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) = A x(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}\cos2x\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{2 T_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)\cos2\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}\cos2\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\le\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^2t^2+t+\frac{1}{3}t^3\,



Partial Differential Equations

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