a




π2

1113874 11138751113874 11138751113874 11138751113877ds

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891 ξσμ minus _aμξσ01113872 11138731113897dτ

(33)






ΔSR 0 (34)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873


(35)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(36)




ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds1113882 1113883

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873 0 (σ 1 2 r)

(37)



ΔSR minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (38)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ

+ RμaDβτa

μminus B1113872 1113873

ddτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113960 1113961

minus RμaDβτa

μminus B1113872 1113873


ddτ


(39)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0

+ RμaDβτa

μminus B1113872 1113873 _ξ

σ0 + Rμ aD

βτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873

minus RμaDβτa

μminus B1113872 1113873




(40)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds1113882 1113883

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873 minus _Gσ (σ 1 2 r)

(41)





ΔSE 0 (42)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113872 1113873 + RμaDβτa

μminus B1113872 1113873


Δτ 0

(43)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873 + RμaDβτa

μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(44)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a




+zR]za

μaDβτa

]minus

zB


αminus 1+ τD



μξσ01113872 1113873 0 (σ 1 2 r)

(45)



ΔSE minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (46)





zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ

+ RμaDβτa

μminus B1113872 1113873

d

dτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113872 1113873

+ RμaDβτa

μminus B1113872 1113873


Δτ minusd


1minus α

(47)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0

+ RμaDβτa

μminus B1113872 1113873 _ξ

σ0 + Rμ aD

βτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873

+ RμaDβτa

μminus B1113872 1113873




(48)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a




+zR]za

μaDβτa

]minus

zB


αminus 1+ τD



μξσ01113872 1113873 minus _Gσ (σ 1 2 r)

(49)




ΔSP 0 (50)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873


Δτ 0

(51)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(52)



ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


π2

1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891 ξσμ minus _aμξσ01113872 1113873 0 (σ 1 2 r)

(53)



ΔSP minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (54)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873




(55)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 minus_Gσ


(56)



ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891


σ (σ 1 2 r)

(57)






Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds c

σ (σ 1 2 r)

(58)


1Γ(α)

1113946b

aεσ

ddτ

RμaDβτa



a




βb Rμ(t minus s)

αminus 11113872 11138731113874 1113875ds1113876 11138771113896

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873dτ1113897 0

(59)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113872 11138731113872 1113873ds1113876 1113877 0 (60)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds

+ Gσ

cσ (σ 1 2 r)

(61)





Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds c

σ (σ 1 2 r)

(62)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds + G

σ c

σ (σ 1 2 r)

(63)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 11138751113876



π2

1113874 11138751113876 1113877]ds cσ (σ 1 2 r)

(64)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD



1113874 11138751113876



π2

1113874 11138751113876 11138771113877ds + Gσ

cσ (σ 1 2 r)

(65)



5 Examples


SR 1Γ(α)

1113946b

aa2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113966 1113967(t minus τ)αminus 1

dτ

(66)



τDβb a

2(t minus τ)

αminus 11113960 1113961 0

aDβτa

1minus a

3 0


αminus 1+ τD

βb a

4(t minus τ)

αminus 11113960 1113961 0

aDβτa

3 0

(67)


aDβτa

1minus a

31113872 1113873ξσ2 minus a

2ξσ3 + aDβτa

3ξσ4 + a2

aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113960 1113961 _ξσ0 minus


middot a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3


1minus α

(68)

Let

ξσ0 1

ξσ1 a1

ξσ2 1

ξσ3 a3

ξσ4 1

Gσ

0

(69)


I a2

aDβτa

1+ a

4aD

βτa

3minus a

3a2

1113872 1113873(t minus τ)αminus 1

const (70)



I a2a1

+ a4a3

minus a2a3

1113872 1113873(t minus τ)αminus 1

const (71)



I a2a1

+ a4a3

minus a2a3

const (72)




SE 1Γ(α)

1113946b

aa2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

+ a3

1113872 11138732


where B a2a3 + (a3)2 R1 a2 + a3 R2 0 R3


τDβb a

2+ a


αminus 11113960 1113961 0

aDβτa

1minus a

3 0

aDβτa

1minus a

2minus 2a


αminus 1+ τD

βb a


αminus 11113960 1113961 0

aDβτa

3 0

(74)


aDβτa

1minus a

31113872 1113873ξσ2 + aD

βτa

1minus a

2minus 2a

31113872 1113873ξσ3 + aD

βτa

3ξσ4 + a2

+ a3

1113872 1113873 aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877 _ξσ0


a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732


1minus α

(75)

Let

ξ0 1

ξ1 a1

ξ2 0

ξ3 a3

ξ4 0

Gσ

0

(76)


I a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877


const(77)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

1113876 1113877


const(78)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

const (79)



SP 1Γ(α)

1113946b

aa3

aDβτa

1+ a

4aD

βτa

2minus12

a3

1113872 11138732

minus12

a4

1113872 11138732


1113874 11138751113882 1113883dτ (80)

where B (12)(a3)2 + (12)(a4)2 R1 a3 R2

a4 andR3 R4 0



τDβb a


1113874 11138751113876 1113877 0

τDβb a


1113874 11138751113876 1113877 0

aDβτa

1minus a

3 0

aDβτa

2minus a

4 0

(81)


aDβτa

1minus a

31113872 1113873ξ3 + aD

βτa

2minus a

41113872 1113873ξ4 + a

3aD

βτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113872 1113873

+ a4

aDβτξ

σ2 minus aD

βτ _a

2ξσ01113872 1113873 + aDβτ _a

2ξσ01113872 1113873 + a3

aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875 _ξσ0

minusα minus 1


aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875ξσ0


(82)

Let

ξ0 0

ξ1 a2

ξ2 minus a1

ξ3 ξ4 0

Gσ

0

(83)


I 1113946τ

aa3

aDβτa


1113874 11138751113882 1113883

minus a4

aDβτa


1113874 1113875

minus a2

sDβb a


1113874 11138751113876 1113877

+a1

sDβb a


1113874 11138751113876 11138771113883ds

const

(84)



I a3a2

minus a4a1


1113874 1113875 const (85)



I a3a2

minus a4a1

const (86)


6 Conclusions






Data Availability




Acknowledgments


References







































































SP 1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113960 1113961sin1113966


1113874 11138751113883dτ

(24)


δSP 0 (25)


δaDβτa

μ aD

βτδa

μ (26)


aμ|τa a

μ1

aμ|τb a

μ2

(27)


zR]za

μaDβτa

]minus

zB


π2

1113874 1113875


π2

1113874 11138751113876 1113877 0 (μ 1 2 2n)

(28)



SP(c) 1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113876 1113877sin


1113874 1113875dτ

(29)

and we have

SP(c) minus SP(c)

1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113876 1113877


1113874 1113875dτ

minus1Γ(α)

1113946b

aRμ τ a

]( 1113857aD

βτa

μminus B τ a

]( 11138571113960 1113961


1113874 1113875dτ

1Γ(α)

1113946b

aRμ τ + Δτ a

]+ Δa]

( 11138571113960 11139611113966 1113967

aDβτa

μ+ aD

βτΔa

μ1113872 1113873

minus aDβτ _a


μ1113872 1113873


+ Δa]( 11138571113859


1113874 1113875

1 +ddτΔτ1113888 1113889

minus Rμ τ a]

( 1113857aDβτa

μminus B τ a

]( 11138571113960 1113961


1113874 11138751113883dτ

(30)

So we have

ΔSP 1Γ(α)

