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^{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

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