Nonlinear self-adjointness, conservation laws and exact solutions of time-fractional Kompaneets equations

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Accepted Manuscript

Nonlinear self-adjointness, conservation laws and exact solutions of time-frac-tional Kompaneets equations

R.K. Gazizov, N.H. Ibragimov, S.Yu. Lukashchuk

PII: S1007-5704(14)00530-9DOI: http://dx.doi.org/10.1016/j.cnsns.2014.11.010Reference: CNSNS 3416

To appear in: Communications in Nonlinear Science and Numer-ical Simulation

Please cite this article as: Gazizov, R.K., Ibragimov, N.H., Lukashchuk, S.Yu., Nonlinear self-adjointness,conservation laws and exact solutions of time-fractional Kompaneets equations, Communications in NonlinearScience and Numerical Simulation (2014), doi: http://dx.doi.org/10.1016/j.cnsns.2014.11.010

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http://dx.doi.org/10.1016/j.cnsns.2014.11.010

http://dx.doi.org/http://dx.doi.org/10.1016/j.cnsns.2014.11.010

Nonlinear self-adjointness, conservation laws andexact solutions of time-fractional Kompaneets

equations

R.K. Gazizova, N.H. Ibragimova,b, S.Yu. Lukashchuka1

aLaboratory ”Group analysis of mathematical models in natural and engineering sci-ences”, Ufa State Aviation Technical University, 450 000 Ufa, Russia.bResearch Centre ALGA: Advances in Lie Group Analysis, Department ofMathematics and Natural Sciences, Blekinge Institute of Technology,SE-371 79 Karlskrona, Sweden.

AbstractFour time-fractional generalizations of the Kompaneets equation are considered. Groupanalysis is performed for physically relevant approximations. It is shown that all approx-imations have nontrivial symmetries and conservation laws. The symmetries are usedfor constructing group invariant solutions, whereas the conservation laws allow to findnon-invariant exact solutions.Keywords: Time-fractional Kompaneets equation, symmetry, nonlinear self-adjointness,conservation law, exact solutionAMS classification numbers: 35Q85, 45K05, 70S10, 70G65

1 Introduction

In this paper, four time-fractional generalizations of the Kompaneets equation areconsidered. The classical Kompaneets equation, known also as the photon diffu-sion equation, was derived independently by A.S. Kompaneets [1] and R. Wey-mann [2] from the Boltzmann equation under the Fokker-Plank approximationsand some additional assumptions. In a dimensionless form, it can be written as

ft =1x2Dx

[x4

(fx + f + f2

)]. (1.1)

This equation describes the evolution of the density function f of the energy ofphotons due to Compton scattering in the homogeneous fully ionized plasma.

Since the equation (1.1) has only one symmetry, namely the time translation,group analysis is rather useless for constructing exact solutions. Therefore, exactsolutions were investigated in the literature for the several approximations of theKompaneets equation. These solutions are collected in the review paper [3].

Recently, a detailed group analysis was done [4] for physically relevant ap-proximations of the Kompaneets equation. As a result, new exact solutions wereobtained for the approximations under consideration. Subsequently, these ap-proximations were investigated in [5] from the point of view of approximate sym-metries and approximate solutions in order to understand the relation of the

1Corresponding author. E-mail: [email protected]

1

exact solutions of the relevant approximation with solutions of the original equa-tion (1.1). The approximate symmetry approach allows to better understand aphysical essence of exact solutions for various approximations to the Kompaneetsequation.

During last two decades several models with time-fractional derivatives wereproposed for describing kinetic processes [6, 7, 8, 9]. In particular, these mod-els contain time-fractional generalizations of the Boltzmann and Fokker-Plankequations. For the case of space-homogeneous plasma, the classical Boltzmannequation can be written in the form

ft = C[f ],

where C[f ] is the collision integral defined by (see, e.g., [10])

C[f ] =∫ ∫ ∫

dp2dp′1dp

′2σ(p1, p2|p′1, p′2)(f1f2 − f ′1f

′2).