1113946b

a

zRμ

za] aD

βτa

μminus

zB


π2

1113874 1113875Δa]+

zRμ

zτ aD

βτa

μminus

zB


π2

1113874 1113875Δτ1113896

+ RμaDβτa


π2

1113874 1113875ddτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a



π2

1113874 1113875


1113874 11138751113883dτ

(31)



ΔSP 1Γ(α)

1113946b

a

ddτ

RμaDβτa


π2

1113874 1113875Δτ + 1113946τ

aRμaD

βs δa


1113874 1113875111387411138761113896

minus δaμ


π2

1113874 11138751113874 11138751113875ds1113877 +zR]za

μaDβτa

]minus

zB


π2

1113874 11138751113890


π2

1113874 11138751113874 11138751113877δaμ1113883dτ

(32)


ΔSP 1Γ(α)

1113946b

aεσ

ddτ

RμaDβτa


π2

1113874 1113875ξσ011138761113896

+ 1113946τ

aRμaD

βs ξσμ minus _a




π2

1113874 11138751113874 11138751113874 11138751113877ds

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891 ξσμ minus _aμξσ01113872 11138731113897dτ

(33)






ΔSR 0 (34)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873


(35)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(36)




ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds1113882 1113883

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873 0 (σ 1 2 r)

(37)



ΔSR minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (38)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ

+ RμaDβτa

μminus B1113872 1113873

ddτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113960 1113961

minus RμaDβτa

μminus B1113872 1113873


ddτ


(39)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0

+ RμaDβτa

μminus B1113872 1113873 _ξ

σ0 + Rμ aD

βτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873

minus RμaDβτa

μminus B1113872 1113873




(40)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds1113882 1113883

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873 minus _Gσ (σ 1 2 r)

(41)





ΔSE 0 (42)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113872 1113873 + RμaDβτa

μminus B1113872 1113873


Δτ 0

(43)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873 + RμaDβτa

μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(44)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a




+zR]za

μaDβτa

]minus

zB


αminus 1+ τD



μξσ01113872 1113873 0 (σ 1 2 r)

(45)



ΔSE minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (46)





zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ

+ RμaDβτa

μminus B1113872 1113873

d

dτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113872 1113873

+ RμaDβτa

μminus B1113872 1113873


Δτ minusd


1minus α

(47)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0

+ RμaDβτa

μminus B1113872 1113873 _ξ

σ0 + Rμ aD

βτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873

+ RμaDβτa

μminus B1113872 1113873




(48)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a




+zR]za

μaDβτa

]minus

zB


αminus 1+ τD



μξσ01113872 1113873 minus _Gσ (σ 1 2 r)

(49)




ΔSP 0 (50)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873


Δτ 0

(51)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(52)



ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


π2

1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891 ξσμ minus _aμξσ01113872 1113873 0 (σ 1 2 r)

(53)



ΔSP minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (54)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873




(55)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 minus_Gσ


(56)



ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891


σ (σ 1 2 r)

(57)






Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds c

σ (σ 1 2 r)

(58)


1Γ(α)

1113946b

aεσ

ddτ

RμaDβτa



a




βb Rμ(t minus s)

αminus 11113872 11138731113874 1113875ds1113876 11138771113896

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873dτ1113897 0

(59)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113872 11138731113872 1113873ds1113876 1113877 0 (60)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds

+ Gσ

cσ (σ 1 2 r)

(61)





Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds c

σ (σ 1 2 r)

(62)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds + G

σ c

σ (σ 1 2 r)

(63)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 11138751113876



π2

1113874 11138751113876 1113877]ds cσ (σ 1 2 r)

(64)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD



1113874 11138751113876



π2

1113874 11138751113876 11138771113877ds + Gσ

cσ (σ 1 2 r)

(65)



5 Examples


SR 1Γ(α)

1113946b

aa2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113966 1113967(t minus τ)αminus 1

dτ

(66)



τDβb a

2(t minus τ)

αminus 11113960 1113961 0

aDβτa

1minus a

3 0


αminus 1+ τD

βb a

4(t minus τ)

αminus 11113960 1113961 0

aDβτa

3 0

(67)


aDβτa

1minus a

31113872 1113873ξσ2 minus a

2ξσ3 + aDβτa

3ξσ4 + a2

aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113960 1113961 _ξσ0 minus


middot a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3


1minus α

(68)

Let

ξσ0 1

ξσ1 a1

ξσ2 1

ξσ3 a3

ξσ4 1

Gσ

0

(69)


I a2

aDβτa

1+ a

4aD

βτa

3minus a

3a2

1113872 1113873(t minus τ)αminus 1

const (70)



I a2a1

+ a4a3

minus a2a3

1113872 1113873(t minus τ)αminus 1

const (71)



I a2a1

+ a4a3

minus a2a3

const (72)




SE 1Γ(α)

1113946b

aa2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

+ a3

1113872 11138732


where B a2a3 + (a3)2 R1 a2 + a3 R2 0 R3


τDβb a

2+ a


αminus 11113960 1113961 0

aDβτa

1minus a

3 0

aDβτa

1minus a

2minus 2a


αminus 1+ τD

βb a


αminus 11113960 1113961 0

aDβτa

3 0

(74)


aDβτa

1minus a

31113872 1113873ξσ2 + aD

βτa

1minus a

2minus 2a

31113872 1113873ξσ3 + aD

βτa

3ξσ4 + a2

+ a3

1113872 1113873 aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877 _ξσ0


a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732


1minus α

(75)

Let

ξ0 1

ξ1 a1

ξ2 0

ξ3 a3

ξ4 0

Gσ

0

(76)


I a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877


const(77)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

1113876 1113877


const(78)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

const (79)



SP 1Γ(α)

1113946b

aa3

aDβτa

1+ a

4aD

βτa

2minus12

a3

1113872 11138732

minus12

a4

1113872 11138732


1113874 11138751113882 1113883dτ (80)

where B (12)(a3)2 + (12)(a4)2 R1 a3 R2

a4 andR3 R4 0



τDβb a


1113874 11138751113876 1113877 0

τDβb a


1113874 11138751113876 1113877 0

aDβτa

1minus a

3 0

aDβτa

2minus a

4 0

(81)


aDβτa

1minus a

31113872 1113873ξ3 + aD

βτa

2minus a

41113872 1113873ξ4 + a

3aD

βτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113872 1113873

+ a4

aDβτξ

σ2 minus aD

βτ _a

2ξσ01113872 1113873 + aDβτ _a

2ξσ01113872 1113873 + a3

aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875 _ξσ0

minusα minus 1


aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875ξσ0


(82)

Let

ξ0 0

ξ1 a2

ξ2 minus a1

ξ3 ξ4 0

Gσ

0

(83)


I 1113946τ

aa3

aDβτa


1113874 11138751113882 1113883

minus a4

aDβτa


1113874 1113875

minus a2

sDβb a


1113874 11138751113876 1113877

+a1

sDβb a


1113874 11138751113876 11138771113883ds

const

(84)



I a3a2

minus a4a1


1113874 1113875 const (85)



I a3a2

minus a4a1

const (86)


6 Conclusions






Data Availability




Acknowledgments


References







































































ΔSP 1Γ(α)

1113946b

a

ddτ

RμaDβτa


π2

1113874 1113875Δτ + 1113946τ

aRμaD

βs δa


1113874 1113875111387411138761113896

minus δaμ


π2

1113874 11138751113874 11138751113875ds1113877 +zR]za

μaDβτa

]minus

zB


π2

1113874 11138751113890


π2

1113874 11138751113874 11138751113877δaμ1113883dτ

(32)


