PDE:Laplaces Equation

From Example Problems
Jump to: navigation, search

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\,

Failed to parse (unknown error):
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\,
Failed to parse (unknown error):
Failed to parse (unknown error):

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\,
Failed to parse (unknown error):
Failed to parse (unknown error):
Failed to parse (unknown error):

  • Failed to parse (unknown error):





solution

Failed to parse (unknown error):

Failed to parse (unknown error):
Failed to parse (unknown error):
Failed to parse (unknown error):
Failed to parse (unknown error):
Failed to parse (unknown error):

  • Failed to parse (unknown error):





solution

Failed to parse (unknown error):

Failed to parse (unknown error):
Failed to parse (unknown error):
Failed to parse (unknown error):

Failed to parse (unknown error):

Failed to parse (unknown error):

  • Failed to parse (unknown error):





solution

Failed to parse (unknown error):


Failed to parse (unknown error):

Failed to parse (unknown error):

  • Failed to parse (unknown error):





solution

Failed to parse (unknown error):


Failed to parse (unknown error):

Failed to parse (unknown error):

  • Failed to parse (unknown error):





solution

Failed to parse (unknown error):


Failed to parse (unknown error):
Failed to parse (unknown error):

Failed to parse (unknown error):

  • Failed to parse (unknown error):






solution

Failed to parse (unknown error):


Failed to parse (unknown error):
Failed to parse (unknown error):

Failed to parse (unknown error):

  • Failed to parse (unknown error):





solution

Failed to parse (unknown error):


Failed to parse (unknown error):

Failed to parse (unknown error):

  • Failed to parse (unknown error):





solution

Failed to parse (unknown error):


Failed to parse (unknown error):
Failed to parse (unknown error):

Failed to parse (unknown error):


  • Failed to parse (unknown error):






solution

Failed to parse (unknown error):


Failed to parse (unknown error):

Failed to parse (unknown error):

  • Failed to parse (unknown error):





solution

Failed to parse (unknown error):


Failed to parse (unknown error):
Failed to parse (unknown error):

Failed to parse (unknown error):

  • Failed to parse (unknown error):





solution

Failed to parse (unknown error):


Failed to parse (unknown error):
Failed to parse (unknown error):

Failed to parse (unknown error):

  • Failed to parse (unknown error):





solution

Failed to parse (unknown error):


Failed to parse (unknown error):
Failed to parse (unknown error):
Failed to parse (unknown error):

Failed to parse (unknown error):

  • Failed to parse (unknown error):





solution

Failed to parse (unknown error):


Failed to parse (unknown error):
Failed to parse (unknown error):

  • Failed to parse (unknown error):





Partial Differential Equations

Main Page