Here we use the standard abbreviations f1 = f(r, p1, t)1, f ′1 = f(r, p′1, t), f2 =f(r, p2, t), f ′2 = f(r, p′2, t), and denote by σ a total cross section for a collisionbetween particles with momenta p1 and p2.

The time-fractional generalizations of the Boltzmann equation can be sum-marized as follows:

ft = Jαt C[f ], (1.2)

ft = D1−αt C[f ], (1.3)

ft = C[Jαt f ], (1.4)

ft = C[D1−αt f ]. (1.5)

Here

Jαt f =

1Γ(α)

∫ t

0

f(τ, . . .)(t− τ)1−α

dτ

is the left-sided fractional integral of order α ∈ (0, 1) with respect to t, and

D1−αt f ≡ DtJ

αt f =

1Γ(α)

∂

∂t

∫ t

0

f(τ, . . .)(t− τ)1−α

dτ

is the left-sided fractional derivative of the Riemann-Liouville type of order 1−αwith respect to t (see, e.g., [11, 12]).

The aim of the present paper is to formulate the time-fractional generaliza-tions of the Kompaneets equation and extend the results of the group analysis[4] to these generalizations.

2

2 Time-fractional generalizations of the Kompaneetsequation

Under the same assumptions that are used for deriving the classical Kompaneetsequation (1.1), one can obtain the time-fractional generalizations of the Kompa-neets equation from the equations (1.2)–(1.5) as

ft =1x2Jα

t Dx

[x4

(fx + f + f2

)], (2.1)

ft =1x2D1−α

t Dx

[x4

(fx + f + f2

)], (2.2)

ft =1x2Dx

[x4

(Jα

t fx + Jαt f + (Jα

t f)2)], (2.3)

ft =1x2Dx

[x4

(D1−α

t fx +D1−αt f + (D1−α

t f)2)]. (2.4)

All these equations can be rewritten so that their right-hand sides will beexactly the same as the right-hand side of the classical Kompaneets equation(1.1). Indeed, if we act on the equation (2.1) by the operator Dα

t of fractionaldifferentiation and denote the dependent variable f by u, then we get the equation

Dαt ut =

1x2Dx

[x4

(ux + u+ u2

)]. (2.5)

Integrating the equation (2.2) with respect to t we obtain

f(t, x) − f(0, x) =1x2Jα

t Dx

[x4

(fx + f + f2

)].

As above, we act on both sides of this equation by the same operator Dαt of

fractional differentiation and denote the dependent variable f by u. Then we getthe equation

CDαt u =

1x2Dx

[x4

(ux + u+ u2

)], (2.6)

whereCDα

t u ≡ J1−αt ut =

1Γ(1 − α)

∫ t

0

uτ (τ, x)(t− τ)α

dτ

is the left-sided time-fractional derivative of the Caputo type of order α ∈ (0, 1)[12].

In the equation (2.3) we can introduce a new nonlocal dependent variableu = Jα

t f. Then this equation takes the form

D1+αt u =

1x2Dx

[x4

(ux + u+ u2

)]. (2.7)

Finally, we make in the equation (2.4) the nonlocal change of the dependentvariable by setting f = J1−α

t u. Since D1−αt J1−α

t u = u, this equation can berewritten as

Dαt u =

1x2Dx

[x4

(ux + u+ u2

)]. (2.8)

3

Thus, we get four different time-fractional generalizations (2.5)–(2.8) of theKompaneets equation. Note that equations (2.6) and (2.8) correspond to theso-called subdiffusion regime because the order of time-fractional differentiationin these equations is less than one. These equations coincide with the classicalKompaneets equation (1.1) in the limiting case of α = 1. The equations (2.5) and(2.7) are intermediate between classical diffusion and wave equations because theorder of time-fractional differentiation in these equations belongs to the interval(1, 2). Such equations are usually known as equations of diffusion-wave type.These equations coincide with the classical Kompaneets equation (1.1) in thelimiting case of α = 0.