ΔSP 1Γ(α)

1113946b

aεσ

ddτ

RμaDβτa


π2

1113874 1113875ξσ011138761113896

+ 1113946τ

aRμaD

βs ξσμ minus _a




π2

1113874 11138751113874 11138751113874 11138751113877ds

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891 ξσμ minus _aμξσ01113872 11138731113897dτ

(33)






ΔSR 0 (34)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873


(35)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(36)




ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds1113882 1113883

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873 0 (σ 1 2 r)

(37)



ΔSR minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (38)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ

+ RμaDβτa

μminus B1113872 1113873

ddτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113960 1113961

minus RμaDβτa

μminus B1113872 1113873


ddτ


(39)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0

+ RμaDβτa

μminus B1113872 1113873 _ξ

σ0 + Rμ aD

βτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873

minus RμaDβτa

μminus B1113872 1113873




(40)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds1113882 1113883

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873 minus _Gσ (σ 1 2 r)

(41)





ΔSE 0 (42)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113872 1113873 + RμaDβτa

μminus B1113872 1113873


Δτ 0

(43)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873 + RμaDβτa

μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(44)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a




+zR]za

μaDβτa

]minus

zB


αminus 1+ τD



μξσ01113872 1113873 0 (σ 1 2 r)

(45)



ΔSE minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (46)





zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ

+ RμaDβτa

μminus B1113872 1113873

d

dτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113872 1113873

+ RμaDβτa

μminus B1113872 1113873


Δτ minusd


1minus α

(47)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0

+ RμaDβτa

μminus B1113872 1113873 _ξ

σ0 + Rμ aD

βτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873

+ RμaDβτa

μminus B1113872 1113873




(48)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a




+zR]za

μaDβτa

]minus

zB


αminus 1+ τD



μξσ01113872 1113873 minus _Gσ (σ 1 2 r)

(49)




ΔSP 0 (50)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873


Δτ 0

(51)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(52)



ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


π2

1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891 ξσμ minus _aμξσ01113872 1113873 0 (σ 1 2 r)

(53)



ΔSP minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (54)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873




(55)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 minus_Gσ


(56)



ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891


σ (σ 1 2 r)

(57)






Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds c

σ (σ 1 2 r)

(58)


1Γ(α)

1113946b

aεσ

ddτ

RμaDβτa



a




βb Rμ(t minus s)

αminus 11113872 11138731113874 1113875ds1113876 11138771113896

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873dτ1113897 0

(59)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113872 11138731113872 1113873ds1113876 1113877 0 (60)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds

+ Gσ

cσ (σ 1 2 r)

(61)





Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds c

σ (σ 1 2 r)

(62)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds + G

σ c

σ (σ 1 2 r)

(63)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 11138751113876



π2

1113874 11138751113876 1113877]ds cσ (σ 1 2 r)

(64)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD



1113874 11138751113876



π2

1113874 11138751113876 11138771113877ds + Gσ

cσ (σ 1 2 r)

(65)



5 Examples


SR 1Γ(α)

1113946b

aa2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113966 1113967(t minus τ)αminus 1

dτ

(66)



τDβb a

2(t minus τ)

αminus 11113960 1113961 0

aDβτa

1minus a

3 0


αminus 1+ τD

βb a

4(t minus τ)

αminus 11113960 1113961 0

aDβτa

3 0

(67)


aDβτa

1minus a

31113872 1113873ξσ2 minus a

2ξσ3 + aDβτa

3ξσ4 + a2

aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113960 1113961 _ξσ0 minus


middot a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3


1minus α

(68)

Let

ξσ0 1

ξσ1 a1

ξσ2 1

ξσ3 a3

ξσ4 1

Gσ

0

(69)


I a2

aDβτa

1+ a

4aD

βτa

3minus a

3a2

1113872 1113873(t minus τ)αminus 1

const (70)



I a2a1

+ a4a3

minus a2a3

1113872 1113873(t minus τ)αminus 1

const (71)



I a2a1

+ a4a3

minus a2a3

const (72)




SE 1Γ(α)

1113946b

aa2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

+ a3

1113872 11138732


where B a2a3 + (a3)2 R1 a2 + a3 R2 0 R3


τDβb a

2+ a


αminus 11113960 1113961 0

aDβτa

1minus a

3 0

aDβτa

1minus a

2minus 2a


αminus 1+ τD

βb a


αminus 11113960 1113961 0

aDβτa

3 0

(74)


aDβτa

1minus a

31113872 1113873ξσ2 + aD

βτa

1minus a

2minus 2a

31113872 1113873ξσ3 + aD

βτa

3ξσ4 + a2

+ a3

1113872 1113873 aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877 _ξσ0


a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732


1minus α

(75)

Let

ξ0 1

ξ1 a1

ξ2 0

ξ3 a3

ξ4 0

Gσ

0

(76)


I a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877


const(77)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

1113876 1113877


const(78)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

const (79)



SP 1Γ(α)

1113946b

aa3

aDβτa

1+ a

4aD

βτa

2minus12

a3

1113872 11138732

minus12

a4

1113872 11138732


1113874 11138751113882 1113883dτ (80)

where B (12)(a3)2 + (12)(a4)2 R1 a3 R2

a4 andR3 R4 0



τDβb a


1113874 11138751113876 1113877 0

τDβb a


1113874 11138751113876 1113877 0

aDβτa

1minus a

3 0

aDβτa

2minus a

4 0

(81)


aDβτa

1minus a

31113872 1113873ξ3 + aD

βτa

2minus a

41113872 1113873ξ4 + a

3aD

βτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113872 1113873

+ a4

aDβτξ

σ2 minus aD

βτ _a

2ξσ01113872 1113873 + aDβτ _a

2ξσ01113872 1113873 + a3

aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875 _ξσ0

minusα minus 1


aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875ξσ0


(82)

Let

ξ0 0

ξ1 a2

ξ2 minus a1

ξ3 ξ4 0

Gσ

0

(83)


I 1113946τ

aa3

aDβτa


1113874 11138751113882 1113883

minus a4

aDβτa


1113874 1113875

minus a2

sDβb a


1113874 11138751113876 1113877

+a1

sDβb a


1113874 11138751113876 11138771113883ds

const

(84)



I a3a2

minus a4a1


1113874 1113875 const (85)



I a3a2

minus a4a1

const (86)


6 Conclusions






Data Availability




Acknowledgments


References








































































ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds1113882 1113883

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873 0 (σ 1 2 r)

(37)



ΔSR minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (38)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ

+ RμaDβτa

μminus B1113872 1113873

ddτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113960 1113961

minus RμaDβτa

μminus B1113872 1113873


ddτ


(39)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0

+ RμaDβτa

μminus B1113872 1113873 _ξ

σ0 + Rμ aD

βτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873

minus RμaDβτa

μminus B1113872 1113873




(40)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds1113882 1113883

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873 minus _Gσ (σ 1 2 r)

(41)





ΔSE 0 (42)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113872 1113873 + RμaDβτa

μminus B1113872 1113873


Δτ 0

(43)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873 + RμaDβτa

μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(44)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a




+zR]za

μaDβτa

]minus

zB


αminus 1+ τD



μξσ01113872 1113873 0 (σ 1 2 r)

(45)



ΔSE minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (46)





zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ

+ RμaDβτa

μminus B1113872 1113873

d

dτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113872 1113873

+ RμaDβτa

μminus B1113872 1113873


Δτ minusd


1minus α

(47)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0

+ RμaDβτa

μminus B1113872 1113873 _ξ

σ0 + Rμ aD

βτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873

+ RμaDβτa

μminus B1113872 1113873




(48)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a




+zR]za

μaDβτa

]minus

zB


αminus 1+ τD



μξσ01113872 1113873 minus _Gσ (σ 1 2 r)

