PDE14

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

u(0,y)=0\,
u(a,y)=0\,
u(x,0)=0\,
u(x,b)=f(x)\,

0<x<a,\,\,\,\,\,0<y<b\,



Seperate variables and plug back into the DE. The equation is now seperable. Separate it and set it equal to a constant. Put a negative sign or power of two on the constant if it makes the later math easier.

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

X''Y+XY''=0\,

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

The ODE for X(x)\, is

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

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

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

X(a)=c_{2}\sin(\lambda a)=0\Rightarrow \lambda ={\frac  {n\pi }{a}},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\,

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

u(x,b)=\sum _{{n=1}}^{\infty }A_{n}\sinh {\frac  {n\pi b}{a}}\sin {\frac  {n\pi x}{a}}=f(x)\,

Through Fourier analysis the equation is solved for the coefficients in the integrand.

A_{n}\sinh {\frac  {n\pi b}{a}}={\frac  {2}{a}}\int _{0}^{a}f(x)\sin {\frac  {n\pi x}{a}}\,dx\,

The final solution is:

u(x,y)=\sum _{{n=1}}^{\infty }{\frac  {{\frac  {2}{a}}\int _{0}^{a}f(x)\sin {\frac  {n\pi x}{a}}\,dx}{\sinh {\frac  {n\pi b}{a}}}}\sin {\frac  {n\pi x}{a}}\sinh {\frac  {n\pi y}{a}}\,

Partial Differential Equations

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