- Email: [email protected]
Group classification of nonlinear time-fractional diffusion equation with a source term
Applied Mathematics and Computation xxx (2014) xxx–xxx
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Group classification of nonlinear time-fractional diffusion equation with a source term S.Yu. Lukashchuk ⇑, A.V. Makunin Laboratory ‘‘Group analysis of mathematical models in natural and engineering sciences’’, Ufa State Aviation Technical University, 12 K. Marx str., Ufa 450000, Russia
a r t i c l e
i n f o
Keywords: Time-fractional diffusion equation Symmetry Equivalence transformation Group classification Invariant solution
a b s t r a c t A complete group classification is presented for a nonlinear time-fractional diffusion equation with a source term. The equation involves the Riemann–Liouville time-fractional derivative of the order a 2 ð0; 2Þ. All coefficients of the equation are sought as a function of the dependent variable. Using the infinitesimal approach, the Lie algebra of infinitesimal generators of equivalence transformations is constructed for the equation under consideration, and group classification is carried out up to the equivalence transformations. Examples of invariant solutions are also presented. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Lie group analysis (or symmetry analysis) is an efficient approach to investigate fundamental properties of differential equations and to construct their solutions (see, e.g., [1–5]). Recently, principal methods of symmetry analysis were extended to fractional differential equations with different types of fractional derivatives [6,7]. Using these methods, symmetry properties of some classes of ordinary and partial fractional differential equations have been investigated [8–13]. Classification of differential equations belonging to a certain class with respect to symmetry groups is one of major tasks of modern symmetry analysis. The problem of group classification can be briefly formulated as follows: it is necessary to determine the Lie symmetry algebra for each differential equation belonging to a given class of equations. Fundamentals of the group classification originated in works by Lie and then methods of group classification were developed by many researchers. An efficient method for group classification was developed by Ovsyannikov [14] (see also [1]). Using this method, he first performed a complete group classification of the nonlinear heat equation, where a thermal conductivity was a function of the temperature. At present, these methods are widely presented in the literature (see, e.g., [1,4,5,15,16]). Results of group classification can be used as a starting point for construction of solutions for nonlinear differential equations using their symmetries. The powerful technique, known as the symmetry reduction, was developed for this purpose. In this approach, a given partial differential equation is reduced using its symmetry to the equation with a less number of independent variables. Solving this reduced equation, the exact (so-called, invariant) solution of the initial equation can be found. In particular, using symmetry reduction, differential equations with two independent variables are transformed to ordinary differential equations. The methods of group classification and symmetry reduction can be also efficiently used for constructing exact solutions of nonlinear fractional differential equations. In [8], the group classification was performed for homogeneous diffusion ⇑ Corresponding author. E-mail address: [email protected] (S.Yu. Lukashchuk). http://dx.doi.org/10.1016/j.amc.2014.11.087 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
Please cite this article in press as: S.Yu. Lukashchuk, A.V. Makunin, Group classification of nonlinear time-fractional diffusion equation with a source term, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.11.087
2
S.Yu. Lukashchuk, A.V. Makunin / Applied Mathematics and Computation xxx (2014) xxx–xxx
equations with the Riemann–Liouville and Caputo time-fractional derivatives. In these equations, a diffusion coefficient was considered as a function of the dependent variable. Using symmetry reduction, several invariant solutions of these nonlinear equations have been constructed and presented in this work. It should be noted that the problem of group classification for a fractional differential equation is much more complicated than a similar problem for an integer-order differential equation. The infinitesimal group generators, admitted by the equations, can be found by solving the appropriate determining equations. Nevertheless, on the contrary to the integer-order differential equations, these determining equations for the fractional differential equations lead to an infinite chain of differential equations of integer and fractional orders [6]. In a general case, solution of such infinite system of differential equations is a non-trivial problem. In this paper, we perform the group classification of the time-fractional diffusion equation with a source term
Dat u ¼ ðkðuÞux Þx þ f ðuÞ;
kðuÞ > 0; a 2 ð0; 2Þ:
ð1:1Þ
Here t; x are independent variables, u ¼ uðt; xÞ is the dependent variable, the diffusion coefficient k and the source term f are sought as a function of the dependent variable u; ux ¼ @u , @x
Dat u ¼
1 @n Cðn aÞ @t n
Z
t
0
uðs; xÞ ðt sÞanþ1
ds;
n ¼ ½a þ 1
is the left-sided Riemann–Liouville fractional derivative of order a with respect to t [17–21], and CðzÞ is the Gamma function. At present, time-fractional diffusion equations are frequently used as mathematical models for describing processes with anomalous kinetics. Such processes are observed in many different areas: heat and mass transfer in solids, hydrodynamics, plasma physics, cosmology, hydrology, quantum optics, chemistry of polymers, biology, computer networks, and many others (see, e.g., [22–28] and references therein). For a 2 ð0; 1Þ, Eq. (1.1) can be used for modeling the subdiffusion processes, whereas for a 2 ð1; 2Þ it describes the diffusion-wave phenomena. In a limiting case of a ¼ 1, Eq. (1.1) coincides with the classical diffusion (or heat) equation. The complete group classification of this classical equation was performed in [29]. Therefore, we do not consider this limiting case in this paper. The paper is organized as follows: in Section 2 the group of equivalence transformations for Eq. (1.1) is constructed; in Section 3 the results of group classification are presented, and Section 4 contains the examples of invariant solutions to Eq. (1.1). 2. Equivalence transformations At the first step of group classification it is necessary to compute the group of equivalence transformations for Eq. (1.1). The equivalence transformations preserve the differential structure of the equation under consideration. In other words, equations which are related by these transformations admit similar groups of the Lie point transformations. We use the infinitesimal approach to obtain the equivalence transformations for Eq. (1.1). Following [1,30], we shall seek the generator of the continuous group of equivalence transformations in the form
E ¼ n0
@ @ @ @ @ þ n1 þ g þl þm : @t @x @u @k @f
ð2:1Þ
Here u; k and f are considered as differential variables: u in the space ðt; xÞ; k and f in the extended space ðt; x; u; Dat u; ux Þ. The coordinates n0 ; n1 and g of the generator (2.1) are sought as a function of t; x and u, whereas the coordinates l and m are sought as functions of t; x; u; Dat u; ux ; k and f. The generator (2.1) can be found from the invariance of the system
Dat u ¼ ku u2x þ kuxx þ f ; kt ¼ 0;
kx ¼ 0;
kDat u ¼ 0;
kux ¼ 0;
f t ¼ 0;
f x ¼ 0;
f Dat u ¼ 0;
f ux ¼ 0:
ð2:2Þ
The infinitesimal invariance criterion for the system (2.2) has the form
fa 2ku ux f1 kf11 luxx l3 u2x m jð2:2Þ ¼ 0;
li jð2:2Þ ¼ 0; mi jð2:2Þ ¼ 0; i ¼ 1; 2; 4; 5:
ð2:3Þ
Here fa ; f1 ; f11 ; li ; mi (i = 1 . . .5) are coordinates of the extended generator
