屈改珠,朱春蓉

(1.渭南師范學院數(shù)學系,陜西渭南 714000;2.安徽師范大學數(shù)學系,安徽蕪湖 214000)

(3+1)維帶有源項的反應擴散方程的不變集和精確解

屈改珠1,朱春蓉2

(1.渭南師范學院數(shù)學系,陜西渭南 714000;2.安徽師范大學數(shù)學系,安徽蕪湖 214000)

反應擴散方程；不變集；精確解

1 引言

到目前為止,求解非線性偏微分方程的方法大致包括:廣義條件對稱群方法[1-3],齊次平衡方法[4-5],分離變量法[6-7],不變集方法[8-14]等等.在文[8-9]中,提出了對伸縮群的推廣,其中的函數(shù)不變集S0={u:ux=(1/x)F(u)},這樣的推廣可以用來解決方程ut= E(x,u,ux,uxx,…,u(k)),其中u(k)是u關(guān)于x的k階導數(shù).文[10-11]推廣了更一般形式的不變集

2 (3+1)維帶有源項的反應擴散方程

其中w是x,y,z的光滑函數(shù),F是由不變條件u(x,y,z,0)∈E0=?u(x,y,z,t)∈E0,t∈(0,1]所確定的函數(shù).當u∈E0時,方程有形如

3 (N+1)維反應擴散方程

同樣地,引入不變集E0={u:uxi=wxif(t)F(u),i=1,2,…,N},其中w是x1,x2,…,xN的光滑函數(shù),f(t)是t的光滑函數(shù),F(u)是u的光滑函數(shù),且由下列不變條件u(x1,x2,…,xN,0) ∈E0=?u(x1,x2,…,xN,t)∈E0t∈(0,1]所確定.

參考文獻

[1]Qu C Z.Group classification and generalized conditional symmetry reduction of the nonlinear diffusionconvection equation with a nonlinear source[J].Stud.Appl.Math.,1997,99(2):107-136.

[2]鄭群珍,姬麗娜.四階非線性發(fā)展方程的精確解和廣義條件對稱[J].純粹數(shù)學與應用數(shù)學,2007,23(3):402-406.

[3]劉秀玲,李金花,胡斌.(2+1)維淺水波方程的新精確解[J].純粹數(shù)學與應用數(shù)學,2008,24(4):666-669.

[4]Zhang J F,Han P.New multisoliton solutions of the(2+1)-dimensional despersive long wave equations[J]. Commun.Non.Sci.Numer.Simul.,2001,6(3):178-182.

[5]Wang M L,Zhou Y B and Li Z B.Applications of homogeneous balance method to exact solutions of nonlinear equations in mathematical physics[J].Phys.Lett.A,1996,216(1):67-75.

[6]Zhang S L,Lou S Y,Qu C Z.Variable separation and exact solutions to generalized nonlinear diffusion equations[J].Chin.Phys.Lett.,2002,19(12):1741-1744.

[7]Zhang S L,Lou S Y,Qu C Z.New variable separation approach:application to nonlinear diffusion equations[J]. J.Phys.A:Math.Gen.,2003,36(49):12223-12242.

[8]Galationov V A.Groups of scaling and invariant sets for higher-order nonlinear evolution equations[J].Diff. Integer Equation.2001,14(8):913-924.

[9]Galationov V A.Ordered invariant sets for nonlinear equations of KdV-type[J].Compute.Math.phys., 1999,39(9):1564-1570.

[10]Qu C Z,Estevez P G.Extended rotation and scaling groups for nonlinear evolution equations[J].Nonlinear Anal.,2003,52(6):1655-1673.

[11]Qu C Z.Symmetries and solutions to the thin film equations[J].J.Math.Anal.Appl.,2006,317(2):381-397.

[12]Zhu C R,Qu C Z.Invariant sets and solutions to higher-dimensional reaction-diffusion equations with source term[J].Phys.Lett.A,2006,354:437-444.

[13]Qu C Z,Zhu C R.Invariant sets and solutions to the generalized thin film equations[J].Sci.China A, 2007,37(4):447-458.

[14]Qu G Z,Zhang S L and Zhu C R.Invariant Sets and Exact Solutions to Higher-dimensional Wave Equations[J].Commun.Theor.Phys.,2008,49(5):1119-1124.

Invariant sets and exact solutions to(3+1)-dimensional reaction-diffusion equations with source term

QU Gai-zhu1,ZHU Chun-rong2
(1.Department of Mathematics,Weinan Teachers University,Weinan714000,China; 2.Department of Mathematics,Anhui Normal University,Wuhu241000,China)

reaction-diffusion equation,invariant sets,exact solutions

O175

A

1008-5513(2009)03-0579-07

2008-12-29.

渭南師范學院科研計劃項目(09YKZ003).

屈改珠(1978-),助教,研究方向：偏微分方程.

2000MSC:35A25