PDE:Integration and Separation of Variables

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

solution z_{xy} = x^2y, 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 k t} \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}}\,c t) + B_{m,n} \sin(\sqrt{\lambda_{m,n}}\,c t)\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, C1 being the innermost circle with radius a, and C2 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

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