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Classification and approximate solutions to perturbed diffusion–convection equations
Applied Mathematics and Computation 219 (2012) 1120–1124
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Classification and approximate solutions to perturbed diffusion–convection equations Zhi-Yong Zhang a,c,⇑, Xue-Lin Yong b,c, Yu-Fu Chen c a
College of Sciences, North China University of Technology, Beijing 100144, PR China School of Mathematical Sciences and Physics, North China Electric Power University, Beijing 102206, PR China c School of Mathematical Sciences, Graduate University of Chinese Academy of Sciences, Beijing 100049, PR China b
a r t i c l e
i n f o
Keywords: Approximate symmetry Classification Approximate solution Lie reduction Diffusion–convection equations
a b s t r a c t Approximate symmetries of the perturbed nonlinear diffusion–convection equations are completely classified by the method originated with Fushchich and Shtelen. Moreover, for some interesting cases, symmetry reductions and approximate solutions are discussed in detail. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction It was Lie’s major goal to develop symmetry group theory for the analytical solutions of differential equations. His integration theorems require solvable group of exact symmetries which leave the differential equations invariant exactly [1,2]. However, many differential equations arising in mathematical physics do not admit nontrivial exact Lie symmetries and any small perturbation of differential equations disturbs the symmetry group properties of the unperturbed equation, which make symmetry group method lose its superiority when discussing the symmetry properties of equations involving a small parameter [3,4]. From the last century, perturbation analysis technique has been widely used especially for nonlinear problems. In order to get the utmost goal from the technique, combination of Lie group theory and perturbations are considered, which leads the notion of approximate symmetry introduced and two methods [4,3] for approximate symmetry have been developed. Baikov et al. [3] employ standard perturbation techniques about the symmetry operator to obtain approximate symmetry. In 1989, Fushchich and Shtelen [4] introduced an effective method to search for approximate symmetry and the method was later followed by Euler et al. [5–7]. They firstly expanded the dependent variables in terms of a small parameter (may be a physical parameter or artificially introduced) as the usual perturbation analysis. Terms are then separated at each order of approximation and a system of equations to be solved in a hierarchy is obtained. The system is assumed to be coupled and the approximate symmetry of the original equation is defined to be the exact symmetry of the system of equations obtained from perturbations. In [8,9], these two methods are applied to potential Burgers equation, non-Newtonian creeping flow equations and advection–diffusion equations and shows that the second method is superior to the first one. Later, approximate condition symmetry, approximate generalized condition symmetry and approximate potential symmetry are developed and applied to several celebrated physical partial differential equations [10,12,11,13]. In this paper, we are mainly concerned with the perturbed nonlinear diffusion–convection equations with a source term
ut ¼ ðAðuÞux Þx þ ðBðuÞux þ CðuÞÞ;
⇑ Corresponding author at: College of Sciences, North China University of Technology, Beijing 100144, PR China. E-mail address: [email protected] (Z.-Y. Zhang). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.07.019
ð1Þ
Z.-Y. Zhang et al. / Applied Mathematics and Computation 219 (2012) 1120–1124
1121
