SUBDIFFUSION MODEL WITH TIME-DEPENDENT …thermalscience.vinca.rs/pdfs/papers-2016/TSCI160427247H.pdf · Hristov, J.: Subdiffusion Model with Time-Dependent Diffusion Coefficient

Hristov J Subdiffusion Model with Time-Dependent Diffusion Coefficient THERMAL SCIENCE Year 2017 Vol 21 No 1A pp 69-80 69

SUBDIFFUSION MODEL WITH TIME-DEPENDENT DIFFUSION COEFFICIENT

Integral-Balance Solution and Analysis

by

Jordan HRISTOV

Department of Chemical Engineering University of Chemical Technology and Metallurgy Sofia Bulgaria

Original scientific paper DOI102298TSCI160427247H

The paper addresses approximate integral-balance approach to a time-fractional diffusion equation of order 0 lt μ lt 1 with a time-dependent diffusion coefficient of power-law type D(t) = D0tβ where 0 lt β lt 1 The form of the solution spreading in a semi-infinite medium through an analysis of the second moment of the approximate solution reveals that depending on the sum μ + β the solution can model subdif-fusive (μ + β lt 1) superdiffusive (μ + β gt 1) or Gaussian (μ + β = 1) process of transport The optimal exponents of the approximate parabolic profiles have been determined by minimization the mean squared error of approximation over the penetration depth Key words time-fractional diffusion integral-balance method time-dependent

diffusivity approximate solutions

Introduction

Fractional differential equations (FDE) are suitable for modelling of anomalous diffu-sive processes such as subdiffusion and superdiffusion in non-homogeneous porous media [1] turbulent transports [2] in both classical no-local [3] and local [4] sense Analytical solutions for FDE are difficult to obtain [5] and most of them are approximate in nature [6-10] resulting in forms unsuitable for post-solution applications and engineering analysis The common ap-proaches are numerical methods such as Galerking method [11] Tau method [12] and spec-tral-method [13] while analytical solutions are rare [14-16]

Most of FDE are with constant [17 18] concentration-dependent [19] or space-de-pendent diffusion coefficients [15 16] depending on the specific features of the transport pro-cess modelled

This communication addresses integral-balance approach to 1-D linear subdiffusion equation with a time-dependent diffusion coefficient of power law type D(t) = D0tβ [m2sndash(1ndashβ]

( ) ( ) ( ) ( ) ( ) ( )2

2

0 0 0 1 0

u x t u x tD t u x u t u t

t x

micro

micro

part part= = = infin =

part part (1a)

( ) 0 with 0 and 0D t D t xβ micro= ge gt (1b)

Authorʼs e-mail jordanhristovmailbg

Hristov J Subdiffusion Model with Time-Dependent Diffusion Coefficient 70 THERMAL SCIENCE Year 2017 Vol 21 No 1A pp 69-80

The time-fractional derivative in eq (1a) is left-sided of either of Riemann-Liouville (RL) eq (2a) or Caputo type eq (2b) [20]

( )( )

( )( )RL 1

0

1 d d 0 11 d

t

t

u x t f x tD z

t t t z

micromicro

micromicro micromicro minus

part= = lt lt

part Γ minus minusint (2a)

( )

( ) ( )( )1

0

1 1 d d 0 11 d

t

C t

u x tD f x z z

t tt z

micromicro


part= = lt lt

part Γ minus minusint (2b)

( ) ( )

for 1u x t u x t

t t

micro

micro micropart part

= =part part

(3)

The problem (1a) and (1b) was conceived by Fa and Lenzi [21] and solved by a series expansion technique The efforts of this analysis were oriented to the first passage time (FPT)in finite and infinite domains Now we address an approximate analytical solution based on the integral-balance approach [22 23] successfully applied previously to solve models with time-fractional [24 25] and space-fractional derivatives [26 27] The solution the Dirichlet problem of eq (1) with zero initial conditions and RL time-fractional derivative is demonstrat-ing the feasibility of the method

Solution of the Dirichlet problem

The integral-balance approach to the subdiffusion equation

Consider eq (1a) with initial and boundary conditions (Dirichlet problem)

( )0 0 (0 ) 1 and ( ) 0 for 0u x u t u t t= = infin = gt (4abc)

The integral-balance method uses the concept of a finite penetration depth δ(t) thus defining a sharp front of propagation of the solution requiring the boundary condition u(infint) = 0 to be replaced by

( ) ( )(0) 1 and 0u u ux

δ δpart= = =

part (5abc)

The single integration approach known as the heat-balance integral method (HBIM) [22 23] suggests integration over the entire penetration depth with respect to the space co-or-dinate x namely

( ) ( ) ( ) ( ) ( ) ( )2

20 0 0

0d d d

u x t u x t u x t u tx D t x x D t

t x t x

micro microδ δ δ

micro micro

part part part part= rArr = minus

part part part partint int int (6ab)

Further the integral-balance method suggests replacement of the function u(xt) by an assumed profile ua(xt) = ua(xδ) as a function of the dimensionless pace variable 0 lt xδ lt 1 This approach leads to an equation about the time evolution of the penetration depth δ(t)

However with HBIM the gradient at x = 0 in the right-hand side of eq (6b) should be determined through the assumed approximate profile This drawback can be avoided by the double-integration method (DIM) conceived for non-linear problems of integer-order (μ = 1) [28 29] and successfully applied to time-fractional equations [24 25]

In accordance with DIM the first step is an integration of eq (1a) from 0 to x namely


( ) ( ) ( ) ( ) ( )

0

0d

x u x t u x t u tx D t D t

t x x

micro

micro

part part part= minus

part part partint (7)

Equation (6b) of HBIM can be expressed equivalently

0

( ) ( ) (0 )d ( )x

x

u x t u x t u tx D tt t x

δmicro micro

micro micro

part part part+ = minus

part part partint int (8)

Then subtracting eq (7) from eq (8) and integrating the result from 0 to δ we get [24 25]

( ) ( ) ( )0

d d 0

x

u x tx x D t u t

t

microδ δ

micro

part= part

int int (9)

Therefore the right-side of eq (9) depends on the boundary condition at x = 0 and in contrast to eq (6b) is independent of the approximation of the gradient at this point

Assumed profile spatial integration and the penetration depth

The solution in this work uses an assumed parabolic profile with unspecified exponent [23-27] ua(xδ) = (1 ndash xδ)n 0 le x le δ n gt 0 The profile satisfies the conditions in eq (5abc) and forms two zones ua(x) gt 0 for x lt δ and ua(x) = 0 for x ge δ The solution based on this as-sumed profile requires n gt 0 but dependent on the fractional order μ and the exponent β Now we turn on the integration of the left-hand side of eq (9) replacing u(xt) by ua(xδ) [24 25]

( )( )2

0

11 d d1 2

n

x

x x xt n n t

δ δ micro micro

micro micro

δδ

part part minus = part + + part int int (10)

Now with D(t) = D0tβ and u(0t) = 1 we get

( )( )2

0 1 2D Nt N n nt

microβ

micro

δpart= = + +

part (11ab)

The Laplace transform of eq (11a) is ( ) ( ) ( )120 1s D N s β microδ β minus + += Γ + Then the in-

verse Laplace transforms yields

( ) ( ) ( ) ( ) ( )20 0 t D t NG t D t NGmicro β micro βδ micro β δ micro β+ += rArr = (12ab)

( ) ( )( )1

1

Gβ

micro βmicro β

Γ +=Γ + +

(12c)

For β = 0 the solution of eq (12b) reduces to DIM solution [24 25] of the time-frac-tional subdiffusion equation namely 12 12

