阿多米安分解法 （Adomian decomposition method，簡稱：ADM法），是1989年美國籍阿馬尼亞數學家George Adomian創建的近似分解法，用以求解非線性偏微分方程[ 1] [ 2]
將非線性偏微分方程寫成如下形式：
L
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u
)
+
R
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u
)
+
N
L
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u
)
=
g
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x
,
t
)
{\displaystyle L(u)+R(u)+NL(u)=g(x,t)}
其中 L、R為線性偏微分算子，NL為非線性項。 將反算子
L
−
1
=
∫
0
t
(
)
{\displaystyle L^{-1}=\int _{0}^{t}()}
L
−
1
L
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u
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=
−
L
−
1
R
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u
)
−
L
−
1
N
L
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u
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+
L
−
1
g
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x
,
t
)
=
{\displaystyle L^{-1}L(u)=-L^{-1}R(u)-L^{-1}NL(u)+L^{-1}g(x,t)=}
得
u
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x
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t
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=
u
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x
,
0
)
−
L
−
1
N
L
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u
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+
L
−
1
g
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x
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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) 為：
u
=
u
0
+
u
1
+
u
2
+
u
3
+
⋯
{\displaystyle u=u_{0}+u_{1}+u_{2}+u_{3}+\cdots }
非線性項
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
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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
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x
,
t
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=
u
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x
,
0
)
+
L
−
1
g
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x
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{\displaystyle u(x,t)=u(x,0)+L_{-1}g(x,t)}
u
1
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x
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t
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=
−
L
−
1
R
u
0
−
L
−
1
A
0
{\displaystyle u_{1}(x,t)=-L^{-1}Ru_{0}-L^{-1}A_{0}}
u
n
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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}}
u
0
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x
,
t
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+
u
1
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x
,
t
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+
u
2
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x
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t
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+
u
3
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x
,
t
)
+
⋯
{\displaystyle u_{0}(x,t)+u_{1}(x,t)+u_{2}(x,t)+u_{3}(x,t)+\cdots }
[ 編輯 ] Burgers-Fisher方程：
∂
u
∂
t
+
u
2
∗
∂
u
∂
x
−
∂
2
u
∂
u
2
=
u
∗
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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
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x
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{\displaystyle u[0]=tanh(x)}
u
[
1
]
=
−
t
a
n
h
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x
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∗
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1
−
t
a
n
h
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x
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2
)
∗
t
{\displaystyle u[1]=-tanh(x)*(1-tanh(x)^{2})*t}
u
[
2
]
=
−
(
1
/
2
)
∗
t
2
∗
t
a
n
h
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x
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∗
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−
1
+
t
a
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h
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2
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∗
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2
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4
∗
t
a
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h
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x
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2
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{\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
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x
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∗
(
3
−
16
∗
t
a
n
h
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2
+
26
∗
t
a
n
h
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x
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4
−
13
∗
t
a
n
h
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x
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6
−
3
∗
t
a
n
h
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x
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2
∗
(
1
−
t
a
n
h
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x
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2
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+
(
2
∗
(
1
−
t
a
n
h
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x
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2
)
)
∗
t
a
n
h
(
x
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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
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=
c
o
s
h
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x
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{\displaystyle u[0]=cosh(x)}
u
[
1
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=
−
c
o
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h
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x
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∗
s
i
n
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∗
t
{\displaystyle u[1]=-cosh(x)*sinh(x)*t}
u
[
5
]
=
−
t
5
∗
c
o
s
h
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x
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∗
s
i
n
h
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5
−
(
20
/
3
)
∗
t
5
∗
c
o
s
h
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x
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3
∗
s
i
n
h
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x
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3
−
(
47
/
15
)
∗
t
5
∗
c
o
s
h
(
x
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5
∗
s
i
n
h
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x
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{\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
![{\displaystyle u[0]=tanh(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e65afe4c2de16c21c54711dba96a5992558962c)
![{\displaystyle u[1]=-tanh(x)*(1-tanh(x)^{2})*t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1258f6d4ca49f5801cfcbc353585746b1558535)
![{\displaystyle u[2]=-(1/2)*t^{2}*tanh(x)*(-1+tanh(x)^{2})*(2-4*tanh(x)^{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4417542ae62d536e2c6857b40205c921ee8e8d1)
![{\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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10012bbedf7f1336fb8cc598f0b8b8be6902f22f)
![{\displaystyle u[0]=cosh(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/973b60d71973c0406fbf327716a37894813b608c)
![{\displaystyle u[1]=-cosh(x)*sinh(x)*t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/228bacb6a3fa31806d5bd250cffa868d09913a63)
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e752c5864ab9f333ebbd54e05c165522424c4074)