where AðuÞ; BðuÞ; CðuÞ are arbitrary and sufficient smooth real-valued functions of their corresponding variables, and AðuÞBðuÞCðuÞ – 0. Eq. (1) without source term was discussed by the Lie’s classical method [1,14], conditional symmetry method [15,16], and conditional Lie-Bäcklund symmetry method (also referred to as the generalized conditional symmetry) [17,18], domain decomposition algorithm [19], etc. Recently, Eq. (1) was studied by Wang and Zhang [20] with the approximate generalized conditional symmetry method and some approximate invariant solutions were obtained as special cases. In this paper, we perform approximate symmetry classification of Eq. (1) and construct some new approximate solutions and reduced equations. It is observed that the calculated approximate symmetries and approximate solutions developed here are new. The outline of the paper is as follows: In Section 2, approximate symmetry classification of nonlinear Eq. (1) is presented. The results are listed in Table 1. In Section 3, reduced equations are constructed and some approximate solutions are calculated. Concluding remarks are discussed in the last section. 2. Approximate symmetry classification In this section, we use the method in [4–6] to obtain complete approximate symmetry classification of Eq. (1) with an accuracy oðÞ. It means that hereinafter in all equalities we neglect the terms of order oðÞ. Expanding the dependent variable to first-order yields
u ¼ v þ w þ oðÞ;
0 < 1;
ð2Þ
then one can expand AðuÞ; BðuÞ; CðuÞ in a series in
0
AðuÞ ¼ Aðv þ wÞ ¼ Aðv Þ þ wA ðv Þ þ oðÞ; BðuÞ ¼ Bðv þ wÞ ¼ Bðv Þ þ wB0 ðv Þ þ oðÞ; CðuÞ ¼ Cðv þ wÞ ¼ Cðv Þ þ wC 0 ðv Þ þ oðÞ:
ð3Þ
Substituting the expansions into Eq. (1) and separating at each order of perturbation parameter, one has the following equations with respect to 0 and 1 :
v t ¼ ðAðv Þv x Þx ; wt ¼ ðwA0 ðv Þv x þ Aðv Þwx Þx þ Bðv Þ v x þ Cðv Þ:
ð4Þ
Approximate symmetries of Eq. (1) correspond to the exact symmetries of system (4). Now, we consider a one-parameter Lie symmetry group of local transformations with an infinitesimal operator of the form
Table 1 Symmetry classifications of system (4). Aðv Þ
Bðv Þ
Cðv Þ
Infinitesimal operators
Arbitrary ev k
Arbitrary ev l
Arbitrary ev q
va
vb
vc
1 1
Arbitrary 2v
v2
1 1 1 1 1
2 þ ln v 1 þ ev
ebv
v þ ev
1 1 1 1
ebv ebv ebv v
ebv
@x ; @t @ x ; @ t ; ðl qÞx@ x þ ð2l 2q kÞt@ t þ @ v þ ð2l q kÞw@ w @ x ; @ t ; ð1 þ b cÞx@ x þ ð2 þ 2b a 2cÞt@ t þ v @ v þð2 þ 2b a cÞw@ w @ x ; @ t ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; @ v ; v @ v þ 2w@ w ; etx ð@ v þ v @ w Þ; @ v þ ð2t xÞv @ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; v @ v þ ½w þ ðt x=2Þv @ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; @ v þ ðt þ x=2Þv @ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; v @ v þ tv @ w ; 2x@ x 4t@ t þ v @ v 2w@ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; v @ v þ tv @ w ; x@ x þ 2t@ t þ ðw þ tv Þ@ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; ðb 1Þx@ x þ 2ðb 1Þt@ t þ @ v þ ½ð2b 1Þw tv @ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; @ v þ bw@ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; bx@ x þ 2bt@ t þ @ v þ ½2bw þ 2tv @ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; @ v þ bðw tv Þ@ w ; x@ x þ 2t@ t þ ðw þ tv Þ@ w ; v @ w ; f ðx; tÞ@ w
v
1 1
v 1
v2 v2
1
1
v þ ev
1
1
v v ln v
where f ðx; tÞ satisfies ft ¼ fxx .
v 1 v 1
Arbitrary
v þ v ln v v þ ev v2 v
v2 v
@ x ; @ t ; @ v xv =2@ w ; v @ v þ ð2w tv Þ@ w ; x@ v þ ðt=2 x2 =4Þv @ w ; x@ x þ 2t@ t þ ðw þ tv Þ@ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; @ v þ ðw t v Þ@ w ; x@ x þ 2t@ t þ ð2w xv =2Þ@ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; @ v þ 2t v @ w ; v @ v þ ð2w xv =2Þ@ w ; x@ x þ 2t@ t þ ð2w xv =2Þ@ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; @ v þ ðw t v þ xv =2Þ@ w ; x@ x þ 2t@ t þ ð2w xv =2Þ@ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; @ v þ ðw þ t v Þ@ w ; x@ x þ 2t@ t þ ð2w þ xv =2Þ@ w ; v @ w ; f ðx; tÞ@ w
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Z.-Y. Zhang et al. / Applied Mathematics and Computation 219 (2012) 1120–1124
X ¼ nðx; t; v ; wÞ@ x þ sðx; t; v ; wÞ@ t þ gðx; t; v ; wÞ@ v þ /ðx; t; v ; wÞ@ w ;
ð5Þ