0 0( ) ( ) [ (1 )]t D t Nmicro βmicro βδ micro+

= = Γ + Further for μ = 1 and β = 0 we get G(10) = 1 and the solution of eq (12b) reduces to the classical DIM solution of the integer-order diffusion equation see comments in [24 25] In the special case when μ + β = 1 and G(μ β) reduces to G(μ + β = 1) = Γ(1 + β) and 12 12

1 0( ) ( ) [ (1 )]t D t Nmicro βδ β+ = = Γ + depends only on β

Hence the front propagates with a speed proportional to t (μ + β)2 and the principle questions arising from this irrespective to the initial stipulation that eq (1a) is a subdiffusion equation are (1) how the speed of the front depends on the mutual effects of the exponent β and


the fractional order μ and (2) are there restrictions on the exponent β taking into account that subdiffusion transport is the problem at issue

The speed of the penetration front and restrictions on the exponent βSince the values of μ and β define the behaviour of the diffusion processes it is im-

portant to explain clearly how they affect the solution and to delineate the two principle regimes of diffusion mentioned earlier Hence the front spread is proportional to t(μ + β)2 with 0 lt μ lt 1 and therefore we have to define the range of variation of β

First since δ is a physically defined length of the diffusion process and it should be pos-itive Therefore this corresponds to the condition μ + β gt 0 which can be satisfied for any β gt 0 With dδdt = [(μ + β)2] t(μ + β)2ndash1 the condition dδdt gt 0 requires μ + β gt 2 ntilde β gt 2 ndash μ and taking into account the range of μ the condition is satisfied for β gt 1 Otherwise the deceler-ating front exists for μ + β lt 2 and therefore this condition and the general one (μ + β gt 0) are satisfied for β lt 1

Futher with 0 lt μ + β lt 1 we have a subdiffusive regime while for μ + β gt 1 the transport is superdiffusive In the light of the previous comments about the effect of the value of β on the rate of the penetration front the condition of accelerating front is μ + β gt 2 ntilde β gt 2 ndash μ which corresponds to the superdiffusion regime In contrary when the front is decelerating that is μ + β lt 2 ntilde β lt 2 ndash μ ntilde β lt 1 we have a transition from subdiffusion (μ + β lt 1) to superdif-fusion transport (μ + β gt 1) The special case μ + β = 0 ntilde β = ndashμ corresponds to δ2 = const and dδdt = 0 and the stationary regime [21] For μ + β = 1 we obtain δ equiv t12 that is the classic Fickrsquos diffusion law corresponding to the Gaussian process

It is worthy to note that β gt 0 means a diffusion coefficient increasing in time while for β lt 0 the diffusion coefficient is decreasing in time The former case could be termed as fast diffusion while the second one is slow diffusion The slow diffusion with β lt 0 the case solved in [21] results in δ equiv t (μ ndash β)2 and dδdt = [(μ ndash β)2] t (μ ndash β)2ndash1 The condition β lt 0 assures the front propagating in one direction that is physically motivated In addition we have for μ ndash β ndash 2 gt 0 ntilde β lt 2 ndash μ with accelerating front while the condition μ ndash β ndash 2 lt 0 ntilde β gt 2 ndash μ corresponds to the decelerating case Both cases require β gt 1 but taking into account that 0 lt μ lt 1 we have the condition μ ndash β ndash 2 lt 0 and a decelerating penetration which is a physically adequate estimation for a slow diffusion process In addition with respect to the mean squared displacement (see the further in this article) we have aacutex2ntilde equiv δ2 equiv t μ ndash β and the slow subdiffusion would exist for μ ndash β lt 1 that is the general condition μ gt β previously established is satisfied The opposite requirement μ ndash β gt 1 ntilde μ gt 1 + β does not satisfy the general condition 0 lt μ lt 1 and therefore the superdiffusion transport with β lt 0 is impossible

In this article we restrict the analysis only for the case β gt 0 since the problems with β lt 0 is out of the scope of the present study and draws future studies

Approximate profile and related issues

Approximate profile

After the development of the penetration depths the approximate solution can be ex-pressed

( ) ( )

0

1 1

n n

ap

xuN nD t NGmicro β

ξmicro βmicro β+

= minus = minus

(13)


( ) ( )( ) ( )( )1

1 21pN NG n n

βmicro β

micro βΓ +

= = + +Γ + +

(14)

This approximate solution defines in a natural way the similarity variable

0

xD tmicro β

ξ+

=

Mean square displacement

Following the work of Fa a Lenzi [21] the mean square displacement (the second moment of the distribution) is eq (15a) while with the assumed profile used in the approximate integral-balance solution (DIM solution) one obtains in eq (15b)

( )

( )( )

( )0 02 2 2

FL DIM

2 1 1

1 1D D

x t x tmicro β micro ββ βδ

micro β micro β+ +Γ + Γ +

= equiv equivΓ + + Γ + +

(15ab)

The factor 2 in eq (15a) appears from the integration in the interval (ndashinfin+ infin) [21] while the integration interval (0+ infin) is equivalent to (0δ) in the framework of the present study However if the integration is performed from (ndash δ+ δ) then the mathematical result is eq (15a) but it is relevant to the case when there is a point source at x = 0 Following the results of the analysis of the penetration speed and the comments in [21] the mean square displacement in eq (15b) clearly states that subdiffusive processes arise when 0 lt μ + β lt 1 and superdiffusion occurs for μ + β gt 1

The first passage time analysis from different viewpoints

If we consider a domain with a finite length L that is 0 le x le L with boundary con-ditions u(Lt) = 0 and ux(Lt) = 0 we may define the time required the penetration depth δ(t) to reach the boundary x = L From eq (12a) with δ(t) = L we get

( ) ( ) ( )

12

20

0

1 1L LL D t NG t L

D NG

micro βmicro β micro βmicro β

micro β

++ +

= rArr =

(16)

This is a practical solution coming from the integer-order solution developed by the integral-balance solutions corresponding to the so-called pre-heating period [22 23]

Further we continue with the classical definition of the FPT distribution [21 29-31] is F(t) = ndashdftdt where ( )

0 d

L

tf u x t x= int Assuming for the purpose of the following analysis that L = δ(t) since no diffusion occurs for x gt δ(t) Hence with the approximate solution (13) and the expression (12b) about δ(t) we have

( ) ( )( ) ( )

22

00

d d 21 d d d 1 2 1

ntx nF t x t D G

t t t n n

δ micro β

δ

δ micro β micro βδ

+ minus + += minus minus = = + +

int (17abc)

From the conditions about the existence of the subdiffusion regime previously com-mented we have the condition μ + β ndash 2 gt 0 ntilde β gt 2 ndash μ corresponds to acceleration penetration front and superdiffusion The stationary solution with a constant speed of δ(t) for μ + β = 0 leads to Fδ(t) = 0 Otherwise when μ + β ndash 2 lt 0 ntilde β lt 2 ndash μ ntilde β lt 1 we have subdiffusive transport Moreover the mean FPT (MFPT) is defined with δ(t) = L [30 31]


( ) ( ) ( )10 2

FPT FPT0 0 0

1 d d1 1

2a

a

L

uu up

DT u x t x t T t

n N

micro β

micro β

infininfin ++

rarr

= rArr = rarrinfin ++ +

int int (18ab)

Therefore the estimation of MFPT developed on the basis of the approximate inte-gral-balance solution (DIM) is consistent with the results of Fa and Lenzi [21 30 31] and the comments of Yuste and Lindenberg [32] to be exact the MFPT for the subdiffusion transport is infinite

