解形如$u_t=g(u_{xx},u_x,u,x,t)$的方程，方程的形式可变为$u(x,t)=u(x,t=0)+\int_0^tg(u_{xx},u_x,u,x,t)\mathrm dt=u(x,t=0)+N(u)$，可利用DJM求解。
例1 一维空间线性抛物型方程
\begin{equation}
\begin{split}
u_t=&x^2tu_{xx} \\
u(x,0)=&x^2
\end{split}
\label{ex1prob}
\end{equation}
方程可化为：
\begin{equation}
u(x,t)=u(x,0)+\int_0^t x^2tu_{xx} \mathrm dt=x^2+N(u)
\label{ex1solform}
\end{equation}
迭代：
\begin{equation}
\begin{split}
u_0=& u(x,0)=x^2 \\
u_1=& \int_0^t x^2tu_{0,xx} \mathrm dt = x^2 t^2 \\
u_2=& \int_0^t x^2t(u_{0,xx}+u_{1,xx})\mathrm dt - \int_0^t x^2tu_{0,xx} \mathrm dt = \frac{x^2 t^4}{2} \\
u_3=& \int_0^t x^2t(u_{0,xx}+u_{1,xx}+u_{2,xx})\mathrm dt - \int_0^t x^2t(u_{0,xx}+u_{1,xx})\mathrm dt = \frac{x^2 t^6}{6} \\
\vdots =& \vdots \\
u_{m+1}=&\int_0^t x^2t(u_{0,xx}+u_{1,xx}+u_{2,xx}+\cdots+u_{m,xx})\mathrm dt -\\
& \int_0^t x^2t(u_{0,xx}+u_{1,xx}+\cdots+u_{m-1,xx})\mathrm dt = \frac{x^2 t^{2m}}{m!}
\end{split}
\label{um}
\end{equation}
Mathematica 代码：
| 1 | | 2 | | 3 | | 4 | | 5 | | 6 | | 7 | | 8 | | 9 | | 10 | | 11 | | 12 | | 13 | | 14 | | 15 | | 16 | | 17 | | 18 | | 19 |
| | In[1]:= u[0] = x^2 | | <p>Out[1]= x^2</p> | | <p>In[2]:= u[1] = Integrate[x^2*D[u[], {x, 2}]*t, {t, 0, t}]</p> | | <p>Out[2]= t^2 x^2</p> | | <p>In[3]:= m = 1;<br /> | | u[m + 1] =<br /> | | Integrate[Sum[x^2<em>D[u[i], {x, 2}]</em>t, {i, 0, m}], {t, 0, t}] -<br /> | | Integrate[Sum[x^2<em>D[u[i], {x, 2}]</em>t, {i, 0, m - 1}], {t, 0, t}]</p> | | <p>Out[4]= (t^4 x^2)/2</p> | | <p>In[5]:= m = 2;<br /> | | u[m + 1] =<br /> | | Integrate[Sum[x^2<em>D[u[i], {x, 2}]</em>t, {i, 0, m}], {t, 0, t}] -<br /> | | Integrate[Sum[x^2<em>D[u[i], {x, 2}]</em>t, {i, 0, m - 1}], {t, 0, t}]</p> | | <p>Out[6]= (t^6 x^2)/6</p> | | <p>In[7]:= m = 3;<br /> | | u[m + 1] =<br /> | | Integrate[Sum[x^2<em>D[u[i], {x, 2}]</em>t, {i, 0, m}], {t, 0, t}] -<br /> | | Integrate[Sum[x^2<em>D[u[i], {x, 2}]</em>t, {i, 0, m - 1}], {t, 0, t}]</p> | | <p>Out[8]= (t^8 x^2)/24</p> |
|
方程的解：
\begin{equation}
u(x,t)=u_0+u_1+u_2+\cdots=x^2+x^2t^2+\frac{x^2 t^4}{2}+\frac{x^2 t^6}{6}+\cdots=x^2e^{t^2}
\label{ex1sol}
\end{equation}
例2 二维空间线性抛物型方程
方程为：
\begin{equation}
\begin{split}
u_t=&y^2tu_{xx}-x^2tu_{yy} \\
u(x,y,0)=&-y^2
\end{split}
\label{ex2prob}
\end{equation}
Mathematica 代码：
| 1 | | 2 | | 3 | | 4 | | 5 | | 6 | | 7 | | 8 | | 9 | | 10 | | 11 | | 12 | | 13 | | 14 | | 15 | | 16 | | 17 | | 18 | | 19 | | 20 | | 21 | | 22 | | 23 | | 24 | | 25 | | 26 | | 27 | | 28 | | 29 | | 30 | | 31 |