@ @ @ @ @ @ @ @ @ @ @ þ f1 þ f11 þ l1 þ l2 þ l3 þ l4 þ l5 þ m1 þ m2 þ m3 @ux @uxx @kt @kx @ku @kDat u @kux @f t @f x @f u @Dat u @ @ þ m4 þ m5 : @f Dat u @f ux
e E ¼ E þ fa
ð2:4Þ
These coordinates are calculated by the appropriate prolongation formulae (see, e.g., [3] for f1 and f11 , [8] for fa , and [30] for
li and mi ). Please cite this article in press as: S.Yu. Lukashchuk, A.V. Makunin, Group classification of nonlinear time-fractional diffusion equation with a source term, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.11.087
S.Yu. Lukashchuk, A.V. Makunin / Applied Mathematics and Computation xxx (2014) xxx–xxx
3
Solving the system (2.3), we find all the generators of equivalence transformations for Eq. (1.1). The obtained results can be summarized as the following proposition. Proposition 1. The fractional diffusion Eq. (1.1) admits a 4-dimensional Lie algebra L4 of infinitesimal generators of equivalence transformations. The algebra L4 is spanned by
E1 ¼
@ ; @x
E2 ¼ t
@ @ @ ak af ; @t @k @f
E3 ¼ x
@ @ þ 2k ; @x @k
E4 ¼ u
@ @ þf : @u @f
The corresponding 4-parameter equivalence transformation group is given by
t ¼ d2 t;
x ¼ cx þ b;
¼ uu; u
¼ c2 d2a k; k
f ¼ ud2a f ;
ð2:5Þ
where the coefficients b; c; d are arbitrary, and u > 0. Remark 1. The reflection x ¼ x is also admitted by Eq. (1.1). The group of equivalence transformations (2.5) contains this reflection. In this paper, the group classification of Eq. (1.1) is performed up to the equivalence transformations (2.5). 3. Group classification For the group classification of Eq. (1.1), we use the approach described in [1]. Let us find the operators
X ¼ n0 ðt; x; uÞ
@ @ @ þ n1 ðt; x; uÞ þ gðt; x; uÞ @t @x @u
ð3:1Þ
admitted by Eq. (1.1) with different pairs of functions fkðuÞ; f ðuÞg. The infinitesimal invariance criterion for Eq. (1.1) has the form
e Da u ku u2 kuxx f j X ð1:1Þ ¼ 0; x t
ð3:2Þ
e of the operator X is written as where the prolongation X
e ¼ n0 @ þ n1 @ þ g @ þ f1 @ þ f11 @ þ fa @ : X @t @x @u @ux @uxx @ðDat uÞ Here the coordinates f1 ; f11 and fa are given by the following prolongation formulae (see, e.g., [3,7]):
f1 ¼ Dx ðgÞ ut Dx ðn0 Þ ux Dx ðn1 Þ; f11 ¼ Dx ðf1 Þ utx Dx ðn0 Þ uxx Dx ðn1 Þ;
ð3:3Þ
fa ¼ Dat ðg n0 ut n1 ux Þ þ n0 Dtaþ1 u þ n1 Dat ux : The invariance criterion (3.2) leads to the determining equation
fa gðkuu u2x þ ku uxx þ f u Þ 2f1 ku ux f11 k ¼ 0: a
We substitute (3.3) into (3.4) and replace uxx by ðDt u rule [17,18]
ð3:4Þ ku u2x
f Þ=k by virtue of Eq. (1.1). Then we use the generalized Leibniz
1 X a an n Dt f Dt g n n¼0
Dat ðg hÞ ¼
and split the obtained equation by ux and by the fractional derivatives and integrals Dtan u; n ¼ 0; 1; . . .. As a result, we obtain the following infinite system of partial differential equations:
n0u ¼ 0;
n0x ¼ 0;
ðKðuÞgÞu ¼ 0;
n1u ¼ 0;
n1t ¼ 0;
2KðuÞgx þ 2gxu
guu ¼ 0;
n1xx
¼ 0;
KðuÞg 2n1x þ an0t ¼ 0;
Dat ðg ugu Þ gðf u KðuÞf Þ kgxx þ gu f 2n1x f ¼ 0; ðn þ 1ÞDnt gu þ ðn aÞDnþ1 n0 ¼ 0; t
ð3:5Þ
n ¼ 1; 2; . . . :
Here we use the condition kðuÞ – 0 and notation KðuÞ ¼ ku =k. From the first five equations of the system (3.5) we find
n0 ¼ n0 ðtÞ;
n1 ¼ n1 ðxÞ;
g ¼ g0 ðt; xÞ þ g1 ðt; xÞu:
Please cite this article in press as: S.Yu. Lukashchuk, A.V. Makunin, Group classification of nonlinear time-fractional diffusion equation with a source term, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.11.087
4
S.Yu. Lukashchuk, A.V. Makunin / Applied Mathematics and Computation xxx (2014) xxx–xxx
Then from the infinite chain of equations given by the last equation of the system (3.5), we obtain
n0 ðtÞ ¼ C 0 þ C 1 t þ C 2 t 2 ;
g1 ðt; xÞ ¼ ða 1ÞC 2 t þ aðxÞ;
where C 0 ; C 1 ; C 2 are arbitrary constants, and aðxÞ is an arbitrary function. Note that the lower limit t ¼ 0 in the fractional differentiation must be invariant under the point transformations admitted by Eq. (1.1) (see [6–8] for more details). This condition leads to the equality nð0Þ ¼ 0 which provides C 0 ¼ 0. Using obtained representations for n0 ; n1 and g, it can be shown that for arbitrary functions kðuÞ and f ðuÞ the system (3.5) has the solution n0 ¼ 0; n1 ¼ const; g ¼ 0. So in this case, the translation in x with the generator
X1 ¼
@ @x
is the only Lie point symmetry of the Eq. (1.1). For the arbitrary function kðuÞ and f ðuÞ ¼ 0, the system (3.5) has the solution n0 ¼ 2C 1 t; n1 ¼ aC 1 x þ C 3 ; g ¼ 0. Therefore, the corresponding Eq. (1.1) additionally admits the group of dilation (scaling transformation) with the generator
X 2 ¼ 2t
@ @ þ ax : @t @x
From the sixth, seventh and eighth equations of the system (3.5) we find that Eq. (1.1) can admit additional symmetries if either KðuÞ ¼ 0, or
00 1 ¼ 0: KðuÞ
ð3:6Þ
This is the classifying relation for the function KðuÞ (or kðuÞ). From the relation (3.6) and system (3.5), in view of the equivalence transformations (2.5), we obtain the preliminary group classification presented in Table 1. In this table, A and C i ði ¼ 1; 2; 3; 4Þ are arbitrary constants, and hðxÞ is an arbitrary function. To complete the group classification, we consider the single equation from the system (3.5) which contains the function f ðuÞ. For each functions kðuÞ from Table 1, this equation provides the classifying relations for the function f ðuÞ as follows: for kðuÞ ¼ 1, the function f ðuÞ either satisfies the equation f uu ¼ 0, or satisfies the equation
fu f uu
¼ 0; uu
for kðuÞ ¼ eu , the function f ðuÞ is either arbitrary and C 1 ¼ C 4 ¼ 0, or it is equal to zero (f ¼ 0) and 2C 4 ¼ aC 1 ; a , the function f ðuÞ is either constant and 2C ¼ að1 þ rÞC , or it satisfies the equation for kðuÞ ¼ ðu þ AÞr r – 0; 43 ; a21 4 1
f ¼ 0; f u uu
2a
aþ2 a
for kðuÞ ¼ ðu þ AÞa1 , there are two additional particular cases of the function f ðuÞ : f ðuÞ ¼ 0 and f ðuÞ ¼ f 0 u an arbitrary constant); 4 for kðuÞ ¼ ðu þ AÞ3 , there are three different cases:
FðuÞ ¼ 0 or
F uu ¼ 0 or
Fu F uu
(here f 0 is
¼ 0;
uu
1
where FðuÞ ¼ u3 f ðuÞ. We compute the solutions of all classifying relations for the function f ðuÞ presented above. Upon substituting these solutions in the system (3.5) and simple algebra, we find all pairs of functions fkðuÞ; f ðuÞg for which the Lie algebra of the point
Table 1 Preliminary group classification. kðuÞ 1 eu a ðu þ AÞr r – 0; 43 ; a21 a a21
ðu þ AÞ
4
ðu þ AÞ3
n0
n1
g0
g1
C1 t C1 t C1 t
aC x þ C 3 2 1
C4 x þ C3 C4 x þ C3
0 2C 4 aC 1 0
C4 C4
C 1 t þ C 2 t2
C4 x þ C3
0
C1 t
hðxÞ
0
1
r ð2C 4 aC 1 Þ a1 ð2aC t 2C þ aC Þ 2 4 1 2a 0
32 h ðxÞ þ 34 aC 1
Please cite this article in press as: S.Yu. Lukashchuk, A.V. Makunin, Group classification of nonlinear time-fractional diffusion equation with a source term, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.11.087
S.Yu. Lukashchuk, A.V. Makunin / Applied Mathematics and Computation xxx (2014) xxx–xxx
5
symmetries is extended. The results of the group classification of Eq. (1.1) up to the equivalence transformations (2.5) are summarized in the following proposition. Proposition 2. The nonlinear fractional diffusion Eq. (1.1) with arbitrary functions kðuÞ > 0 and f ðuÞ has a single symmetry with the generator
X1 ¼
@ : @x
The homogeneous equation (f ðuÞ ¼ 0) with an arbitrary function kðuÞ admits an additional symmetry
X 2 ¼ 2t
@ @ þ ax : @t @x
The algebra extends for the following cases. I. kðuÞ ¼ 1. I.1. f ðuÞ ¼ 0:
X3 ¼ u
@ ; @u
X 1 ¼ gðt; xÞ
@ ; @u
where the function gðt; xÞ is an arbitrary solution of the equation Dat g ¼ g xx . I.2. f ðuÞ ¼ d ðd ¼ 1Þ:
X2 ¼ t
@ a @ ta @ þ x þd ; @t 2 @x CðaÞ @u
X 1 ¼ gðt; xÞ
X3 ¼
ud
ta @ ; Cða þ 1Þ @u
@ ; @u
where the function gðt; xÞ is an arbitrary solution of the equation Dat g ¼ g xx . I.3. f ðuÞ ¼ du þ b ðd ¼ 1; b ¼ 1Þ:
X 2 ¼ ½u bt a Ea;aþ1 ðdt a Þ
@ ; @u
X 1 ¼ gðt; xÞ
@ ; @u
where the function gðt; xÞ is an arbitrary solution of the equation Dat g ¼ g xx þ dg, and Ea;aþ1 ðtÞ is the Mittag–Leffler function (see, e.g., [18]). I.4. f ðuÞ ¼ duc ðd ¼ 1; c – 0; 1Þ:
X2 ¼ t
@ a @ 1 a @ þ x þ u : @t 2 @x d 1 c @u
a . II. kðuÞ ¼ ur r – 0; 43 ; a21 II.1. f ðuÞ ¼ 0:
X3 ¼ II.2.