Refinement of the approximate solution

Residual function and approximation of the time-fractional derivative

The exponent n of the assumed profile is the only parameter which can not be defined by the boundary conditions at x = 0 and (5abc) [22-27] and therefore we have to impose addi-tional constraints The approach used here minimises the residual function of eq (1a) precisely the mean squared error of approximation over the penetration depth δ(t) as a procedure allow-ing to calculating the optimal value of the exponent n In this context the residual function of eq (1a) is

( ) ( ) ( ) ( )2

2

a a

a

u x t u x tR u x t D t

t x

micro

micro

part part= minus part part

(19)

Now the problem is how the time-fractional derivative partμua(xt)parttμ of the approxi-mate profile can be expressed in a way easy for calculations To resolve this problem we will apply the technique developed in [25-27] Let us express the assumed profile as V(η) = (1 ndash η)n

where 0 le η = xδ le 1 Moreover the dimensionless profile can be approximated as a converg-ing series ( ) 0

N ka kk

V bη η=

asympsum where 0 le μ le 1 see [25-27] for details Therefore the residual function can be approximated

( ) ( ) ( ) ( )( ) 2

RL RL 20

1 1

nNk

a t kk

n nR u x t R t D b D tmicro η

η micro β ηδ

minus

=

minus minus = asymp minus

sum (20)

Taking into account that η = ξNP see eq (14) and ξ = [x(Dμ)12]tndash(β+μ)2 all terms in 0

( ) N ka kk

V bη η=

asympsum are power-law functions of time and can be easily differentiated

( ) ( ) ( ) ( )RL 1 1 2t k kD b t b t kmicro λ λ microλ λ micro λ micro βminus= Γ + Γ + minus = minus + (21ab)

Consequently the time-fractional derivative of order μ from the series 0

( ) N ka kk

V bη η=

asympsum is [25]

( ) ( ) ( )RL0 0

n n

k kt k k

k kD V c t n t n cmicro micro microΦ Φη η η micro η micro β ηminus minus

= =

asymp asymp =sum sum (22ab)

( ) ( )1 1k kc b k k micro= Γ + Γ + minus (23)

Thus the residual function (19) can be expressed


( ) ( ) ( )( )( )

2

RL RL

1 11

n

p

n nR t n

t N nmicro Φη

η micro β η micromicro β

minus minus minusasymp minus

(24)

With the Caputo derivative applied to 0

( ) N ka kk

V bη η=

asympsum we have ( ) C t a CD V n tmicro microΦη η micro β minus( )asymp The difference RL ( ) ( )t C t aD V D Vmicro microη ηminus is independent of n

see [25] for comments namely

( ) ( ) ( ) ( ) ( )RL RL 1 1t C t a CD V D V n nmicro microη η Φ η micro Φ η micro microminus = minus = Γ minus (25)

The residual function has the same construction as the one presented by eq (24)

Constraints on the exponent n at the boundaries

With the assumed profile ua(xδ) = (1 ndash xδ)n and u(0t) = 1 the residual function with RL ( )t aD u x tmicro (see [25] for details and comments for the similar case with D(t) = D0 = const) is

( ) ( ) ( )( ) ( ) 2

RL 20

11 d 1 1 d 11 d

n nt n nx xR x t D tt t micromicro β τ

micro δ δ δτ

minus minus = minus minus minus Γ minus minus int (26)

The conditions (5bc) ie ua(δt) = 0 and partua(δt)partx = 0 lead to see eq (26)

( ) ( ) ( ) 2

RL RL 2

1 lim lim 1

n

x x

n n xR t R tδ δ

δ δδ δ

minus

rarr rarr

minus = = minus minus

(27)

Hence the condition (27) is satisfied at x = δ when n gt 2Further at x = 0 with ua(0t) = 1 we have

( ) ( )( )

( )RL 00

1101

n nR t t D t

D t NGmicro β

micro βmicro micro βminus

+

minus= minusΓ minus

(28)

Setting RLR(0t) = 0 we get the following equation relating n(x = 0) with μ and β

( )( ) ( ) ( ) ( )

( ) ( )1 2 1 1

01

n n G n nNG

micro β micromicro micro β

+ + minus minus Γ minus=

Γ minus (29)

From the denominator in eq (29) it is hard to establish exact values of n because we have simultaneous effects of μ and β

Optimal exponents of the approximate solutions

Applying the Langford criterion [33] for the integral-balance method we need

( ) ( )2

L RL0

d minE n t R t xδ

micro β η micro β= rarr int (30)

In terms of the dimensionless variable ξ the squared error function (30) can be ex-pressed as a ratio decaying in time with a rate t2μ namely EL(n α β t) = eL(n μ β)t2μ Here eL(n μ β) is time-independent function defined by


( ) ( ) ( )( )( )

221

L RL0

1 1 d min

n

p

n ne n n

N nη

micro β Φ η micro β ηmicro β

minus minus minus= minus rarr

int (31)

The estimation of the optimal n through minimization of eL(n μ β) was carried out in accordance with the procedure explained in [26 27] where it was established that an expansion up to nine terms of the series is enough to calculate RLΦ(n η μ β) and evaluate the integral in eq (31) by help of MAPLE for instance The optimal exponents determined with this technol-ogy are summarized in tab 1 Regarding the values of the optimal exponents summarized in tab 1 reveal some principle features

Table 1 Optimal exponents for μisin[0109] and βisin[0109]α darr βrarr 00 01 02 03 04 05 06 07 08 0901 3806 3049 3160 3969 3642 3712 3929 3934 4116 4691EL∙103 3961 2840 1525 0081 0789 1972 2070 0141 283702 3477 3470 3736 3908 3969 4074 4312 3917 4894 4888EL∙103 5100 472 2861 4341 0653 8211 0299 0276 178403 3529 3624 2894 3871 4175 4390 3893 2989 2812 2812EL∙103 1700 1543 5253 0662 0324 1334 2264 0508 086004 3150 3703 2920 3857 4808 4779 2844 2970 2812 2835EL∙103 0333 1340 2131 1400 4981 2850 4823 0750 083505 3477 2920 3908 3934 2971 2989 2750 2824 2865 2920EL∙103 2133 3640 5452 1519 1953 1290 0124 0125 013606 2596 3926 3917 3087 2611 2763 2862 2906 3060 3554EL∙103 0124 0243 2408 8119 1174 6016 0127 0126 074407 2022 3012 2989 2985 2799 2880 2988 2989 3752 4021EL∙103 6948 1601 1892 0931 0520 0936 0160 0893 031708 2643 2997 3006 2989 2925 3105 2447 2447 4235 3977EL∙103 2629 2111 1512 0154 2230 0386 1300 2111 360409 2310 2776 2897 2968 2997 3777 2400 2235 3997 4647EL∙103 3298 4712 2467 3611 0725 0451 4746 0801 0584