which leaves Eq. (4) invariant. Then, with the aid of differential characteristic set method [21,22], performing the standard Lie group algebra [1,2,23], one finally has the symmetry classification results in Table 1. 3. Reduced equation and approximate solution One of the main purpose for calculating symmetries is to use them for obtaining symmetry reductions and similarity solutions. The goal of this section is to use the symmetries calculated in the previous section to obtain symmetry reductions and approximate solutions whenever it is possible. Hence, first-order approximate solutions of Eq. (1) can be achieved by (2) and similarity solutions of system (4). Case 1. Choosing X ¼ x@ x þ 2t@ t þ ðw þ t v Þ@ w for Aðv Þ ¼ 1; Bðv Þ ¼ 1=v ; Cðv Þ ¼ v , solving the determining equations [1,2,15]
dx dt dv dw ¼ ¼ ; ¼ x 2t w þ tv 0
ð6Þ
2 pffi v . we get the similarity variables y ¼ xt ; U ¼ v ; T ¼ wt t Substituting these variables into Eq. (4), one finally converts it into ordinary differential equations
ð2 þ yÞU y þ 4yU yy ¼ 0; pffiffiffi 4UT y þ 8yUT yy þ 4 yU y UT þ 2yUT y ¼ 0:
ð7Þ
After inserting solutions U; T of (7) into (2), Eq. (1) admits following first-order precision approximate solutions pffiffiffi pffiffi pffiffiffiffi pffiffiffi pffiffi y y u ¼ 1 þ ðx þ tÞ;u ¼ 1 þ t 2e4 þ t p y Erf ; 2 0 1 pffiffi pffiffiffi pffiffiffi Z pffiffiffi pffiffi pffiffiffi pffiffiffiffi pffiffiffi y y y t @ pffiffi 2 1 3 hpffiffii dyA t y ln Erf u ¼ 2 p Erf þ ; 4 t y2 þ y pffiffiffiffi þ yErf 2 2 2 4 e4 p Erf 2y 0 1 1 rffiffiffiffi 0 pffiffiffi pffiffiffi Z pffiffiffi! Z pffiffiffiffi y pffiffiffi t @ 4y pffiffiffiffi pffiffi 1 x 1 y y y A hpffiffii dyA þ @Erf hpffiffii dy 4 ln Erf þ u ¼ 2 pErf e p 4 t ðy 4Þ y þ 2e4 ; 4 4 p 2 2 2 Erf 2y Erf 2y rffiffiffiffi pffiffi pffiffiffi pffiffiffi Z pffiffiffi pffiffiffi!!! pffiffi pffiffiffiffi pffiffiffiffi pffiffiffi y y pffiffiffi pffiffiffiffi y y y y t t 1 3 hpffiffii dy ; Erf p 2e4 þ p y Erf þ þ ln Erf 8e4 4 t þ y2 þ 4 y u ¼ 2 p Erf 4 p 2 2 2 2 4 Erf 2y ð8Þ
where y ¼ x2 =t and
2 Erf½z ¼ pffiffiffiffi
p
Z
z
2
ex dx
ð9Þ
0
is standard error function. In what follows, we skip the intermediate cumbersome calculations and just list immediately the final results. The similarity variables are listed in bracket. 2
2v
3
Case 2. For Aðv Þ ¼ 1; Bðv Þ ¼ e2v ; Cðv Þ ¼ v þ ev , Eq. (4) can be reduced to (y ¼ xt ; U ¼ e t ; T ¼ t 2 ðw tv Þ)
ð2 þ yÞUU y þ 4yUU yy 2yU 2y U 2 ¼ 0; pffiffiffiffi pffiffiffi 3 yT yy þ 2T y þ yU y þ U þ yT y T ¼ 0: 2
ð10Þ
2
2
Case 3. For Aðv Þ ¼ 1; Bðv Þ ¼ 1=v ; Cðv Þ ¼ v 2 , reduced equations are (y ¼ xt ; U ¼ t v 4 ; T ¼ wt )
ð2 þ yÞUU y þ 4y2 UU yy 16yU 2y þ U 2 ¼ 0; 2T y þ 4yT yy
ð11Þ
pffiffiffiffiffiffiffi 4U y pffiffiffiffiffiffi 2yT 2y þ 2 UT þ yT T þ yT y ¼ 0: T 3U 2
pffi v ) Case 4. For Aðv Þ ¼ 1; Bðv Þ ¼ ebv ; Cðv Þ ¼ v , reduced equations are (y ¼ xt ; U ¼ v ; T ¼ wt t
ð2 þ yÞU y þ 4yU yy ¼ 0; pffiffiffi 4T y þ 8yT yy þ 4 yebU U y T þ 2yT y ¼ 0:
ð12Þ
Z.-Y. Zhang et al. / Applied Mathematics and Computation 219 (2012) 1120–1124
1123
Hence, we get first-order of precision approximate solutions for Eq. (1) y pffiffi u ¼ e4 t W; u ¼ 1 þ t; u ¼ x; 0 1
pffiffiffi pffiffiffi pffiffi Z 4yþ2 pffiffipffi b Erf p2ffiy Z pffiy
pffiffiffi pffiffiffiffi pffiffiffiffi y y y W x e W t dyA; e4 e2 p b Erf 2 dy u ¼ 2 pErf þ @2 p t Erf þ pffiffiffi y 2 2 2 2 y
where W ¼ 2 þ e4
ð13Þ
hpffiffii pffiffiffiffiffiffi py Erf 2y and Erf½z is the standard error function defined as (9).
Case 5. For Aðv Þ ¼ 1; Bðv Þ ¼ ebv ; Cðv Þ ¼ ebv , reduced equations are (y ¼ x; U ¼ v t; T ¼ webt )
U yy ¼ 1;
ð14Þ
T yy þ ebU U y bT þ ebU ¼ 0:
The first-order of precision approximate solutions of Eq. (1) can be sought in the form pffiffi pffiffi pffiffi pffiffi 3 pffiffiffiffiffiffiffi pffiffiffi pffiffi pffiffiffi pffiffiffi 1 1 1 1þ bx bx 1 e2 bx þbt 4 e2þ2 bx b2 þ 2 p b 1 Erfi pffiffi2 e2þ bx þbt p2 b þ 1 Erfi pffiffi2 2 x u¼tþ þ ; 3 3 2 4 b2 2 b2 pffiffi pffiffi pffiffi pffiffi pffiffiffi pffiffiffi pffiffiffi 3 pffiffiffiffiffiffiffi pffiffiffi 1 1 1þ bx bx 1 e2 bx þbt 4 e b2 þ 2 p b 1 Erfi pffiffi2 e2þ bx þbt p2 b þ 1 Erfi pffiffi2 2 x u¼tþ þ ; 3 3 2 4 b2 2 b2 pffiffi pffiffi pffiffi pffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 1 1 1 1þ bð1þxÞ 1þ bð1þxÞ pffiffi pffiffi e2 bð1þxÞ þbð2þtÞ p2 b 1 Erfi e2þ bð1þxÞ þbð2þtÞ p2 b þ 1 Erfi 2 2 1 u ¼ t þ x ð2 þ xÞ þ ; 3 3 2 2 b2 2 b2 pffiffi pffiffi pffiffi pffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 1 bx þ1 bx 1 e2 bx þbð1þtÞ p2 b 1 Erfi pffiffi2 e2þ bx þbð1þtÞ p2 b þ 1 Erfi pffiffi2 2 x u¼ 1þtþ þ ; 3 3 2 2 b2 2 b2
