# PDE:Laplaces Equation

## Laplace's Equation

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

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

solution

 $\displaystyle \Delta u = \frac{1}{r}(ru_r)_r + \frac{1}{r^2}u_{\theta\theta}=0\,$ $\displaystyle u(a,\theta) = 0\,$ $\displaystyle u(b,\theta) = 0\,$ $\displaystyle u(r,0) = f(r)\,$ $\displaystyle u(r,\alpha) = 0\,$ $\displaystyle a $\displaystyle u(r,\theta) = \sum_{n=1}^\infty c_n \frac{\sin(\frac{n\pi}{c}(\ln r - \ln a))\sinh(\frac{n\pi}{c}(\alpha-\theta))}{\sinh(\frac{n\pi\alpha}{c})}\,$

 $\displaystyle \Delta u = 0\,$ $\displaystyle u(0,y) = 0\,$ $\displaystyle u(a,y) = 0\,$ $\displaystyle u(x,0) = 0\,$ $\displaystyle u(x,b) = f(x)\,$ $\displaystyle 0 $\displaystyle u(x,y) = \sum_{n=1}^\infty \frac{ \frac{2}{a}\int_0^a f(x)\sin\frac{n\pi x}{a}\,dx}{\sinh\frac{n\pi b}{a}}\sin\frac{n\pi x}{a}\sinh\frac{n\pi y}{a}\,$

 $\displaystyle \Delta u =0\,$ $\displaystyle u(0,y) = 0\,$ $\displaystyle u(1,y) = 0\,$ $\displaystyle u(x,0) = 0\,$ $\displaystyle u(x,1) = A x(1-x)\,$ $\displaystyle t>0,\,\,0 $\displaystyle u(x,y) = \frac{8A}{\pi^3}\sum_{n=1}^\infty \frac{\sinh(2n-1)\pi y \sin(2n-1)\pi x}{(2n-1)^3\sinh(2n-1)\pi}\,$

 $\displaystyle \Delta u=0\,$ $\displaystyle u_x(0,y) = 0\,$ $\displaystyle u_x(\pi,y) = 0\,$ $\displaystyle u(x,0) = K \cos x\,$ $\displaystyle u(x,1) = K \cos^2 x\,$ $\displaystyle t>0,\,\,0 $\displaystyle u(x,y) = K\left[\frac{1}{2}y + \frac{\sinh(1-y)}{\sinh 1}\cos x + \frac{\sinh 2y}{2\sinh 2}\cos2x\right]\,$

 $\displaystyle \Delta u = u_{rr} + \frac{1}{r}u_r + \frac{1}{r^2}u_{\theta\theta}=0\,$ $\displaystyle u(r,-\pi) = u(r,\pi)\,$ $\displaystyle u_\theta(r,-\pi) = u_\theta(r,\pi)\,$ $\displaystyle \lim_{r->0^+}u(r,\theta) < \infty\,$ $\displaystyle u(\rho,\theta) = f(\theta)\,$ $\displaystyle 0 $\displaystyle u(r,\theta) = \frac{\rho^2-r^2}{2\pi}\int_{-\pi}^\pi \frac{f(x)}{\rho^2-2\rho r \cos(x-\theta) + r^2}\,dx\,$

 $\displaystyle \Delta u =0\,$ $\displaystyle u(1,\theta) = \begin{cases} 0 & -\pi<\theta<0 \\ T_0 & 0<\theta<\pi \end{cases}\,$ $\displaystyle 0 $\displaystyle u(r,\theta) = \frac{1}{2}T_0 + \frac{2 T_0}{\pi}\sum_{n=1}^\infty \frac{r^{2n-1}}{2n-1}\sin(2n-1)\theta\,$

 $\displaystyle \Delta u = 0\,$ $\displaystyle u_r(\rho,\theta) = f(\theta)\,$ $\displaystyle 0 $\displaystyle u(r,\theta) = \frac{1}{2}a_0 + \sum_{n=1}^\infty \left(r/\rho\right)^n(a_n\cos n\theta + b_n \sin n \theta)\,$