-established in [26] Bold (at the diagonal of the table) the case μ + β = 1

ndash The exponent of the approximate solution in the case of subdiffusive regime (μ + β lt 1) decreases with increase in μ if β is stipulated Alternatively when μ is stipulated then the optimal exponent increases with increase in the value of β

ndash The diagonal of the tab 1 is formed by cells (in bold) containing the optimal exponents in the case when μ + β = 1 In this special case we have δ2 equiv t and the optimal exponents decrease from the top right corner (μ = 01 β = 09) to the left bottom corner (μ = 09 β = 01) The diagonal divides the data summarized in tab 1 into two groups subdiffusive group (LG) to the left of the diagonal and superdiffusive group (RG) to the right of the diagonal

ndash For the LG the general behaviour is that mentioned in point 1 This is well demonstrated for μ = 01 and 01 le β le 09 with increase in μ the right boundaries of the LG is defined by diagonal of the table (the case μ + β = 1) It is worthy to note that close to the diagonal there is anomalous behaviour of the exponents which become close to that when μ + β = 1 The transition through the diagonal μ + β = 1 is demonstrated by a similar behaviour but after that the values of the optimal exponents increase with increase in β This transient be-


haviour of the exponents of the profile could be related to change in the speed of the front dδdt namely from decelerating front (subdiffusion) to accelerating front (superdiffusion)

ndash The behaviour of the optimal exponents previously commented can be attributed the factor G(μβ) in the penetration depth see eqs (12c) and (14) The G(μβ) varies in a narrow range 05 le G(μβ) le 1 fig 1 but its behaviour is strongly non-linear

Numerical experiments

The approximate solutions are presented graphically in fig 2 For μ = 01 fig 2(a) we have entirely subdiffusive profiles while for β = 09 fig 2(b) the distributions correspond to the superdiffusive process In these limiting cases as well in the intermediate situations

Figure 1 Behaviour of the factor

( ) ( )( )+

= + +

le le le le

1

1

001 099 and 001 099

Gβ

micro βmicro β

micro β

Γ

Γ

G(micro

β)

micro [minus] β [minus]08

0604

020

0806

0402

0

11

10

09

08

07

06

05

Figure 2 Dimensionless approximate profiles at various fixed values of the fractional order μ and the exponent β as parameter

u a [minus

]

ξ [minus]

β = 09

β = 05

β = 01

micro = 01

0 1 2 3 4

10

08

06

04

02

0

u a [minus

]

(a)

u a [minus

]

10

08

06

04

02

0

micro = 05

β = 01

β = 05

β = 09

β = 07

(b)

micro = 08 micro = 09

β = 01

β = 03

β = 06β = 09

ξ [minus]0 1 2 3 4

10

08

06

04

02

0

ξ [minus]0 1 2 3 4

ξ [minus]0 1 2 3 4

u a [minus

]

10

08

06

04

02

0

(d)

β = 01

β = 03

β = 05

β = 07

β = 09

(c)


figs 2(b) and 2(c) the arrangement of the curves from right to the left strongly follows the in-crease in the value of β This arrangement could be explained by the effect of β on the derivative dudξ namely ndash the derivative 1d d ( )(1 )n

p pu n N Nξ ξ minus= minus minus is a function of ξ since for given μ and β the denominator Np is a constant However as it was demonstrated the exponent n increases with increase in β and

ndash hence if ξ is fixed (a certain point of the profile) as a result of the increase in β and the nu-merical value of Np and the difference (1 ndash ξNp) will increase too Consequently the higher value of β the higher value of the derivative dudξ (absolute value since really dudξ lt 0) and graphically the curves corresponding to large β are more steep and located to the left of the profiles corresponding to small β

The simultaneous effect of μ and β can be visually demonstrated by the 3-D plots for fixed values of the dimensionless approximate profiles fig 3 close to the middle of the profile fig 3(a) and at for large ξ figs 3(b) and 3(c) The behaviour of the surfaces in fig 3 resem-bles that of the function G(μβ) fig 1 In fact the approximate solution ua can be expressed as ua(ξ) = 1 ndash kn[G(μβ)]12n where kn = ξN For the sake of the simplicity the plots in fig 3 are developed for stipulated value n = 4

Figure 3 The 3-D approximate profiles at fixed values of ξ as function of μ and β

β [minus] β [minus]micro [minus]

09

0705

0301

0705

03

01

ξ = 1

u a [minus

] 046044042040038036034

(a) (b) (c)

micro [minus]

09

0705

0301

0705

03

01

ξ = 3

u a [minus

] 004

003

002

001

ξ = 45

micro [minus]

09

0705

0301

β [minus]07

05

03

01

β [minus]

u a 1

05 [minus] 18

16141210

8642

The numerical results clearly demonstrate the general behaviour of the approximate solution developed by the integral-balance approach The upper limit of β was chosen β = 09 since there is no reason to use large values where in general the regime will be superdiffusive

The integral-balance method demonstrates its feasibility in solution of a subdiffusion equation with a time-dependent of a power-law type diffusion coefficient This study is part of a large project toward application of integral balance methods and developing approximate physically motivated solution to non-linear diffusion models as it was already done in [24-27 34 35]

Conclusions

The integral balance approach to the time-fractional diffusion equation with a time-de-pendent diffusion power-law coefficient D(t) = D0tβ confirms the analytical results of Fa and Lenzi [21] that its second moment is of power-law type with exponent μ + β The solution can present subdiffusive (0 lt μ + β lt 1) superdiffusive (μ + β gt 1) and Gaussian transport (μ + β = 1)

The solution was performed in the infinite domain but with the integral-balance ap-proach a front propagation of the solution is defined This allows investigating the principle


problem of the FPT in infinite domain The results confirm the principle statement [21 31] that in case of subdiffusive regime the FPT is infinite

References[1] Nigmatullin R R The Realization of the Generalized Transfer Equation in a Medium with Fractal Ge-

ometry Physica Status Solidi (B) Basic Research 133 (1986) 1 pp 425-430[2] Bakunin O G Description of the Anomalous Diffusion of Fast Electrons by a Kinetic Equation with a

Fractional Spatial Derivative Plasma Physics Reports 30 (2004) 4 pp 338-342[3] Hilfer R Applications of Fractional Calculus in Physics World Scientific Hackensack N J USA 2000[4] Yang X J et al Local Fractional Integral Transforms and their Applications Academic Press London

2015[5] Luchko Yu Srivastava H M The Exact Solution of Certain Differential Equations of Fractional Order

by Using Operational Calculus Comp Math Appl 29 (1995) 8 pp 73-85[6] Nakagawa J et al Overview to Mathematical Analysis for Fractional Diffusion Equations ndash New Math-

ematical Aspects Motivated by Industrial Collaboration J Math-for-Indust 2 (2010A-10) Jan pp 99-108

[7] Das S et al An Approximate Analytical Solution of Nonlinear Fractional Diffusion Equation Appl Math Model 35 (2011) 8 pp 4071-4076

[8] Wu G-C Variational Iteration Method for Solving the Time-Fractional Diffusion Equations in Porous Medium Chin Phys B 21 (2012a) 22 ID 120504

[9] Wu G-C Applications of the Variational Iteration Method to Fractional Diffusion Equations Local versus Nonlocal Ones Int Rev ChemEng 4 (2012b) 5 pp 505-510

[10] Wu G-C Baleanu D Variational Iteration Method for Fractional Calculus ndash A Universal Approach by Laplace Transform Advances in Difference Equations 18 (2013) Dec pp 1-9

[11] El-Kady M et al El-Gendi Nodal Galerkin Method for Solving Linear and Nonlinear Partial Fractional Space Equations Int J Latest Res Sci Technol 2 (2013) 6 pp 10-17

[12] Saadatmandi A Deghan M A Tau Approach for Solution of the Space Fractional Diffusion Equation Comp Math Appl 62 (2011) 3 pp 1135-1142

[13] Nie N et al Solving Spatial-Fractional Partial Differential Diffusion Equations by Spectral Method J Stat Comp Sim 84 (2014) 6 pp 1173-1189

[14] Ray S S Analytical Solution for the Space Fractional Diffusion Equation by Two-Step Adomian De-composition Method Comp Nonlinear Sci Num Simul 14 (2009) 4 pp 1295-1306