ð15Þ where the imaginary error function Erfi½z is given by
2 Erfi½z ¼ pffiffiffiffi i p
Z
iz
2
ex dx:
ð16Þ
0
v
Case 6. For Aðv Þ ¼ ev ; Bðv Þ ¼ e2v ; Cðv Þ ¼ ev , reduced equations are (y ¼ xt ; U ¼ et ; T ¼ tw2 )
UU yy þ yU y U ¼ 0; UT yy þ TU yy þ yT y þ 2U y T y þ UU y þ U 2T ¼ 0:
ð17Þ
From the first equation, we can get special solutions U ¼ c1 y, then the second equation gives T ¼ ðc1 þ c21 Þy þ c1 c2 , so a special first-order of precision approximate solution of Eq. (1) is
u ¼ ln jc1 xj þ ðc1 c2 t 2 þ ðc1 þ c21 ÞxtÞ;
ð18Þ
where c1 ð – 0Þ; c2 are arbitrary constants. Case 7. For Aðv Þ ¼ v ; Bðv Þ ¼ v ; Cðv Þ ¼ v , reduced equations are (y ¼ xt ; U ¼ vt ; T ¼ tw2 )
UU yy þ U 2y þ yU y U ¼ 0; UT yy þ TU yy þ yT y þ 2U y T y þ UU y 2T þ U ¼ 0:
ð19Þ
From Eq. (19), we obtain following special first-order of precision approximate solutions for Eq. (1)
pffiffiffi x pffiffiffi x 6 t2 11 þ 8 2 ; u ¼ 2t 6 þ 2 4 t t pffiffiffi pffiffiffi x x pffiffiffi x t 2 66 þ 48 2 10 þ 6 2 : u ¼ 2t 6 þ 2 4 t t t Case 8. For Aðv Þ; Bðv Þ; Cðv Þ are arbitrary functions, reduced equations are (y ¼ x t; U ¼ v ; T ¼ w)
ð20Þ
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Z.-Y. Zhang et al. / Applied Mathematics and Computation 219 (2012) 1120–1124
U y þ A0 ðUÞU 2y þ AðUÞU yy ¼ 0; V y þ 2V y A0 ðUÞU y þ VA00 ðUÞðU y Þ2 þ VA0 ðUÞ þ AðUÞV yy þ BðUÞU y þ CðUÞ ¼ 0:
ð21Þ
We get special solution U ¼ c1 from the first equation, then Eq. (1) has special first-order of precision approximate solutions xt Aðc Þ
u ¼ c1 ðCðc1 Þðx tÞ þ c2 Aðc1 Þe
1
þ c3 Þ;
ð22Þ
where c1 ; c2 ; c3 are arbitrary constants and Aðc1 Þ – 0 . 4. Conclusion In this paper, we concentrate on approximate symmetry group analysis of the perturbed nonlinear diffusion–convection equation with nonlinear source term by employing the dependent variables expansion in a perturbation series. Some new symmetry reductions and approximate solutions are obtained. Furthermore, because the method uses only standard Lie algorithms and can be implemented in computer algebra system, then this approximate symmetry approach may readily be extended to determine approximate nonclassical symmetries and approximate potential symmetries. It would be interesting to investigate them in detail in our future work. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
P.J. Olver, Application of Lie Groups to Differential Equation, second ed., Springer, New York, 1993. G.W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, New York, 1989. V.A. Baikov, R.K. Gazizov, N.H. Ibragimov, Perturbation methods in group analysis, J. Sov. Math. 55 (1991) 1450–1490. W.I. Fushchich, W.H. Shtelen, On approximate symmetry and approximate solutions of the non-linear wave equation with a small parameter, J. Phys. A: Math. Gen. 22 (1989) 887–890. N. Euler, M.W. Shulga, W.H. Steeb, Approximate symmetries and approximate solutions for a multi-dimensional Landau–Ginzburg equation, J. Phys. A: Math. Gen. 25 (1992) L1095–1103. M. Euler, N. Euler, A. KHohler, On the construction of approximate solutions for a multi-dimensional nonlinear heat equation, J. Phys. A: Math. Gen. 27 (1994) 2083–2092. N. Euler, M. Euler, Symmetry properties of the approximations of multidimensional generalized Van der Pol equations, Nonlinear Math. Phys. 1 (1994) 41–59. M. Pakdemirli, M. Yurusoy, T. Dolapc, Comparison of approximate symmetry methods for differential equations, Acta Appl. Math. 80 (2004) 243–271. R. Wiltshire, Two approaches to the calculation of approximate symmetry exemplified using a system of advection-diffusion equations, J. Comput. Appl. Math. 197 (2006) 287–301. A.F. Kara, F.M. Mahomed, C.Z. Qu, Approximate potential symmetries for partial differential equations, J. Phys. A: Math. Gen. 33 (2000) 6601–6613. S.L. Zhang, P.Z. Wang, C.Z. Qu, Approximate generalized conditional symmetries for the perturbed general KdV–Burgers equation, Chin. Phys. Lett. 23 (2006) 2625–2628. S.L. Zhang, Y. Wang, S.Y. Lou, Approximate generalized conditional symmetries for perturbed evolution equations, Commun. Theor. Phys. 47 (2007) 975–980. F.M. Mahomed, C.Z. Qu, Approximate conditional symmetries for partial differential equation, J. Phys. A: Math. Gen. 33 (2000) 