r @
@ x þu : @u 2 @x
f ðuÞ ¼ duc ðd ¼ 1Þ:
X 2 ¼ ð1 cÞt
@ að1 þ r cÞ @ @ þ x þ au : @t 2 @x @u
2a
III. kðuÞ ¼ ua1 . III.1. f ðuÞ ¼ 0:
X 3 ¼ ax III.2.
@ @ þ ða 1Þu ; @x @u
f ðuÞ ¼ duc d ¼ 1; c – aaþ1 : 1
X 2 ¼ ð1 cÞt III.3.
X 4 ¼ t2
@ @ þ ða 1Þtu : @t @u
@ a 2a @ @ x þ au : þ 1c a 1 @x @t 2 @u
aþ1
f ðuÞ ¼ dua1 ðd ¼ 1Þ:
X 2 ¼ 2t
@ @ þ ða 1Þu ; @t @u
X 3 ¼ t2
@ @ þ ða 1Þtu : @t @u
4
IV. kðuÞ ¼ u3 . IV.1. f ðuÞ ¼ 0:
Please cite this article in press as: S.Yu. Lukashchuk, A.V. Makunin, Group classification of nonlinear time-fractional diffusion equation with a source term, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.11.087
6
S.Yu. Lukashchuk, A.V. Makunin / Applied Mathematics and Computation xxx (2014) xxx–xxx
@ @ 3u ; @x @u
X 3 ¼ 2x IV.2.
@ @ 3u ; @x @u
@ @ 3xu : @x @u
1
@ @ þ 3au ; @t @u
X 3;4 ¼ 2exx
@ @ 3xexx u ; @x @u
pffiffiffi pffiffiffi where x ¼ 2= 3 for d ¼ 1, and x ¼ 2i= 3 for d ¼ 1 (here i is the imaginary unit). f ðuÞ ¼ duc ðd ¼ 1; c – 1Þ:
X 2 ¼ ð1 cÞt IV.5.
X 3 ¼ x2
f ðuÞ ¼ du3 ðd ¼ 1Þ:
X 2 ¼ 4t
IV.4.
@ @ 3xu : @x @u
f ðuÞ ¼ du ðd ¼ 1Þ:
X 2 ¼ 2x IV.3.
X 4 ¼ x2
@ a 1 @ @ c þ x þ au : @t 2 3 @x @u
1
f ðuÞ ¼ du3 þ bu ðd ¼ 1; b ¼ 1Þ:
X 2;3 ¼ 2exx
@ @ 3xexx u ; @x @u
pffiffiffi pffiffiffi where x ¼ 2= 3 for d ¼ 1, and x ¼ 2i= 3 for d ¼ 1. Thus, there are 14 cases of extensions of the Lie algebra for Eq. (1.1). The maximal dimension of the Lie algebra is equal to four. Remark 2. Contrary to the classical integer-order diffusion equation, the case kðuÞ ¼ eu from Table 1 does not provide the extension of the Lie algebra. Remark 3. Calculations show that the Lie algebra extends if and only if A ¼ 0 for all functions kðuÞ presented in Table 1. Nevertheless, the constant A can not be set equal to zero by virtue of the equivalence transformations (2.5).
4. Examples of invariant solutions Examples of invariant solutions to Eq. (1.1) with f ¼ 0 have been presented in [6]. In this section, we give several invariant solutions of the nonhomogeneous Eq. (1.1). 4.1. Example 1 Let us consider Eq. (1.1) with kðuÞ ¼ ur and f ðuÞ ¼ urþ1 operator
X ¼ rt
a . As it follows from Item II.2 in Proposition 2, the r – 0; 43 ; a21
@ @ þ au @t @u
is admitted by the equation under consideration. The invariant solution, corresponding to this operator, has the form a
uðt; xÞ ¼ t r uðxÞ; where the function uðxÞ is a solution of the second order ordinary differential equation
u00 þ ru1 ðu0 Þ2 þ u au1r ¼ 0;
a¼
Cð1 a=rÞ : Cð1 a a=rÞ
ð4:7Þ
The solution of Eq. (4.7) can be written as
Z
du pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ C 2 x; wðu; C 1 Þ
where C 1 and C 2 are the constants of integration, and the function wðu; C 1 Þ depends on
8 2r 1 þ r2a u2r rþ1 u2 ; > þ2 < C1u 2 3 2 wðu; C 1 Þ ¼ C 1 u þ 2au 2u lnðuÞ; > : C 1 u4 þ 2au4 lnðuÞ þ u2 ;
r:
r – 1; 2; r ¼ 1; r ¼ 2:
Please cite this article in press as: S.Yu. Lukashchuk, A.V. Makunin, Group classification of nonlinear time-fractional diffusion equation with a source term, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.11.087
7
S.Yu. Lukashchuk, A.V. Makunin / Applied Mathematics and Computation xxx (2014) xxx–xxx
4.2. Example 2 4
1
Eq. (1.1) with kðuÞ ¼ u3 and f ðuÞ ¼ u3 has the symmetries
X ¼ 2exx
@ @ 3xexx u @x @u
(see Item IV.3 in Proposition 2). The corresponding invariant solutions have the form 3
uðt; xÞ ¼ e2xx uðtÞ; where the function uðtÞ is a solution of the homogeneous equation
Dat u ¼ 0: This equation has a well-known solution:
(
uðtÞ ¼
C 1 ta1 ; C1t
a1
þ C2t
a2
a 2 ð0; 1Þ; ; a 2 ð1; 2Þ:
4.3. Example 3 4
Eq. (1.1) with kðuÞ ¼ u3 and f ðuÞ ¼ u has the symmetry
@ @ 3xu @x @u
X ¼ x2
(see Item IV.2 in Proposition 2). The corresponding invariant solution has the form
uðt; xÞ ¼ x3 uðtÞ;
ð4:8Þ
where the function uðtÞ is a solution of the equation
Dat u ¼ u:
ð4:9Þ
The common solution of Eq. (4.9) is known (see, e.g., [17,18]):
uðtÞ ¼
n X C k tak Ea;aþ1k ðt a Þ;
n ¼ ½a þ 1:
ð4:10Þ
k¼1
Here C k are the constants of integration, and Ea;aþ1k ðtÞ is the Mittag–Leffler function. Substituting (4.10) into (4.8), one can get the final form of the invariant solution. 4.4. Example 4 2a
aþ1
Eq. (1.1) with kðuÞ ¼ ua1 and f ðuÞ ¼ ua1 has the symmetry
X ¼ t2
@ @ þ ða 1Þtu @t @u
(see Item III.3 in Proposition 2). The corresponding invariant solution has the form
uðt; xÞ ¼ t a1 uðxÞ;
ð4:11Þ
where the function uðxÞ is a solution of the equation
u00
2a u1 ðu0 Þ2 þ u ¼ 0: a1
This equation has the common solution 2
uðxÞ ¼ ½C 1 x sinðxxÞ þ C 2 x cosðxxÞx ; x ¼
1þa : 1a
Substituting this solution into (4.11), one can write the invariant solution in the final form. 4.5. Example 5 As the final example, let us consider Eq. (1.1) with kðuÞ ¼ ur and f ðuÞ ¼ uc from Item II.2 in Proposition 2, this equation has the symmetry
a and c – r þ 1 . As it follows r – 0; 43 ; a21
Please cite this article in press as: S.Yu. Lukashchuk, A.V. Makunin, Group classification of nonlinear time-fractional diffusion equation with a source term, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.11.087
8
S.Yu. Lukashchuk, A.V. Makunin / Applied Mathematics and Computation xxx (2014) xxx–xxx
X ¼ ð1 cÞt
@ að1 þ r cÞ @ @ þ x þ au : @t 2 @x @u
The corresponding invariant solution has the form
uðt; xÞ ¼ xk uðtxq Þ; with
k¼
2 ; 1þrc
q¼
c1 k: a
The function uðyÞ ðy ¼ txq Þ is a solution of the ordinary fractional differential equation 2
