非线性扩散方程(Nonlinear Diffusion equation)是一个非线性偏微分方程:[1]
解析解
![{\displaystyle u(x,t)=-(1/2)*\gamma /\beta -(1/2)*\gamma *coth(_{C}1-(1/2)*\gamma *x/{\sqrt {(}}-2*\beta *\alpha )-(1/4)*(-2*\gamma *\beta /{\sqrt {(}}-2*\beta *\alpha )+\gamma ^{2})*t/\beta )/\beta }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b2e7909e649d7218eacf872f9f1995127c9d0531)
![{\displaystyle {u(x,t)=-(1/2)*\gamma /\beta +(1/2)*\gamma *coth(_{C}1-(1/2)*\gamma *x/{\sqrt {(}}-2*\beta *\alpha )+(1/4)*(2*\gamma *\beta /{\sqrt {(}}-2*\beta *\alpha )+\gamma ^{2})*t/\beta )/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8089c7dd9424d73f720b253f20ab3b37793d61e9)
![{\displaystyle {u(x,t)=-(1/2)*\gamma /\beta +(1/2)*\gamma *coth(_{C}1+(1/2)*\gamma *x/{\sqrt {(}}-2*\beta *\alpha )+(1/4)*(-2*\gamma *\beta /{\sqrt {(}}-2*\beta *\alpha )+\gamma ^{2})*t/\beta )/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bac4f90dce18207c08454ff51ae33badf98f209d)
![{\displaystyle {u(x,t)=-(1/2)*\gamma /\beta -(1/2)*\gamma *coth(_{C}1+(1/2)*\gamma *x/{\sqrt {(}}-2*\beta *\alpha )-(1/4)*(2*\gamma *\beta /{\sqrt {(}}-2*\beta *\alpha )+\gamma ^{2})*t/\beta )/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c7ab642d0b17539ca6c6467ab574086e4bcdcd4)
![{\displaystyle {u(x,t)=-(1/2)*\gamma /\beta -(1/2)*\gamma *tanh(_{C}1-(1/2)*\gamma *x/{\sqrt {(}}-2*\beta *\alpha )-(1/4)*(-2*\gamma *\beta /{\sqrt {(}}-2*\beta *\alpha )+\gamma ^{2})*t/\beta )/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f5527119c023bcc1d0e3cc411e9c871f600c99a)
![{\displaystyle {u(x,t)=-(1/2)*\gamma /\beta +(1/2)*\gamma *tanh(_{C}1-(1/2)*\gamma *x/{\sqrt {(}}-2*\beta *\alpha )+(1/4)*(2*\gamma *\beta /{\sqrt {(}}-2*\beta *\alpha )+\gamma ^{2})*t/\beta )/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/481b186dd1f5c165cdbca7f88d9be10aa54606e3)
![{\displaystyle {u(x,t)=-(1/2)*\gamma /\beta +(1/2)*\gamma *tanh(_{C}1+(1/2)*\gamma *x/{\sqrt {(}}-2*\beta *\alpha )+(1/4)*(-2*\gamma *\beta /{\sqrt {(}}-2*\beta *\alpha )+\gamma ^{2})*t/\beta )/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e223001f622f128b41d224634258b4b5d097351c)
![{\displaystyle {u(x,t)=-(1/2)*\gamma /\beta -(1/2)*\gamma *tanh(_{C}1+(1/2)*\gamma *x/{\sqrt {(}}-2*\beta *\alpha )-(1/4)*(2*\gamma *\beta /{\sqrt {(}}-2*\beta *\alpha )+\gamma ^{2})*t/\beta )/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c46f7fc4f2ef36badede112673e18e2d59742fd0)
![{\displaystyle {u(x,t)=-(1/2)*\gamma /\beta +(1/2*I)*\gamma *tan(_{C}1+(1/4)*\gamma *{\sqrt {(}}2)*x/{\sqrt {(}}\beta *\alpha )-(1/4)*(I*\gamma ^{2}+\gamma *{\sqrt {(}}2)*\beta /{\sqrt {(}}\beta *\alpha ))*t/\beta )/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4cfffe4a6eab5a3ad8056034ede65f27326123)
![{\displaystyle {u(x,t)=-(1/2)*\gamma /\beta +(1/2*I)*\gamma *cot(_{C}1+(1/4)*\gamma *{\sqrt {(}}2)*x/{\sqrt {(}}\beta *\alpha )+(1/4)*(I*\gamma ^{2}-\gamma *{\sqrt {(}}2)*\beta /{\sqrt {(}}\beta *\alpha ))*t/\beta )/\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3e6a716237d9391c1151efea1e3e4de9a31de48)
行波图
非线性扩散方程行波图
|
非线性扩散方程行波图
|
非线性扩散方程行波图
|
非线性扩散方程行波图|
|
非线性扩散方程行波图
|
非线性扩散方程行波图
|
非线性扩散方程行波图
|
非线性扩散方程行波图|
|
非线性扩散方程行波图
|
非线性扩散方程行波图
|
非线性扩散方程行波图
|
非线性扩散方程行波图|
|
非线性扩散方程行波图
|
非线性扩散方程行波图
|
参考文献
- ^ Graham Griffiths Chapter 5 p67-110
- *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