# 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 $u(r,\theta )=\sum _{n=1}^{\infty }c_{n}{\frac {\sin({\frac {n\pi }{c}}(\ln r-\ln a))\sinh({\frac {n\pi }{c}}(\alpha -\theta ))}{\sinh({\frac {n\pi \alpha }{c}})}}\,$ $\Delta u=0\,$ $u(0,y)=0\,$ $u(a,y)=0\,$ $u(x,0)=0\,$ $u(x,b)=f(x)\,$ $0 $u(x,y)=\sum _{n=1}^{\infty }{\frac {{\frac {2}{a}}\int _{0}^{a}f(x)\sin {\frac {n\pi x}{a}}\,dx}{\sinh {\frac {n\pi b}{a}}}}\sin {\frac {n\pi x}{a}}\sinh {\frac {n\pi y}{a}}\,$ $\Delta u=0\,$ $u(0,y)=0\,$ $u(1,y)=0\,$ $u(x,0)=0\,$ $u(x,1)=Ax(1-x)\,$ $t>0,\,\,0 $u(x,y)={\frac {8A}{\pi ^{3}}}\sum _{n=1}^{\infty }{\frac {\sinh(2n-1)\pi y\sin(2n-1)\pi x}{(2n-1)^{3}\sinh(2n-1)\pi }}\,$ $\Delta u=0\,$ $u_{x}(0,y)=0\,$ $u_{x}(\pi ,y)=0\,$ $u(x,0)=K\cos x\,$ $u(x,1)=K\cos ^{2}x\,$ $t>0,\,\,0 $u(x,y)=K\left[{\frac {1}{2}}y+{\frac {\sinh(1-y)}{\sinh 1}}\cos x+{\frac {\sinh 2y}{2\sinh 2}}\cos 2x\right]\,$ $\Delta u=u_{rr}+{\frac {1}{r}}u_{r}+{\frac {1}{r^{2}}}u_{\theta \theta }=0\,$ $u(r,-\pi )=u(r,\pi )\,$ $u_{\theta }(r,-\pi )=u_{\theta }(r,\pi )\,$ $\lim _{r->0^{+}}u(r,\theta )<\infty \,$ $u(\rho ,\theta )=f(\theta )\,$ $0 $u(r,\theta )={\frac {\rho ^{2}-r^{2}}{2\pi }}\int _{-\pi }^{\pi }{\frac {f(x)}{\rho ^{2}-2\rho r\cos(x-\theta )+r^{2}}}\,dx\,$ $\Delta u=0\,$ $u(1,\theta )={\begin{cases}0&-\pi <\theta <0\\T_{0}&0<\theta <\pi \end{cases}}\,$ $0 $u(r,\theta )={\frac {1}{2}}T_{0}+{\frac {2T_{0}}{\pi }}\sum _{n=1}^{\infty }{\frac {r^{2n-1}}{2n-1}}\sin(2n-1)\theta \,$ $\Delta u=0\,$ $u_{r}(\rho ,\theta )=f(\theta )\,$ $0 $u(r,\theta )={\frac {1}{2}}a_{0}+\sum _{n=1}^{\infty }\left(r/\rho \right)^{n}(a_{n}\cos n\theta +b_{n}\sin n\theta )\,$ $\Delta u=0\,$ $u(a,\theta )=f(\theta )\,$ $u(b,\theta )=g(\theta )\,$ $a $u(r,\theta )={\frac {1}{2}}(A_{0}+B_{0}\log r)+\sum _{n=1}^{\infty }\left[(A_{n}r^{n}+B_{n}r^{-n})\cos n\theta +(C_{n}r^{n}+D_{n}r^{-n})\sin n\theta \right]\,$ $\Delta u=0\,$ $u_{r}(2,\theta )=0\,$ $u_{r}(1,\theta )=\sin ^{2}\theta \,$ $1 $u(r,\theta )={\frac {1}{2}}-{\frac {1}{17}}\left({\frac {r^{2}}{2}}+{\frac {8}{r^{2}}}\right)\cos 2\theta \,$ $\Delta u=0\,$ $u(\rho ,\theta )=\cos ^{2}\theta \,$ $0 $u(r,\theta )={\frac {1}{2}}-{\frac {1}{2}}\cos 2\theta \left({\frac {r}{\rho }}\right)^{2}\,$ $\Delta u=0\,$ $u(10,\theta )=15\cos \theta \,$ $u(20,\theta )=30\sin \theta \,$ $10 $u(r,\theta )=\left(-{\frac {r}{2}}+{\frac {200}{r}}\right)\cos \theta +\left(2r-{\frac {200}{r}}\right)\sin \theta \,$ $\Delta u=0\,$ $u_{r}(1,\theta )={\begin{cases}-1&-\pi <\theta <0\\1&0<\theta <\pi \end{cases}}\,$ $0 $u(r,\theta )={\frac {1}{2}}A_{0}+{\frac {4}{\pi }}\sum _{n=1}^{\infty }{\frac {\sin(2n-1)\theta }{(2n-1)^{2}\pi }}r^{2n-1}\,$ $\Delta u=0\,$ $u(x,0)=f(x)\,$ $\lim _{x^{2}+y^{2}\rightarrow \infty }u(x,y)=0\,$ $-\infty $u(x,y)=\int _{0}^{\infty }\left[A_{\lambda }\cos \lambda x+B_{\lambda }\sin \lambda x\right]e^{-\lambda x}\,d\lambda \,$ $\Delta u=0\,$ $u(x,0)={\begin{cases}T_{0}&|x|b\end{cases}}\,$ $\lim _{x^{2}+y^{2}\rightarrow \infty }u(x,y)=0\,$ $-\infty $u(x,y)={\frac {2T_{0}}{\pi }}\int _{0}^{\infty }{\frac {1}{\lambda }}\sin \lambda b\cos \lambda x\,d\lambda \,e^{-\lambda y}\,$ $\Delta u=0\,$ $u_{y}(x,a)=0\,$ $u_{x}(0,y)=0\,$ $u(x,0)=f(x)\,$ $0 $u(x,y)={\frac {1}{\sqrt {2\pi }}}\int _{0}^{\infty }f(\xi )\left[g(x+\xi ,y)+g(|x-\xi |,y)\right]\,d\xi \,$ $u_{tt}-c^{2}u_{xx}=2t\,$ $u(x,0)=x^{2}\,$ $u_{t}(x,0)=1\,$ $u(x,t)=x^{2}+c^{2}t^{2}+t+{\frac {1}{3}}t^{3}\,$ 