(49)




ΔSP 0 (50)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873


Δτ 0

(51)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(52)



ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


π2

1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891 ξσμ minus _aμξσ01113872 1113873 0 (σ 1 2 r)

(53)



ΔSP minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (54)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873




(55)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 minus_Gσ


(56)



ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891


σ (σ 1 2 r)

(57)






Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds c

σ (σ 1 2 r)

(58)


1Γ(α)

1113946b

aεσ

ddτ

RμaDβτa



a




βb Rμ(t minus s)

αminus 11113872 11138731113874 1113875ds1113876 11138771113896

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873dτ1113897 0

(59)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113872 11138731113872 1113873ds1113876 1113877 0 (60)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds

+ Gσ

cσ (σ 1 2 r)

(61)





Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds c

σ (σ 1 2 r)

(62)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds + G

σ c

σ (σ 1 2 r)

(63)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 11138751113876



π2

1113874 11138751113876 1113877]ds cσ (σ 1 2 r)

(64)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD



1113874 11138751113876



π2

1113874 11138751113876 11138771113877ds + Gσ

cσ (σ 1 2 r)

(65)



5 Examples


SR 1Γ(α)

1113946b

aa2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113966 1113967(t minus τ)αminus 1

dτ

(66)



τDβb a

2(t minus τ)

αminus 11113960 1113961 0

aDβτa

1minus a

3 0


αminus 1+ τD

βb a

4(t minus τ)

αminus 11113960 1113961 0

aDβτa

3 0

(67)


aDβτa

1minus a

31113872 1113873ξσ2 minus a

2ξσ3 + aDβτa

3ξσ4 + a2

aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113960 1113961 _ξσ0 minus


middot a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3


1minus α

(68)

Let

ξσ0 1

ξσ1 a1

ξσ2 1

ξσ3 a3

ξσ4 1

Gσ

0

(69)


I a2

aDβτa

1+ a

4aD

βτa

3minus a

3a2

1113872 1113873(t minus τ)αminus 1

const (70)



I a2a1

+ a4a3

minus a2a3

1113872 1113873(t minus τ)αminus 1

const (71)



I a2a1

+ a4a3

minus a2a3

const (72)




SE 1Γ(α)

1113946b

aa2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

+ a3

1113872 11138732


where B a2a3 + (a3)2 R1 a2 + a3 R2 0 R3


τDβb a

2+ a


αminus 11113960 1113961 0

aDβτa

1minus a

3 0

aDβτa

1minus a

2minus 2a


αminus 1+ τD

βb a


αminus 11113960 1113961 0

aDβτa

3 0

(74)


aDβτa

1minus a

31113872 1113873ξσ2 + aD

βτa

1minus a

2minus 2a

31113872 1113873ξσ3 + aD

βτa

3ξσ4 + a2

+ a3

1113872 1113873 aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877 _ξσ0


a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732


1minus α

(75)

Let

ξ0 1

ξ1 a1

ξ2 0

ξ3 a3

ξ4 0

Gσ

0

(76)


I a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877


const(77)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

1113876 1113877


const(78)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

const (79)



SP 1Γ(α)

1113946b

aa3

aDβτa

1+ a

4aD

βτa

2minus12

a3

1113872 11138732

minus12

a4

1113872 11138732


1113874 11138751113882 1113883dτ (80)

where B (12)(a3)2 + (12)(a4)2 R1 a3 R2

a4 andR3 R4 0



τDβb a


1113874 11138751113876 1113877 0

τDβb a


1113874 11138751113876 1113877 0

aDβτa

1minus a

3 0

aDβτa

2minus a

4 0

(81)


aDβτa

1minus a

31113872 1113873ξ3 + aD

βτa

2minus a

41113872 1113873ξ4 + a

3aD

βτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113872 1113873

+ a4

aDβτξ

σ2 minus aD

βτ _a

2ξσ01113872 1113873 + aDβτ _a

2ξσ01113872 1113873 + a3

aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875 _ξσ0

minusα minus 1


aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875ξσ0


(82)

Let

ξ0 0

ξ1 a2

ξ2 minus a1

ξ3 ξ4 0

Gσ

0

(83)


I 1113946τ

aa3

aDβτa


1113874 11138751113882 1113883

minus a4

aDβτa


1113874 1113875

minus a2

sDβb a


1113874 11138751113876 1113877

+a1

sDβb a


1113874 11138751113876 11138771113883ds

const

(84)



I a3a2

minus a4a1


1113874 1113875 const (85)



I a3a2

minus a4a1

const (86)


6 Conclusions






Data Availability




Acknowledgments


References








































































ΔSE 0 (42)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113872 1113873 + RμaDβτa

μminus B1113872 1113873


Δτ 0

(43)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873 + RμaDβτa

μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(44)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a




+zR]za

μaDβτa

]minus

zB


αminus 1+ τD



μξσ01113872 1113873 0 (σ 1 2 r)

(45)



ΔSE minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (46)





zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ

+ RμaDβτa

μminus B1113872 1113873

d

dτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113872 1113873

+ RμaDβτa

μminus B1113872 1113873


Δτ minusd


1minus α

(47)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0

+ RμaDβτa

μminus B1113872 1113873 _ξ

σ0 + Rμ aD

βτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873

+ RμaDβτa

μminus B1113872 1113873




(48)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a




+zR]za

μaDβτa

]minus

zB


αminus 1+ τD



μξσ01113872 1113873 minus _Gσ (σ 1 2 r)

(49)




ΔSP 0 (50)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873


Δτ 0

(51)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(52)



ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


π2

1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891 ξσμ minus _aμξσ01113872 1113873 0 (σ 1 2 r)

(53)



ΔSP minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (54)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873




(55)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 minus_Gσ


(56)



ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891


σ (σ 1 2 r)

(57)






Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds c

σ (σ 1 2 r)

(58)


1Γ(α)

1113946b

aεσ

ddτ

RμaDβτa



a




βb Rμ(t minus s)

αminus 11113872 11138731113874 1113875ds1113876 11138771113896

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873dτ1113897 0

(59)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113872 11138731113872 1113873ds1113876 1113877 0 (60)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds

+ Gσ

cσ (σ 1 2 r)

(61)





Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds c

σ (σ 1 2 r)

(62)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds + G

σ c

σ (σ 1 2 r)

(63)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 11138751113876



π2

1113874 11138751113876 1113877]ds cσ (σ 1 2 r)

(64)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD



1113874 11138751113876



π2

1113874 11138751113876 11138771113877ds + Gσ

cσ (σ 1 2 r)

(65)



5 Examples


SR 1Γ(α)

1113946b

aa2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113966 1113967(t minus τ)αminus 1

dτ

(66)



τDβb a

2(t minus τ)

αminus 11113960 1113961 0

aDβτa

1minus a

3 0


αminus 1+ τD

βb a

4(t minus τ)

αminus 11113960 1113961 0

aDβτa

3 0

(67)


aDβτa

1minus a

31113872 1113873ξσ2 minus a

2ξσ3 + aDβτa

3ξσ4 + a2

aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113960 1113961 _ξσ0 minus


middot a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3


1minus α

(68)

Let

ξσ0 1

ξσ1 a1

ξσ2 1

ξσ3 a3

ξσ4 1

Gσ

0

(69)


