21 Wave PDE in 2D Polar coordinates

 21.1 In circular disk. fixed edge of disk, no \(\theta \) dependency, with initial position and velocity given
 21.2 In circular disk. fixed edge of disk, with \(\theta \) dependency, zero initial velocity

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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

\[ \text{ sol=() } \]