屈改珠,朱春蓉

(1.渭南師范學(xué)院數(shù)學(xué)系,陜西渭南 714000;2.安徽師范大學(xué)數(shù)學(xué)系,安徽蕪湖 214000)

(3+1)維帶有源項(xiàng)的反應(yīng)擴(kuò)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(xué)院數(shù)學(xué)系,陜西渭南 714000;2.安徽師范大學(xué)數(shù)學(xué)系,安徽蕪湖 214000)

反應(yīng)擴(kuò)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(duì)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(duì)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(dǎo)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(xiàng)的反應(yīng)擴(kuò)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(dāng)u∈E0時(shí),方程有形如

3 (N+1)維反應(yīng)擴(kuò)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.

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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(xué)院科研計(jì)劃項(xiàng)目(09YKZ003).

屈改珠(1978-),助教,研究方向：偏微分方程.

2000MSC:35A25