# Heat Equation IBVP on the Quarter plane

I have come across the following Heat equation IBVP but I am not quite sure how to solve it:

$$v_t = kv_{xx} \ \ \ \ \ \ ( 1 < x < \infty, \ \ 0 < t < \infty) ,$$ $$v(x,0) = \delta (x - x_0) \ \ \ \ \ \ \mathrm{for} \ \ t = 0, \ \ x_0 > 1$$ $$v(1,t) = 0 \ \ \ \ \ \ \mathrm{for} \ \ x = 1$$ $$v(\infty,t) = 0 \ \ \ \ \ \ \mathrm{for} \ \ x = \infty.$$

Due to the Dirac delta initial condition, I was thinking some form of fourier like transformation is needed, but I am not sure given the boundary conditions if that will work.

Is there a way to solve this problem ? Thanks.

Start from the 1D heat kernel: $$v(x,t) = \frac{1}{\sqrt{4\pi k t}} e^{-x^2/(4 k t)}.$$ First, shift it so that the initial conditions line up: $$v(x,t) = \frac{1}{\sqrt{4\pi k t}} e^{-(x-x_0)^2/(4 k t)}.$$ This does not satisfy your Dirichlet boundary conditions; however you can fix it by applying the reflection principle:
$$v(x,t) = \frac{1}{\sqrt{4\pi k t}} e^{-(x-x_0)^2/(4 k t)}-\frac{1}{\sqrt{4\pi k t}} e^{-(x-[2-x_0])^2/(4 k t)}.$$