I a2

aDβτa

1+ a

4aD

βτa

3minus a

3a2

1113872 1113873(t minus τ)αminus 1

const (70)



I a2a1

+ a4a3

minus a2a3

1113872 1113873(t minus τ)αminus 1

const (71)



I a2a1

+ a4a3

minus a2a3

const (72)




SE 1Γ(α)

1113946b

aa2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

+ a3

1113872 11138732


where B a2a3 + (a3)2 R1 a2 + a3 R2 0 R3


τDβb a

2+ a


αminus 11113960 1113961 0

aDβτa

1minus a

3 0

aDβτa

1minus a

2minus 2a


αminus 1+ τD

βb a


αminus 11113960 1113961 0

aDβτa

3 0

(74)


aDβτa

1minus a

31113872 1113873ξσ2 + aD

βτa

1minus a

2minus 2a

31113872 1113873ξσ3 + aD

βτa

3ξσ4 + a2

+ a3

1113872 1113873 aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877 _ξσ0


a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732


1minus α

(75)

Let

ξ0 1

ξ1 a1

ξ2 0

ξ3 a3

ξ4 0

Gσ

0

(76)


I a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877


const(77)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

1113876 1113877


const(78)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

const (79)



SP 1Γ(α)

1113946b

aa3

aDβτa

1+ a

4aD

βτa

2minus12

a3

1113872 11138732

minus12

a4

1113872 11138732


1113874 11138751113882 1113883dτ (80)

where B (12)(a3)2 + (12)(a4)2 R1 a3 R2

a4 andR3 R4 0



τDβb a


1113874 11138751113876 1113877 0

τDβb a


1113874 11138751113876 1113877 0

aDβτa

1minus a

3 0

aDβτa

2minus a

4 0

(81)


aDβτa

1minus a

31113872 1113873ξ3 + aD

βτa

2minus a

41113872 1113873ξ4 + a

3aD

βτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113872 1113873

+ a4

aDβτξ

σ2 minus aD

βτ _a

2ξσ01113872 1113873 + aDβτ _a

2ξσ01113872 1113873 + a3

aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875 _ξσ0

minusα minus 1


aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875ξσ0


(82)

Let

ξ0 0

ξ1 a2

ξ2 minus a1

ξ3 ξ4 0

Gσ

0

(83)


I 1113946τ

aa3

aDβτa


1113874 11138751113882 1113883

minus a4

aDβτa


1113874 1113875

minus a2

sDβb a


1113874 11138751113876 1113877

+a1

sDβb a


1113874 11138751113876 11138771113883ds

const

(84)



I a3a2

minus a4a1


1113874 1113875 const (85)



I a3a2

minus a4a1

const (86)


6 Conclusions






Data Availability




Acknowledgments


References







































































zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ

+ RμaDβτa

μminus B1113872 1113873

d

dτΔτ + Rμ aD

βτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a

μΔτ1113872 1113873

+ RμaDβτa

μminus B1113872 1113873


Δτ minusd


1minus α

(47)


zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0

+ RμaDβτa

μminus B1113872 1113873 _ξ

σ0 + Rμ aD

βτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a

μξσ01113872 1113873

+ RμaDβτa

μminus B1113872 1113873




(48)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a




+zR]za

μaDβτa

]minus

zB


αminus 1+ τD



μξσ01113872 1113873 minus _Gσ (σ 1 2 r)

(49)




ΔSP 0 (50)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873


Δτ 0

(51)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(52)



ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


π2

1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891 ξσμ minus _aμξσ01113872 1113873 0 (σ 1 2 r)

(53)



ΔSP minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (54)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873




(55)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 minus_Gσ


(56)



ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891


σ (σ 1 2 r)

(57)






Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds c

σ (σ 1 2 r)

(58)


1Γ(α)

1113946b

aεσ

ddτ

RμaDβτa



a




βb Rμ(t minus s)

αminus 11113872 11138731113874 1113875ds1113876 11138771113896

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873dτ1113897 0

(59)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113872 11138731113872 1113873ds1113876 1113877 0 (60)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds

+ Gσ

cσ (σ 1 2 r)

(61)





Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds c

σ (σ 1 2 r)

(62)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds + G

σ c

σ (σ 1 2 r)

(63)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 11138751113876



π2

1113874 11138751113876 1113877]ds cσ (σ 1 2 r)

(64)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD



1113874 11138751113876



π2

1113874 11138751113876 11138771113877ds + Gσ

cσ (σ 1 2 r)

(65)



5 Examples


SR 1Γ(α)

1113946b

aa2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113966 1113967(t minus τ)αminus 1

dτ

(66)



τDβb a

2(t minus τ)

αminus 11113960 1113961 0

aDβτa

1minus a

3 0


αminus 1+ τD

βb a

4(t minus τ)

αminus 11113960 1113961 0

aDβτa

3 0

(67)


aDβτa

1minus a

31113872 1113873ξσ2 minus a

2ξσ3 + aDβτa

3ξσ4 + a2

aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113960 1113961 _ξσ0 minus


middot a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3


1minus α

(68)

Let

ξσ0 1

ξσ1 a1

ξσ2 1

ξσ3 a3

ξσ4 1

Gσ

0

(69)


I a2

aDβτa

1+ a

4aD

βτa

3minus a

3a2

1113872 1113873(t minus τ)αminus 1

const (70)



I a2a1

+ a4a3

minus a2a3

1113872 1113873(t minus τ)αminus 1

const (71)



I a2a1

+ a4a3

minus a2a3

const (72)




SE 1Γ(α)

1113946b

aa2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

+ a3

1113872 11138732


where B a2a3 + (a3)2 R1 a2 + a3 R2 0 R3


τDβb a

2+ a


αminus 11113960 1113961 0

aDβτa

1minus a

3 0

aDβτa

1minus a

2minus 2a


αminus 1+ τD

βb a


αminus 11113960 1113961 0

aDβτa

3 0

(74)


aDβτa

1minus a

31113872 1113873ξσ2 + aD

βτa

1minus a

2minus 2a

31113872 1113873ξσ3 + aD

βτa

3ξσ4 + a2

+ a3

1113872 1113873 aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877 _ξσ0


a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732


1minus α

(75)

Let

ξ0 1

ξ1 a1

ξ2 0

ξ3 a3

ξ4 0

Gσ

0

(76)


I a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877


const(77)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

1113876 1113877


const(78)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

const (79)



SP 1Γ(α)

1113946b

aa3

aDβτa

1+ a

4aD

βτa

2minus12

a3

1113872 11138732

minus12

a4

1113872 11138732


1113874 11138751113882 1113883dτ (80)

where B (12)(a3)2 + (12)(a4)2 R1 a3 R2

a4 andR3 R4 0



τDβb a


1113874 11138751113876 1113877 0

τDβb a


1113874 11138751113876 1113877 0

aDβτa

1minus a

3 0

aDβτa

2minus a

4 0

(81)


aDβτa

1minus a

31113872 1113873ξ3 + aD

βτa

2minus a

41113872 1113873ξ4 + a

3aD

βτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113872 1113873

+ a4

aDβτξ

σ2 minus aD

βτ _a

2ξσ01113872 1113873 + aDβτ _a

2ξσ01113872 1113873 + a3

aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875 _ξσ0

minusα minus 1


aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875ξσ0


(82)

Let

ξ0 0

ξ1 a2

ξ2 minus a1

ξ3 ξ4 0

Gσ

0

(83)


I 1113946τ

aa3

aDβτa


1113874 11138751113882 1113883

minus a4

aDβτa


1113874 1113875

minus a2

sDβb a


1113874 11138751113876 1113877

+a1

sDβb a


1113874 11138751113876 11138771113883ds

const

(84)



