4.4  Helmholtz in 2D

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4.4.0.1 [306] In rectangle

problem number 306

Taken from Mathematica DSolve help pages.

Solve for \(u\left ( x,y\right ) \) \begin{align*} u_{xx}+ u_{yy} + 5 u(x,y) & = 0 \end{align*}

Boundary conditions \begin{align*} u(x,0) &= \text{UnitTriangle[x-2]} \\ u(x,2) &= 0 \\ u(0,y) &= 0 \\ u(4,y) &=0 \end{align*}

pict
Figure 4.40:PDE specification

Mathematica

\[ \left \{\left \{u(x,y)\to \frac{1}{2} \underset{n=1}{\overset{\infty }{\sum }}\frac{128 \left (\cos \left (\frac{n \pi }{8}\right )+\cos \left (\frac{3 n \pi }{8}\right )\right ) \text{csch}\left (\frac{1}{2} \sqrt{n^2 \pi ^2-80}\right ) \sin ^3\left (\frac{n \pi }{8}\right ) \sin \left (\frac{n \pi x}{4}\right ) \sinh \left (\frac{1}{4} \sqrt{n^2 \pi ^2-80} (2-y)\right )}{n^2 \pi ^2}\right \}\right \} \]

Maple

\[ u \left ( x,y \right ) =\sum _{n=1}^{\infty }32\,{\frac{\sin \left ( 1/4\,\pi \,nx \right ) \left ( 1/2\, \left ( \sin \left ( 1/2\,\pi \,n \right ) -1/2\,\sin \left ( 1/4\,\pi \,n \right ) -1/2\,\sin \left ( 3/4\,\pi \,n \right ) \right ) \sin \left ( 1/2\,\sqrt{-{\pi }^{2}{n}^{2}+80} \right ) \cos \left ( 1/4\,\sqrt{-{\pi }^{2}{n}^{2}+80}y \right ) +\cos \left ( 1/2\,\sqrt{-{\pi }^{2}{n}^{2}+80} \right ) \sin \left ( 1/4\,\pi \,n \right ) \cos \left ( 1/4\,\pi \,n \right ) \sin \left ( 1/4\,\sqrt{-{\pi }^{2}{n}^{2}+80}y \right ) \left ( \cos \left ( 1/4\,\pi \,n \right ) -1 \right ) \right ) }{\sin \left ( 1/2\,\sqrt{-{\pi }^{2}{n}^{2}+80} \right ){n}^{2}{\pi }^{2}}} \]

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4.4.0.2 [307] On whole plane

problem number 307

Added December 27, 2018.

Solve for \(u\left ( x,y\right ) \) \begin{align*} u_{xx}+u_{yy} + 5 u(x,y) & = 0 \end{align*}

pict
Figure 4.41:PDE specification

Mathematica

\[ \text{Failed} \] why? It solved earlier with BC?

Maple

\[ u \left ( x,y \right ) ={\frac{ \left ( \left ({{\rm e}^{\sqrt{{\it \_c}_{{1}}}x}} \right ) ^{2}{\it \_C1}+{\it \_C2} \right ) \left ({\it \_C3}\,\sin \left ( \sqrt{{\it \_c}_{{1}}+5}y \right ) +{\it \_C4}\,\cos \left ( \sqrt{{\it \_c}_{{1}}+5}y \right ) \right ) }{{{\rm e}^{\sqrt{{\it \_c}_{{1}}}x}}}} \]

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4.4.0.3 [308] Reduced Helmholtz Inside square

problem number 308

Added December 20, 2018.

Example 24, taken from https://www.mapleprimes.com/posts/209970-Exact-Solutions-For-PDE-And-Boundary--Initial-Conditions-2018

Solve for \(u\left ( x,y\right ) \) \begin{align*} \frac{\partial ^{2}u}{\partial x^{2}}+\frac{\partial ^{2}u}{\partial y^2} - k u(x,y) & = 0 \end{align*}

With \(k>0\). It is called reduced Helmholtz, because of the minus sign above. Otherwise, standard Helmholtz has a positive sign.

Boundary conditions \begin{align*} u(x,0) &= 0 \\ u(x,\pi ) &= 0 \\ u(0,y) &= 1 \\ u(\pi ,y) &=0 \end{align*}

pict
Figure 4.42:PDE specification

Mathematica

\[ \text{Failed} \]

Maple

\[ u \left ( x,y \right ) =\sum _{n=1}^{\infty }2\,{\frac{\sin \left ( ny \right ) \left ( -1+ \left ( -1 \right ) ^{n} \right ) \left ( -{{\rm e}^{- \left ( -2\,\pi +x \right ) \sqrt{{n}^{2}+k}}}+{{\rm e}^{\sqrt{{n}^{2}+k}x}} \right ) }{ \left ({{\rm e}^{2\,\sqrt{{n}^{2}+k}\pi }}-1 \right ) \pi \,n}} \]