| | In[10]:= Clear["Global`*"] (*Clear all variables*) | | <p>In[13]:= u[</p> | | <p>Out[13]= -y^2</p> | | <p>Out[14]= t^2 x^2</p> | | <p>In[15]:= m = 1;<br /> | | u[m + 1] =<br /> | | Integrate[<br /> | | Sum[D[u[i], {x, 2}]<em>y^2</em>t - D[u[i], {y, 2}]<em>x^2</em>t, {i, 0, m}], {t,<br /> | | 0, t}] -<br /> | | Integrate[<br /> | | Sum[D[u[i], {x, 2}]<em>y^2</em>t - D[u[i], {y, 2}]<em>x^2</em>t, {i, 0,<br /> | | m - 1}], {t, 0, t}]</p> | | <p>Out[16]= (t^4 y^2)/2</p> | | <p>In[17]:= m = 2;<br /> | | u[m + 1] =<br /> | | Integrate[<br /> | | Sum[D[u[i], {x, 2}]<em>y^2</em>t - D[u[i], {y, 2}]<em>x^2</em>t, {i, 0, m}], {t,<br /> | | 0, t}] -<br /> | | Integrate[<br /> | | Sum[D[u[i], {x, 2}]<em>y^2</em>t - D[u[i], {y, 2}]<em>x^2</em>t, {i, 0,<br /> | | m - 1}], {t, 0, t}]</p> | | <p>Out[18]= -(1/6) t^6 x^2</p> | | <p>In[19]:= m = 3;<br /> | | u[m + 1] =<br /> | | Integrate[<br /> | | Sum[D[u[i], {x, 2}]<em>y^2</em>t - D[u[i], {y, 2}]<em>x^2</em>t, {i, 0, m}], {t,<br /> | | 0, t}] -<br /> | | Integrate[<br /> | | Sum[D[u[i], {x, 2}]<em>y^2</em>t - D[u[i], {y, 2}]<em>x^2</em>t, {i, 0,<br /> | | m - 1}], {t, 0, t}]</p> | | <p>Out[20]= -(1/24) t^8 y^2</p> |
|
最后，得方程的解为：
\begin{equation}
u(x,t)=u_0+u_1+u_2+\cdots=x^2\sin t^2-y^2\cos t^2
\label{ex2sol}
\end{equation}
例3 一维非线性抛物型方程
方程为：
\begin{equation}
\begin{split}
u_t=&\frac{1}{x}u_{xx}+\frac{1}{x}u^2 \\
u(x,y,0)=&x
\end{split}
\label{ex3prob}
\end{equation}
迭代：
\begin{equation}
\begin{split}
u_0=& u(x,0)=x \\
u_1=& \frac{1}{x}\int_0^t (u_{0,xx}+u_0^2) \mathrm dt = x t \\
u_2=& \frac{1}{x}\int_0^t [u_{0,xx}+u_{1,xx}+(u_0+u_1)^2] \mathrm dt-\frac{1}{x}\int_0^t (u_{0,xx}+u_0^2) \mathrm dt= \frac{t^3 x}{3}+t^2 x\\
u_3=& \frac{1}{x}\int_0^t [u_{0,xx}+u_{1,xx}+u_{2,xx}+(u_0+u_1+u_2)^2] \mathrm dt-\\
& \frac{1}{x}\int_0^t [u_{0,xx}+u_{1,xx}+(u_0+u_1)^2] \mathrm dt\\
=&\frac{t^7 x}{63}+\frac{t^6 x}{9}+\frac{t^5 x}{3}+\frac{2 t^4 x}{3}+\frac{2 t^3 x}{3} \\
u_4=& \frac{t^{15} x}{59535}+\frac{t^{14} x}{3969}+\frac{t^{13} x}{567}+\frac{t^{12} x}{126}+\frac{5 t^{11} x}{189}+\frac{22 t^{10} x}{315}+\frac{86 t^9 x}{567}+\frac{71 t^8 x}{252}+\frac{4 t^7 x}{9}+\frac{5 t^6 x}{9}+\frac{8 t^5 x}{15}+\frac{t^4 x}{3} \\
\vdots =& \vdots \\
u_{m+1}=&\frac{1}{x}\int_0^t [u_{0,xx}+u_{1,xx}+u_{2,xx}+\cdots+u_{m,xx}+(u_0+u_1+u_2+\cdots+u_m)^2] \mathrm dt -\\
& \int_0^t x^2t(u_{0,xx}+u_{1,xx}+\cdots+u_{m-1,xx})\mathrm dt =
\end{split}
\label{u3m}
\end{equation}
Mathematica 代码：
| 1 | | 2 | | 3 | | 4 | | 5 | | 6 | | 7 | | 8 | | 9 | | 10 | | 11 | | 12 | | 13 | | 14 | | 15 | | 16 | | 17 | | 18 | | 19 | | 20 | | 21 | | 22 | | 23 | | 24 | | 25 | | 26 | | 27 | | 28 | | 29 | | 30 | | 31 | | 32 | | 33 | | 34 | | 35 | | 36 | | 37 | | 38 | | 39 | | 40 | | 41 | | 42 | | 43 | | 44 | | 45 | | 46 | | 47 | | 48 | | 49 | | 50 | | 51 | | 52 | | 53 | | 54 | | 55 | | 56 | | 57 | | 58 | | 59 | | 60 | | 61 | | 62 |
