非線性擴散方程

非線性擴散方程（Nonlinear Diffusion equation）是一個非線性偏微分方程：^[1]

${\displaystyle u_{t}=\alpha *u_{xx}-\beta *u^{3}-\gamma *u^{2}}$

解析解

${\displaystyle u(x,t)=-(1/2)*\gamma /\beta -(1/2)*\gamma *coth(_{C}1-(1/2)*\gamma *x/{\sqrt {(}}-2*\beta *\alpha )-(1/4)*(-2*\gamma *\beta /{\sqrt {(}}-2*\beta *\alpha )+\gamma ^{2})*t/\beta )/\beta }$

${\displaystyle {u(x,t)=-(1/2)*\gamma /\beta +(1/2)*\gamma *coth(_{C}1-(1/2)*\gamma *x/{\sqrt {(}}-2*\beta *\alpha )+(1/4)*(2*\gamma *\beta /{\sqrt {(}}-2*\beta *\alpha )+\gamma ^{2})*t/\beta )/\beta }}$

${\displaystyle {u(x,t)=-(1/2)*\gamma /\beta +(1/2)*\gamma *coth(_{C}1+(1/2)*\gamma *x/{\sqrt {(}}-2*\beta *\alpha )+(1/4)*(-2*\gamma *\beta /{\sqrt {(}}-2*\beta *\alpha )+\gamma ^{2})*t/\beta )/\beta }}$

${\displaystyle {u(x,t)=-(1/2)*\gamma /\beta -(1/2)*\gamma *coth(_{C}1+(1/2)*\gamma *x/{\sqrt {(}}-2*\beta *\alpha )-(1/4)*(2*\gamma *\beta /{\sqrt {(}}-2*\beta *\alpha )+\gamma ^{2})*t/\beta )/\beta }}$

${\displaystyle {u(x,t)=-(1/2)*\gamma /\beta -(1/2)*\gamma *tanh(_{C}1-(1/2)*\gamma *x/{\sqrt {(}}-2*\beta *\alpha )-(1/4)*(-2*\gamma *\beta /{\sqrt {(}}-2*\beta *\alpha )+\gamma ^{2})*t/\beta )/\beta }}$

${\displaystyle {u(x,t)=-(1/2)*\gamma /\beta +(1/2)*\gamma *tanh(_{C}1-(1/2)*\gamma *x/{\sqrt {(}}-2*\beta *\alpha )+(1/4)*(2*\gamma *\beta /{\sqrt {(}}-2*\beta *\alpha )+\gamma ^{2})*t/\beta )/\beta }}$

${\displaystyle {u(x,t)=-(1/2)*\gamma /\beta +(1/2)*\gamma *tanh(_{C}1+(1/2)*\gamma *x/{\sqrt {(}}-2*\beta *\alpha )+(1/4)*(-2*\gamma *\beta /{\sqrt {(}}-2*\beta *\alpha )+\gamma ^{2})*t/\beta )/\beta }}$

${\displaystyle {u(x,t)=-(1/2)*\gamma /\beta -(1/2)*\gamma *tanh(_{C}1+(1/2)*\gamma *x/{\sqrt {(}}-2*\beta *\alpha )-(1/4)*(2*\gamma *\beta /{\sqrt {(}}-2*\beta *\alpha )+\gamma ^{2})*t/\beta )/\beta }}$

${\displaystyle {u(x,t)=-(1/2)*\gamma /\beta +(1/2*I)*\gamma *tan(_{C}1+(1/4)*\gamma *{\sqrt {(}}2)*x/{\sqrt {(}}\beta *\alpha )-(1/4)*(I*\gamma ^{2}+\gamma *{\sqrt {(}}2)*\beta /{\sqrt {(}}\beta *\alpha ))*t/\beta )/\beta }}$

${\displaystyle {u(x,t)=-(1/2)*\gamma /\beta +(1/2*I)*\gamma *cot(_{C}1+(1/4)*\gamma *{\sqrt {(}}2)*x/{\sqrt {(}}\beta *\alpha )+(1/4)*(I*\gamma ^{2}-\gamma *{\sqrt {(}}2)*\beta /{\sqrt {(}}\beta *\alpha ))*t/\beta )/\beta }}$

行波圖

非線性擴散方程行波圖

非線性擴散方程行波圖

非線性擴散方程行波圖

非線性擴散方程行波圖 |

非線性擴散方程行波圖

非線性擴散方程行波圖

非線性擴散方程行波圖

非線性擴散方程行波圖 |

非線性擴散方程行波圖

非線性擴散方程行波圖

非線性擴散方程行波圖

非線性擴散方程行波圖 |

非線性擴散方程行波圖

非線性擴散方程行波圖

參考文獻

1. Graham Griffiths Chapter 5 p67-110

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