We can formally rewrite the equations (2.5)–(2.8) as

F (t, x, u,Dγ(α)t u, ux, uxx) ≡ Dγ(α)

t u− x2Dxh(u, ux) − 4xh(u, ux) = 0. (2.9)

Here we denote by Dγ(α)t any time-fractional differential operator in equations

(2.5)–(2.8), and the function h is

h(u, ux) = ux + u+ u2. (2.10)

Every term in the function h(u, ux) describes a certain physical effect. Thediffusion term ux is responsible for the Doppler effect, the term u describes theCompton scattering, and the nonlinear term u2 describes the induced scatter-ing. If in a certain physical situation one of these effects is negligible compareto the other effects, then one drops the corresponding term in the function h.The obtained equation (2.9) is considered as a (physical) approximation of thecorresponding time-fractional Kompaneets equation. In this paper, different ap-proximations will be discussed.

3 Symmetries

In [13, 14, 15], the basic methods of the group analysis have been extendedto the fractional differential equations. Using these methods, we compute thesymmetries of the time-fractional Kompaneets equations (2.5)–(2.8) and of theirapproximations.

It is known [4] that the time-translations with the generator

X =∂

∂t

is the only Lie point symmetry of the Kompaneets equation (1.1).The time-fractional generalizations of the Kompaneets equation do not admit

the translation in time. The calculation shows that the equations (2.5)–(2.8) haveno Lie point symmetries. Nevertheless, the physically relevant approximations ofthese equations have nontrivial Lie point symmetries.

4

Proposition 1 Let the equation (2.9) is a diffusion-type time-fractional equa-tion, i.e. the function h have the term ux. Then the following approximations ofthis equation has the nontrivial Lie point symmetries:

• the equation (2.9) with h = ux admits

X1 = u∂

∂u, X2 = x

∂

∂x, Xg = g(t, x)

∂

∂u; (3.1)

• the equation (2.9) with h = ux + u admits

X1 = u∂

∂u, Xg = g(t, x)

∂

∂u; (3.2)

• the equation (2.9) with h = ux + u2 admits

X3 = X2 −X1 ≡ x∂

∂x− u

∂

∂u, (3.3)

where the function g(t, x) is a solution of the equation

Dγ(α)t g = x2Dxh(g, gx) + 4xh(g, gx).

The above symmetries are valid for all considered types of the time-fractionalderivatives Dγ(α)

t u.

4 Nonlinear self-adjointness

The concept of nonlinear self-adjointness for differential equations of integralorders was developed in [16]. It has been proved there that using this conceptand the Noether’s theorem, the conservation laws can be constructed for theequations that do not have Lagrangians in a classical sense. It was shown in [17]that this concept is also applicable to the time-fractional differential equations.Moreover, conservation laws have been constructed in [17] for the linear andnonlinear subdiffusion and diffusion-wave equations.

In this section, we prove that equations (2.5)–(2.8) are all nonlinearly self-adjoint and therefore the conservation laws can be constructed for them usingtheir symmetries.

In the same manner as it was done for the equations of integral orders [16],we define the formal Lagrangian L for the equation (2.9) by

L = vF (t, x, u,Dγ(α)t u, ux, uxx). (4.1)

Here v = v(t, x) is a new dependent variable.The adjoint equation to the equation (2.9) is defined by

F ∗(t, x, u, v,Dγ(α)

t u,(Dγ(α)

t

)∗v, ux, vx, uxx, vxx

)≡ δLδu

= 0. (4.2)

5

Here(Dγ(α)

t

)∗denotes the adjoint operator to Dγ(α)

t . It is defined below for eachparticular cases of fractional differential operators used in the equation (2.9).

If we consider the equation (2.9) for a finite time interval t ∈ [0, T ], then thecorresponding Euler-Lagrange operator δ/δu in (4.2) has the form

δ

δu=

∂

∂u−Dx

∂

∂ux+D2

x

∂

∂uxx+ (Dγ(α)

t )∗∂

∂ (Dαt u)

, (4.3)

where (Dγ(α)

t

)∗≡ (Dα

t Dt)∗ = tDαTDt

for the equation (2.5), (Dγ(α)

t

)∗≡ ( CDα

t )∗ = tDαT

for the equation (2.6), (Dγ(α)

t

)∗≡ (D1+α

t )∗ = Ct D

1+αT

for the equation (2.7), and (Dγ(α)

t

)∗≡ (Dα

t )∗ = Ct D

αT

for the equation (2.8). Here

tDβTu =

(−1)n

Γ(n− β)∂n

∂tn

∫ T

t

u(τ, x)(τ − t)β−n+1

dτ

is the right-sided Riemann-Liouville time-fractional derivative of order β ∈ R+,n = [β] + 1, and

Ct D

βTu =

(−1)n

Γ(n− β)

∫ T

t

Dnτ u(τ, x)

(τ − t)β−n+1dτ

is the right-sided Caputo time-fractional derivative of order β ∈ R+, n = [β] + 1(see, e.g., [12]).