Day u ¼ ay2 ur u00 þ by ur1 ðu0 Þ þ cyur u0 þ durþ1 þ uc ; 2
ð4:12Þ
where
a ¼ aq2 ;
b ¼ arq2 ;
c ¼ q 1 þ k þ 2að1 þ qÞ þ ak þ a2 q ;
d ¼ kð1 þ k þ aqÞ:
5. Conclusion The group classification of Eq. (1.1), presented in the paper, permits to construct exact solutions of this equation. Some examples of the invariant solutions are presented in the paper. At the same time, a new problem arises during construction of the invariant solutions: after symmetry reduction one can get a nonlinear ordinary fractional differential equation (see, e.g, Example 5). It is an open problem to develop the methods of integration for such equations. The given group classification also can be used for classification of invariant solutions of the time-fractional diffusion equation. The optimal systems of subalgebras have to be constructed for this purpose. This is a possible direction for future work. Acknowledgement The authors acknowledge a financial support of the Government of Russian Federation through Resolution No. 220, contract No. 11.G34.31.0042 with Ufa State Aviation Technical University and leading scientist professor N. H. Ibragimov. References [1] [2] [3] [4] [5]
[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
L.V. Ovsyannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982. L.V. Ovsyannikov, Lectures on the Theory of Group Properties of Differential Equations, World Scientific, 2013. N.H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, John Wiley & Sons, Chichester, 1999. P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1993. N.H. Ibragimov (Ed.), CRC Handbook of Lie Group Analysis of Differential Equations, Symmetries, Exact Solutions and Conservation Laws, vol. 1, CRC Press, Boca Raton, Florida, 1994 (Application in Engineering and Physical Sciences, vol. 2, 1995. New Trends in Theoretical Developments and Computational Methods, vol.3, 1996). R.K. Gazizov, A.A. Kasatkin, S.Y. Lukashchuk, Continuous transformation groups of fractional differential equations, Vestnik UGATU 9 (2007) 125–135 (in Russian). R.K. Gazizov, A.A. Kasatkin, S.Y. Lukashchuk, Fractional differential equations: change of variables and nonlocal symmetries, Ufa Math. J. 4 (4) (2012) 54–67. R.K. Gazizov, A.A. Kasatkin, S.Y. Lukashchuk, Symmetry properties of fractional diffusion equations, Phys. Scr. T136 (2009) 014016. R.K. Gazizov, A.A. Kasatkin, Construction of exact solutions for fractional order differential equations by the invariant subspace method, Comput. Math. Appl. 66 (5) (2013) 576–584. G.-C. Wu, Lie group classifications and non-differentiable solutions for time-fractional Burgers equation, Commun. Theor. Phys. 55 (6) (2011) 1073– 1076. R. Sahadevan, T. Bakkyaraj, Invariant analysis of time fractional generalized Burgers and Korteweg-de Vries equations, J. Math. Anal. Appl. 393 (2) (2012) 341–347. G.-W. Wang, X.-Q. Liu, Y.-Y. Zhang, Lie symmetry analysis to the time fractional generalized fifth-order KdV equation, Commun. Nonlinear Sci. Numer. Simulat. 18 (9) (2013) 2321–2326. G.-W. Wang, T.-Z. Xu, Invariant analysis and exact solutions of nonlinear time fractional Sharma-Tasso-Olver equation by Lie group analysis, Nonlinear Dyn. 76 (1) (2014) 571–580. L.V. Ovsyannikov, The group properties of nonlinear heat conduction equations, Dokl. Akad. Nauk. SSSR 125 (3) (1959) 492–495 (in Russian). N.H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Reidel, Boston, 1985. G. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, Berlin, 1989. S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, 1993. A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, Singapore, 2012. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Background and Theory, vol. I, Springer-Higher Education Press, Beijing, 2013. Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (1) (2000) 1–77. R. Klages, G. Radons, I.M. Sokolov (Eds.), Anomalous Transport: Foundations and Applications, Willey-VCH, Berlin, 2008. J. Klafter, S.C. Lim, R. Metzler, Fractional Dynamics: Recent Advances, World Scientific, Singapore, 2011. Yu. Luchko, Anomalous diffusion: models, their analysis, and interpretation, in: S. Rogosin, A. Koroleva (Eds.), Advances in Applied Analysis, Birkhauser Verlag, Boston, 2012.
Please cite this article in press as: S.Yu. Lukashchuk, A.V. Makunin, Group classification of nonlinear time-fractional diffusion equation with a source term, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.11.087
S.Yu. Lukashchuk, A.V. Makunin / Applied Mathematics and Computation xxx (2014) xxx–xxx
9
[26] M. Meerschaert, A. Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter, Berlin, 2012. [27] V. Uchaikin, R. Sibatov, Fractional Kinetics in Solids: Anomalous Charge Transport in Semiconductors, Dielectrics and Nanosystems, World Scientific, Singapore, 2013. [28] Yu. Luchko, Fractional diffusion and wave propagation, in: W. Freeden, M.Z. Nashed, T. Sonar (Eds.), Handbook of Geomathematics, Springer, Berlin, 2014, pp. 1–36. [29] V. Dorodnitsyn, On invariant solutions of the equation of non-linear heat conduction with a source, U.S.S.R. Comput. Maths. Math. Phys. 22 (6) (1982) 115–122. [30] I. Akhatov, R. Gazizov, N. Ibragimov, Nonlocal symmetries. Heuristic approach, J. Sov. Math. 55 (1) (1991) 1401–1450.
