### 20 Wave PDE in 2D Cartesian coordinates

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#### 20.1 In square, given initial position but with zero initial velocity

problem number 144

Taken from Maple PDE help pages. This wave PDE inside square with free to move on left edge and right edge, and top and bottom edges are ﬁxed. It has zero initial velocity, but given a non-zero initial position. Where $$0<x<\pi$$ and $$0<y<\pi$$ and $$t>0$$.

Solve $\frac{\partial ^2 u}{\partial t^2} = \frac{1}{4} \left ( \frac{\partial ^2 u}{\partial x^2}+ \frac{\partial ^2 u}{\partial y^2} \right )$

With boundary conditions

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

With initial conditions

\begin{align*} \frac{\partial u}{\partial t}(x,y,0) &=0 \\ u(x,0) &= x y (\pi -y) \end{align*}

Mathematica

$\text{DSolve}\left [\left \{u^{(0,0,2)}(x,y,t)=\frac{1}{4} \left (u^{(0,2,0)}(x,y,t)+u^{(2,0,0)}(x,y,t)\right ),\left \{u^{(1,0,0)}(0,y,t)=0,u^{(1,0,0)}(\pi ,y,t)=0,u(x,0,t)=0,u(x,\pi ,t)=0\right \},\left \{u^{(0,0,1)}(x,y,0)=0,u(x,y,0)=x (\pi -y) y\right \}\right \},u(x,y,t),\{x,y,t\}\right ]$

Maple

$u \left ( x,y,t \right ) =\sum _{n=1}^{\infty }-2\,{\frac{ \left ( \left ( -1 \right ) ^{n}-1 \right ) \sin \left ( ny \right ) \cos \left ( 1/2\,nt \right ) }{{n}^{3}}}+\sum _{n=1}^{\infty } \left ( \sum _{m=1}^{\infty }8\,{\frac{ \left ( - \left ( -1 \right ) ^{n+m}+ \left ( -1 \right ) ^{n}+ \left ( -1 \right ) ^{m}-1 \right ) \cos \left ( mx \right ) \sin \left ( ny \right ) \cos \left ( 1/2\,\sqrt{{m}^{2}+{n}^{2}}t \right ) }{{\pi }^{2}{m}^{2}{n}^{3}}} \right )$

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#### 20.2 In square with damping. Given zero initial position but with non-zero initial velocity

problem number 145

Taken from Maple PDE help pages. This wave PDE inside square with damping present.

Membrane is free to move on the right edge and also on top edge. But ﬁxed at left edge and bottom edge.

It has zero initial position, but given a non-zero initial velocity. Where $$0<x<1$$ and $$0<y<1$$ and $$t>0$$.

Solve $\frac{\partial ^2 u}{\partial t^2} = \frac{1}{4} \left ( \frac{\partial ^2 u}{\partial x^2}+ \frac{\partial ^2 u}{\partial y^2} \right ) -\frac{1}{10} \frac{\partial u}{\partial t}$

With boundary conditions

\begin{align*} u(0,y,t) &=0\\ \frac{\partial u}{\partial x}u(1,y,t) &= 0 \\ u(x,0,t) &=0 \\ \frac{\partial u}{\partial y}u(x,1,t) &= 0 \end{align*}

With initial conditions

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

Mathematica

$\text{DSolve}\left [\left \{u^{(0,0,2)}(x,y,t)=\frac{1}{4} \left (u^{(0,2,0)}(x,y,t)+u^{(2,0,0)}(x,y,t)\right )-\frac{1}{10} u^{(0,0,1)}(x,y,t),\left \{u(0,y,t)=0,u^{(1,0,0)}(1,y,t)=0,u(x,0,t)=0,u^{(0,1,0)}(x,1,t)=0\right \},\left \{u(x,y,0)=0,u^{(0,0,1)}(x,y,0)=\left (1-\frac{x}{2}\right ) x \left (1-\frac{y}{2}\right ) y\right \}\right \},u(x,y,t),\{x,y,t\}\right ]$