After simple calculations in (4.2), we obtain the following adjoint time-fractionalKompaneets equation

(Dγ(α)t )∗v − x2vxx + x2(1 + 2u)vx + 2(1 − x− 2xu)v = 0 (4.4)

for the equation (2.9).The definition of the nonlinear self-adjointness (see, e.g., definition 2 from

[16]) can be extended to the time-fractional Kompaneets equations. Namely, theequation (2.9) is said to be nonlinearly self-adjoint if the adjoint equation (4.4)is satisfied for all solutions u of the equation (2.9) upon a substitution

v = φ(t, x, u) (4.5)

6

satisfying the condition φ(t, x, u) ̸= 0.We find all substitutions (4.5) that provide the nonlinear self-adjointness of

the time-fractional Kompaneets equations (2.5)–(2.8) and their approximations,and arrive at the following result.

Proposition 2 The time-fractional Kompaneets equation (2.9) and their diffusion-type approximations are all nonlinarly self-adjoint and the substitution (4.5) hasthe form

φ = Φ(t)Ψ(x). (4.6)

Here the function Φ(t) depends on the type of fractional differential operatorDγ(α)

t , namely

Dγ(α)t = Dα

t Dt : Φ(t) = ϕ1(T − t)α + ϕ2; (4.7)

Dγ(α)t = CDα

t : Φ(t) = ϕ1(T − t)α−1; (4.8)

Dγ(α)t = D1+α

t : Φ(t) = ϕ1t+ ϕ2; (4.9)

Dγ(α)t = Dα

t : Φ(t) = ϕ1, (4.10)

where ϕ1 and ϕ2 are arbitrary constants. The function Ψ(x) depends on theapproximation of the function h(u, ux) defined by (2.10), namely

h = ux + u+ u2, h = ux + u2 : Ψ(x) = ψ1x2; (4.11)

h = ux : Ψ(x) = ψ1x2 + ψ2x

−1; (4.12)

h = ux + u : Ψ(x) = ψ1x2 + ψ2Θ(x), (4.13)

where ψ1 and ψ2 are arbitrary constants, and the function Θ(x) is

Θ(x) = exx−1[e−xx3Ei(x) − x2 − x− 2

]. (4.14)

Remark 1 We do not present here the derivation of the substitutions (4.6). In-stead, we note that the term ψ1x

2 in each substitution is preciesly the similar sub-stitution for the classical Kompaneets equation (1.1) (see [18], Eq. (2.10)). Thefunctions Φ(t) defined by (4.7)–(4.10) are the solutions of the fractional equationD

γ(α)t φ = 0. The functions Ψ(x) defined by (4.11)–(4.13) are the solutions of the

equationx2Ψ′′ − x2huΨ′ + 2(xhu − 1)Ψ = 0

for arbitrary function u.

5 Conservation laws

5.1 Generalities

Here we extend the usual notion of a conserved vector to the time-fractionalequation (2.9). A vector C = (Ct, Cx) is called a conserved vector for the equation

7

(2.9) if it satisfies the conservation equation[Dt(Ct) +Dx(Cx)

](2.9)

= 0.

Let the equation (2.9) be nonlinearly self-adjoint and admit a one-parameterpoint transformation group with the generator

X = ξ0(t, x, u)∂

∂t+ ξ1(t, x, u)

∂

∂x+ η(t, x, u)

∂

∂u.