343–356. N.H. Ibragimov, Transformation Groups Applied to Mathematical Physics, Reidel, Boston, MA, 1985. G.W. Bluman, J.D. Cole, The general similarity solution of the heat equation, J. Math. Mech. 18 (1969) 1025–1042. P.A. Clarkson, E.L. Mansfield, Symmetry reductions and exact solutions of a class of nonlinear heat equations, Phys. D. 70 (1994) 250–288. C.Z. Qu, P.G. Estvez, On nonlinear diffusion equations with x-dependent convection and absorption, Nonlinear Anal. 57 (2004) 549–577. C.Z. Qu, L.N. Ji, J.H. Du, Exact solutions and generalized conditional symmetries to (n + 1)-dimensional nonlinear diffusion equations with source term, Phys. Lett. A 343 (2005) 139–147. I. Boglaev, The solution of a semilinear evolutionary convection–diffusion problem by a monotone domain decomposition algorithm, Appl. Math. Comput. 197 (2008) 536–547. Y. Wang, S.L. Zhang, Classification and approximate solutions to perturbed nonlinear diffusion–convection equations, Commun. Theor. Phys. 49 (2008) 17–21. Temuerchaolu, An Algorithmic Theory of Reduction of Differential Polynomial System, Adv. Math. 32 (2003) 208–220. Temuerchaolu, X.S. Gao, Nearly characteristic set of differential polynomial system, Acta. Math. Sin. 45 (2002) 1041–1050. Z.L. Yan, X.Q. Liu, Symmetry and similarity solutions of variable coefficients gen eralized Zakharov–Kuznetsov equation, Appl. Math. Comput. 180 (2006) 288–294.
Contents lists available at SciVerse ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Classification and approximate solutions to perturbed diffusion–convection equations Zhi-Yong Zhang a,c,⇑, Xue-Lin Yong b,c, Yu-Fu Chen c a
College of Sciences, North China University of Technology, Beijing 100144, PR China School of Mathematical Sciences and Physics, North China Electric Power University, Beijing 102206, PR China c School of Mathematical Sciences, Graduate University of Chinese Academy of Sciences, Beijing 100049, PR China b
a r t i c l e
i n f o
Keywords: Approximate symmetry Classification Approximate solution Lie reduction Diffusion–convection equations
a b s t r a c t Approximate symmetries of the perturbed nonlinear diffusion–convection equations are completely classified by the method originated with Fushchich and Shtelen. Moreover, for some interesting cases, symmetry reductions and approximate solutions are discussed in detail. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction It was Lie’s major goal to develop symmetry group theory for the analytical solutions of differential equations. His integration theorems require solvable group of exact symmetries which leave the differential equations invariant exactly [1,2]. However, many differential equations arising in mathematical physics do not admit nontrivial exact Lie symmetries and any small perturbation of differential equations disturbs the symmetry group properties of the unperturbed equation, which make symmetry group method lose its superiority when discussing the symmetry properties of equations involving a small parameter [3,4]. From the last century, perturbation analysis technique has been widely used especially for nonlinear problems. In order to get the utmost goal from the technique, combination of Lie group theory and perturbations are considered, which leads the notion of approximate symmetry introduced and two methods [4,3] for approximate symmetry have been developed. Baikov et al. [3] employ standard perturbation techniques about the symmetry operator to obtain approximate symmetry. In 1989, Fushchich and Shtelen [4] introduced an effective method to search for approximate symmetry and the method was later followed by Euler et al. [5–7]. They firstly expanded the dependent variables in terms of a small parameter (may be a physical parameter or artificially introduced) as the usual perturbation analysis. Terms are then separated at each order of approximation and a system of equations to be solved in a hierarchy is obtained. The system is assumed to be coupled and the approximate symmetry of the original equation is defined to be the exact symmetry of the system of equations obtained from perturbations. In [8,9], these two methods are applied to potential Burgers equation, non-Newtonian creeping flow equations and advection–diffusion equations and shows that the second method is superior to the first one. Later, approximate condition symmetry, approximate generalized condition symmetry and approximate potential symmetry are developed and applied to several celebrated physical partial differential equations [10,12,11,13]. In this paper, we are mainly concerned with the perturbed nonlinear diffusion–convection equations with a source term
