PDE:Laplaces Equation

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Laplace's Equation


solution Derive the Green's function for Laplace's equation with homogeneous Dirichlet boundary condition in the unit ball in {\mathbb  {R}}^{n}\,

solution Derive the Green's function for Laplace's equation with homogeneous Neumann boundary condition in the unit ball in {\mathbb  {R}}^{n}\,

solution

\Delta u={\frac  {1}{r}}(ru_{r})_{r}+{\frac  {1}{r^{2}}}u_{{\theta \theta }}=0\,

u(a,\theta )=0\,
u(b,\theta )=0\,
u(r,0)=f(r)\,
u(r,\alpha )=0\,

a<r<b,\,\,\,\,\,0<\theta <\alpha \,

  • u(r,\theta )=\sum _{{n=1}}^{\infty }c_{n}{\frac  {\sin({\frac  {n\pi }{c}}(\ln r-\ln a))\sinh({\frac  {n\pi }{c}}(\alpha -\theta ))}{\sinh({\frac  {n\pi \alpha }{c}})}}\,



solution

\Delta u=0\,

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

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

  • 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}}\,




solution

\Delta u=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)={\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 }}\,



solution

\Delta u=0\,

u_{x}(0,y)=0\,
u_{x}(\pi ,y)=0\,
u(x,0)=K\cos x\,
u(x,1)=K\cos ^{2}x\,
t>0,\,\,0<x<\pi \,

  • u(x,y)=K\left[{\frac  {1}{2}}y+{\frac  {\sinh(1-y)}{\sinh 1}}\cos x+{\frac  {\sinh 2y}{2\sinh 2}}\cos 2x\right]\,



solution

\Delta u=u_{{rr}}+{\frac  {1}{r}}u_{r}+{\frac  {1}{r^{2}}}u_{{\theta \theta }}=0\,

u(r,-\pi )=u(r,\pi )\,
u_{\theta }(r,-\pi )=u_{\theta }(r,\pi )\,
\lim _{{r->0^{+}}}u(r,\theta )<\infty \,

u(\rho ,\theta )=f(\theta )\,

0<r<\rho ,\,\,\,\,\,-\pi <\theta <\pi \,

  • u(r,\theta )={\frac  {\rho ^{2}-r^{2}}{2\pi }}\int _{{-\pi }}^{\pi }{\frac  {f(x)}{\rho ^{2}-2\rho r\cos(x-\theta )+r^{2}}}\,dx\,



solution

\Delta u=0\,


u(1,\theta )={\begin{cases}0&-\pi <\theta <0\\T_{0}&0<\theta <\pi \end{cases}}\,

0<r<1,\,\,\,\,\,-\pi <\theta <\pi \,

  • u(r,\theta )={\frac  {1}{2}}T_{0}+{\frac  {2T_{0}}{\pi }}\sum _{{n=1}}^{\infty }{\frac  {r^{{2n-1}}}{2n-1}}\sin(2n-1)\theta \,



solution

\Delta u=0\,


u_{r}(\rho ,\theta )=f(\theta )\,

0<r<\rho ,\,\,\,\,\,-\pi <\theta <\pi \,

  • u(r,\theta )={\frac  {1}{2}}a_{0}+\sum _{{n=1}}^{\infty }\left(r/\rho \right)^{n}(a_{n}\cos n\theta +b_{n}\sin n\theta )\,



solution

\Delta u=0\,


u(a,\theta )=f(\theta )\,
u(b,\theta )=g(\theta )\,

a<r<b,\,\,\,\,\,-\pi <\theta <\pi \,

  • u(r,\theta )={\frac  {1}{2}}(A_{0}+B_{0}\log r)+\sum _{{n=1}}^{\infty }\left[(A_{n}r^{n}+B_{n}r^{{-n}})\cos n\theta +(C_{n}r^{n}+D_{n}r^{{-n}})\sin n\theta \right]\,




solution

\Delta u=0\,


u_{r}(2,\theta )=0\,
u_{r}(1,\theta )=\sin ^{2}\theta \,

1<r<2,\,\,\,\,\,-\pi <\theta <\pi \,

  • u(r,\theta )={\frac  {1}{2}}-{\frac  {1}{17}}\left({\frac  {r^{2}}{2}}+{\frac  {8}{r^{2}}}\right)\cos 2\theta \,



solution

\Delta u=0\,


u(\rho ,\theta )=\cos ^{2}\theta \,

0<r<\rho ,\,\,\,\,\,-\pi <\theta <\pi \,

  • u(r,\theta )={\frac  {1}{2}}-{\frac  {1}{2}}\cos 2\theta \left({\frac  {r}{\rho }}\right)^{2}\,



solution

\Delta u=0\,


u(10,\theta )=15\cos \theta \,
u(20,\theta )=30\sin \theta \,

10<r<20,\,\,\,\,\,-\pi <\theta \leq \pi \,


  • u(r,\theta )=\left(-{\frac  {r}{2}}+{\frac  {200}{r}}\right)\cos \theta +\left(2r-{\frac  {200}{r}}\right)\sin \theta \,




solution

\Delta u=0\,


u_{r}(1,\theta )={\begin{cases}-1&-\pi <\theta <0\\1&0<\theta <\pi \end{cases}}\,

0<r<1,\,\,\,\,\,-\pi <\theta <\pi \,

  • u(r,\theta )={\frac  {1}{2}}A_{0}+{\frac  {4}{\pi }}\sum _{{n=1}}^{\infty }{\frac  {\sin(2n-1)\theta }{(2n-1)^{2}\pi }}r^{{2n-1}}\,



solution

\Delta u=0\,


u(x,0)=f(x)\,
\lim _{{x^{2}+y^{2}\rightarrow \infty }}u(x,y)=0\,

-\infty <x<\infty ,\,\,\,\,\,0<y<\infty \,

  • u(x,y)=\int _{0}^{\infty }\left[A_{\lambda }\cos \lambda x+B_{\lambda }\sin \lambda x\right]e^{{-\lambda x}}\,d\lambda \,



solution

\Delta u=0\,


u(x,0)={\begin{cases}T_{0}&|x|<b\\0&|x|>b\end{cases}}\,
\lim _{{x^{2}+y^{2}\rightarrow \infty }}u(x,y)=0\,

-\infty <x<\infty ,\,\,\,\,\,0<y<\infty \,

  • u(x,y)={\frac  {2T_{0}}{\pi }}\int _{0}^{\infty }{\frac  {1}{\lambda }}\sin \lambda b\cos \lambda x\,d\lambda \,e^{{-\lambda y}}\,



solution

\Delta u=0\,


u_{y}(x,a)=0\,
u_{x}(0,y)=0\,
u(x,0)=f(x)\,

0<x<\infty ,\,\,\,\,\,0<y<a\,

  • u(x,y)={\frac  {1}{{\sqrt  {2\pi }}}}\int _{0}^{\infty }f(\xi )\left[g(x+\xi ,y)+g(|x-\xi |,y)\right]\,d\xi \,



solution

u_{{tt}}-c^{2}u_{{xx}}=2t\,


u(x,0)=x^{2}\,
u_{t}(x,0)=1\,

  • u(x,t)=x^{2}+c^{2}t^{2}+t+{\frac  {1}{3}}t^{3}\,



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