# PDE:Integration and Separation of Variables

solution $\displaystyle 3u_x+4u_y-2u=1, u(x,0)=e^x\,$

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

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

solution $\displaystyle u_{xx}-u_t=0\,$

 solution $\displaystyle u_t=ku_{xx}\,$ $\displaystyle u(x,0) = f(x)\,$ $\displaystyle u(0,t) = 0\,$ $\displaystyle u(l,t) = 0\,$ $\displaystyle u(x,t) = \frac{2}{l} \sum_{n=1}^\infty A_n e^{-\left(\frac{n\pi}{l}\right)^2 k t} \sin(\frac{n\pi x}{l})\,$

 solution $\displaystyle u_t=ku_{xx}\,$ $\displaystyle u_x(0,t) = 0\,$ $\displaystyle u_x(1,t) = 0\,$ $\displaystyle u(x,0) = \phi(x)\,$ $\displaystyle u(x,t) = \sum_{n=1}^\infty A_n e^{-\lambda_n t} \cos(\sqrt{\lambda_n}x)\,$

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

 $\displaystyle u_t=u_{xx} + w(x,t)\,$ $\displaystyle u_x(0,t) = \alpha(t)\,$ $\displaystyle u_x(1,t) = \beta(t)\,$ $\displaystyle u(x,0) = \phi(x)\,$ $\displaystyle 0 $\displaystyle t>0\,$

 solution $\displaystyle u_{tt}=c^2(u_{xx}+u_{yy})\,$ $\displaystyle u(x,0,t) = 0\,$ $\displaystyle u(x,b,t) = 0\,$ $\displaystyle u(0,y,t) = 0\,$ $\displaystyle u(a,y,t) = 0\,$ $\displaystyle u(x,y,0) = f(x,y)\,$ $\displaystyle u_t(x,y,0) = g(x,y)\,$ $\displaystyle 00\,$

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

 solution $\displaystyle u_{t}=k(\Delta u) + q(x,y,t)\,$ $\displaystyle u(x,0,t) = 0\,$ $\displaystyle u(x,b,t) = 0\,$ $\displaystyle u(0,y,t) = 0\,$ $\displaystyle u(a,y,t) = 0\,$ $\displaystyle u(x,y,0) = f(x,y)\,$ $\displaystyle 00\,$

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

solution $\displaystyle u_x + 2u_y = 0,\,\,\,u(0,y) = 3e^{-2y}\,$

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

 $\displaystyle u(0,t) = 10\,$ $\displaystyle u(10,t) = 30\,$ $\displaystyle u(x,0) = 0\,$ $\displaystyle 00\,$ $\displaystyle u(x,t) = 2x+10+{20\over\pi}\sum_{n=1}^\infty {3(-1)^n-1\over n}\sin({n\pi x\over 10}) e^{-a^2{n^2\pi^2\over 10^2}t}\,$

 Solve Dirichlet's problem for a circular annulus. The domain $\displaystyle D\,$ is the space between two concentric circles, $\displaystyle C_1$ being the innermost circle with radius $\displaystyle a$ , and $\displaystyle C_2$ being the outermost circle with radius $\displaystyle b$ . $\displaystyle \nabla^2 u = 0\,$ in $\displaystyle D\,$ $\displaystyle u = g\,$ on $\displaystyle C_1\,$ $\displaystyle u = f\,$ on $\displaystyle C_2\,$

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