PDE15

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\Delta u=u_{{xx}}+u_{{yy}}=0\,

u(0,y)=0\,
u(1,y)=0\,
u(x,0)=0\,

u(x,1)=Ax(1-x)\,
t>0,\,\,0<x<\infty \,


u(x,y)=X(x)Y(y)\,

{\frac  {X''}{X}}=-{\frac  {Y''}{Y}}=-\lambda ^{2}\,

X''+\lambda ^{2}X=0\,

X(x)=c_{1}\cos \lambda x+c_{2}\sin \lambda x\,

X(0)=c_{1}=0\,

X(1)=c_{2}\sin \lambda =0,\lambda =n\pi ,n=1,2,...\,

  • X(x)=c_{2}\sin n\pi x,n=1,2,...\,

Y''-\lambda ^{2}Y=0\,

Y(y)=c_{3}\cosh \lambda y+c_{4}\sinh \lambda y\,

Y(0)=c_{3}=0\,

Y(y)=c_{4}\sinh n\pi y,n=1,2,...\,

u(x,y)=\sum _{{n=1}}^{\infty }A_{n}\sin n\pi x\sinh n\pi y\,

u(x,1)=\sum _{{n=1}}^{\infty }A_{n}\sin n\pi x\sinh n\pi =Ax(1-x)\,

From Fourier series coefficients,

A_{n}\sinh n\pi =2A\int _{0}^{1}x(1-x)\sin n\pi x\,dx\,

Which can be integrated by parts and results in

={\begin{cases}0&n=2,4,...\\{\frac  {8A}{n^{3}\pi ^{3}}}&{\frac  {n-1}{2}}\in {\mathbb  {Z}}\end{cases}},n=1,2,...\,

u(x,y)=\sum _{{n=1}}^{\infty }c_{n}\sinh n\pi y\sin n\pi x,\,\,\,c_{n}={\begin{cases}{\frac  {8A}{n^{3}\pi ^{3}\sinh n\pi }}&n=1,3,5,...\\0&n=2,4,...\end{cases}}\,

Making a change of index variable to only odd numbers, the final solution is

u(x,y)={\frac  {8A}{\pi ^{3}}}\sum _{{n=1}}^{\infty }{\frac  {\sinh(2n-1)\pi y\sin(2n-1)\pi x}{(2n-1)^{3}\sinh(2n-1)\pi }}\,


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