[15] Huang F Liu F The Space-Time Fractional Diffusion Equation with Caputo Derivatives J Appl Math amp Computing 19 (2005) 1-2 pp 179-190

[16] Huang F Liu F The Fundamental Solution of the Space-Time Fractional Advection-Diffusion Equa-tion J Appl Math amp Computing 19 (2005) 1-2 pp 339-350

[17] Meerschaert M M Tadjeran C Finite Difference Approximations for Two-Sided Space-Fractional Par-tial Differential Equations Appl Numer Math 56 (2006) 1 pp80-90

[18] Ervin V J et al Numerical Approximation of a Time Dependent Nonlinear Space Fractional Diffusion Equation SIAM J Num Anal 45 (2008) 2 pp 572-591

[19] Das S A Note on Fractional Diffusion Equation Chaos Solitons and Fractals 42 (2009) 4 pp 2074-2079

[20] Podlubny I Fractional Differential Equations Academic Press New York USA 1999[21] Fa K S Lenzi E K Time-Fractional Diffusion Equation with Time Dependent Diffusion Coefficient

Physica A 72 (2005) 1 011107 [22] Goodman T R Application of Integral Methods to Transient Nonlinear Heat Transfer in Advances in

Heat Transfer (Eds Irvine T F Hartnett J P) Academic Press San Diego Cal USA Vol 1 1964 pp 51-122

[23] Hristov J The Heat-Balance Integral Method by a Parabolic Profile with Unspecified Exponent Analysis and Benchmark Exercises Thermal Science 13 (2009) 2 pp 22-48

[24] Hristov J Approximate Solutions to Time-Fractional Models by Integral Balance Approach in Fractals and Fractional Dynamics (Eds Cattani C Srivastava H M Yang Xia-Jun) De Gruyter Open War-saw 2015 pp 78-109

[25] Hristov J Double Integral-Balance Method to the Fractional Subdiffusion Equation Approximate Solutions Optimization Problems to be Resolved and Numerical Simulations J Vibration and Control On-line first doi 1011771077546315622773


[26] Hristov J An Approximate Solution to the Transient Space-Fractional Diffusion Equation Integral-Bal-ance Approach Optimization Problems and Analysis Thermal Science On-line first doi102298TSCI160113075H

[27] Hristov J Transient Space-Fractional Diffusion with a Power-Law Super Diffusivity Approximate Inte-gral-Balance Approach Fundamenta Informaticae (IOS Press) 151 (2017) 1-4 pp 371-388

[28] Volkov V N Li-Orlov V K A Refinement of the Integral Method in Solving the Heat Conduction Equation Heat Transfer Sov Res 2 (1970) pp 41-47

[29] Sadoun N et al On the Refined Integral Method for One-Phase Stefan Problem with Time-Dependent Boundary Conditions Appl Math Model 30 (2006) 6 pp 531-544

[30] Fa K S Lenzi E K Anomalous Diffusion Solutions and the First Passage Time Influence of Diffu-sion Coefficient Physical Review E 71 (2005) Jan 012101

[31] Fa K S Lenzi E K Power Law Diffusion Coefficient and Anomalous Diffusion Analysis of Solutions and the First Passage Time Physical Review E 67 (2003) Jun 061105

[32] Yuste S B Lindenberg K Comments on Mean First Passage Time for Anomalous Diffusion Physical Review E 69 (2004) 3 033101

[33] Langford D The Heat Balance Integral Method Int J Heat Mass Transfer 16 (1973) 12 pp 2424-2428 [34] Hristov J Integral Solutions to Transient Nonlinear Heat (Mass) Diffusion with a Power-Law Diffu-

sivity A Semi-Infinite Medium with Fixed Boundary Conditions Heat Mass Transfer 52 (2016) 3 pp 635-655

[35] Fabre A Hristov J On the Integral-Balance Approach to the Transient Heat Conduction with Linearly Temperature-Dependent Thermal Diffusivity Heat Mass Transfer 53 (2017) 1 pp 177-204

Paper submitted April 27 2016Paper revised May 30 2016Paper accepted June 27 2016

copy 2017 Society of Thermal Engineers of SerbiaPublished by the Vinča Institute of Nuclear Sciences Belgrade Serbia

This is an open access article distributed under the CC BY-NC-ND 40 terms and conditions


The time-fractional derivative in eq (1a) is left-sided of either of Riemann-Liouville (RL) eq (2a) or Caputo type eq (2b) [20]

( )( )

( )( )RL 1

0

1 d d 0 11 d

t

t

u x t f x tD z

t t t z

micromicro


part= = lt lt

part Γ minus minusint (2a)

( )

( ) ( )( )1

0

1 1 d d 0 11 d

t

C t

u x tD f x z z

t tt z

micromicro


part= = lt lt

part Γ minus minusint (2b)

( ) ( )

for 1u x t u x t

t t

micro

micro micropart part

= =part part

(3)

The problem (1a) and (1b) was conceived by Fa and Lenzi [21] and solved by a series expansion technique The efforts of this analysis were oriented to the first passage time (FPT)in finite and infinite domains Now we address an approximate analytical solution based on the integral-balance approach [22 23] successfully applied previously to solve models with time-fractional [24 25] and space-fractional derivatives [26 27] The solution the Dirichlet problem of eq (1) with zero initial conditions and RL time-fractional derivative is demonstrat-ing the feasibility of the method

Solution of the Dirichlet problem

The integral-balance approach to the subdiffusion equation

Consider eq (1a) with initial and boundary conditions (Dirichlet problem)

( )0 0 (0 ) 1 and ( ) 0 for 0u x u t u t t= = infin = gt (4abc)

The integral-balance method uses the concept of a finite penetration depth δ(t) thus defining a sharp front of propagation of the solution requiring the boundary condition u(infint) = 0 to be replaced by

( ) ( )(0) 1 and 0u u ux

δ δpart= = =

part (5abc)

The single integration approach known as the heat-balance integral method (HBIM) [22 23] suggests integration over the entire penetration depth with respect to the space co-or-dinate x namely

( ) ( ) ( ) ( ) ( ) ( )2

20 0 0

0d d d

u x t u x t u x t u tx D t x x D t

t x t x

micro microδ δ δ

micro micro

part part part part= rArr = minus

part part part partint int int (6ab)

Further the integral-balance method suggests replacement of the function u(xt) by an assumed profile ua(xt) = ua(xδ) as a function of the dimensionless pace variable 0 lt xδ lt 1 This approach leads to an equation about the time evolution of the penetration depth δ(t)

However with HBIM the gradient at x = 0 in the right-hand side of eq (6b) should be determined through the assumed approximate profile This drawback can be avoided by the double-integration method (DIM) conceived for non-linear problems of integer-order (μ = 1) [28 29] and successfully applied to time-fractional equations [24 25]

In accordance with DIM the first step is an integration of eq (1a) from 0 to x namely


( ) ( ) ( ) ( ) ( )

0

0d


t x x

micro

micro




0

( ) ( ) (0 )d ( )x

x


δmicro micro

micro micro




( ) ( ) ( )0

d d 0

x

u x tx x D t u t

t

microδ δ

micro

part= part

int int (9)




( )( )2

0

11 d d1 2

n

x

x x xt n n t

δ δ micro micro

micro micro

δδ



( )( )2

0 1 2D Nt N n nt

microβ

micro

δpart= = + +

part (11ab)




( ) ( )( )1

1

Gβ

micro βmicro β

Γ +=Γ + +

(12c)















Approximate profile


( ) ( )

0

1 1

n n

ap

xuN nD t NGmicro β

ξmicro βmicro β+

= minus = minus

(13)


