### 32 miscellaneous PDE’s

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#### 32.1 A second order PDE

problem number 170

Taken from Maple pdsolve help pages, problem 4.

Solve for $$S \left ( x,y \right )$$ \begin{align*} S(x,y) \left ( \frac{\partial ^2 S}{\partial x \partial y} \right ) + \frac{\partial S}{\partial x} \frac{\partial S}{\partial y} &=1 \end{align*}

Mathematica

$\text{DSolve}\left [s^{(0,1)}(x,y) s^{(1,0)}(x,y)+s(x,y) s^{(1,1)}(x,y)=1,s(x,y),\{x,y\}\right ]$

Maple

$S \left ( x,y \right ) ={\frac{\sqrt{2\,{\it \_c}_{{1}}x+{\it \_C1}}\sqrt{{\it \_C2}\,{{\it \_c}_{{1}}}^{2}+{\it \_c}_{{1}}y}}{{\it \_c}_{{1}}}}$

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#### 32.2 second order PDE in Polar coordinates

problem number 171

Solve for $$u(r,\theta )$$

$\frac{\partial ^2 u}{\partial r^2} + \frac{\partial ^2 u}{\partial \theta ^2} = 0$

With boundary conditions

\begin{align*} u(2,\theta )&=3 \sin (2 \theta )+1 \end{align*}

Mathematica

$\text{DSolve}\left [\left \{u^{(0,2)}(r,\theta )+u^{(2,0)}(r,\theta )=0,u(2,\theta )=3 \sin (2 \theta )+1\right \},u(r,\theta ),\{r,\theta \}\right ]$

Maple

$u \left ( r,\theta \right ) =-3/2\,i{{\rm e}^{-2\,r+4+2\,i\theta }}+3/2\,i{{\rm e}^{2\,r-4-2\,i\theta }}+1$

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#### 32.3 Laplace like PDE with polynomial solution

problem number 172

Solve for $$u(x,y)$$

$\frac{\partial ^2 u}{\partial x^2} + y \frac{\partial ^2 u}{\partial y^2} = 0$

With boundary conditions

\begin{align*} u(x,0)&=0 \\ \frac{\partial u}{\partial y}(x,0) &=x^2 \end{align*}

Mathematica

$\left \{\left \{u(x,y)\to -y \left (y-x^2\right )\right \}\right \}$

Maple

$u \left ( x,y \right ) =y \left ({x}^{2}-y \right )$

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#### 32.4 Third oder PDE

problem number 173

Solve for $$u(x,y)$$

$\frac{\partial u}{\partial t} = - \frac{\partial ^3 u}{\partial x^2}$

With initial conditions

\begin{align*} u(x,0)&=f(x) \end{align*}

Mathematica

$\text{DSolve}\left [\left \{u^{(0,1)}(x,t)=-u^{(3,0)}(x,t),u(x,0)=f(x)\right \},u(x,t),\{x,t\}\right ]$

Maple

$u \left ( x,t \right ) =1/4\,{\frac{1}{{\pi }^{2}}\int _{-\infty }^{\infty }\!4/3\,{\frac{\pi \,f \left ( -\zeta \right ) }{\sqrt [3]{-t}}\sqrt{-{\frac{x+\zeta }{\sqrt [3]{-t}}}}\BesselK \left ( 1/3,-2/9\,{\frac{\sqrt{3} \left ( x+\zeta \right ) }{\sqrt [3]{-t}}\sqrt{-{\frac{x+\zeta }{\sqrt [3]{-t}}}}} \right ) }\,{\rm d}\zeta }$

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#### 32.5 PDE solved by Laplace transform

problem number 174

Solve for $$u(x,y)$$

$\frac{\partial u^2}{\partial x y} = \sin (x) \sin (y)$

With boundary conditions

\begin{align*} u(x,0)&=1+\cos (x) \\ \frac{\partial u}{\partial y}(0,y) &= -2 \sin y \end{align*}

Mathematica

$\text{DSolve}\left [\left \{u^{(1,1)}(x,y)=\sin (x) \sin (y),\left \{u(x,0)=\cos (x)+1,u^{(0,1)}(0,y)=-2 \sin (y)\right \}\right \},u(x,y),x,y\right ]$

Maple

$u \left ( x,y \right ) =1/2\,\cos \left ( x-y \right ) +1/2\,\cos \left ( x+y \right ) +\cos \left ( y \right )$

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#### 32.6 Linear PDE, initial conditions at $$t=1$$

problem number 175

Solve for $$w(x_1,x_2,x_3,t)$$

$\frac{\partial w}{\partial t} = \frac{\partial w^2}{\partial x_1^2} + \frac{\partial w^2}{\partial x_2^2} + \frac{\partial w^2}{\partial x_3^2}$

With initial condition $$w(x_1,x_2,x_3,1) = e^a x_1^2 +x_2 x_3$$

Mathematica

$\text{DSolve}\left [\left \{w^{(0,0,0,1)}(\text{x1},\text{x2},\text{x3},t)=w^{(0,0,2,0)}(\text{x1},\text{x2},\text{x3},t)+w^{(0,2,0,0)}(\text{x1},\text{x2},\text{x3},t)+w^{(2,0,0,0)}(\text{x1},\text{x2},\text{x3},t),w(\text{x1},\text{x2},\text{x3},1)=e^a \text{x1}^2+\text{x2} \text{x3}\right \},w(\text{x1},\text{x2},\text{x3},t),\{\text{x1},\text{x2},\text{x3},t\}\right ]$

