Diffusion equation on the positive half-line with Dirac delta-function boundary condition. The diffusion equation $u_{t} = k^{2}u_{xx}$ has initial boundary conditions $u(x,0) = A \delta(x - x_{0})$,  $u(0,t) = 0$ where $A\neq0$ and $x_{0} > 0$ are given constants, and $\delta(\cdot)$ is the Dirac delta-function. I am not sure how to solve this without having another boundary condition. I believe I am supposed to use an (odd) extension to solve this but I'm not sure how to go about doing that.

Consider the same equation on $(-\infty,\infty)\times[0,\infty)$ with initial value $u(x,0)=-A\,\delta(x+x_0)+A\,\delta(x-x_0)$. Since the initial value is odd, the solution will be odd and satisfy the boundary condition $u(0,t)=0$.