Please cite this article in press as: S.Yu. Lukashchuk, A.V. Makunin, Group classification of nonlinear time-fractional diffusion equation with a source term, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.11.087
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Group classification of nonlinear time-fractional diffusion equation with a source term S.Yu. Lukashchuk ⇑, A.V. Makunin Laboratory ‘‘Group analysis of mathematical models in natural and engineering sciences’’, Ufa State Aviation Technical University, 12 K. Marx str., Ufa 450000, Russia
a r t i c l e
i n f o
Keywords: Time-fractional diffusion equation Symmetry Equivalence transformation Group classification Invariant solution
a b s t r a c t A complete group classification is presented for a nonlinear time-fractional diffusion equation with a source term. The equation involves the Riemann–Liouville time-fractional derivative of the order a 2 ð0; 2Þ. All coefficients of the equation are sought as a function of the dependent variable. Using the infinitesimal approach, the Lie algebra of infinitesimal generators of equivalence transformations is constructed for the equation under consideration, and group classification is carried out up to the equivalence transformations. Examples of invariant solutions are also presented. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Lie group analysis (or symmetry analysis) is an efficient approach to investigate fundamental properties of differential equations and to construct their solutions (see, e.g., [1–5]). Recently, principal methods of symmetry analysis were extended to fractional differential equations with different types of fractional derivatives [6,7]. Using these methods, symmetry properties of some classes of ordinary and partial fractional differential equations have been investigated [8–13]. Classification of differential equations belonging to a certain class with respect to symmetry groups is one of major tasks of modern symmetry analysis. The problem of group classification can be briefly formulated as follows: it is necessary to determine the Lie symmetry algebra for each differential equation belonging to a given class of equations. Fundamentals of the group classification originated in works by Lie and then methods of group classification were developed by many researchers. An efficient method for group classification was developed by Ovsyannikov [14] (see also [1]). Using this method, he first performed a complete group classification of the nonlinear heat equation, where a thermal conductivity was a function of the temperature. At present, these methods are widely presented in the literature (see, e.g., [1,4,5,15,16]). Results of group classification can be used as a starting point for construction of solutions for nonlinear differential equations using their symmetries. The powerful technique, known as the symmetry reduction, was developed for this purpose. In this approach, a given partial differential equation is reduced using its symmetry to the equation with a less number of independent variables. Solving this reduced equation, the exact (so-called, invariant) solution of the initial equation can be found. In particular, using symmetry reduction, differential equations with two independent variables are transformed to ordinary differential equations. The methods of group classification and symmetry reduction can be also efficiently used for constructing exact solutions of nonlinear fractional differential equations. In [8], the group classification was performed for homogeneous diffusion ⇑ Corresponding author. E-mail address: [email protected] (S.Yu. Lukashchuk). http://dx.doi.org/10.1016/j.amc.2014.11.087 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
Please cite this article in press as: S.Yu. Lukashchuk, A.V. Makunin, Group classification of nonlinear time-fractional diffusion equation with a source term, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.11.087
2
S.Yu. Lukashchuk, A.V. Makunin / Applied Mathematics and Computation xxx (2014) xxx–xxx
equations with the Riemann–Liouville and Caputo time-fractional derivatives. In these equations, a diffusion coefficient was considered as a function of the dependent variable. Using symmetry reduction, several invariant solutions of these nonlinear equations have been constructed and presented in this work. It should be noted that the problem of group classification for a fractional differential equation is much more complicated than a similar problem for an integer-order differential equation. The infinitesimal group generators, admitted by the equations, can be found by solving the appropriate determining equations. Nevertheless, on the contrary to the integer-order differential equations, these determining equations for the fractional differential equations lead to an infinite chain of differential equations of integer and fractional orders [6]. In a general case, solution of such infinite system of differential equations is a non-trivial problem. In this paper, we perform the group classification of the time-fractional diffusion equation with a source term
Dat u ¼ ðkðuÞux Þx þ f ðuÞ;
kðuÞ > 0; a 2 ð0; 2Þ:
ð1:1Þ
Here t; x are independent variables, u ¼ uðt; xÞ is the dependent variable, the diffusion coefficient k and the source term f are sought as a function of the dependent variable u; ux ¼ @u , @x
Dat u ¼
1 @n Cðn aÞ @t n
Z
t
0
uðs; xÞ ðt sÞanþ1
ds;
n ¼ ½a þ 1
is the left-sided Riemann–Liouville fractional derivative of order a with respect to t [17–21], and CðzÞ is the Gamma function. At present, time-fractional diffusion equations are frequently used as mathematical models for describing processes with anomalous kinetics. Such processes are observed in many different areas: heat and mass transfer in solids, hydrodynamics, plasma physics, cosmology, hydrology, quantum optics, chemistry of polymers, biology, computer networks, and many others (see, e.g., [22–28] and references therein). For a 2 ð0; 1Þ, Eq. (1.1) can be used for modeling the subdiffusion processes, whereas for a 2 ð1; 2Þ it describes the diffusion-wave phenomena. In a limiting case of a ¼ 1, Eq. (1.1) coincides with the classical diffusion (or heat) equation. The complete group classification of this classical equation was performed in [29]. Therefore, we do not consider this limiting case in this paper. The paper is organized as follows: in Section 2 the group of equivalence transformations for Eq. (1.1) is constructed; in Section 3 the results of group classification are presented, and Section 4 contains the examples of invariant solutions to Eq. (1.1). 2. Equivalence transformations At the first step of group classification it is necessary to compute the group of equivalence transformations for Eq. (1.1). The equivalence transformations preserve the differential structure of the equation under consideration. In other words, equations which are related by these transformations admit similar groups of the Lie point transformations. We use the infinitesimal approach to obtain the equivalence transformations for Eq. (1.1). Following [1,30], we shall seek the generator of the continuous group of equivalence transformations in the form
E ¼ n0
@ @ @ @ @ þ n1 þ g þl þm : @t @x @u @k @f
ð2:1Þ
Here u; k and f are considered as differential variables: u in the space ðt; xÞ; k and f in the extended space ðt; x; u; Dat u; ux Þ. The coordinates n0 ; n1 and g of the generator (2.1) are sought as a function of t; x and u, whereas the coordinates l and m are sought as functions of t; x; u; Dat u; ux ; k and f. The generator (2.1) can be found from the invariance of the system
Dat u ¼ ku u2x þ kuxx þ f ; kt ¼ 0;
kx ¼ 0;
kDat u ¼ 0;
kux ¼ 0;
f t ¼ 0;
f x ¼ 0;
f Dat u ¼ 0;
f ux ¼ 0:
ð2:2Þ
The infinitesimal invariance criterion for the system (2.2) has the form
fa 2ku ux f1 kf11 luxx l3 u2x m jð2:2Þ ¼ 0;
li jð2:2Þ ¼ 0; mi jð2:2Þ ¼ 0; i ¼ 1; 2; 4; 5:
ð2:3Þ
Here fa ; f1 ; f11 ; li ; mi (i = 1 . . .5) are coordinates of the extended generator
