查看“︁阿多米安分解法”︁的源代码

'''阿多米安分解法'''（Adomian decomposition method，简称：ADM法），是1989年美国籍阿马尼亚数学家George Adomian创建的近似分解法，用以求解非线性偏微分方程<ref>George Adomian, Nonlinear Stochastic Systems and Application to Physics,Kluwer Academic Publisher</ref><ref>George Adomian,Solving Frontier Problems of Physics,The Decomposition Method,Boston, Kluwer Academic Publisher 1994</ref>

将非线性偏微分方程写成如下形式：

<math>L(u)+R(u)+NL(u)=g(x,t)</math>  

其中 L、R为线性偏微分算子，NL为非线性项。 将反算子<math>L^{-1}=\int_{0}^{t}() </math>. 用于上式

<math>L^{-1}L(u)=-L^{-1}R(u)-L^{-1}NL(u)+L^{-1}g(x,t)=</math>.

得

<math>u(x,t)=u(x,0)-L^{-1}NL(u)+L^{-1}g(x,t)</math>.

令方程的解u（x,t) 为：

:<math> u = u_{0} + u_{1} + u_{2} + u_{3} + \cdots</math>

非线性项

NL（u）=<math>A_{0} +A_{1} +  A_{2} +\cdots</math>

其中

:<math>
A_{n} = \frac{1}{n!} \frac{\mathrm{d}^{n}}{\mathrm{d}\lambda^{n}} f(u(\lambda))\mid_{\lambda=0},
</math>

<math>\frac{\mathrm{d}^{n}}{\mathrm{d}\lambda^{n}} u(\lambda)\mid_{\lambda=0} = n! u_{n}</math>

由此得

:<math>u(x,t)=u(x,0)+L_{-1}g(x,t)</math>
:<math>u_{1}(x,t)=-L^{-1}Ru_{0}-L^{-1}A_{0}</math>

:<math>u_{n}(x,t)=-L^{-1}Ru_{n-1}-L^{-1}A_{n-1}</math>


近似解=<math>u_{0}(x,t)+u_{1}(x,t)+u_{2}(x,t)+u_{3}(x,t)+\cdots</math>

==Burgers-Fisher 方程 ADM解==
[[File:Burgers Fisher equation tanh Adomian plot.gif|thumb|right|300px]]
Burgers-Fisher方程：

:<math> \frac{\partial u}{\partial t}+u^2*\frac{\partial u}{\partial x}-\frac{\partial^2 u}{\partial u^2}=u*(1-u^2) </math>

:<math> u[0] = tanh(x)</math>
:<math>u[1] = -tanh(x)*(1-tanh(x)^2)*t</math>
:<math>u[2] = -(1/2)*t^2*tanh(x)*(-1+tanh(x)^2)*(2-4*tanh(x)^2)</math>
:<math>u[3] = -(1/3)*t^3*tanh(x)*(3-16*tanh(x)^2+26*tanh(x)^4-13*tanh(x)^6-3*tanh(x)^2*(1-tanh(x)^2)+(2*(1-tanh(x)^2))*tanh(x)^4)</math>
近似解：

pa := (-1.*tanh(x)-82360.*tanh(x)^13+73.*tanh(x)^3-1195.*tanh(x)^5+8233.*tanh(x)^7-29990.*tanh(x)^9+63510.*tanh(x)^15-26980.*tanh(x)^17+4862.*tanh(x)^19+63850.*tanh(x)^11)*t^9+(14650.*tanh(x)^13-16170.*tanh(x)^11+tanh(x)+1430.*tanh(x)^17+688.8*tanh(x)^5+10230.*tanh(x)^9-7102.*tanh(x)^15-54.67*tanh(x)^3-3672.*tanh(x)^7)*t^8+(-373.8*tanh(x)^5+1491.*tanh(x)^7-1.*tanh(x)+39.67*tanh(x)^3+3333.*tanh(x)^11+429.*tanh(x)^15-3036.*tanh(x)^9-1881.*tanh(x)^13)*t^7+(132.*tanh(x)^13+187.8*tanh(x)^5-502.*tanh(x)^11+743.5*tanh(x)^9-27.67*tanh(x)^3+tanh(x)-534.6*tanh(x)^7)*t^6+(-135.3*tanh(x)^9+161.1*tanh(x)^7-1.*tanh(x)+42.*tanh(x)^11-85.13*tanh(x)^5+18.33*tanh(x)^3)*t^5+(-37.*tanh(x)^7+33.33*tanh(x)^5+14.*tanh(x)^9-11.33*tanh(x)^3+tanh(x))*t^4+(5.*tanh(x)^7-10.33*tanh(x)^5+6.333*tanh(x)^3-1.*tanh(x))*t^3+(-3.*tanh(x)^3+tanh(x)+2.*tanh(x)^5)*t^2+(-1.*tanh(x)+tanh(x)^3)*t+tanh(x)

==迪姆方程ADM解==
[[File:Dym equation Admain plot cosh.gif|thumb|right|300px]]
迪姆方程：

:<math>u_t = u^3u_{xxx}.\,</math>

:<math>u[0] = cosh(x)</math>
:<math>u[1] = -cosh(x)*sinh(x)*t</math>
:<math>u[5] = -t^5*cosh(x)*sinh(x)^5-(20/3)*t^5*cosh(x)^3*sinh(x)^3-(47/15)*t^5*cosh(x)^5*sinh(x)</math>
:<math></math>

ADM近似：

u（x,t)~pa := (-.5382*sinh(10.*x)-.7224*sinh(8.*x)-.2441*sinh(6.*x)-0.5787e-4*sinh(2.*x)-0.1693e-1*sinh(4.*x))*t^9+(.4634*cosh(9.*x)+0.5933e-2*cosh(3.*x)+.5585*cosh(7.*x)+.1514*cosh(5.*x)+0.1356e-5*cosh(x))*t^8+(-.4063*sinh(8.*x)-0.8889e-1*sinh(4.*x)-.4339*sinh(6.*x)-0.1389e-2*sinh(2.*x))*t^7+(0.1085e-3*cosh(x)+0.4746e-1*cosh(3.*x)+.3647*cosh(7.*x)+.3391*cosh(5.*x))*t^6+(-0.2083e-1*sinh(2.*x)-.2667*sinh(4.*x)-.3375*sinh(6.*x))*t^5+(.3255*cosh(5.*x)+0.5208e-2*cosh(x)+.2109*cosh(3.*x))*t^4+(-.3333*sinh(4.*x)-.1667*sinh(2.*x))*t^3+(.3750*cosh(3.*x)+.1250*cosh(x))*t^2-.5000*t*sinh(2.*x)+cosh(x)



==参考文献==
<references/>

{{非线性偏微分方程理论与解法}}

[[category:非线性偏微分方程]]