( ) ( )( ) ( )( )1

1 21pN NG n n

βmicro β

micro βΓ +

= = + +Γ + +

(14)


0

xD tmicro β

ξ+

=



( )

( )( )

( )0 02 2 2

FL DIM

2 1 1

1 1D D




(15ab)




( ) ( ) ( )

12

20

0

1 1L LL D t NG t L

D NG


micro β

++ +

= rArr =

(16)



0 d

L


( ) ( )( ) ( )

22

00

d d 21 d d d 1 2 1

ntx nF t x t D G

t t t n n

δ micro β

δ



int (17abc)



( ) ( ) ( )10 2

FPT FPT0 0 0

1 d d1 1

2a

a

L

uu up

DT u x t x t T t

n N

micro β

micro β

infininfin ++

rarr


int int (18ab)





( ) ( ) ( ) ( )2

2

a a

a


t x

micro

micro


(19)



N ka kk

V bη η=


( ) ( ) ( ) ( )( ) 2

RL RL 20

1 1

nNk

a t kk


η micro β ηδ

minus

=


sum (20)


( ) N ka kk

V bη η=




( ) N ka kk

V bη η=

asympsum is [25]

( ) ( ) ( )RL0 0

n n

k kt k k


= =





( ) ( ) ( )( )( )

2

RL RL

1 11

n

p

n nR t n

t N nmicro Φη



(24)


( ) N ka kk

V bη η=







( ) ( ) ( )( ) ( ) 2

RL 20

11 d 1 1 d 11 d


micro δ δ δτ



( ) ( ) ( ) 2

RL RL 2

1 lim lim 1

n

x x

n n xR t R tδ δ

δ δδ δ

minus

rarr rarr


(27)


( ) ( )( )

( )RL 00

1101

n nR t t D t

D t NGmicro β


+


(28)


( )( ) ( ) ( ) ( )

( ) ( )1 2 1 1

01

n n G n nNG



Γ minus (29)




( ) ( )2

L RL0

d minE n t R t xδ




( ) ( ) ( )( )( )

221

L RL0

1 1 d min

n

p

n ne n n

N nη



int (31)













( ) ( )( )+

= + +

le le le le

1

1

001 099 and 001 099

Gβ

micro βmicro β

micro β

Γ

Γ

G(micro

β)


0604

020

0806

0402

0

11

10

09

08

07

06

05


u a [minus

]

ξ [minus]

β = 09

β = 05

β = 01

micro = 01

0 1 2 3 4

10

08

06

04

02

0

u a [minus

]

(a)

u a [minus

]

10

08

06

04

02

0

micro = 05

β = 01

β = 05

β = 09

β = 07

(b)


β = 01

β = 03

β = 06β = 09

ξ [minus]0 1 2 3 4

10

08

06

04

02

0

ξ [minus]0 1 2 3 4

ξ [minus]0 1 2 3 4

u a [minus

]

10

08

06

04

02

0

(d)

β = 01

β = 03

β = 05

β = 07

β = 09

(c)








09

0705

0301

0705

03

01

ξ = 1

u a [minus

] 046044042040038036034

(a) (b) (c)

micro [minus]

09

0705

0301

0705

03

01

ξ = 3

u a [minus

] 004

003

002

001

ξ = 45

micro [minus]

09

0705

0301

β [minus]07

05

03

01

β [minus]

u a 1

05 [minus] 18

16141210

8642



Conclusions













































( ) ( ) ( ) ( ) ( )

0

0d


t x x

micro

micro




0

( ) ( ) (0 )d ( )x

x


δmicro micro

micro micro




( ) ( ) ( )0

d d 0

x

u x tx x D t u t

t

microδ δ

micro

part= part

int int (9)




( )( )2

0

11 d d1 2

n

x

x x xt n n t

δ δ micro micro

micro micro

δδ



( )( )2

0 1 2D Nt N n nt

microβ

micro

δpart= = + +

part (11ab)




( ) ( )( )1

1

Gβ

micro βmicro β

Γ +=Γ + +

(12c)















Approximate profile


( ) ( )

0

1 1

n n

ap

xuN nD t NGmicro β

ξmicro βmicro β+

= minus = minus

(13)


( ) ( )( ) ( )( )1

1 21pN NG n n

βmicro β

micro βΓ +

= = + +Γ + +

(14)


0

xD tmicro β

ξ+

=



( )

( )( )

( )0 02 2 2

FL DIM

2 1 1

1 1D D




(15ab)




( ) ( ) ( )

12

20

0

1 1L LL D t NG t L

D NG


micro β

++ +

= rArr =

(16)



0 d

L


( ) ( )( ) ( )

22

00

d d 21 d d d 1 2 1

ntx nF t x t D G

t t t n n

δ micro β

δ



int (17abc)



( ) ( ) ( )10 2

FPT FPT0 0 0

1 d d1 1

2a

a

L

uu up

DT u x t x t T t

n N

micro β

micro β

infininfin ++

rarr


int int (18ab)





( ) ( ) ( ) ( )2

2

a a

a


t x

micro

micro


(19)



N ka kk

V bη η=


( ) ( ) ( ) ( )( ) 2

RL RL 20

1 1

nNk

a t kk


η micro β ηδ

minus

=


sum (20)


( ) N ka kk

V bη η=




( ) N ka kk

V bη η=

asympsum is [25]

( ) ( ) ( )RL0 0

n n

k kt k k


= =





( ) ( ) ( )( )( )

2

RL RL

1 11

n

p

n nR t n

t N nmicro Φη



(24)


( ) N ka kk

V bη η=







( ) ( ) ( )( ) ( ) 2

RL 20

11 d 1 1 d 11 d


micro δ δ δτ



( ) ( ) ( ) 2

RL RL 2

1 lim lim 1

n

x x

n n xR t R tδ δ

δ δδ δ

minus

rarr rarr


(27)


( ) ( )( )

( )RL 00

1101

n nR t t D t

D t NGmicro β


+


(28)


( )( ) ( ) ( ) ( )

( ) ( )1 2 1 1

01

n n G n nNG



Γ minus (29)




( ) ( )2

L RL0

d minE n t R t xδ




( ) ( ) ( )( )( )

221

L RL0

1 1 d min

n

p

n ne n n

N nη



int (31)













( ) ( )( )+

= + +

le le le le

1

1

001 099 and 001 099

Gβ

micro βmicro β

micro β

Γ

Γ

G(micro

β)


0604

020

0806

0402

0

11

10

09

08

07

06

05


u a [minus

]

ξ [minus]

β = 09

β = 05

β = 01

micro = 01

0 1 2 3 4

10

08

06

04

02

0

u a [minus

]

(a)

u a [minus

]

10

08

06

04

02

0

micro = 05

β = 01

β = 05

β = 09

β = 07

(b)


β = 01

β = 03

β = 06β = 09

ξ [minus]0 1 2 3 4

10

08

06

04

02

0

ξ [minus]0 1 2 3 4

ξ [minus]0 1 2 3 4

u a [minus

]

10

08

06

04

02

0

(d)

β = 01

β = 03

β = 05

β = 07

β = 09

(c)








09

0705

0301

0705

03

01

ξ = 1

u a [minus

] 046044042040038036034

(a) (b) (c)

micro [minus]

09

0705

0301

0705

03

01

ξ = 3

u a [minus

] 004

003

002

001

ξ = 45

micro [minus]

09

0705

0301

β [minus]07

05

03

01

β [minus]

u a 1

05 [minus] 18

16141210

8642



Conclusions





















































Approximate profile


( ) ( )