ut ¼ ðAðuÞux Þx þ ðBðuÞux þ CðuÞÞ;
⇑ Corresponding author at: College of Sciences, North China University of Technology, Beijing 100144, PR China. E-mail address: [email protected] (Z.-Y. Zhang). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.07.019
ð1Þ
Z.-Y. Zhang et al. / Applied Mathematics and Computation 219 (2012) 1120–1124
1121
where AðuÞ; BðuÞ; CðuÞ are arbitrary and sufficient smooth real-valued functions of their corresponding variables, and AðuÞBðuÞCðuÞ – 0. Eq. (1) without source term was discussed by the Lie’s classical method [1,14], conditional symmetry method [15,16], and conditional Lie-Bäcklund symmetry method (also referred to as the generalized conditional symmetry) [17,18], domain decomposition algorithm [19], etc. Recently, Eq. (1) was studied by Wang and Zhang [20] with the approximate generalized conditional symmetry method and some approximate invariant solutions were obtained as special cases. In this paper, we perform approximate symmetry classification of Eq. (1) and construct some new approximate solutions and reduced equations. It is observed that the calculated approximate symmetries and approximate solutions developed here are new. The outline of the paper is as follows: In Section 2, approximate symmetry classification of nonlinear Eq. (1) is presented. The results are listed in Table 1. In Section 3, reduced equations are constructed and some approximate solutions are calculated. Concluding remarks are discussed in the last section. 2. Approximate symmetry classification In this section, we use the method in [4–6] to obtain complete approximate symmetry classification of Eq. (1) with an accuracy oðÞ. It means that hereinafter in all equalities we neglect the terms of order oðÞ. Expanding the dependent variable to first-order yields
u ¼ v þ w þ oðÞ;
0 < 1;
ð2Þ
then one can expand AðuÞ; BðuÞ; CðuÞ in a series in
0
AðuÞ ¼ Aðv þ wÞ ¼ Aðv Þ þ wA ðv Þ þ oðÞ; BðuÞ ¼ Bðv þ wÞ ¼ Bðv Þ þ wB0 ðv Þ þ oðÞ; CðuÞ ¼ Cðv þ wÞ ¼ Cðv Þ þ wC 0 ðv Þ þ oðÞ:
ð3Þ
Substituting the expansions into Eq. (1) and separating at each order of perturbation parameter, one has the following equations with respect to 0 and 1 :
v t ¼ ðAðv Þv x Þx ; wt ¼ ðwA0 ðv Þv x þ Aðv Þwx Þx þ Bðv Þ v x þ Cðv Þ:
ð4Þ
Approximate symmetries of Eq. (1) correspond to the exact symmetries of system (4). Now, we consider a one-parameter Lie symmetry group of local transformations with an infinitesimal operator of the form
Table 1 Symmetry classifications of system (4). Aðv Þ
Bðv Þ
Cðv Þ
Infinitesimal operators
Arbitrary ev k
Arbitrary ev l
Arbitrary ev q
va
vb
vc
1 1
Arbitrary 2v
v2
1 1 1 1 1
2 þ ln v 1 þ ev
ebv
v þ ev
1 1 1 1
ebv ebv ebv v
ebv
@x ; @t @ x ; @ t ; ðl qÞx@ x þ ð2l 2q kÞt@ t þ @ v þ ð2l q kÞw@ w @ x ; @ t ; ð1 þ b cÞx@ x þ ð2 þ 2b a 2cÞt@ t þ v @ v þð2 þ 2b a cÞw@ w @ x ; @ t ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; @ v ; v @ v þ 2w@ w ; etx ð@ v þ v @ w Þ; @ v þ ð2t xÞv @ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; v @ v þ ½w þ ðt x=2Þv @ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; @ v þ ðt þ x=2Þv @ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; v @ v þ tv @ w ; 2x@ x 4t@ t þ v @ v 2w@ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; v @ v þ tv @ w ; x@ x þ 2t@ t þ ðw þ tv Þ@ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; ðb 1Þx@ x þ 2ðb 1Þt@ t þ @ v þ ½ð2b 1Þw tv @ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; @ v þ bw@ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; bx@ x þ 2bt@ t þ @ v þ ½2bw þ 2tv @ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; @ v þ bðw tv Þ@ w ; x@ x þ 2t@ t þ ðw þ tv Þ@ w ; v @ w ; f ðx; tÞ@ w
v
1 1
v 1
v2 v2
1
1
v þ ev
1
1
v v ln v
where f ðx; tÞ satisfies ft ¼ fxx .