Since the equation (2.9) does not involve the fractional derivatives with respect tox, the x-component of the conserved vector can be found by the general formula[16] for calculating conseved vectors associated with symmetries. This formulagives

Cx = W

(∂L∂ux

−Dx∂L∂uxx

)+Dx(W )

∂L∂uxx

. (5.1)

HereW = η − ξ0ut − ξ1ux,

and L is the formal Lagrangian (4.1) where the variable v is eliminated by usinga suitable substitution v = φ(t, x, u) from Proposition 2.

The formula for the t-component of conserved vector depends on the type oftime-fractional derivative in the equation (2.9):

Ct = J1−αt Dt(W )

∂L∂(Dα

t ut)−W tJ

1−αT Dt

∂L∂(Dα

t ut)+ I

(Dt(W ), Dt

∂L∂(Dα

t ut)

)(5.2)

for the equation (2.5),

Ct = W tJ1−αT

∂L∂ (CDα

t u)− I

(Dt(W ),

∂L∂ (CDα

t u)

)(5.3)

for the equation (2.6),

Ct = Dαt (W )

∂L∂(D1+α

t u)− J1−α

t (W )Dt∂L

∂(D1+αt u)

− I

(W,D2

t

∂L∂(D1+α

t u)

)(5.4)

for the equation (2.7), and

Ct = J1−αt (W )

∂L∂(Dα

t u)+ I

(W,Dt

∂L∂(Dα

t u)

)(5.5)

for the equation (2.8). Here we denote by I the integral

I(f, g) =1

Γ(1 − α)

∫ t

0

∫ T

t

f(τ, x)g(µ, x)(µ− τ)α

dµdτ. (5.6)

Note that the integral (5.6) has a useful property

DtI(f, g) = f tJ1−αT g − g J1−α

t f.

8

We obtain the formulae (5.2)–(5.5) from the fundamental identity

X̃ +Dt(ξ0) +Dx(ξ1) = Wδ

δu+DtN

t +DxNx, (5.7)

where δ/δu is the Euler-Lagrange operator defined by (4.3), and X̃ is a prolonga-tion of the Lie point group generator to the derivatives of the dependent variableinvolved in the equation (2.9).

Now we consider separately each of the equations (2.5)–(2.8).

5.2 Conservation laws for approximations of the equation (2.5)

In this case we substitute Dγ(α)t = Dα

t Dt in the equation (2.9). Then this equationhas the conservation form with the conserved vector having the components

Ct = x2J1−αt ut, Cx = −x4h(u, ux). (5.8)

The formal Lagrangian (4.1) upon the substitution v = φ(t, x), with thefunction φ(t, x) defined by (4.6), (4.7), takes the form

L = (ϕ1(T − t)α + ϕ2)Ψ(x)(Dαt ut − x2Dx(h) − 4xh). (5.9)

Remind that the function Ψ(x) depends on the approximation of the function has it follows from Proposition 2.

Substituting the formal Lagrangian (5.9) and the symmetries (3.1)–(3.3) in(5.1) and (5.2) we find new conservation laws.

For h = ux, h = ux + u, h = ux + u2 we obtain the conserved vector

Ct = x2[(T − t)αJ1−α

t ut + Γ(1 + α)u− αI(ut, (T − t)α−1)],

Cx = −x4(T − t)αh(u, ux).(5.10)

The conserved vector (5.10) corresponds to the product ϕ1ψ1 of the constantsϕ1 and ψ1 in the substitution (4.6) with (4.7), (4.11). The conserved vectorcorresponding to the product ϕ2ψ1 of the constants ϕ2 and ψ1 in (4.6) coincideswith the conserved vector (5.8).

For h = ux, using the symmetry X1 from (3.1), we find two additional con-served vectors:

Ct = x−1[(T − t)αJ1−α

t ut + Γ(1 + α)u− αI(ut, (T − t)α−1)],

Cx = −(T − t)α(xux + 3u),(5.11)

Ct = x−1J1−αt ut, Cx = −(xux + 3u). (5.12)

The conserved vector (5.11) corresponds to the product ϕ1ψ2 of the constants ϕ1

and ψ2 in (4.6) with (4.7), (4.12), and conserved vector (5.12) corresponds to theproduct ϕ2ψ2 of the constants ϕ2 and ψ2. Note that the symmetry X2 from (3.1)provides the trivial conservation law only.