0

1 1

n n

ap

xuN nD t NGmicro β

ξmicro βmicro β+

= minus = minus

(13)


( ) ( )( ) ( )( )1

1 21pN NG n n

βmicro β

micro βΓ +

= = + +Γ + +

(14)


0

xD tmicro β

ξ+

=



( )

( )( )

( )0 02 2 2

FL DIM

2 1 1

1 1D D




(15ab)




( ) ( ) ( )

12

20

0

1 1L LL D t NG t L

D NG


micro β

++ +

= rArr =

(16)



0 d

L


( ) ( )( ) ( )

22

00

d d 21 d d d 1 2 1

ntx nF t x t D G

t t t n n

δ micro β

δ



int (17abc)



( ) ( ) ( )10 2

FPT FPT0 0 0

1 d d1 1

2a

a

L

uu up

DT u x t x t T t

n N

micro β

micro β

infininfin ++

rarr


int int (18ab)





( ) ( ) ( ) ( )2

2

a a

a


t x

micro

micro


(19)



N ka kk

V bη η=


( ) ( ) ( ) ( )( ) 2

RL RL 20

1 1

nNk

a t kk


η micro β ηδ

minus

=


sum (20)


( ) N ka kk

V bη η=




( ) N ka kk

V bη η=

asympsum is [25]

( ) ( ) ( )RL0 0

n n

k kt k k


= =





( ) ( ) ( )( )( )

2

RL RL

1 11

n

p

n nR t n

t N nmicro Φη



(24)


( ) N ka kk

V bη η=







( ) ( ) ( )( ) ( ) 2

RL 20

11 d 1 1 d 11 d


micro δ δ δτ



( ) ( ) ( ) 2

RL RL 2

1 lim lim 1

n

x x

n n xR t R tδ δ

δ δδ δ

minus

rarr rarr


(27)


( ) ( )( )

( )RL 00

1101

n nR t t D t

D t NGmicro β


+


(28)


( )( ) ( ) ( ) ( )

( ) ( )1 2 1 1

01

n n G n nNG



Γ minus (29)




( ) ( )2

L RL0

d minE n t R t xδ




( ) ( ) ( )( )( )

221

L RL0

1 1 d min

n

p

n ne n n

N nη



int (31)













( ) ( )( )+

= + +

le le le le

1

1

001 099 and 001 099

Gβ

micro βmicro β

micro β

Γ

Γ

G(micro

β)


0604

020

0806

0402

0

11

10

09

08

07

06

05


u a [minus

]

ξ [minus]

β = 09

β = 05

β = 01

micro = 01

0 1 2 3 4

10

08

06

04

02

0

u a [minus

]

(a)

u a [minus

]

10

08

06

04

02

0

micro = 05

β = 01

β = 05

β = 09

β = 07

(b)


β = 01

β = 03

β = 06β = 09

ξ [minus]0 1 2 3 4

10

08

06

04

02

0

ξ [minus]0 1 2 3 4

ξ [minus]0 1 2 3 4

u a [minus

]

10

08

06

04

02

0

(d)

β = 01

β = 03

β = 05

β = 07

β = 09

(c)








09

0705

0301

0705

03

01

ξ = 1

u a [minus

] 046044042040038036034

(a) (b) (c)

micro [minus]

09

0705

0301

0705

03

01

ξ = 3

u a [minus

] 004

003

002

001

ξ = 45

micro [minus]

09

0705

0301

β [minus]07

05

03

01

β [minus]

u a 1

05 [minus] 18

16141210

8642



Conclusions













































( ) ( )( ) ( )( )1

1 21pN NG n n

βmicro β

micro βΓ +

= = + +Γ + +

(14)


0

xD tmicro β

ξ+

=



( )

( )( )

( )0 02 2 2

FL DIM

2 1 1

1 1D D




(15ab)




( ) ( ) ( )

12

20

0

1 1L LL D t NG t L

D NG


micro β

++ +

= rArr =

(16)



0 d

L


( ) ( )( ) ( )

22

00

d d 21 d d d 1 2 1

ntx nF t x t D G

t t t n n

δ micro β

δ



int (17abc)



( ) ( ) ( )10 2

FPT FPT0 0 0

1 d d1 1

2a

a

L

uu up

DT u x t x t T t

n N

micro β

micro β

infininfin ++

rarr


int int (18ab)





( ) ( ) ( ) ( )2

2

a a

a


t x

micro

micro


(19)



N ka kk

V bη η=


( ) ( ) ( ) ( )( ) 2

RL RL 20

1 1

nNk

a t kk


η micro β ηδ

minus

=


sum (20)


( ) N ka kk

V bη η=




( ) N ka kk

V bη η=

asympsum is [25]

( ) ( ) ( )RL0 0

n n

k kt k k


= =





( ) ( ) ( )( )( )

2

RL RL

1 11

n

p

n nR t n

t N nmicro Φη



(24)


( ) N ka kk

V bη η=







( ) ( ) ( )( ) ( ) 2

RL 20

11 d 1 1 d 11 d


micro δ δ δτ



( ) ( ) ( ) 2

RL RL 2

1 lim lim 1

n

x x

n n xR t R tδ δ

δ δδ δ

minus

rarr rarr


(27)


( ) ( )( )

( )RL 00

1101

n nR t t D t

D t NGmicro β


+


(28)


( )( ) ( ) ( ) ( )

( ) ( )1 2 1 1

01

n n G n nNG



Γ minus (29)




( ) ( )2

L RL0

d minE n t R t xδ




( ) ( ) ( )( )( )

221

L RL0

1 1 d min

n

p

n ne n n

N nη



int (31)













( ) ( )( )+

= + +

le le le le

1

1

001 099 and 001 099

Gβ

micro βmicro β

micro β

Γ

Γ

G(micro

β)


0604

020

0806

0402

0

11

10

09

08

07

06

05


u a [minus

]

ξ [minus]

β = 09

β = 05

β = 01

micro = 01

0 1 2 3 4

10

08

06

04

02

0

u a [minus

]

(a)

u a [minus

]

10

08

06

04

02

0

micro = 05

β = 01

β = 05

β = 09

β = 07

(b)


β = 01

β = 03

β = 06β = 09

ξ [minus]0 1 2 3 4

10

08

06

04

02

0

ξ [minus]0 1 2 3 4

ξ [minus]0 1 2 3 4

u a [minus

]

10

08

06

04

02

0

(d)

β = 01

β = 03

β = 05

β = 07

β = 09

(c)








09

0705

0301

0705

03

01

ξ = 1

u a [minus

] 046044042040038036034

(a) (b) (c)

micro [minus]

09

0705

0301

0705

03

01

ξ = 3

u a [minus

] 004

003

002

001

ξ = 45

micro [minus]

09

0705

0301

β [minus]07

05

03

01

β [minus]

u a 1

05 [minus] 18

16141210

8642



Conclusions













































( ) ( ) ( )10 2

FPT FPT0 0 0

1 d d1 1

2a

a

L

uu up

DT u x t x t T t

n N

micro β

micro β

infininfin ++

rarr


int int (18ab)





( ) ( ) ( ) ( )2

2

a a

a


t x

micro

micro


(19)



N ka kk

V bη η=


( ) ( ) ( ) ( )( ) 2

RL RL 20

1 1

nNk

a t kk


η micro β ηδ

minus

=


sum (20)


( ) N ka kk

V bη η=




( ) N ka kk

V bη η=

asympsum is [25]

( ) ( ) ( )RL0 0

n n

k kt k k


= =





( ) ( ) ( )( )( )

