21 Wave PDE in 2D Polar coordinates

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21.1 In circular disk. ﬁxed edge of disk, no $$\theta$$ dependency, with initial position and velocity given

problem number 148

Taken from Mathematica helps pages on DSolve

Solve for $$u(r,t)$$ with $$0<r<1$$ and $$t>0$$.

$\frac{\partial ^2 u}{\partial t^2} = c^2 \left ( \frac{\partial ^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} \right )$

With boundary conditions

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

With initial conditions

\begin{align*} u(r,0) &=1 \\ \frac{\partial u}{\partial t}(r,0) &= \frac{r}{3} \end{align*}

Mathematica

$\left \{\left \{u(r,t)\to \underset{n=1}{\overset{\infty }{\sum }}\frac{2 \text{BesselJ}(0,r \text{BesselJZero}(0,n)) \left (9 \sqrt{c^2} \text{BesselJ}(1,\text{BesselJZero}(0,n)) \cos (c t \text{BesselJZero}(0,n))+\text{HypergeometricPFQ}\left (\left \{\frac{3}{2}\right \},\left \{1,\frac{5}{2}\right \},-\frac{1}{4} \text{BesselJZero}(0,n)^2\right ) \sin \left (\sqrt{c^2} t \text{BesselJZero}(0,n)\right )\right )}{9 \sqrt{c^2} \left (\text{BesselJ}(0,\text{BesselJZero}(0,n))^2+\text{BesselJ}(1,\text{BesselJZero}(0,n))^2\right ) \text{BesselJZero}(0,n)}\right \}\right \}$

Maple

$u \left ( r,t \right ) =-{\it invlaplace} \left ({\frac{1}{s}\BesselI \left ( 0,{\frac{sr}{c}} \right ) \left ( \BesselI \left ( 0,{\frac{s}{c}} \right ) \right ) ^{-1}},s,t \right ) -1/3\,{\it invlaplace} \left ({\frac{1}{{s}^{2}}\BesselI \left ( 0,{\frac{sr}{c}} \right ) \left ( \BesselI \left ( 0,{\frac{s}{c}} \right ) \right ) ^{-1}},s,t \right ) +1/6\,\pi \,c{\it invlaplace} \left ({\frac{1}{{s}^{3}}\BesselI \left ( 0,{\frac{sr}{c}} \right ) \StruveL \left ( 0,{\frac{s}{c}} \right ) \left ( \BesselI \left ( 0,{\frac{s}{c}} \right ) \right ) ^{-1}},s,t \right ) -1/6\,\pi \,c{\it invlaplace} \left ({\frac{1}{{s}^{3}}\StruveL \left ( 0,{\frac{sr}{c}} \right ) },s,t \right ) +1+1/3\,tr$ Has unresolved Invlaplace calls

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21.2 In circular disk. ﬁxed edge of disk, with $$\theta$$ dependency, zero initial velocity

problem number 149

Solve for $$u(r,\theta ,t)$$ with $$0<r<a$$ and $$t>0$$ and $$-\pi <\theta <\pi$$

$\frac{\partial ^2 u}{\partial t^2} = c^2 \left ( \frac{\partial ^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} +\frac{1}{r^2} \frac{\partial ^2 u}{\partial \theta ^2} \right )$

With boundary conditions

\begin{align*} u(a,\theta ,t) &=0 \\ |u(0,\theta ,t)| < \infty \\ u(r,-\pi ,t) &= u(r,\pi ,t) \\ \frac{\partial u}{\partial \theta }(r,-\pi ,t) &= \frac{\partial u}{\partial \theta }(r,\pi ,t)\\ \end{align*}

With initial conditions

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

Mathematica

$\text{DSolve}\left [\left \{u^{(0,0,2)}(r,\theta ,t)=c^2 \left (\frac{u^{(0,2,0)}(r,\theta ,t)}{r^2}+\frac{u^{(1,0,0)}(r,\theta ,t)}{r}+u^{(2,0,0)}(r,\theta ,t)\right ),\left \{u(r,\theta ,0)=f(r,\theta ),u^{(0,0,1)}(r,\theta ,0)=0\right \},\left \{u(a,\theta ,t)=0,u(r,-\pi ,t)=u(r,\pi ,t),u^{(0,1,0)}(r,-\pi ,t)=u^{(0,1,0)}(r,\pi ,t)\right \}\right \},u(r,\theta ,t),\{r,\theta ,t\},\text{Assumptions}\to \{0<r<a,a>0,t>0,-\pi <\theta <\pi \}\right ]$

Maple

$\text{ sol=() }$