 $\displaystyle \Delta u =0\,$ $\displaystyle u(a,\theta) = f(\theta)\,$ $\displaystyle u(b,\theta) = g(\theta)\,$ $\displaystyle a $\displaystyle u(r,\theta) = \frac{1}{2}(A_0 + B_0\log r) + \sum_{n=1}^\infty\left[(A_n r^n + B_n r^{-n}) \cos n\theta + (C_n r^n + D_n r^{-n})\sin n\theta\right]\,$

 $\displaystyle \Delta u = 0\,$ $\displaystyle u_r(2,\theta) = 0\,$ $\displaystyle u_r(1,\theta) = \sin^2\theta\,$ $\displaystyle 1 $\displaystyle u(r,\theta) = \frac{1}{2} - \frac{1}{17}\left(\frac{r^2}{2}+\frac{8}{r^2}\right)\cos2\theta\,$

 $\displaystyle \Delta u = 0\,$ $\displaystyle u(\rho,\theta) = \cos^2\theta\,$ $\displaystyle 0 $\displaystyle u(r,\theta) = \frac{1}{2}-\frac{1}{2}\cos2\theta\left(\frac{r}{\rho}\right)^2\,$

 $\displaystyle \Delta u = 0\,$ $\displaystyle u(10,\theta) = 15\cos\theta\,$ $\displaystyle u(20,\theta) = 30\sin\theta\,$ $\displaystyle 10 $\displaystyle u(r,\theta) = \left(-\frac{r}{2} + \frac{200}{r}\right)\cos\theta + \left(2r - \frac{200}{r}\right)\sin\theta\,$

 $\displaystyle \Delta u = 0\,$ $\displaystyle u_r(1,\theta) = \begin{cases}-1&-\pi<\theta<0 \\ 1 & 0<\theta<\pi\end{cases}\,$ $\displaystyle 0 $\displaystyle u(r,\theta) = \frac{1}{2}A_0 + \frac{4}{\pi}\sum_{n=1}^\infty \frac{\sin(2n-1)\theta}{(2n-1)^2\pi}r^{2n-1}\,$

 $\displaystyle \Delta u = 0\,$ $\displaystyle u(x,0) = f(x)\,$ $\displaystyle \lim_{x^2+y^2\rightarrow\infty} u(x,y) = 0\,$ $\displaystyle -\infty $\displaystyle u(x,y) = \int_0^\infty\left[A_\lambda\cos\lambda x + B_\lambda\sin\lambda x\right] e^{-\lambda x}\,d\lambda\,$

 $\displaystyle \Delta u = 0\,$ $\displaystyle u(x,0) = \begin{cases}T_0 & |x|b\end{cases}\,$ $\displaystyle \lim_{x^2+y^2\rightarrow\infty} u(x,y) = 0\,$ $\displaystyle -\infty $\displaystyle u(x,y) = \frac{2T_0}{\pi}\int_0^\infty \frac{1}{\lambda}\sin\lambda b \cos \lambda x\,d\lambda\,e^{-\lambda y}\,$

 $\displaystyle \Delta u = 0\,$ $\displaystyle u_y(x,a)=0\,$ $\displaystyle u_x(0,y)=0\,$ $\displaystyle u(x,0) = f(x)\,$ $\displaystyle 0 $\displaystyle u(x,y) = \frac{1}{\sqrt{2\pi}}\int_0^\infty f(\xi)\left[g(x+\xi,y)+g(|x-\xi|,y)\right]\,d\xi\,$

 $\displaystyle u_{tt} - c^2 u_{xx}=2t\,$ $\displaystyle u(x,0)=x^2\,$ $\displaystyle u_t(x,0)=1\,$ $\displaystyle u(x,t)=x^2+c^2t^2+t+\frac{1}{3}t^3\,$