阿多米安分解法

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 将非线性偏微分方程写成如下形式： L ( u ) + R ( u ) + N L ( u ) = g ( x , t ) {\displaystyle L(u)+R(u)+NL(u)=g(x,t)} 其中 L、R为线性偏微分算子，NL为非线性项。 将反算子 L − 1 = ∫ 0 t ( ) {\displaystyle L^{-1}=\int _{0}^{t}()} . 用于上式 L − 1 L ( u ) = − L − 1 R ( u ) − L − 1 N L ( u ) + L − 1 g ( x , t ) = {\displaystyle L^{-1}L(u)=-L^{-1}R(u)-L^{-1}NL(u)+L^{-1}g(x,t)=} . 得 u ( x , t ) = u ( x , 0 ) − L − 1 N L ( u ) + L − 1 g ( x , t ) {\displaystyle u(x,t)=u(x,0)-L^{-1}NL(u)+L^{-1}g(x,t)} . 令方程的解u（x,t) 为： u = u 0 + u 1 + u 2 + u 3 + ⋯ {\displaystyle u=u_{0}+u_{1}+u_{2}+u_{3}+\cdots } 非线性项 NL（u）= A 0 + A 1 + A 2 + ⋯ {\displaystyle A_{0}+A_{1}+A_{2}+\cdots } 其中 A n = 1 n ! d n d λ n f ( u ( λ ) ) ∣ λ = 0 , {\displaystyle A_{n}={\frac {1}{n!}}{\frac {\mathrm {d} ^{n}}{\mathrm {d} \lambda ^{n}}}f(u(\lambda ))\mid _{\lambda =0},} d n d λ n u ( λ ) ∣ λ = 0 = n ! u n {\displaystyle {\frac {\mathrm {d} ^{n}}{\mathrm {d} \lambda ^{n}}}u(\lambda )\mid _{\lambda =0}=n!u_{n}} 由此得 u ( x , t ) = u ( x , 0 ) + L − 1 g ( x , t ) {\displaystyle u(x,t)=u(x,0)+L_{-1}g(x,t)} u 1 ( x , t ) = − L − 1 R u 0 − L − 1 A 0 {\displaystyle u_{1}(x,t)=-L^{-1}Ru_{0}-L^{-1}A_{0}} u n ( x , t ) = − L − 1 R u n − 1 − L − 1 A n − 1 {\displaystyle u_{n}(x,t)=-L^{-1}Ru_{n-1}-L^{-1}A_{n-1}} 近似解= u 0 ( x , t ) + u 1 ( x , t ) + u 2 ( x , t ) + u 3 ( x , t ) + ⋯ {\displaystyle u_{0}(x,t)+u_{1}(x,t)+u_{2}(x,t)+u_{3}(x,t)+\cdots } Burgers-Fisher方程： ∂ u ∂ t + u 2 ∗ ∂ u ∂ x − ∂ 2 u ∂ u 2 = u ∗ ( 1 − u 2 ) {\displaystyle {\frac {\partial u}{\partial t}}+u^{2}*{\frac {\partial u}{\partial x}}-{\frac {\partial ^{2}u}{\partial u^{2}}}=u*(1-u^{2})} u [ 0 ] = t a n h ( x ) {\displaystyle u[0]=tanh(x)} u [ 1 ] = − t a n h ( x ) ∗ ( 1 − t a n h ( x ) 2 ) ∗ t {\displaystyle u[1]=-tanh(x)*(1-tanh(x)^{2})*t} u [ 2 ] = − ( 1 / 2 ) ∗ t 2 ∗ t a n h ( x ) ∗ ( − 1 + t a n h ( x ) 2 ) ∗ ( 2 − 4 ∗ t a n h ( x ) 2 ) {\displaystyle u[2]=-(1/2)*t^{2}*tanh(x)*(-1+tanh(x)^{2})*(2-4*tanh(x)^{2})} u [ 3 ] = − ( 1 / 3 ) ∗ t 3 ∗ t a n h ( x ) ∗ ( 3 − 16 ∗ t a n h ( x ) 2 + 26 ∗ t a n h ( x ) 4 − 13 ∗ t a n h ( x ) 6 − 3 ∗ t a n h ( x ) 2 ∗ ( 1 − t a n h ( x ) 2 ) + ( 2 ∗ ( 1 − t a n h ( x ) 2 ) ) ∗ t a n h ( x ) 4 ) {\displaystyle 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})} 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： u t = u 3 u x x x . {\displaystyle u_{t}=u^{3}u_{xxx}.\,} u [ 0 ] = c o s h ( x ) {\displaystyle u[0]=cosh(x)} u [ 1 ] = − c o s h ( x ) ∗ s i n h ( x ) ∗ t {\displaystyle u[1]=-cosh(x)*sinh(x)*t} u [ 5 ] = − t 5 ∗ c o s h ( x ) ∗ s i n h ( x ) 5 − ( 20 / 3 ) ∗ t 5 ∗ c o s h ( x ) 3 ∗ s i n h ( x ) 3 − ( 47 / 15 ) ∗ t 5 ∗ c o s h ( x ) 5 ∗ s i n h ( x ) {\displaystyle 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)} {\displaystyle } 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) George Adomian, Nonlinear Stochastic Systems and Application to Physics,Kluwer Academic Publisher George Adomian,Solving Frontier Problems of Physics,The Decomposition Method,Boston, Kluwer Academic Publisher 1994