I a3a2

minus a4a1


1113874 1113875 const (85)



I a3a2

minus a4a1

const (86)


6 Conclusions






Data Availability




Acknowledgments


References







































































zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 0 (σ 1 2 r)

(52)



ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


π2

1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891 ξσμ minus _aμξσ01113872 1113873 0 (σ 1 2 r)

(53)



ΔSP minus1Γ(α)

1113946b

a

ddτ

(ΔG)dτ (54)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889Δa]

+zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889Δτ + RμaD

βτa

μminus B1113872 1113873

ddτΔτ

+ Rμ aDβτΔa

μminus aD

βτ _a

μΔτ( 1113857 + aDβτ _a


μminus B1113872 1113873




(55)




zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 minus_Gσ


(56)



ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891


σ (σ 1 2 r)

(57)






Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds c

σ (σ 1 2 r)

(58)


1Γ(α)

1113946b

aεσ

ddτ

RμaDβτa



a




βb Rμ(t minus s)

αminus 11113872 11138731113874 1113875ds1113876 11138771113896

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873dτ1113897 0

(59)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113872 11138731113872 1113873ds1113876 1113877 0 (60)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds

+ Gσ

cσ (σ 1 2 r)

(61)





Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds c

σ (σ 1 2 r)

(62)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds + G

σ c

σ (σ 1 2 r)

(63)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 11138751113876



π2

1113874 11138751113876 1113877]ds cσ (σ 1 2 r)

(64)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD



1113874 11138751113876



π2

1113874 11138751113876 11138771113877ds + Gσ

cσ (σ 1 2 r)

(65)



5 Examples


SR 1Γ(α)

1113946b

aa2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113966 1113967(t minus τ)αminus 1

dτ

(66)



τDβb a

2(t minus τ)

αminus 11113960 1113961 0

aDβτa

1minus a

3 0


αminus 1+ τD

βb a

4(t minus τ)

αminus 11113960 1113961 0

aDβτa

3 0

(67)


aDβτa

1minus a

31113872 1113873ξσ2 minus a

2ξσ3 + aDβτa

3ξσ4 + a2

aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113960 1113961 _ξσ0 minus


middot a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3


1minus α

(68)

Let

ξσ0 1

ξσ1 a1

ξσ2 1

ξσ3 a3

ξσ4 1

Gσ

0

(69)


I a2

aDβτa

1+ a

4aD

βτa

3minus a

3a2

1113872 1113873(t minus τ)αminus 1

const (70)



I a2a1

+ a4a3

minus a2a3

1113872 1113873(t minus τ)αminus 1

const (71)



I a2a1

+ a4a3

minus a2a3

const (72)




SE 1Γ(α)

1113946b

aa2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

+ a3

1113872 11138732


where B a2a3 + (a3)2 R1 a2 + a3 R2 0 R3


τDβb a

2+ a


αminus 11113960 1113961 0

aDβτa

1minus a

3 0

aDβτa

1minus a

2minus 2a


αminus 1+ τD

βb a


αminus 11113960 1113961 0

aDβτa

3 0

(74)


aDβτa

1minus a

31113872 1113873ξσ2 + aD

βτa

1minus a

2minus 2a

31113872 1113873ξσ3 + aD

βτa

3ξσ4 + a2

+ a3

1113872 1113873 aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877 _ξσ0


a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732


1minus α

(75)

Let

ξ0 1

ξ1 a1

ξ2 0

ξ3 a3

ξ4 0

Gσ

0

(76)


I a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877


const(77)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

1113876 1113877


const(78)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

const (79)



SP 1Γ(α)

1113946b

aa3

aDβτa

1+ a

4aD

βτa

2minus12

a3

1113872 11138732

minus12

a4

1113872 11138732


1113874 11138751113882 1113883dτ (80)

where B (12)(a3)2 + (12)(a4)2 R1 a3 R2

a4 andR3 R4 0



τDβb a


1113874 11138751113876 1113877 0

τDβb a


1113874 11138751113876 1113877 0

aDβτa

1minus a

3 0

aDβτa

2minus a

4 0

(81)


aDβτa

1minus a

31113872 1113873ξ3 + aD

βτa

2minus a

41113872 1113873ξ4 + a

3aD

βτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113872 1113873

+ a4

aDβτξ

σ2 minus aD

βτ _a

2ξσ01113872 1113873 + aDβτ _a

2ξσ01113872 1113873 + a3

aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875 _ξσ0

minusα minus 1


aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875ξσ0


(82)

Let

ξ0 0

ξ1 a2

ξ2 minus a1

ξ3 ξ4 0

Gσ

0

(83)


I 1113946τ

aa3

aDβτa


1113874 11138751113882 1113883

minus a4

aDβτa


1113874 1113875

minus a2

sDβb a


1113874 11138751113876 1113877

+a1

sDβb a


1113874 11138751113876 11138771113883ds

const

(84)



I a3a2

minus a4a1


1113874 1113875 const (85)



I a3a2

minus a4a1

const (86)


6 Conclusions






Data Availability




Acknowledgments


References







































































zRμ

za] aD

βτa

μminus

zB

za]1113888 1113889ξσ] +

zRμ

zτ aDβτa

μminus

zB

zτ1113888 1113889ξσ0 + RμaD

βτa

μminus B1113872 1113873 _ξ

σ0

+ Rμ aDβτξ

σμ minus aD

βτ _a

μξσ0( 1113857 + aDβτ _a


μminus B1113872 1113873


ξσ0 minus_Gσ


(56)



ddτ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 111387511138761113882



π2

1113874 11138751113876 11138771113877ds1113883

+zR]za

μaDβτa

]minus

zB


π2


π2

1113874 11138751113874 11138751113890 1113891


σ (σ 1 2 r)

(57)






Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds c

σ (σ 1 2 r)

(58)


1Γ(α)

1113946b

aεσ

ddτ

RμaDβτa



a




βb Rμ(t minus s)

αminus 11113872 11138731113874 1113875ds1113876 11138771113896

+zR]za

μaDβτa

]minus

zB

zaμ1113888 1113889(t minus τ)

αminus 1+ τD

βb Rμ(t minus τ)


μξσ01113872 1113873dτ1113897 0

(59)



ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113872 11138731113872 1113873ds1113876 1113877 0 (60)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds

+ Gσ

cσ (σ 1 2 r)

(61)





Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds c

σ (σ 1 2 r)

(62)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds + G

σ c

σ (σ 1 2 r)

(63)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 11138751113876



π2

1113874 11138751113876 1113877]ds cσ (σ 1 2 r)

(64)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD



1113874 11138751113876



π2

1113874 11138751113876 11138771113877ds + Gσ

cσ (σ 1 2 r)

(65)



5 Examples


SR 1Γ(α)

1113946b

aa2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113966 1113967(t minus τ)αminus 1

dτ

(66)



τDβb a

2(t minus τ)

αminus 11113960 1113961 0

aDβτa

1minus a

3 0


αminus 1+ τD

βb a

4(t minus τ)

αminus 11113960 1113961 0

aDβτa

3 0

(67)


aDβτa

1minus a

31113872 1113873ξσ2 minus a

2ξσ3 + aDβτa

3ξσ4 + a2

aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113960 1113961 _ξσ0 minus


middot a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3


1minus α

(68)

Let

ξσ0 1

ξσ1 a1

ξσ2 1

ξσ3 a3

ξσ4 1

Gσ

0

(69)


