32 miscellaneous PDE’s

 32.1 A second order PDE
 32.2 second order PDE in Polar coordinates
 32.3 Laplace like PDE with polynomial solution
 32.4 Third oder PDE
 32.5 PDE solved by Laplace transform
 32.6 Linear PDE, initial conditions at \(t=1\)
 32.7 Linear PDE, initial conditions at \(t=t_0\)
 32.8 second order in time, Linear PDE, initial conditions at \(t=t_0\)
 32.9 Einstein-Weiner PDE

_______________________________________________________________________________________

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

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

_______________________________________________________________________________________

32.2 second order PDE in Polar coordinates

problem number 171

Added December 20, 2018.

Taken from https://www.mapleprimes.com/posts/209970-Exact-Solutions-For-PDE-And-Boundary--Initial-Conditions-2018

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

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

_______________________________________________________________________________________

32.3 Laplace like PDE with polynomial solution

problem number 172

Added December 20, 2018.

Taken from https://www.mapleprimes.com/posts/209970-Exact-Solutions-For-PDE-And-Boundary--Initial-Conditions-2018

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

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

_______________________________________________________________________________________

32.4 Third oder PDE

problem number 173

Added December 20, 2018.

Taken from https://www.mapleprimes.com/posts/209970-Exact-Solutions-For-PDE-And-Boundary--Initial-Conditions-2018

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

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

_______________________________________________________________________________________

32.5 PDE solved by Laplace transform

problem number 174

Added December 20, 2018.

Taken from https://www.mapleprimes.com/posts/209970-Exact-Solutions-For-PDE-And-Boundary--Initial-Conditions-2018

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

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

_______________________________________________________________________________________

32.6 Linear PDE, initial conditions at \(t=1\)

problem number 175

Added December 20, 2018.

Example 25, Taken from https://www.mapleprimes.com/posts/209970-Exact-Solutions-For-PDE-And-Boundary--Initial-Conditions-2018

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

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

_______________________________________________________________________________________

32.7 Linear PDE, initial conditions at \(t=t_0\)

problem number 176

Added December 20, 2018.

Example 26, Taken from https://www.mapleprimes.com/posts/209970-Exact-Solutions-For-PDE-And-Boundary--Initial-Conditions-2018

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

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

_______________________________________________________________________________________

32.8 second order in time, Linear PDE, initial conditions at \(t=t_0\)

problem number 177

Added December 20, 2018.

Example 27, Taken from https://www.mapleprimes.com/posts/209970-Exact-Solutions-For-PDE-And-Boundary--Initial-Conditions-2018

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

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

_______________________________________________________________________________________

32.9 Einstein-Weiner PDE

problem number 178

Added January 2, 2018.

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

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