@ @ @ @ @ @ @ @ @ @ @ þ f1 þ f11 þ l1 þ l2 þ l3 þ l4 þ l5 þ m1 þ m2 þ m3 @ux @uxx @kt @kx @ku @kDat u @kux @f t @f x @f u @Dat u @ @ þ m4 þ m5 : @f Dat u @f ux
e E ¼ E þ fa
ð2:4Þ
These coordinates are calculated by the appropriate prolongation formulae (see, e.g., [3] for f1 and f11 , [8] for fa , and [30] for
li and mi ). Please cite this article in press as: S.Yu. Lukashchuk, A.V. Makunin, Group classification of nonlinear time-fractional diffusion equation with a source term, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.11.087
S.Yu. Lukashchuk, A.V. Makunin / Applied Mathematics and Computation xxx (2014) xxx–xxx
3
Solving the system (2.3), we find all the generators of equivalence transformations for Eq. (1.1). The obtained results can be summarized as the following proposition. Proposition 1. The fractional diffusion Eq. (1.1) admits a 4-dimensional Lie algebra L4 of infinitesimal generators of equivalence transformations. The algebra L4 is spanned by
E1 ¼
@ ; @x
E2 ¼ t
@ @ @ ak af ; @t @k @f
E3 ¼ x
@ @ þ 2k ; @x @k
E4 ¼ u
@ @ þf : @u @f
The corresponding 4-parameter equivalence transformation group is given by
t ¼ d2 t;
x ¼ cx þ b;
¼ uu; u
¼ c2 d2a k; k
f ¼ ud2a f ;
ð2:5Þ
where the coefficients b; c; d are arbitrary, and u > 0. Remark 1. The reflection x ¼ x is also admitted by Eq. (1.1). The group of equivalence transformations (2.5) contains this reflection. In this paper, the group classification of Eq. (1.1) is performed up to the equivalence transformations (2.5). 3. Group classification For the group classification of Eq. (1.1), we use the approach described in [1]. Let us find the operators
X ¼ n0 ðt; x; uÞ
@ @ @ þ n1 ðt; x; uÞ þ gðt; x; uÞ @t @x @u
ð3:1Þ
admitted by Eq. (1.1) with different pairs of functions fkðuÞ; f ðuÞg. The infinitesimal invariance criterion for Eq. (1.1) has the form
e Da u ku u2 kuxx f j X ð1:1Þ ¼ 0; x t
ð3:2Þ
e of the operator X is written as where the prolongation X
e ¼ n0 @ þ n1 @ þ g @ þ f1 @ þ f11 @ þ fa @ : X @t @x @u @ux @uxx @ðDat uÞ Here the coordinates f1 ; f11 and fa are given by the following prolongation formulae (see, e.g., [3,7]):
f1 ¼ Dx ðgÞ ut Dx ðn0 Þ ux Dx ðn1 Þ; f11 ¼ Dx ðf1 Þ utx Dx ðn0 Þ uxx Dx ðn1 Þ;
ð3:3Þ
fa ¼ Dat ðg n0 ut n1 ux Þ þ n0 Dtaþ1 u þ n1 Dat ux : The invariance criterion (3.2) leads to the determining equation
fa gðkuu u2x þ ku uxx þ f u Þ 2f1 ku ux f11 k ¼ 0: a
We substitute (3.3) into (3.4) and replace uxx by ðDt u rule [17,18]
ð3:4Þ ku u2x
f Þ=k by virtue of Eq. (1.1). Then we use the generalized Leibniz
1 X a an n Dt f Dt g n n¼0
Dat ðg hÞ ¼
and split the obtained equation by ux and by the fractional derivatives and integrals Dtan u; n ¼ 0; 1; . . .. As a result, we obtain the following infinite system of partial differential equations:
n0u ¼ 0;
n0x ¼ 0;
ðKðuÞgÞu ¼ 0;
n1u ¼ 0;
n1t ¼ 0;
2KðuÞgx þ 2gxu
guu ¼ 0;
n1xx
¼ 0;
KðuÞg 2n1x þ an0t ¼ 0;
Dat ðg ugu Þ gðf u KðuÞf Þ kgxx þ gu f 2n1x f ¼ 0; ðn þ 1ÞDnt gu þ ðn aÞDnþ1 n0 ¼ 0; t
ð3:5Þ
n ¼ 1; 2; . . . :
Here we use the condition kðuÞ – 0 and notation KðuÞ ¼ ku =k. From the first five equations of the system (3.5) we find
n0 ¼ n0 ðtÞ;
n1 ¼ n1 ðxÞ;
g ¼ g0 ðt; xÞ þ g1 ðt; xÞu:
Please cite this article in press as: S.Yu. Lukashchuk, A.V. Makunin, Group classification of nonlinear time-fractional diffusion equation with a source term, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.11.087
4
S.Yu. Lukashchuk, A.V. Makunin / Applied Mathematics and Computation xxx (2014) xxx–xxx
Then from the infinite chain of equations given by the last equation of the system (3.5), we obtain
n0 ðtÞ ¼ C 0 þ C 1 t þ C 2 t 2 ;
g1 ðt; xÞ ¼ ða 1ÞC 2 t þ aðxÞ;
where C 0 ; C 1 ; C 2 are arbitrary constants, and aðxÞ is an arbitrary function. Note that the lower limit t ¼ 0 in the fractional differentiation must be invariant under the point transformations admitted by Eq. (1.1) (see [6–8] for more details). This condition leads to the equality nð0Þ ¼ 0 which provides C 0 ¼ 0. Using obtained representations for n0 ; n1 and g, it can be shown that for arbitrary functions kðuÞ and f ðuÞ the system (3.5) has the solution n0 ¼ 0; n1 ¼ const; g ¼ 0. So in this case, the translation in x with the generator
X1 ¼
@ @x
is the only Lie point symmetry of the Eq. (1.1). For the arbitrary function kðuÞ and f ðuÞ ¼ 0, the system (3.5) has the solution n0 ¼ 2C 1 t; n1 ¼ aC 1 x þ C 3 ; g ¼ 0. Therefore, the corresponding Eq. (1.1) additionally admits the group of dilation (scaling transformation) with the generator
X 2 ¼ 2t
@ @ þ ax : @t @x
From the sixth, seventh and eighth equations of the system (3.5) we find that Eq. (1.1) can admit additional symmetries if either KðuÞ ¼ 0, or
00 1 ¼ 0: KðuÞ
ð3:6Þ
This is the classifying relation for the function KðuÞ (or kðuÞ). From the relation (3.6) and system (3.5), in view of the equivalence transformations (2.5), we obtain the preliminary group classification presented in Table 1. In this table, A and C i ði ¼ 1; 2; 3; 4Þ are arbitrary constants, and hðxÞ is an arbitrary function. To complete the group classification, we consider the single equation from the system (3.5) which contains the function f ðuÞ. For each functions kðuÞ from Table 1, this equation provides the classifying relations for the function f ðuÞ as follows: for kðuÞ ¼ 1, the function f ðuÞ either satisfies the equation f uu ¼ 0, or satisfies the equation
fu f uu
¼ 0; uu
for kðuÞ ¼ eu , the function f ðuÞ is either arbitrary and C 1 ¼ C 4 ¼ 0, or it is equal to zero (f ¼ 0) and 2C 4 ¼ aC 1 ; a , the function f ðuÞ is either constant and 2C ¼ að1 þ rÞC , or it satisfies the equation for kðuÞ ¼ ðu þ AÞr r – 0; 43 ; a21 4 1
f ¼ 0; f u uu
2a
aþ2 a
for kðuÞ ¼ ðu þ AÞa1 , there are two additional particular cases of the function f ðuÞ : f ðuÞ ¼ 0 and f ðuÞ ¼ f 0 u an arbitrary constant); 4 for kðuÞ ¼ ðu þ AÞ3 , there are three different cases:
FðuÞ ¼ 0 or
F uu ¼ 0 or
Fu F uu
(here f 0 is
¼ 0;
uu
1
where FðuÞ ¼ u3 f ðuÞ. We compute the solutions of all classifying relations for the function f ðuÞ presented above. Upon substituting these solutions in the system (3.5) and simple algebra, we find all pairs of functions fkðuÞ; f ðuÞg for which the Lie algebra of the point
Table 1 Preliminary group classification. kðuÞ 1 eu a ðu þ AÞr r – 0; 43 ; a21 a a21
ðu þ AÞ
4
ðu þ AÞ3
n0
n1
g0
g1
C1 t C1 t C1 t
aC x þ C 3 2 1
C4 x þ C3 C4 x þ C3
0 2C 4 aC 1 0
C4 C4
C 1 t þ C 2 t2
C4 x þ C3
0
C1 t
hðxÞ
0
1
r ð2C 4 aC 1 Þ a1 ð2aC t 2C þ aC Þ 2 4 1 2a 0
32 h ðxÞ þ 34 aC 1
Please cite this article in press as: S.Yu. Lukashchuk, A.V. Makunin, Group classification of nonlinear time-fractional diffusion equation with a source term, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.11.087
S.Yu. Lukashchuk, A.V. Makunin / Applied Mathematics and Computation xxx (2014) xxx–xxx
5
symmetries is extended. The results of the group classification of Eq. (1.1) up to the equivalence transformations (2.5) are summarized in the following proposition. Proposition 2. The nonlinear fractional diffusion Eq. (1.1) with arbitrary functions kðuÞ > 0 and f ðuÞ has a single symmetry with the generator
X1 ¼
@ : @x
The homogeneous equation (f ðuÞ ¼ 0) with an arbitrary function kðuÞ admits an additional symmetry
X 2 ¼ 2t
@ @ þ ax : @t @x
The algebra extends for the following cases. I. kðuÞ ¼ 1. I.1. f ðuÞ ¼ 0:
X3 ¼ u
@ ; @u
X 1 ¼ gðt; xÞ
@ ; @u
where the function gðt; xÞ is an arbitrary solution of the equation Dat g ¼ g xx . I.2. f ðuÞ ¼ d ðd ¼ 1Þ:
X2 ¼ t
@ a @ ta @ þ x þd ; @t 2 @x CðaÞ @u
X 1 ¼ gðt; xÞ
X3 ¼
ud
ta @ ; Cða þ 1Þ @u
@ ; @u
where the function gðt; xÞ is an arbitrary solution of the equation Dat g ¼ g xx . I.3. f ðuÞ ¼ du þ b ðd ¼ 1; b ¼ 1Þ:
X 2 ¼ ½u bt a Ea;aþ1 ðdt a Þ
@ ; @u
X 1 ¼ gðt; xÞ
@ ; @u
where the function gðt; xÞ is an arbitrary solution of the equation Dat g ¼ g xx þ dg, and Ea;aþ1 ðtÞ is the Mittag–Leffler function (see, e.g., [18]). I.4. f ðuÞ ¼ duc ðd ¼ 1; c – 0; 1Þ:
X2 ¼ t
@ a @ 1 a @ þ x þ u : @t 2 @x d 1 c @u
a . II. kðuÞ ¼ ur r – 0; 43 ; a21 II.1. f ðuÞ ¼ 0:
X3 ¼ II.2.