Maple

$w \left ({\it x1},{\it x2},{\it x3},t \right ) = \left ({{\it x1}}^{2}+2\,t-2 \right ){{\rm e}^{a}}+{\it x2}\,{\it x3}$

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#### 32.7 Linear PDE, initial conditions at $$t=t_0$$

problem number 176

Solve for $$w(x_1,x_2,x_3,t)$$

$\frac{\partial w}{\partial t} = \frac{\partial w^2}{\partial x_1 x_2} + \frac{\partial w^2}{\partial x_1 x_3} + \frac{\partial w^2}{\partial x_3^2} + \frac{\partial w^2}{\partial x_2 x_3}$

With initial condition $$w(x_1,x_2,x_3,t_0) = e^{x_1} +x_2 -3 x_3$$

Mathematica

$\text{DSolve}\left [\left \{w^{(0,0,0,1)}(\text{x1},\text{x2},\text{x3},t)=w^{(0,0,2,0)}(\text{x1},\text{x2},\text{x3},t)-w^{(0,1,1,0)}(\text{x1},\text{x2},\text{x3},t)+w^{(1,0,1,0)}(\text{x1},\text{x2},\text{x3},t)+w^{(1,1,0,0)}(\text{x1},\text{x2},\text{x3},t),w(\text{x1},\text{x2},\text{x3},\text{t0})=e^{\text{x1}}+\text{x2}-3 \text{x3}\right \},w(\text{x1},\text{x2},\text{x3},t),\{\text{x1},\text{x2},\text{x3},t\}\right ]$

Maple

$w \left ({\it x1},{\it x2},{\it x3},t \right ) ={{\rm e}^{{\it x1}}}+{\it x2}-3\,{\it x3}$

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#### 32.8 second order in time, Linear PDE, initial conditions at $$t=t_0$$

problem number 177

Solve for $$w(x_1,x_2,x_3,t)$$

$\frac{\partial w^2}{\partial t^2} = \frac{\partial w^2}{\partial x_1 x_2} + \frac{\partial w^2}{\partial x_1 x_3} + \frac{\partial w^2}{\partial x_3^2} - \frac{\partial w^2}{\partial x_2 x_3}$

With initial condition \begin{align*} w(x_1,x_2,x_3,t_0) &= x_1^3 x_2^2 + x_3 \\ \frac{\partial w}{\partial t}(x_1,x_2,x_3,t_0) &= -x_2 x_3 + x_1 \end{align*}

Mathematica

$\text{DSolve}\left [\left \{w^{(0,0,0,2)}(\text{x1},\text{x2},\text{x3},t)=w^{(0,0,2,0)}(\text{x1},\text{x2},\text{x3},t)-w^{(0,1,1,0)}(\text{x1},\text{x2},\text{x3},t)+w^{(1,0,1,0)}(\text{x1},\text{x2},\text{x3},t)+w^{(1,1,0,0)}(\text{x1},\text{x2},\text{x3},t),\left \{w(\text{x1},\text{x2},\text{x3},\text{t0})=\text{x1}^3 \text{x2}^2+\text{x3},w^{(0,0,0,1)}(\text{x1},\text{x2},\text{x3},\text{t0})=\text{x1}-\text{x2} \text{x3}\right \}\right \},w(\text{x1},\text{x2},\text{x3},t),\{\text{x1},\text{x2},\text{x3},t\}\right ]$

Maple

$w \left ({\it x1},{\it x2},{\it x3},t \right ) =1/2\,{{\it t0}}^{4}{\it x1}+1/6\, \left ( -12\,{\it x1}\,t-1 \right ){{\it t0}}^{3}+1/6\, \left ( 18\,{\it x1}\,{t}^{2}+18\,{{\it x1}}^{2}{\it x2}+3\,t \right ){{\it t0}}^{2}+1/6\, \left ( -36\,t{{\it x1}}^{2}{\it x2}+ \left ( -12\,{t}^{3}-6 \right ){\it x1}-3\,{t}^{2}+6\,{\it x2}\,{\it x3} \right ){\it t0}+{{\it x1}}^{3}{{\it x2}}^{2}+3\,{t}^{2}{{\it x1}}^{2}{\it x2}+1/6\, \left ( 3\,{t}^{4}+6\,t \right ){\it x1}+1/6\,{t}^{3}-{\it x2}\,{\it x3}\,t+{\it x3}$

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#### 32.9 Einstein-Weiner PDE

problem number 178

Solve for $$u(x,t)$$ with $$x>0,t>0$$

$u_t = -\beta u_x + D u_{xx}$

Assuming $$\beta >0,D>0$$

Mathematica

$\text{DSolve}\left [u^{(0,1)}(x,t)=\beta u^{(1,0)}(x,t)+d u^{(2,0)}(x,t),u(x,t),\{x,t\},\text{Assumptions}\to \{\beta >0,d>0,x>0,t>0\}\right ]$

Maple

$u \left ( x,t \right ) ={\frac{{\it \_C3}\,{\it \_C1}}{{{\rm e}^{{\it \_c}_{{1}}t}}}\sqrt{{{\rm e}^{{\frac{x\beta }{d}}}}}{{\rm e}^{1/2\,{\frac{x\sqrt{{\beta }^{2}-4\,d{\it \_c}_{{1}}}}{d}}}}}+{\frac{{\it \_C3}\,{\it \_C2}}{{{\rm e}^{{\it \_c}_{{1}}t}}}\sqrt{{{\rm e}^{{\frac{x\beta }{d}}}}}{{\rm e}^{-1/2\,{\frac{x\sqrt{{\beta }^{2}-4\,d{\it \_c}_{{1}}}}{d}}}}}$