| | In[21]:= (*Appl.Math.Comput.162 (2005) 687\[Dash]693 Example 3*) | | Clear["Global`*"] (*Clear all variables*) | | <p>In[22]:= u[</p> | | <p>Out[22]= x</p> | | <p>In[23]:= u[1] = Integrate[D[u[</p> | | <p>Out[23]= t x</p> | | <p>In[28]:= m = 1;<br /> | | u[m + 1] =<br /> | | Integrate[<br /> | | D[Sum[u[i], {i, 0, m}], {x, 2}]/x + (Sum[u[i], {i, 0, m}])^2/x, {t,<br /> | | 0, t}] -<br /> | | Integrate[<br /> | | D[Sum[u[i], {i, 0, m - 1}], {x, 2}]/<br /> | | x + (Sum[u[i], {i, 0, m - 1}])^2/x, {t, 0, t}]</p> | | <p>Out[29]= t^2 x + (t^3 x)/3</p> | | <p>In[30]:= m = 2;<br /> | | u[m + 1] =<br /> | | Integrate[<br /> | | D[Sum[u[i], {i, 0, m}], {x, 2}]/x + (Sum[u[i], {i, 0, m}])^2/x, {t,<br /> | | 0, t}] -<br /> | | Integrate[<br /> | | D[Sum[u[i], {i, 0, m - 1}], {x, 2}]/<br /> | | x + (Sum[u[i], {i, 0, m - 1}])^2/x, {t, 0, t}]</p> | | <p>Out[31]= (2 t^3 x)/3 + (2 t^4 x)/3 + (t^5 x)/3 + (t^6 x)/9 + (t^7 x)/63</p> | | <p>In[32]:= m = 3;<br /> | | u[m + 1] =<br /> | | Integrate[<br /> | | D[Sum[u[i], {i, 0, m}], {x, 2}]/x + (Sum[u[i], {i, 0, m}])^2/x, {t,<br /> | | 0, t}] -<br /> | | Integrate[<br /> | | D[Sum[u[i], {i, 0, m - 1}], {x, 2}]/<br /> | | x + (Sum[u[i], {i, 0, m - 1}])^2/x, {t, 0, t}]</p> | | <p>Out[33]= (t^4 x)/3 + (8 t^5 x)/15 + (5 t^6 x)/9 + (4 t^7 x)/9 + (<br /> | | 71 t^8 x)/252 + (86 t^9 x)/567 + (22 t^10 x)/315 + (5 t^11 x)/189 + (<br /> | | t^12 x)/126 + (t^13 x)/567 + (t^14 x)/3969 + (t^15 x)/59535</p> | | <p>In[34]:= m = 4;<br /> | | u[m + 1] =<br /> | | Integrate[<br /> | | D[Sum[u[i], {i, 0, m}], {x, 2}]/x + (Sum[u[i], {i, 0, m}])^2/x, {t,<br /> | | 0, t}] -<br /> | | Integrate[<br /> | | D[Sum[u[i], {i, 0, m - 1}], {x, 2}]/<br /> | | x + (Sum[u[i], {i, 0, m - 1}])^2/x, {t, 0, t}]</p> | | <p>Out[35]= (2 t^5 x)/15 + (13 t^6 x)/45 + (128 t^7 x)/315 + (<br /> | | 7 t^8 x)/15 + (293 t^9 x)/630 + (23563 t^10 x)/56700 + (<br /> | | 9589 t^11 x)/28350 + (17239 t^12 x)/68040 + (4337 t^13 x)/24570 + (<br /> | | 9151 t^14 x)/79380 + (63268 t^15 x)/893025 + (<br /> | | 43363 t^16 x)/1058400 + (1080013 t^17 x)/48580560 + (<br /> | | 2588 t^18 x)/229635 + (162179 t^19 x)/30541455 + (<br /> | | 16511 t^20 x)/7144200 + (207509 t^21 x)/225042300 + (<br /> | | 557 t^22 x)/1666980 + (2447 t^23 x)/22504230 + (<br /> | | 16927 t^24 x)/540101520 + (5309 t^25 x)/675126900 + (<br /> | | t^26 x)/595350 + (2 t^27 x)/6751269 + (13 t^28 x)/315059220 + (<br /> | | t^29 x)/236294415 + (t^30 x)/3544416225 + (t^31 x)/109876902975</p> | | <p>In[36]:= m = 5;<br /> | | u[m + 1] =<br /> | | Integrate[<br /> | | D[Sum[u[i], {i, 0, m}], {x, 2}]/x + (Sum[u[i], {i, 0, m}])^2/x, {t,<br /> | | 0, t}] -<br /> | | Integrate[<br /> | | D[Sum[u[i], {i, 0, m - 1}], {x, 2}]/<br /> | | x + (Sum[u[i], {i, 0, m - 1}])^2/x, {t, 0, t}]</p> |
|
方程的解：
\begin{equation}
u(x,t)=u_0+u_1+u_2+\cdots=\frac{x}{1-t}
\label{ex3sol}
\end{equation}