I a2

aDβτa

1+ a

4aD

βτa

3minus a

3a2

1113872 1113873(t minus τ)αminus 1

const (70)



I a2a1

+ a4a3

minus a2a3

1113872 1113873(t minus τ)αminus 1

const (71)



I a2a1

+ a4a3

minus a2a3

const (72)




SE 1Γ(α)

1113946b

aa2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

+ a3

1113872 11138732


where B a2a3 + (a3)2 R1 a2 + a3 R2 0 R3


τDβb a

2+ a


αminus 11113960 1113961 0

aDβτa

1minus a

3 0

aDβτa

1minus a

2minus 2a


αminus 1+ τD

βb a


αminus 11113960 1113961 0

aDβτa

3 0

(74)


aDβτa

1minus a

31113872 1113873ξσ2 + aD

βτa

1minus a

2minus 2a

31113872 1113873ξσ3 + aD

βτa

3ξσ4 + a2

+ a3

1113872 1113873 aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877 _ξσ0


a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732


1minus α

(75)

Let

ξ0 1

ξ1 a1

ξ2 0

ξ3 a3

ξ4 0

Gσ

0

(76)


I a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877


const(77)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

1113876 1113877


const(78)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

const (79)



SP 1Γ(α)

1113946b

aa3

aDβτa

1+ a

4aD

βτa

2minus12

a3

1113872 11138732

minus12

a4

1113872 11138732


1113874 11138751113882 1113883dτ (80)

where B (12)(a3)2 + (12)(a4)2 R1 a3 R2

a4 andR3 R4 0



τDβb a


1113874 11138751113876 1113877 0

τDβb a


1113874 11138751113876 1113877 0

aDβτa

1minus a

3 0

aDβτa

2minus a

4 0

(81)


aDβτa

1minus a

31113872 1113873ξ3 + aD

βτa

2minus a

41113872 1113873ξ4 + a

3aD

βτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113872 1113873

+ a4

aDβτξ

σ2 minus aD

βτ _a

2ξσ01113872 1113873 + aDβτ _a

2ξσ01113872 1113873 + a3

aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875 _ξσ0

minusα minus 1


aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875ξσ0


(82)

Let

ξ0 0

ξ1 a2

ξ2 minus a1

ξ3 ξ4 0

Gσ

0

(83)


I 1113946τ

aa3

aDβτa


1113874 11138751113882 1113883

minus a4

aDβτa


1113874 1113875

minus a2

sDβb a


1113874 11138751113876 1113877

+a1

sDβb a


1113874 11138751113876 11138771113883ds

const

(84)



I a3a2

minus a4a1


1113874 1113875 const (85)



I a3a2

minus a4a1

const (86)


6 Conclusions






Data Availability




Acknowledgments


References







































































ddτ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113872 11138731113872 1113873ds1113876 1113877 0 (60)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a



βb Rμ(t minus s)

αminus 11113960 11139611113876 1113877ds

+ Gσ

cσ (σ 1 2 r)

(61)





Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds c

σ (σ 1 2 r)

(62)


Iσ

RμaDβτa



aRμaD

βs ξσμ minus _a


1113876



αminus 11113960 11139611113877ds + G

σ c

σ (σ 1 2 r)

(63)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD

βs ξσμ minus _a


1113874 11138751113876



π2

1113874 11138751113876 1113877]ds cσ (σ 1 2 r)

(64)




Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD



1113874 11138751113876



π2

1113874 11138751113876 11138771113877ds + Gσ

cσ (σ 1 2 r)

(65)



5 Examples


SR 1Γ(α)

1113946b

aa2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113966 1113967(t minus τ)αminus 1

dτ

(66)



τDβb a

2(t minus τ)

αminus 11113960 1113961 0

aDβτa

1minus a

3 0


αminus 1+ τD

βb a

4(t minus τ)

αminus 11113960 1113961 0

aDβτa

3 0

(67)


aDβτa

1minus a

31113872 1113873ξσ2 minus a

2ξσ3 + aDβτa

3ξσ4 + a2

aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113960 1113961 _ξσ0 minus


middot a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3


1minus α

(68)

Let

ξσ0 1

ξσ1 a1

ξσ2 1

ξσ3 a3

ξσ4 1

Gσ

0

(69)


I a2

aDβτa

1+ a

4aD

βτa

3minus a

3a2

1113872 1113873(t minus τ)αminus 1

const (70)



I a2a1

+ a4a3

minus a2a3

1113872 1113873(t minus τ)αminus 1

const (71)



I a2a1

+ a4a3

minus a2a3

const (72)




SE 1Γ(α)

1113946b

aa2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

+ a3

1113872 11138732


where B a2a3 + (a3)2 R1 a2 + a3 R2 0 R3


τDβb a

2+ a


αminus 11113960 1113961 0

aDβτa

1minus a

3 0

aDβτa

1minus a

2minus 2a


αminus 1+ τD

βb a


αminus 11113960 1113961 0

aDβτa

3 0

(74)


aDβτa

1minus a

31113872 1113873ξσ2 + aD

βτa

1minus a

2minus 2a

31113872 1113873ξσ3 + aD

βτa

3ξσ4 + a2

+ a3

1113872 1113873 aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877 _ξσ0


a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732


1minus α

(75)

Let

ξ0 1

ξ1 a1

ξ2 0

ξ3 a3

ξ4 0

Gσ

0

(76)


I a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877


const(77)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

1113876 1113877


const(78)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

const (79)



SP 1Γ(α)

1113946b

aa3

aDβτa

1+ a

4aD

βτa

2minus12

a3

1113872 11138732

minus12

a4

1113872 11138732


1113874 11138751113882 1113883dτ (80)

where B (12)(a3)2 + (12)(a4)2 R1 a3 R2

a4 andR3 R4 0



τDβb a


1113874 11138751113876 1113877 0

τDβb a


1113874 11138751113876 1113877 0

aDβτa

1minus a

3 0

aDβτa

2minus a

4 0

(81)


aDβτa

1minus a

31113872 1113873ξ3 + aD

βτa

2minus a

41113872 1113873ξ4 + a

3aD

βτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113872 1113873

+ a4

aDβτξ

σ2 minus aD

βτ _a

2ξσ01113872 1113873 + aDβτ _a

2ξσ01113872 1113873 + a3

aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875 _ξσ0

minusα minus 1


aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875ξσ0


(82)

Let

ξ0 0

ξ1 a2

ξ2 minus a1

ξ3 ξ4 0

Gσ

0

(83)


I 1113946τ

aa3

aDβτa


1113874 11138751113882 1113883

minus a4

aDβτa


1113874 1113875

minus a2

sDβb a


1113874 11138751113876 1113877

+a1

sDβb a


1113874 11138751113876 11138771113883ds

const

(84)



I a3a2

minus a4a1


1113874 1113875 const (85)



I a3a2

minus a4a1

const (86)


6 Conclusions






Data Availability




Acknowledgments


References







































































Iσ

RμaDβτa


π2

1113874 1113875ξσ0 + 1113946τ

aRμaD



1113874 11138751113876



π2

1113874 11138751113876 11138771113877ds + Gσ

cσ (σ 1 2 r)

(65)



5 Examples


SR 1Γ(α)

1113946b

aa2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113966 1113967(t minus τ)αminus 1

dτ

(66)



τDβb a

2(t minus τ)

αminus 11113960 1113961 0

aDβτa

1minus a

3 0


αminus 1+ τD

βb a

4(t minus τ)

