查看“︁非线性扩散方程”︁的源代码

'''非线性扩散方程'''（Nonlinear Diffusion equation）是一个非线性偏微分方程：<ref name=g>Graham Griffiths Chapter 5 p67-110</ref>

<math> u_{t}=\alpha*u_{xx}-\beta*u^3-\gamma*u^2   </math>

==解析解==
:<math> u(x, t) = -(1/2)*\gamma/\beta-(1/2)*\gamma*coth(_C1-(1/2)*\gamma*x/\sqrt(-2*\beta*\alpha)-(1/4)*(-2*\gamma*\beta/\sqrt(-2*\beta*\alpha)+\gamma^2)*t/\beta)/\beta  </math>
:<math> {u(x, t) = -(1/2)*\gamma/\beta+(1/2)*\gamma*coth(_C1-(1/2)*\gamma*x/\sqrt(-2*\beta*\alpha)+(1/4)*(2*\gamma*\beta/\sqrt(-2*\beta*\alpha)+\gamma^2)*t/\beta)/\beta}  </math>
:<math>  {u(x, t) = -(1/2)*\gamma/\beta+(1/2)*\gamma*coth(_C1+(1/2)*\gamma*x/\sqrt(-2*\beta*\alpha)+(1/4)*(-2*\gamma*\beta/\sqrt(-2*\beta*\alpha)+\gamma^2)*t/\beta)/\beta} </math>
:<math>{u(x, t) = -(1/2)*\gamma/\beta-(1/2)*\gamma*coth(_C1+(1/2)*\gamma*x/\sqrt(-2*\beta*\alpha)-(1/4)*(2*\gamma*\beta/\sqrt(-2*\beta*\alpha)+\gamma^2)*t/\beta)/\beta}   </math>
:<math>  {u(x, t) = -(1/2)*\gamma/\beta-(1/2)*\gamma*tanh(_C1-(1/2)*\gamma*x/\sqrt(-2*\beta*\alpha)-(1/4)*(-2*\gamma*\beta/\sqrt(-2*\beta*\alpha)+\gamma^2)*t/\beta)/\beta} </math>
:<math> {u(x, t) = -(1/2)*\gamma/\beta+(1/2)*\gamma*tanh(_C1-(1/2)*\gamma*x/\sqrt(-2*\beta*\alpha)+(1/4)*(2*\gamma*\beta/\sqrt(-2*\beta*\alpha)+\gamma^2)*t/\beta)/\beta}  </math>
:<math>  {u(x, t) = -(1/2)*\gamma/\beta+(1/2)*\gamma*tanh(_C1+(1/2)*\gamma*x/\sqrt(-2*\beta*\alpha)+(1/4)*(-2*\gamma*\beta/\sqrt(-2*\beta*\alpha)+\gamma^2)*t/\beta)/\beta} </math>
:<math> {u(x, t) = -(1/2)*\gamma/\beta-(1/2)*\gamma*tanh(_C1+(1/2)*\gamma*x/\sqrt(-2*\beta*\alpha)-(1/4)*(2*\gamma*\beta/\sqrt(-2*\beta*\alpha)+\gamma^2)*t/\beta)/\beta}  </math>
:<math> {u(x, t) = -(1/2)*\gamma/\beta+(1/2*I)*\gamma*tan(_C1+(1/4)*\gamma*\sqrt(2)*x/\sqrt(\beta*\alpha)-(1/4)*(I*\gamma^2+\gamma*\sqrt(2)*\beta/\sqrt(\beta*\alpha))*t/\beta)/\beta}  </math>
:<math> {u(x, t) = -(1/2)*\gamma/\beta+(1/2*I)*\gamma*cot(_C1+(1/4)*\gamma*\sqrt(2)*x/\sqrt(\beta*\alpha)+(1/4)*(I*\gamma^2-\gamma*\sqrt(2)*\beta/\sqrt(\beta*\alpha))*t/\beta)/\beta}  </math>

==行波图==
{|
|[[File:Nonlinear Diffusion equation traveling wave plot 1.gif|thumb|非线性扩散方程行波图]]
|[[File:Nonlinear Diffusion equation traveling wave plot 2.gif|thumb|非线性扩散方程行波图]]
|[[File:Nonlinear Diffusion equation traveling wave plot 3.gif|thumb|非线性扩散方程行波图]]
|[[File:Nonlinear Diffusion equation traveling wave plot 4.gif|thumb|非线性扩散方程行波图]]|
|}
{|
|[[File:Nonlinear Diffusion equation traveling wave plot 5.gif|thumb|非线性扩散方程行波图]]
|[[File:Nonlinear Diffusion equation traveling wave plot 6.gif|thumb|非线性扩散方程行波图]]
|[[File:Nonlinear Diffusion equation traveling wave plot 7.gif|thumb|非线性扩散方程行波图]]
|[[File:Nonlinear Diffusion equation traveling wave plot 8.gif|thumb|非线性扩散方程行波图]]|
|}
{|
|[[File:Nonlinear Diffusion equation traveling wave plot 9.gif|thumb|非线性扩散方程行波图]]
|[[File:Nonlinear Diffusion equation traveling wave plot 10.gif|thumb|非线性扩散方程行波图]]
|[[File:Nonlinear Diffusion equation traveling wave plot 11.gif|thumb|非线性扩散方程行波图]]
|[[File:Nonlinear Diffusion equation traveling wave plot 12.gif|thumb|非线性扩散方程行波图]]|
|}
{|
|[[File:Nonlinear Diffusion equation traveling wave plot 13.gif|thumb|非线性扩散方程行波图]]
|[[File:Nonlinear Diffusion equation traveling wave plot 14.gif|thumb|非线性扩散方程行波图]]
|}

==参考文献==
<references/>
# *谷超豪 《[[孤立子]]理论中的[[达布变换]]及其几何应用》 上海科学技术出版社
# *阎振亚著 《复杂非线性波的构造性理论及其应用》 科学出版社 2007年
# 李志斌编著 《非线性数学物理方程的行波解》 科学出版社
#王东明著 《消去法及其应用》 科学出版社 2002
# *何青 王丽芬编著 《[[Maple]] 教程》 科学出版社 2010 ISBN 9787030177445
#Graham W. Griffiths William E.Shiesser Traveling Wave Analysis of Partial Differential p135 Equations Academy Press
# Richard H. Enns George C. McCGuire, Nonlinear Physics Birkhauser,1997
#Inna Shingareva, Carlos Lizárraga-Celaya,Solving Nonlinear Partial Differential Equations with Maple Springer.
#Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
#Saber Elaydi,An Introduction to Difference Equationns, Springer 2000
#Dongming Wang, Elimination Practice,Imperial College Press 2004
# David Betounes, Partial Differential Equations for Computational Science: With Maple and Vector Analysis Springer, 1998 ISBN 9780387983004
# George Articolo Partial Differential Equations & Boundary Value Problems with Maple V Academic Press 1998 ISBN 9780120644759

{{非线性偏微分方程}}
[[category:非线性偏微分方程]]