非线性扩散方程

非线性扩散方程（Nonlinear Diffusion equation）是一个非线性偏微分方程：

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

解析解

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

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

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

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

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

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

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

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

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

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

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

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3. 李志斌编著 《非线性数学物理方程的行波解》 科学出版社
4. 王东明著 《消去法及其应用》 科学出版社 2002
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9. Eryk Infeld and George Rowlands,Nonlinear Waves,Solitons and Chaos,Cambridge 2000
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