r @
@ x þu : @u 2 @x
f ðuÞ ¼ duc ðd ¼ 1Þ:
X 2 ¼ ð1 cÞt
@ að1 þ r cÞ @ @ þ x þ au : @t 2 @x @u
2a
III. kðuÞ ¼ ua1 . III.1. f ðuÞ ¼ 0:
X 3 ¼ ax III.2.
@ @ þ ða 1Þu ; @x @u
f ðuÞ ¼ duc d ¼ 1; c – aaþ1 : 1
X 2 ¼ ð1 cÞt III.3.
X 4 ¼ t2
@ @ þ ða 1Þtu : @t @u
@ a 2a @ @ x þ au : þ 1c a 1 @x @t 2 @u
aþ1
f ðuÞ ¼ dua1 ðd ¼ 1Þ:
X 2 ¼ 2t
@ @ þ ða 1Þu ; @t @u
X 3 ¼ t2
@ @ þ ða 1Þtu : @t @u
4
IV. kðuÞ ¼ u3 . IV.1. f ðuÞ ¼ 0:
Please cite this article in press as: S.Yu. Lukashchuk, A.V. Makunin, Group classification of nonlinear time-fractional diffusion equation with a source term, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.11.087
6
S.Yu. Lukashchuk, A.V. Makunin / Applied Mathematics and Computation xxx (2014) xxx–xxx
@ @ 3u ; @x @u
X 3 ¼ 2x IV.2.
@ @ 3u ; @x @u
@ @ 3xu : @x @u
1
@ @ þ 3au ; @t @u
X 3;4 ¼ 2exx
@ @ 3xexx u ; @x @u
pffiffiffi pffiffiffi where x ¼ 2= 3 for d ¼ 1, and x ¼ 2i= 3 for d ¼ 1 (here i is the imaginary unit). f ðuÞ ¼ duc ðd ¼ 1; c – 1Þ:
X 2 ¼ ð1 cÞt IV.5.
X 3 ¼ x2
f ðuÞ ¼ du3 ðd ¼ 1Þ:
X 2 ¼ 4t
IV.4.
@ @ 3xu : @x @u
f ðuÞ ¼ du ðd ¼ 1Þ:
X 2 ¼ 2x IV.3.
X 4 ¼ x2
@ a 1 @ @ c þ x þ au : @t 2 3 @x @u
1
f ðuÞ ¼ du3 þ bu ðd ¼ 1; b ¼ 1Þ:
X 2;3 ¼ 2exx
@ @ 3xexx u ; @x @u
pffiffiffi pffiffiffi where x ¼ 2= 3 for d ¼ 1, and x ¼ 2i= 3 for d ¼ 1. Thus, there are 14 cases of extensions of the Lie algebra for Eq. (1.1). The maximal dimension of the Lie algebra is equal to four. Remark 2. Contrary to the classical integer-order diffusion equation, the case kðuÞ ¼ eu from Table 1 does not provide the extension of the Lie algebra. Remark 3. Calculations show that the Lie algebra extends if and only if A ¼ 0 for all functions kðuÞ presented in Table 1. Nevertheless, the constant A can not be set equal to zero by virtue of the equivalence transformations (2.5).
4. Examples of invariant solutions Examples of invariant solutions to Eq. (1.1) with f ¼ 0 have been presented in [6]. In this section, we give several invariant solutions of the nonhomogeneous Eq. (1.1). 4.1. Example 1 Let us consider Eq. (1.1) with kðuÞ ¼ ur and f ðuÞ ¼ urþ1 operator
X ¼ rt
a . As it follows from Item II.2 in Proposition 2, the r – 0; 43 ; a21
@ @ þ au @t @u
is admitted by the equation under consideration. The invariant solution, corresponding to this operator, has the form a
uðt; xÞ ¼ t r uðxÞ; where the function uðxÞ is a solution of the second order ordinary differential equation
u00 þ ru1 ðu0 Þ2 þ u au1r ¼ 0;
a¼
Cð1 a=rÞ : Cð1 a a=rÞ
ð4:7Þ
The solution of Eq. (4.7) can be written as
Z
du pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ C 2 x; wðu; C 1 Þ
where C 1 and C 2 are the constants of integration, and the function wðu; C 1 Þ depends on
8 2r 1 þ r2a u2r rþ1 u2 ; > þ2 < C1u 2 3 2 wðu; C 1 Þ ¼ C 1 u þ 2au 2u lnðuÞ; > : C 1 u4 þ 2au4 lnðuÞ þ u2 ;
r:
r – 1; 2; r ¼ 1; r ¼ 2:
Please cite this article in press as: S.Yu. Lukashchuk, A.V. Makunin, Group classification of nonlinear time-fractional diffusion equation with a source term, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.11.087
7
S.Yu. Lukashchuk, A.V. Makunin / Applied Mathematics and Computation xxx (2014) xxx–xxx
4.2. Example 2 4
1
Eq. (1.1) with kðuÞ ¼ u3 and f ðuÞ ¼ u3 has the symmetries
X ¼ 2exx
@ @ 3xexx u @x @u
(see Item IV.3 in Proposition 2). The corresponding invariant solutions have the form 3
uðt; xÞ ¼ e2xx uðtÞ; where the function uðtÞ is a solution of the homogeneous equation
Dat u ¼ 0: This equation has a well-known solution:
(
uðtÞ ¼
C 1 ta1 ; C1t
a1
þ C2t
a2
a 2 ð0; 1Þ; ; a 2 ð1; 2Þ:
4.3. Example 3 4
Eq. (1.1) with kðuÞ ¼ u3 and f ðuÞ ¼ u has the symmetry
@ @ 3xu @x @u
X ¼ x2
(see Item IV.2 in Proposition 2). The corresponding invariant solution has the form
uðt; xÞ ¼ x3 uðtÞ;
ð4:8Þ
where the function uðtÞ is a solution of the equation
Dat u ¼ u:
ð4:9Þ
The common solution of Eq. (4.9) is known (see, e.g., [17,18]):
uðtÞ ¼
n X C k tak Ea;aþ1k ðt a Þ;
n ¼ ½a þ 1:
ð4:10Þ
k¼1
Here C k are the constants of integration, and Ea;aþ1k ðtÞ is the Mittag–Leffler function. Substituting (4.10) into (4.8), one can get the final form of the invariant solution. 4.4. Example 4 2a
aþ1
Eq. (1.1) with kðuÞ ¼ ua1 and f ðuÞ ¼ ua1 has the symmetry
X ¼ t2
@ @ þ ða 1Þtu @t @u
(see Item III.3 in Proposition 2). The corresponding invariant solution has the form
uðt; xÞ ¼ t a1 uðxÞ;
ð4:11Þ
where the function uðxÞ is a solution of the equation
u00
2a u1 ðu0 Þ2 þ u ¼ 0: a1
This equation has the common solution 2
uðxÞ ¼ ½C 1 x sinðxxÞ þ C 2 x cosðxxÞx ; x ¼
1þa : 1a
Substituting this solution into (4.11), one can write the invariant solution in the final form. 4.5. Example 5 As the final example, let us consider Eq. (1.1) with kðuÞ ¼ ur and f ðuÞ ¼ uc from Item II.2 in Proposition 2, this equation has the symmetry
a and c – r þ 1 . As it follows r – 0; 43 ; a21
Please cite this article in press as: S.Yu. Lukashchuk, A.V. Makunin, Group classification of nonlinear time-fractional diffusion equation with a source term, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.11.087
8
S.Yu. Lukashchuk, A.V. Makunin / Applied Mathematics and Computation xxx (2014) xxx–xxx
X ¼ ð1 cÞt
@ að1 þ r cÞ @ @ þ x þ au : @t 2 @x @u
The corresponding invariant solution has the form
uðt; xÞ ¼ xk uðtxq Þ; with
k¼
2 ; 1þrc
q¼
c1 k: a
The function uðyÞ ðy ¼ txq Þ is a solution of the ordinary fractional differential equation 2
Day u ¼ ay2 ur u00 þ by ur1 ðu0 Þ þ cyur u0 þ durþ1 þ uc ; 2
ð4:12Þ
where
a ¼ aq2 ;
b ¼ arq2 ;
c ¼ q 1 þ k þ 2að1 þ qÞ þ ak þ a2 q ;
