[Math] PDE with a Dirac Delta function as boundary condition

Question

I would like to have some information how to solve this PDE:
$\partial_tu(x,t)=k^2\partial_{xx}u(x,t)$
with the following boundary and initial conditions:
$u(x,0)=u_0(x)$,$u(0,t)=\delta(t-t_0)$,$u(L,t)=u_L(t)$
Thanks.

Answer 1

This is a classical Diffusion PDE, if I have understood your question correctly.

In fact its solution was marking the birthday of Fourier technique. The straight forward solution is really standard literature since 190 years and you will find it >>> here for the so called Heat Equation as invented by Fourier. It would be quite redundant to copy/paste all hereto.

When applying the technique you particularly arrive to the point where the constants A, B, C need to be at final steps identified, there you should apply instead, your own boundary/initial conditions.

This should equip you best for the solution of the problem.