v 1 v 1
Arbitrary
v þ v ln v v þ ev v2 v
v2 v
@ x ; @ t ; @ v xv =2@ w ; v @ v þ ð2w tv Þ@ w ; x@ v þ ðt=2 x2 =4Þv @ w ; x@ x þ 2t@ t þ ðw þ tv Þ@ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; @ v þ ðw t v Þ@ w ; x@ x þ 2t@ t þ ð2w xv =2Þ@ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; @ v þ 2t v @ w ; v @ v þ ð2w xv =2Þ@ w ; x@ x þ 2t@ t þ ð2w xv =2Þ@ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; @ v þ ðw t v þ xv =2Þ@ w ; x@ x þ 2t@ t þ ð2w xv =2Þ@ w ; v @ w ; f ðx; tÞ@ w @ x ; @ t ; @ v þ ðw þ t v Þ@ w ; x@ x þ 2t@ t þ ð2w þ xv =2Þ@ w ; v @ w ; f ðx; tÞ@ w
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X ¼ nðx; t; v ; wÞ@ x þ sðx; t; v ; wÞ@ t þ gðx; t; v ; wÞ@ v þ /ðx; t; v ; wÞ@ w ;
ð5Þ
which leaves Eq. (4) invariant. Then, with the aid of differential characteristic set method [21,22], performing the standard Lie group algebra [1,2,23], one finally has the symmetry classification results in Table 1. 3. Reduced equation and approximate solution One of the main purpose for calculating symmetries is to use them for obtaining symmetry reductions and similarity solutions. The goal of this section is to use the symmetries calculated in the previous section to obtain symmetry reductions and approximate solutions whenever it is possible. Hence, first-order approximate solutions of Eq. (1) can be achieved by (2) and similarity solutions of system (4). Case 1. Choosing X ¼ x@ x þ 2t@ t þ ðw þ t v Þ@ w for Aðv Þ ¼ 1; Bðv Þ ¼ 1=v ; Cðv Þ ¼ v , solving the determining equations [1,2,15]
dx dt dv dw ¼ ¼ ; ¼ x 2t w þ tv 0
ð6Þ
2 pffi v . we get the similarity variables y ¼ xt ; U ¼ v ; T ¼ wt t Substituting these variables into Eq. (4), one finally converts it into ordinary differential equations
ð2 þ yÞU y þ 4yU yy ¼ 0; pffiffiffi 4UT y þ 8yUT yy þ 4 yU y UT þ 2yUT y ¼ 0:
ð7Þ
After inserting solutions U; T of (7) into (2), Eq. (1) admits following first-order precision approximate solutions pffiffiffi pffiffi pffiffiffiffi pffiffiffi pffiffi y y u ¼ 1 þ ðx þ tÞ;u ¼ 1 þ t 2e4 þ t p y Erf ; 2 0 1 pffiffi pffiffiffi pffiffiffi Z pffiffiffi pffiffi pffiffiffi pffiffiffiffi pffiffiffi y y y t @ pffiffi 2 1 3 hpffiffii dyA t y ln Erf u ¼ 2 p Erf þ ; 4 t y2 þ y pffiffiffiffi þ yErf 2 2 2 4 e4 p Erf 2y 0 1 1 rffiffiffiffi 0 pffiffiffi pffiffiffi Z pffiffiffi! Z pffiffiffiffi y pffiffiffi t @ 4y pffiffiffiffi pffiffi 1 x 1 y y y A hpffiffii dyA þ @Erf hpffiffii dy 4 ln Erf þ u ¼ 2 pErf e p 4 t ðy 4Þ y þ 2e4 ; 4 4 p 2 2 2 Erf 2y Erf 2y rffiffiffiffi pffiffi pffiffiffi pffiffiffi Z pffiffiffi pffiffiffi!!! pffiffi pffiffiffiffi pffiffiffiffi pffiffiffi y y pffiffiffi pffiffiffiffi y y y y t t 1 3 hpffiffii dy ; Erf p 2e4 þ p y Erf þ þ ln Erf 8e4 4 t þ y2 þ 4 y u ¼ 2 p Erf 4 p 2 2 2 2 4 Erf 2y ð8Þ
where y ¼ x2 =t and
2 Erf½z ¼ pffiffiffiffi
p
Z
z
2
ex dx
ð9Þ
0
is standard error function. In what follows, we skip the intermediate cumbersome calculations and just list immediately the final results. The similarity variables are listed in bracket. 2
2v
3
Case 2. For Aðv Þ ¼ 1; Bðv Þ ¼ e2v ; Cðv Þ ¼ v þ ev , Eq. (4) can be reduced to (y ¼ xt ; U ¼ e t ; T ¼ t 2 ðw tv Þ)
ð2 þ yÞUU y þ 4yUU yy 2yU 2y U 2 ¼ 0; pffiffiffiffi pffiffiffi 3 yT yy þ 2T y þ yU y þ U þ yT y T ¼ 0: 2
ð10Þ
2
2
Case 3. For Aðv Þ ¼ 1; Bðv Þ ¼ 1=v ; Cðv Þ ¼ v 2 , reduced equations are (y ¼ xt ; U ¼ t v 4 ; T ¼ wt )
ð2 þ yÞUU y þ 4y2 UU yy 16yU 2y þ U 2 ¼ 0; 2T y þ 4yT yy
ð11Þ
pffiffiffiffiffiffiffi 4U y pffiffiffiffiffiffi 2yT 2y þ 2 UT þ yT T þ yT y ¼ 0: T 3U 2
pffi v ) Case 4. For Aðv Þ ¼ 1; Bðv Þ ¼ ebv ; Cðv Þ ¼ v , reduced equations are (y ¼ xt ; U ¼ v ; T ¼ wt t
ð2 þ yÞU y þ 4yU yy ¼ 0; pffiffiffi 4T y þ 8yT yy þ 4 yebU U y T þ 2yT y ¼ 0:
ð12Þ
Z.-Y. Zhang et al. / Applied Mathematics and Computation 219 (2012) 1120–1124
1123
Hence, we get first-order of precision approximate solutions for Eq. (1) y pffiffi u ¼ e4 t W; u ¼ 1 þ t; u ¼ x; 0 1
pffiffiffi pffiffiffi pffiffi Z 4yþ2 pffiffipffi b Erf p2ffiy Z pffiy
pffiffiffi pffiffiffiffi pffiffiffiffi y y y W x e W t dyA; e4 e2 p b Erf 2 dy u ¼ 2 pErf þ @2 p t Erf þ pffiffiffi y 2 2 2 2 y
where W ¼ 2 þ e4
ð13Þ
hpffiffii pffiffiffiffiffiffi py Erf 2y and Erf½z is the standard error function defined as (9).