d ¼ kð1 þ k þ aqÞ:
5. Conclusion The group classification of Eq. (1.1), presented in the paper, permits to construct exact solutions of this equation. Some examples of the invariant solutions are presented in the paper. At the same time, a new problem arises during construction of the invariant solutions: after symmetry reduction one can get a nonlinear ordinary fractional differential equation (see, e.g, Example 5). It is an open problem to develop the methods of integration for such equations. The given group classification also can be used for classification of invariant solutions of the time-fractional diffusion equation. The optimal systems of subalgebras have to be constructed for this purpose. This is a possible direction for future work. Acknowledgement The authors acknowledge a financial support of the Government of Russian Federation through Resolution No. 220, contract No. 11.G34.31.0042 with Ufa State Aviation Technical University and leading scientist professor N. H. Ibragimov. References [1] [2] [3] [4] [5]
[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
L.V. Ovsyannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982. L.V. Ovsyannikov, Lectures on the Theory of Group Properties of Differential Equations, World Scientific, 2013. N.H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, John Wiley & Sons, Chichester, 1999. P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1993. N.H. Ibragimov (Ed.), CRC Handbook of Lie Group Analysis of Differential Equations, Symmetries, Exact Solutions and Conservation Laws, vol. 1, CRC Press, Boca Raton, Florida, 1994 (Application in Engineering and Physical Sciences, vol. 2, 1995. New Trends in Theoretical Developments and Computational Methods, vol.3, 1996). R.K. Gazizov, A.A. Kasatkin, S.Y. Lukashchuk, Continuous transformation groups of fractional differential equations, Vestnik UGATU 9 (2007) 125–135 (in Russian). R.K. Gazizov, A.A. Kasatkin, S.Y. Lukashchuk, Fractional differential equations: change of variables and nonlocal symmetries, Ufa Math. J. 4 (4) (2012) 54–67. R.K. Gazizov, A.A. Kasatkin, S.Y. Lukashchuk, Symmetry properties of fractional diffusion equations, Phys. Scr. T136 (2009) 014016. R.K. Gazizov, A.A. Kasatkin, Construction of exact solutions for fractional order differential equations by the invariant subspace method, Comput. Math. Appl. 66 (5) (2013) 576–584. G.-C. Wu, Lie group classifications and non-differentiable solutions for time-fractional Burgers equation, Commun. Theor. Phys. 55 (6) (2011) 1073– 1076. R. Sahadevan, T. Bakkyaraj, Invariant analysis of time fractional generalized Burgers and Korteweg-de Vries equations, J. Math. Anal. Appl. 393 (2) (2012) 341–347. G.-W. Wang, X.-Q. Liu, Y.-Y. Zhang, Lie symmetry analysis to the time fractional generalized fifth-order KdV equation, Commun. Nonlinear Sci. Numer. Simulat. 18 (9) (2013) 2321–2326. G.-W. Wang, T.-Z. Xu, Invariant analysis and exact solutions of nonlinear time fractional Sharma-Tasso-Olver equation by Lie group analysis, Nonlinear Dyn. 76 (1) (2014) 571–580. L.V. Ovsyannikov, The group properties of nonlinear heat conduction equations, Dokl. Akad. Nauk. SSSR 125 (3) (1959) 492–495 (in Russian). N.H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Reidel, Boston, 1985. G. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, Berlin, 1989. S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, 1993. A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus: Models and Numerical Methods, World Scientific, Singapore, 2012. V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Background and Theory, vol. I, Springer-Higher Education Press, Beijing, 2013. Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014. R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (1) (2000) 1–77. R. Klages, G. Radons, I.M. Sokolov (Eds.), Anomalous Transport: Foundations and Applications, Willey-VCH, Berlin, 2008. J. Klafter, S.C. Lim, R. Metzler, Fractional Dynamics: Recent Advances, World Scientific, Singapore, 2011. Yu. Luchko, Anomalous diffusion: models, their analysis, and interpretation, in: S. Rogosin, A. Koroleva (Eds.), Advances in Applied Analysis, Birkhauser Verlag, Boston, 2012.
Please cite this article in press as: S.Yu. Lukashchuk, A.V. Makunin, Group classification of nonlinear time-fractional diffusion equation with a source term, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.11.087
S.Yu. Lukashchuk, A.V. Makunin / Applied Mathematics and Computation xxx (2014) xxx–xxx
9
[26] M. Meerschaert, A. Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter, Berlin, 2012. [27] V. Uchaikin, R. Sibatov, Fractional Kinetics in Solids: Anomalous Charge Transport in Semiconductors, Dielectrics and Nanosystems, World Scientific, Singapore, 2013. [28] Yu. Luchko, Fractional diffusion and wave propagation, in: W. Freeden, M.Z. Nashed, T. Sonar (Eds.), Handbook of Geomathematics, Springer, Berlin, 2014, pp. 1–36. [29] V. Dorodnitsyn, On invariant solutions of the equation of non-linear heat conduction with a source, U.S.S.R. Comput. Maths. Math. Phys. 22 (6) (1982) 115–122. [30] I. Akhatov, R. Gazizov, N. Ibragimov, Nonlocal symmetries. Heuristic approach, J. Sov. Math. 55 (1) (1991) 1401–1450.
Please cite this article in press as: S.Yu. Lukashchuk, A.V. Makunin, Group classification of nonlinear time-fractional diffusion equation with a source term, Appl. Math. Comput. (2014), http://dx.doi.org/10.1016/j.amc.2014.11.087