Maple

$u \left ( x,y,t \right ) =\sum _{m=0}^{\infty } \left ( \sum _{n=0}^{\infty }5120\,{\frac{\sin \left ( 1/2\, \left ( 1+2\,m \right ) \pi \,y \right ) \sin \left ( 1/2\, \left ( 1+2\,n \right ) \pi \,x \right ){{\rm e}^{-t/20}}\sin \left ( 1/20\,t\sqrt{-1+ \left ( 100\,{m}^{2}+100\,{n}^{2}+100\,m+100\,n+50 \right ){\pi }^{2}} \right ) }{\sqrt{-1+ \left ( 100\,{m}^{2}+100\,{n}^{2}+100\,m+100\,n+50 \right ){\pi }^{2}}{\pi }^{6} \left ( 1+2\,m \right ) ^{3} \left ( 1+2\,n \right ) ^{3}}} \right )$

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#### 20.3 In rectangle. All 4 edges are ﬁxed and given non-zero initial position with zero initial velocity

problem number 146

Taken from Mathematica helps pages on DSolve

Solve for $$u(x,y,t)$$ with $$0<x<1$$ and $$0<y<2$$ and $$t>0$$.

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

With boundary conditions

\begin{align*} u(x,0,t) &=0 \\ u(0,y,t) &= 0 \\ u(1,y,t) &=0 \\ u(x,2,t) &= 0 \end{align*}

With initial conditions

\begin{align*} u(x,y,0) &=\frac{1}{10} (x-x^2)(2 y-y^2) \\ \frac{\partial u}{\partial t}(x,y,0) &= 0 \end{align*}

Mathematica

$\left \{\left \{u(x,y,t)\to \underset{n=1}{\overset{\infty }{\sum }}\underset{m=1}{\overset{\infty }{\sum }}\frac{32 \left (-1+(-1)^m\right ) \left (-1+(-1)^n\right ) \cos \left (\sqrt{\frac{m^2}{4}+n^2} \pi t\right ) \sin (n \pi x) \sin \left (\frac{m \pi y}{2}\right )}{5 m^3 n^3 \pi ^6}\right \}\right \}$

Maple

$u \left ( x,y,t \right ) =\sum _{m=1}^{\infty } \left ( \sum _{n=1}^{\infty }-{\frac{32\,\sin \left ( n\pi \,x \right ) \sin \left ( 1/2\,m\pi \,y \right ) \cos \left ( 1/2\,\pi \,\sqrt{{m}^{2}+4\,{n}^{2}}t \right ) \left ( - \left ( -1 \right ) ^{n+m}+ \left ( -1 \right ) ^{n}+ \left ( -1 \right ) ^{m}-1 \right ) }{5\,{n}^{3}{\pi }^{6}{m}^{3}}} \right )$

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#### 20.4 In rectangle. All 4 edges are ﬁxed and given non-zero initial position with zero initial velocity (Haberman 8.5.5 (a))

problem number 147

This is problem 8.5.5 part(a) from Richard Haberman applied partial diﬀerential equations 5th edition.

Solve the initial value problem for membrane with time-dependent forcing and ﬁxed boundaries $$u=0$$.

$\frac{\partial ^2 u(x,y,t)}{\partial t^2} = c^2 \nabla (u) + Q(x,y,t)$

If the memberane is rectangle $$(0<x<L,0<y<H)$$.

With initial conditions

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

See my HW9, Math 322, UW Madison.

Mathematica

$\text{DSolve}\left [\left \{u^{(0,0,2)}(x,y,t)=c^2 \left (u^{(0,2,0)}(x,y,t)+u^{(2,0,0)}(x,y,t)\right )+Q(x,y,t),\left \{u(x,y,0)=f(x,y),u^{(0,0,1)}(x,y,0)=0\right \},\{u(0,y,t)=0,u(L,y,t)=0,u(x,0,t)=0,u(x,H,t)=0\}\right \},u(x,y,t),\{x,y,t\},\text{Assumptions}\to \{L>0,H>0,t>0,c>0\}\right ]$

Maple

$\text{ sol=() }$