Case 5. For Aðv Þ ¼ 1; Bðv Þ ¼ ebv ; Cðv Þ ¼ ebv , reduced equations are (y ¼ x; U ¼ v t; T ¼ webt )
U yy ¼ 1;
ð14Þ
T yy þ ebU U y bT þ ebU ¼ 0:
The first-order of precision approximate solutions of Eq. (1) can be sought in the form pffiffi pffiffi pffiffi pffiffi 3 pffiffiffiffiffiffiffi pffiffiffi pffiffi pffiffiffi pffiffiffi 1 1 1 1þ bx bx 1 e2 bx þbt 4 e2þ2 bx b2 þ 2 p b 1 Erfi pffiffi2 e2þ bx þbt p2 b þ 1 Erfi pffiffi2 2 x u¼tþ þ ; 3 3 2 4 b2 2 b2 pffiffi pffiffi pffiffi pffiffi pffiffiffi pffiffiffi pffiffiffi 3 pffiffiffiffiffiffiffi pffiffiffi 1 1 1þ bx bx 1 e2 bx þbt 4 e b2 þ 2 p b 1 Erfi pffiffi2 e2þ bx þbt p2 b þ 1 Erfi pffiffi2 2 x u¼tþ þ ; 3 3 2 4 b2 2 b2 pffiffi pffiffi pffiffi pffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 1 1 1 1þ bð1þxÞ 1þ bð1þxÞ pffiffi pffiffi e2 bð1þxÞ þbð2þtÞ p2 b 1 Erfi e2þ bð1þxÞ þbð2þtÞ p2 b þ 1 Erfi 2 2 1 u ¼ t þ x ð2 þ xÞ þ ; 3 3 2 2 b2 2 b2 pffiffi pffiffi pffiffi pffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 1 1 bx þ1 bx 1 e2 bx þbð1þtÞ p2 b 1 Erfi pffiffi2 e2þ bx þbð1þtÞ p2 b þ 1 Erfi pffiffi2 2 x u¼ 1þtþ þ ; 3 3 2 2 b2 2 b2
ð15Þ where the imaginary error function Erfi½z is given by
2 Erfi½z ¼ pffiffiffiffi i p
Z
iz
2
ex dx:
ð16Þ
0
v
Case 6. For Aðv Þ ¼ ev ; Bðv Þ ¼ e2v ; Cðv Þ ¼ ev , reduced equations are (y ¼ xt ; U ¼ et ; T ¼ tw2 )
UU yy þ yU y U ¼ 0; UT yy þ TU yy þ yT y þ 2U y T y þ UU y þ U 2T ¼ 0:
ð17Þ
From the first equation, we can get special solutions U ¼ c1 y, then the second equation gives T ¼ ðc1 þ c21 Þy þ c1 c2 , so a special first-order of precision approximate solution of Eq. (1) is
u ¼ ln jc1 xj þ ðc1 c2 t 2 þ ðc1 þ c21 ÞxtÞ;
ð18Þ
where c1 ð – 0Þ; c2 are arbitrary constants. Case 7. For Aðv Þ ¼ v ; Bðv Þ ¼ v ; Cðv Þ ¼ v , reduced equations are (y ¼ xt ; U ¼ vt ; T ¼ tw2 )
UU yy þ U 2y þ yU y U ¼ 0; UT yy þ TU yy þ yT y þ 2U y T y þ UU y 2T þ U ¼ 0:
ð19Þ
From Eq. (19), we obtain following special first-order of precision approximate solutions for Eq. (1)
pffiffiffi x pffiffiffi x 6 t2 11 þ 8 2 ; u ¼ 2t 6 þ 2 4 t t pffiffiffi pffiffiffi x x pffiffiffi x t 2 66 þ 48 2 10 þ 6 2 : u ¼ 2t 6 þ 2 4 t t t Case 8. For Aðv Þ; Bðv Þ; Cðv Þ are arbitrary functions, reduced equations are (y ¼ x t; U ¼ v ; T ¼ w)
ð20Þ
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Z.-Y. Zhang et al. / Applied Mathematics and Computation 219 (2012) 1120–1124
U y þ A0 ðUÞU 2y þ AðUÞU yy ¼ 0; V y þ 2V y A0 ðUÞU y þ VA00 ðUÞðU y Þ2 þ VA0 ðUÞ þ AðUÞV yy þ BðUÞU y þ CðUÞ ¼ 0:
ð21Þ
We get special solution U ¼ c1 from the first equation, then Eq. (1) has special first-order of precision approximate solutions xt Aðc Þ
u ¼ c1 ðCðc1 Þðx tÞ þ c2 Aðc1 Þe
1
þ c3 Þ;
ð22Þ
where c1 ; c2 ; c3 are arbitrary constants and Aðc1 Þ – 0 . 4. Conclusion In this paper, we concentrate on approximate symmetry group analysis of the perturbed nonlinear diffusion–convection equation with nonlinear source term by employing the dependent variables expansion in a perturbation series. Some new symmetry reductions and approximate solutions are obtained. Furthermore, because the method uses only standard Lie algorithms and can be implemented in computer algebra system, then this approximate symmetry approach may readily be extended to determine approximate nonclassical symmetries and approximate potential symmetries. It would be interesting to investigate them in detail in our future work. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
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