9

For h = ux+u, using the symmetry X1 from (3.2), we also find two additionalconserved vectors:

Ct = Θ(x)[(T − t)αJ1−α

t ut + Γ(1 + α)u− αI(ut, (T − t)α−1)],

Cx = −(T − t)αx2[Θ(x)ux + (Θ(x) + 2x−1Θ(x) − Θ′(x))u

];

(5.13)

Ct = Θ(x)J1−αt ut,

Cx = −x2[Θ(x)ux + (Θ(x) + 2x−1Θ(x) − Θ′(x))u

],

(5.14)

where the function Θ(x) is defined by (4.14). The conserved vector (5.13) corre-sponds to the product ϕ1ψ2 of the constants ϕ1 and ψ2 in (4.6) with (4.7), (4.13),and conserved vector (5.14) corresponds to the product ϕ2ψ2 of the constants ϕ2

and ψ2.


In this case we substitute Dγ(α)t = CDα

t in the equation (2.9). Unlike the previuoscase, the equation under consideration does not have a conservation form. Theformal Lagrangian (4.1) upon the substitution v = φ(t, x), with the functionφ(t, x) defined by (4.6), (4.8), takes the form

L = ϕ1(T − t)α−1Ψ(x)(CDαt u− x2Dx(h) − 4xh). (5.15)

Substituting the formal Lagrangian (5.15) and the symmetries (3.1)–(3.3) in(5.1) and (5.3) we find new conservation laws.


Ct = x2[Γ(α)u− I(ut, (T − t)α−1)

],

Cx = −x4(T − t)α−1h(u, ux).(5.16)


and ψ1 in the substitution (4.6) with (4.8), (4.11).For h = ux, using the symmetry X1 from (3.1), we find the conserved vector

Ct = x−1[Γ(α)u− I(ut, (T − t)α−1)

],

Cx = −(T − t)α(xux + 3u).(5.17)

The conserved vector (5.17) corresponds to the product ϕ1ψ2 of the constantsϕ1 and ψ2 in (4.6) with (4.8), (4.12). The symmetry X2 from (3.1) provides thetrivial conservation law only.

For h = ux +u, using the symmetry X1 from (3.2), we also find an additionalconserved vector, namely:

Ct = Θ(x)[Γ(α)u− I(ut, (T − t)α−1)

],

Cx = −(T − t)αx2[Θ(x)ux + (Θ(x) + 2x−1Θ(x) − Θ′(x))u

],

(5.18)

where the function Θ(x) is defined by (4.14). The conserved vector (5.18) cor-responds to the product ϕ1ψ2 of the constants ϕ1 and ψ2 in (4.6) with (4.8),(4.13).

10


Now we consider the equation (2.7) upon substitution Dγ(α)t = D1+α

t . Thenthis equation has the conservation form with the conserved vector having thecomponents

Ct = x2Dαt u, Cx = −x4h(u, ux). (5.19)


L = (ϕ1t+ ϕ2)Ψ(x)(D1+αt u− x2Dx(h) − 4xh). (5.20)

Substituting the formal Lagrangian (5.20) and the symmetries (3.1)–(3.3) in(5.1) and (5.4), we find new conserved vectors for all three particular types of thefunction h(u, ux).


Ct = x2[tDα

t u− J1−αt u

],

Cx = −tx4h(u, ux).(5.21)

The conserved vector (5.21) corresponds to the product ϕ1ψ1 of the constantsϕ1 and ψ1 in the substitution (4.6) with (4.9), (4.11). The conserved vectorcorresponding to the product ϕ2ψ1 of the constants ϕ2 and ψ1 in (4.6) coincideswith the conserved vector (5.19).

For h = ux, using the symmetry X1 from (3.10), we find two additionalconserved vectors:

Ct = x−1[tDα

t u− J1−αt u

],

Cx = −t(xux + 3u),(5.22)

Ct = x−1Dαt u, Cx = −(xux + 3u). (5.23)


and ψ2 in (4.6) with (4.9), (4.12), and conserved vector (5.23) corresponds to theproduct ϕ2ψ2 of the constants ϕ2 and ψ2. As previously, the symmetry X2 from(3.1) provides the trivial conservation law only.

