PDE:Integration and Separation of Variables

solution $3u_{x}+4u_{y}-2u=1,u(x,0)=e^{x}\,$

solution $z_{{xy}}=x^{2}y,z(x,0)=x^{2},z(1,y)=\cos(y)\,$

solution $z_{{xy}}={\frac {1}{2}}xy^{2},z(x,0)=e^{x},z(0,y)=\sin(y)\,$

solution $u_{{xx}}-u_{t}=0\,$

solution $u_{t}=ku_{{xx}}\,$

$u(x,0)=f(x)\,$ $u(0,t)=0\,$ $u(l,t)=0\,$ $u(x,t)={\frac {2}{l}}\sum _{{n=1}}^{\infty }A_{n}e^{{-\left({\frac {n\pi }{l}}\right)^{2}kt}}\sin({\frac {n\pi x}{l}})\,$

solution $u_{t}=ku_{{xx}}\,$

$u_{x}(0,t)=0\,$ $u_{x}(1,t)=0\,$ $u(x,0)=\phi (x)\,$ $u(x,t)=\sum _{{n=1}}^{\infty }A_{n}e^{{-\lambda _{n}t}}\cos({\sqrt {\lambda _{n}}}x)\,$

solution Transform this initial boundary value problem into one with homogeneous boundary conditions.

$u_{t}=u_{{xx}}+w(x,t)\,$

$u_{x}(0,t)=\alpha (t)\,$ $u_{x}(1,t)=\beta (t)\,$ $u(x,0)=\phi (x)\,$ $0<x<1\,$ $t>0\,$

solution $u_{{tt}}=c^{2}(u_{{xx}}+u_{{yy}})\,$

$u(x,0,t)=0\,$ $u(x,b,t)=0\,$ $u(0,y,t)=0\,$ $u(a,y,t)=0\,$ $u(x,y,0)=f(x,y)\,$ $u_{t}(x,y,0)=g(x,y)\,$ $0<x<a,\,\,\,\,\,0<y<b,\,\,\,\,\,t>0\,$

• $u(x,y,t)=\sum _{{m,n=1}}^{\infty }\sin({\frac {m\pi x}{a}})\sin({\frac {n\pi y}{b}})\left[A_{{m,n}}\cos({\sqrt {\lambda _{{m,n}}}}\,ct)+B_{{m,n}}\sin({\sqrt {\lambda _{{m,n}}}}\,ct)\right]\,$
  
  

solution $u_{{t}}=k(\Delta u)+q(x,y,t)\,$

$u(x,0,t)=0\,$ $u(x,b,t)=0\,$ $u(0,y,t)=0\,$ $u(a,y,t)=0\,$ $u(x,y,0)=f(x,y)\,$ $0<x<a,\,\,\,\,\,0<y<b,\,\,\,\,\,t>0\,$

solution Transform this equation: $u_{{t}}=\nu u_{{xx}}+\lambda u_{x}+\mu u\,$ into the standard heat equation: $v_{t}=v_{{xx}}\,$

solution $u_{x}+2u_{y}=0,\,\,\,u(0,y)=3e^{{-2y}}\,$

solution $u_{{xx}}=a^{{-2}}u_{t}\,$

$u(0,t)=10\,$ $u(10,t)=30\,$ $u(x,0)=0\,$ $0<x<10,\,\,\,\,\,t>0\,$ $u(x,t)=2x+10+{20 \over \pi }\sum _{{n=1}}^{\infty }{3(-1)^{n}-1 \over n}\sin({n\pi x \over 10})e^{{-a^{2}{n^{2}\pi ^{2} \over 10^{2}}t}}\,$

solution

Solve Dirichlet's problem for a circular annulus. The domain $D\,$ is the space between two concentric circles, $C_{1}$ being the innermost circle with radius $a$, and $C_{2}$ being the outermost circle with radius $b$.
	$\nabla ^{2}u=0\,$ in $D\,$ $u=g\,$ on $C_{1}\,$ $u=f\,$ on $C_{2}\,$

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

Partial Differential Equations

Main Page