2

RL RL

1 11

n

p

n nR t n

t N nmicro Φη



(24)


( ) N ka kk

V bη η=







( ) ( ) ( )( ) ( ) 2

RL 20

11 d 1 1 d 11 d


micro δ δ δτ



( ) ( ) ( ) 2

RL RL 2

1 lim lim 1

n

x x

n n xR t R tδ δ

δ δδ δ

minus

rarr rarr


(27)


( ) ( )( )

( )RL 00

1101

n nR t t D t

D t NGmicro β


+


(28)


( )( ) ( ) ( ) ( )

( ) ( )1 2 1 1

01

n n G n nNG



Γ minus (29)




( ) ( )2

L RL0

d minE n t R t xδ




( ) ( ) ( )( )( )

221

L RL0

1 1 d min

n

p

n ne n n

N nη



int (31)













( ) ( )( )+

= + +

le le le le

1

1

001 099 and 001 099

Gβ

micro βmicro β

micro β

Γ

Γ

G(micro

β)


0604

020

0806

0402

0

11

10

09

08

07

06

05


u a [minus

]

ξ [minus]

β = 09

β = 05

β = 01

micro = 01

0 1 2 3 4

10

08

06

04

02

0

u a [minus

]

(a)

u a [minus

]

10

08

06

04

02

0

micro = 05

β = 01

β = 05

β = 09

β = 07

(b)


β = 01

β = 03

β = 06β = 09

ξ [minus]0 1 2 3 4

10

08

06

04

02

0

ξ [minus]0 1 2 3 4

ξ [minus]0 1 2 3 4

u a [minus

]

10

08

06

04

02

0

(d)

β = 01

β = 03

β = 05

β = 07

β = 09

(c)








09

0705

0301

0705

03

01

ξ = 1

u a [minus

] 046044042040038036034

(a) (b) (c)

micro [minus]

09

0705

0301

0705

03

01

ξ = 3

u a [minus

] 004

003

002

001

ξ = 45

micro [minus]

09

0705

0301

β [minus]07

05

03

01

β [minus]

u a 1

05 [minus] 18

16141210

8642



Conclusions













































( ) ( ) ( )( )( )

2

RL RL

1 11

n

p

n nR t n

t N nmicro Φη



(24)


( ) N ka kk

V bη η=







( ) ( ) ( )( ) ( ) 2

RL 20

11 d 1 1 d 11 d


micro δ δ δτ



( ) ( ) ( ) 2

RL RL 2

1 lim lim 1

n

x x

n n xR t R tδ δ

δ δδ δ

minus

rarr rarr


(27)


( ) ( )( )

( )RL 00

1101

n nR t t D t

D t NGmicro β


+


(28)


( )( ) ( ) ( ) ( )

( ) ( )1 2 1 1

01

n n G n nNG



Γ minus (29)




( ) ( )2

L RL0

d minE n t R t xδ




( ) ( ) ( )( )( )

221

L RL0

1 1 d min

n

p

n ne n n

N nη



int (31)













( ) ( )( )+

= + +

le le le le

1

1

001 099 and 001 099

Gβ

micro βmicro β

micro β

Γ

Γ

G(micro

β)


0604

020

0806

0402

0

11

10

09

08

07

06

05


u a [minus

]

ξ [minus]

β = 09

β = 05

β = 01

micro = 01

0 1 2 3 4

10

08

06

04

02

0

u a [minus

]

(a)

u a [minus

]

10

08

06

04

02

0

micro = 05

β = 01

β = 05

β = 09

β = 07

(b)


β = 01

β = 03

β = 06β = 09

ξ [minus]0 1 2 3 4

10

08

06

04

02

0

ξ [minus]0 1 2 3 4

ξ [minus]0 1 2 3 4

u a [minus

]

10

08

06

04

02

0

(d)

β = 01

β = 03

β = 05

β = 07

β = 09

(c)








09

0705

0301

0705

03

01

ξ = 1

u a [minus

] 046044042040038036034

(a) (b) (c)

micro [minus]

09

0705

0301

0705

03

01

ξ = 3

u a [minus

] 004

003

002

001

ξ = 45

micro [minus]

09

0705

0301

β [minus]07

05

03

01

β [minus]

u a 1

05 [minus] 18

16141210

8642



Conclusions













































( ) ( ) ( )( )( )

221

L RL0

1 1 d min

n

p

n ne n n

N nη



int (31)













( ) ( )( )+

= + +

le le le le

1

1

001 099 and 001 099

Gβ

micro βmicro β

micro β

Γ

Γ

G(micro

β)


0604

020

0806

0402

0

11

10

09

08

07

06

05


u a [minus

]

ξ [minus]

β = 09

β = 05

β = 01

micro = 01

0 1 2 3 4

10

08

06

04

02

0

u a [minus

]

(a)

u a [minus

]

10

08

06

04

02

0

micro = 05

β = 01

β = 05

β = 09

β = 07

(b)


β = 01

β = 03

β = 06β = 09

ξ [minus]0 1 2 3 4

10

08

06

04

02

0

ξ [minus]0 1 2 3 4

ξ [minus]0 1 2 3 4

u a [minus

]

10

08

06

04

02

0

(d)

β = 01

β = 03

β = 05

β = 07

β = 09

(c)








09

0705

0301

0705

03

01

ξ = 1

u a [minus

] 046044042040038036034

(a) (b) (c)

micro [minus]

09

0705

0301

0705

03

01

ξ = 3

u a [minus

] 004

003

002

001

ξ = 45

micro [minus]

09

0705

0301

β [minus]07

05

03

01

β [minus]

u a 1

05 [minus] 18

16141210

8642



Conclusions


















































( ) ( )( )+

= + +

le le le le

1

1

001 099 and 001 099

Gβ

micro βmicro β

micro β

Γ

Γ

G(micro

β)


0604

020

0806

0402

0

11

10

09

08

07

06

05


u a [minus

]

ξ [minus]

β = 09

β = 05

β = 01

micro = 01

0 1 2 3 4

10

08

06

04

02

0

u a [minus

]

(a)

u a [minus

]

10

08

06

04

02

0

micro = 05

β = 01

β = 05

β = 09

β = 07

(b)


β = 01

β = 03

β = 06β = 09

ξ [minus]0 1 2 3 4

10

08

06

04

02

0

ξ [minus]0 1 2 3 4

ξ [minus]0 1 2 3 4

u a [minus

]

10

08

06

04

02

0

(d)

β = 01

β = 03

β = 05

β = 07

β = 09

(c)








09

0705

0301

0705

03

01

ξ = 1

u a [minus

] 046044042040038036034

(a) (b) (c)

micro [minus]

09

0705

0301

0705

03

01

ξ = 3

u a [minus

] 004

003

002

001

ξ = 45

micro [minus]

09

0705

0301

β [minus]07

05

03

01

β [minus]

u a 1

05 [minus] 18

16141210

8642



Conclusions



















































09

0705

0301

0705

03

01

ξ = 1

u a [minus

] 046044042040038036034

(a) (b) (c)

micro [minus]

09

0705

0301

0705

03

01

ξ = 3

u a [minus

] 004

003

002

001

ξ = 45

micro [minus]

09

0705

0301

β [minus]07

05

03

01

β [minus]

u a 1

05 [minus] 18

16141210

8642



Conclusions



































































































Documents

SUBDIFFUSION MODEL WITH TIME-DEPENDENT …thermalscience.vinca.rs/pdfs/papers-2016/TSCI160427247H.pdf · Hristov, J.: Subdiffusion Model with Time-Dependent Diffusion Coefficient