For h = ux + u, using the symmetry X1 from (3.2), we also construct twoadditional conserved vectors:

Ct = Θ(x)[tDα

t u− J1−αt u

],

Cx = −tx2[Θ(x)ux + (Θ(x) + 2x−1Θ(x) − Θ′(x))u

];

(5.24)

Ct = Θ(x)Dαt u,


],

(5.25)

where the function Θ(x) is defined by (4.14). The conserved vector (5.24) corre-sponds to the product ϕ1ψ2 of the constants ϕ1 and ψ2 in (4.6) with (4.9), (4.13),and conserved vector (5.25) corresponds to the product ϕ2ψ2 of the constants ϕ2

and ψ2.

11


Now we substitute Dαt = Dα

t in the equation (2.9). Then this equation has theconservation form with the conserved vector having the components

Ct = x2J1−αt u, Cx = −x4h(u, ux). (5.26)


L = ϕ1Ψ(x)(Dαt u− x2Dx(h) − 4xh). (5.27)

Substituting the formal Lagrangian (5.27) and the symmetries (3.1)–(3.3) in(5.1) and (5.5) we construct two new conservation laws.

For h = ux, using the symmetry X1 from (3.1), we find a conserved vector

Ct = x−1J1−αt u, Cx = −(xux + 3u). (5.28)


and ψ2 in (4.6) with (4.10), (4.12). The symmetry X2 from (3.1) provides thetrivial conservation law only.

For h = ux + u, using the symmetry X1 from (3.2), we find the additionalconserved vector

Ct = Θ(x)J1−αt u,


],

(5.29)

where the function Θ(x) is defined by (4.14). The conserved vector (5.29) cor-responds to the product ϕ1ψ2 of the constants ϕ1 and ψ2 in (4.6) with (4.10),(4.13).

It is interesting to note that for all three particular types of the functionh(u, ux), i.e. h = ux, h = ux + u, h = ux + u2, the symmetries (3.1)–(3.3) givethe conserved vector with the components (5.26) that corresponds to the productϕ1ψ1 of the constants ϕ1 and ψ1 in the substitution (4.6) with (4.10), (4.11).

6 Particular solutions

6.1 Invariant solutions

We can construct invariant solutions only for two types of approximations of thetime-fractional Kompaneets equations that correspond to the following types ofthe function h: h = ux and h = ux + u2.

Let h = ux. Then we can find the invariant solutions of the equationq (2.9)using a linear combination of the operators X1 and X2:

X = βX1 +X2 = βu∂

∂u+ x

∂

∂x.

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The corresponding invariant solution has the form

u(t, x) = xβz(t).

Substituting this expression in the equation (2.9), we obtain the ordinary frac-tional differential equation for the function z(t):

Dγ(α)t z = β(β + 3)z. (6.1)

For β = 0 or β = −3 this equation takes very simple form

Dγ(α)t z = 0. (6.2)

For different time-fractional differential operators, the equation (6.2) has thefollowing common solutions:

Dγ(α)t = Dα

t Dt : z(t) = c1tα + c2;

Dγ(α)t = CDα

t : z(t) = c1;Dγ(α)

t = D1+αt : z(t) = c1t

α + c2tα−1;

Dγ(α)t = Dα

t : z(t) = c1tα−1.

(6.3)

Here c1 and c2 are arbitrary constants. Then, we get the following invariantsolutions:

• for the approximation of the equation (2.5)

u = c1tα + c2, u = x−3(c1tα + c2);


u = c1, u = x−3c1;


u = tα−1(c1t+ c2), u = x−3tα−1(c1t+ c2);


u = c1tα−1, u = c1x

−3tα−1.

Remark 2 To give a physical interpretation of the obtained solutions, it is nec-essary to make the inverse change of variables u→ f in accordance with the def-initions of u for different generalizations of the Kompaneets equation presentedin Section 2. It is easy to show that after such inverse change of variables, for allobtained solutions we have ft = 0, i.e. these solutions are the stationary solutionsof the corresponding approximations of the equations (2.1)–(2.4).