αminus 11113960 1113961 0

aDβτa

3 0

(67)


aDβτa

1minus a

31113872 1113873ξσ2 minus a

2ξσ3 + aDβτa

3ξσ4 + a2

aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3

1113960 1113961 _ξσ0 minus


middot a2

aDβτa

1+ a

4aD

βτa

3minus a

2a3


1minus α

(68)

Let

ξσ0 1

ξσ1 a1

ξσ2 1

ξσ3 a3

ξσ4 1

Gσ

0

(69)


I a2

aDβτa

1+ a

4aD

βτa

3minus a

3a2

1113872 1113873(t minus τ)αminus 1

const (70)



I a2a1

+ a4a3

minus a2a3

1113872 1113873(t minus τ)αminus 1

const (71)



I a2a1

+ a4a3

minus a2a3

const (72)




SE 1Γ(α)

1113946b

aa2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

+ a3

1113872 11138732


where B a2a3 + (a3)2 R1 a2 + a3 R2 0 R3


τDβb a

2+ a


αminus 11113960 1113961 0

aDβτa

1minus a

3 0

aDβτa

1minus a

2minus 2a


αminus 1+ τD

βb a


αminus 11113960 1113961 0

aDβτa

3 0

(74)


aDβτa

1minus a

31113872 1113873ξσ2 + aD

βτa

1minus a

2minus 2a

31113872 1113873ξσ3 + aD

βτa

3ξσ4 + a2

+ a3

1113872 1113873 aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877 _ξσ0


a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732


1minus α

(75)

Let

ξ0 1

ξ1 a1

ξ2 0

ξ3 a3

ξ4 0

Gσ

0

(76)


I a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877


const(77)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

1113876 1113877


const(78)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

const (79)



SP 1Γ(α)

1113946b

aa3

aDβτa

1+ a

4aD

βτa

2minus12

a3

1113872 11138732

minus12

a4

1113872 11138732


1113874 11138751113882 1113883dτ (80)

where B (12)(a3)2 + (12)(a4)2 R1 a3 R2

a4 andR3 R4 0



τDβb a


1113874 11138751113876 1113877 0

τDβb a


1113874 11138751113876 1113877 0

aDβτa

1minus a

3 0

aDβτa

2minus a

4 0

(81)


aDβτa

1minus a

31113872 1113873ξ3 + aD

βτa

2minus a

41113872 1113873ξ4 + a

3aD

βτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113872 1113873

+ a4

aDβτξ

σ2 minus aD

βτ _a

2ξσ01113872 1113873 + aDβτ _a

2ξσ01113872 1113873 + a3

aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875 _ξσ0

minusα minus 1


aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875ξσ0


(82)

Let

ξ0 0

ξ1 a2

ξ2 minus a1

ξ3 ξ4 0

Gσ

0

(83)


I 1113946τ

aa3

aDβτa


1113874 11138751113882 1113883

minus a4

aDβτa


1113874 1113875

minus a2

sDβb a


1113874 11138751113876 1113877

+a1

sDβb a


1113874 11138751113876 11138771113883ds

const

(84)



I a3a2

minus a4a1


1113874 1113875 const (85)



I a3a2

minus a4a1

const (86)


6 Conclusions






Data Availability




Acknowledgments


References






































































SE 1Γ(α)

1113946b

aa2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

+ a3

1113872 11138732


where B a2a3 + (a3)2 R1 a2 + a3 R2 0 R3


τDβb a

2+ a


αminus 11113960 1113961 0

aDβτa

1minus a

3 0

aDβτa

1minus a

2minus 2a


αminus 1+ τD

βb a


αminus 11113960 1113961 0

aDβτa

3 0

(74)


aDβτa

1minus a

31113872 1113873ξσ2 + aD

βτa

1minus a

2minus 2a

31113872 1113873ξσ3 + aD

βτa

3ξσ4 + a2

+ a3

1113872 1113873 aDβτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113960 1113961

+ a4

aDβτξ

σ3 minus aD

βτ _a

3ξσ01113872 1113873 + aDβτ _a

3ξσ01113960 1113961 + a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877 _ξσ0


a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732


1minus α

(75)

Let

ξ0 1

ξ1 a1

ξ2 0

ξ3 a3

ξ4 0

Gσ

0

(76)


I a2

+ a3

1113872 1113873aDβτa

1+ a

4aD

βτa

3minus a

2a3

minus a3

1113872 11138732

1113876 1113877


const(77)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

1113876 1113877


const(78)



I a2

+ a3

1113872 1113873a1

+ a4a3

minus a2a3

minus a3

1113872 11138732

const (79)



SP 1Γ(α)

1113946b

aa3

aDβτa

1+ a

4aD

βτa

2minus12

a3

1113872 11138732

minus12

a4

1113872 11138732


1113874 11138751113882 1113883dτ (80)

where B (12)(a3)2 + (12)(a4)2 R1 a3 R2

a4 andR3 R4 0



τDβb a


1113874 11138751113876 1113877 0

τDβb a


1113874 11138751113876 1113877 0

aDβτa

1minus a

3 0

aDβτa

2minus a

4 0

(81)


aDβτa

1minus a

31113872 1113873ξ3 + aD

βτa

2minus a

41113872 1113873ξ4 + a

3aD

βτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113872 1113873

+ a4

aDβτξ

σ2 minus aD

βτ _a

2ξσ01113872 1113873 + aDβτ _a

2ξσ01113872 1113873 + a3

aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875 _ξσ0

minusα minus 1


aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875ξσ0


(82)

Let

ξ0 0

ξ1 a2

ξ2 minus a1

ξ3 ξ4 0

Gσ

0

(83)


I 1113946τ

aa3

aDβτa


1113874 11138751113882 1113883

minus a4

aDβτa


1113874 1113875

minus a2

sDβb a


1113874 11138751113876 1113877

+a1

sDβb a


1113874 11138751113876 11138771113883ds

const

(84)



I a3a2

minus a4a1


1113874 1113875 const (85)



I a3a2

minus a4a1

const (86)


6 Conclusions






Data Availability




Acknowledgments


References







































































τDβb a


1113874 11138751113876 1113877 0

τDβb a


1113874 11138751113876 1113877 0

aDβτa

1minus a

3 0

aDβτa

2minus a

4 0

(81)


aDβτa

1minus a

31113872 1113873ξ3 + aD

βτa

2minus a

41113872 1113873ξ4 + a

3aD

βτξ

σ1 minus aD

βτ _a

1ξσ01113872 1113873 + aDβτ _a

1ξσ01113872 1113873

+ a4

aDβτξ

σ2 minus aD

βτ _a

2ξσ01113872 1113873 + aDβτ _a

2ξσ01113872 1113873 + a3

aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875 _ξσ0

minusα minus 1


aDβτa

1+ a

4aD

βτa

2minus

12

a3

1113872 11138732

+12

a4

1113872 11138732

1113874 11138751113874 1113875ξσ0


(82)

Let

ξ0 0

ξ1 a2

ξ2 minus a1

ξ3 ξ4 0

Gσ

0

(83)


I 1113946τ

aa3

aDβτa


1113874 11138751113882 1113883

minus a4

aDβτa


1113874 1113875

minus a2

sDβb a


1113874 11138751113876 1113877

+a1

sDβb a


1113874 11138751113876 11138771113883ds

const

(84)



I a3a2

minus a4a1


1113874 1113875 const (85)



I a3a2

minus a4a1

const (86)


6 Conclusions






Data Availability




Acknowledgments


References








































































Data Availability




Acknowledgments


References











































































































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FractionalBirkhoffianMechanicsBasedonQuasi-Fractional