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My question is, how to solve an inhomogeneous heat equation on finite interval. The problem is $u_{t} - ku_{xx} = f(x, t) $ on finite interval with homogeneous Dirichlet boundary condition $ u(0, t) = u(L, t) = 0$ and initial condition $ u(x, 0) = \phi(x) $ in $ 0 < x < L$. My approach is using extended function that, solving $ u_x - ku_{xx} = f_{ext}(x, t)$ with homogeneous DC and initial condition $ u(x, 0) = \phi_{ext}(x)$ where $$ f_{ext} = \begin{cases} f(x, t) & \mbox{if } 0<x<L \\ -f(-x, t) &\mbox{if } -L < x< 0\\ \mbox{extended to be of period 2L} \end{cases}$$ and $\phi_{ext}$ defined likely. I think that the solution is, $$ u (x, t) = \int_{-\infty}^{\infty} S(x-y, t) \phi_{ext}(y) \, dy \, + \int_0^t \!\! \int_{-\infty}^{\infty} S(x-y, t-s) f_{ext}(y, s) \, dy \,ds.$$ $S(x, t) = \frac{1}{\sqrt{4\pi kt}} \exp(-\frac{x^2}{4kt})$ is heat kernel. I think that this solution satisfies boundary condition, since two extended functions are odd-symmetric to $x = 0$ and $x = L$. I wonder that whether this argument is valid. Further, can this method be applied for Neumann boundary and Robin boundary condition?

Summary

  • How to solve 1-D inhomogeneous heat equation in finite interval?
  • Is the method above valid?
  • If it is valid, can it be applied for other boundary conditions?
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