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Now we consider the equation (6.1) with β ̸= 0, β ̸= −3, and denote λ =β(β+3). The common solutions of the equation (6.1) for different time-fractionaldifferential operators can be obtained using the Laplace transform. As a result,we arrive at the following invariant solutions:


u = xβ(c1E1+α,1(λt1+α) + c2t

αE1+α,1+α(λt1+α));


u = c1xβEα,1(λtα);


u = xβtα−1(c1E1+α,α(λtα+1) + c2tEα+1,α+1(λtα)

)• for the approximation of the equation (2.8)

u = c1xβtα−1Eα,α(λtα).

Here

Eα,β(z) =∞∑

k=0

zk

Γ(αk + β)

is a two parameter function of the Mittag-Leffler type.

Remark 3 To give a physical interpretation, we note that after inverse change ofvariables u→ f in these solutions, one can conclude that these solutions describethe dynamic regimes in which the width of the corresponding diffusion packetsincreases according to the power law.

Now let us consider another case when h = ux + u2. The invariant solutioncorresponding to the operator

X = x∂

∂x− u

∂

∂u

has the formu = x−1z(t).

Substituting it into the equation (2.9), we get the following ordinary fractionaldifferential equation for the function f(t):

Dγ(α)t z = 2(z2 − z). (6.4)

We can not present here any exact solution of this nonlinear equation for anytime-fractional differential operators considered in this paper. Note that theequation (6.4) does not have Lie point symmetries.

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6.2 Non-invariant particular solutions

One can use the method of conservation laws (see [19], chapter 2) for construct-ing the particular solutions for the diffusion-type approximations of the time-fractional Kompaneets equation. According to this method, a particular solutionis obtained by letting

Ct = p(x), Cx = q(t),

where Ct and Cx are the components of a conserved vector. The calculationslead to the following results.

Calculations show that solutions of any approximation of the equation (2.9),obtained by using conservation laws presented in the previous section and corre-sponding this approximation, coincide.

For all approximations of the time-fractional Kompaneets equations (2.9) withh = ux, this approach gives the linear combinations of the invariant solutionscorresponding to β = 0 and β = −3 that have been presented in the previoussubsection.

For h = ux + u and h = ux + u2, using the method of conservation laws, weobtain the following solutions:


u = z(x, c1, c2)tα + z(x, c3, c4),


u = z(x, c1, c2),


u = z(x, c1, c2)tα−1 + z(x, c3, c4)tα,

• and the approximation of the equation (2.8)

u = z(x, c1, c2)tα−1,

where c1, c2, c3, c4 are arbitrary constants. The function z depends on thefunction h: for h = ux + u we have

z(x, a, b) = e−x[ax−2Θ(x) + b

],

where the function Θ(x) is defined by (4.14). For h = ux + u2 we find two typesof the function f :

z(x, a, b) =1x

+a

x2tanh

(b− a

x

), z(x, a, b) =

1x− a

x2tan

(b− a

x

).

Remark 4 Proceeding as in Remark 2, one can see that all these solutions arethe stationary solutions of the appropriate approximations of the equations (2.1)–(2.4).

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7 Conclusion

The time-fractional generalizations of the Komapneets equation considered in thepaper can be used for investigating scattering processes in plasma whose kineticbehavior demonstrates different memory effects. The obtained conservation lawsgive explicit expressions of conserved quantities. This fact is particularly signifi-cant for fractions differential models. The particular solutions constructed in thepaper permit to analyze in detail several physical effects in plasma.

Acknowledgements

We acknowledge a financial support of the Government of Russian Federationthrough Resolution No. 220, Agreement No. 11.G34.31.0042.

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Bibliography

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[12] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractionaldifferential equations. Amsterdam: Elsevier; 2006.

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Highlights

• Four time-fractional generalizations of the Kompaneets equation are

proposed.

• The nonlinear self-adjointness of these equations is established.

• Conservation laws are constructed using symmetries.

• Invariant and noninvariant exact solutions are obtained.

Documents

Nonlinear self-adjointness, conservation laws and exact solutions of time-fractional Kompaneets equations