Given the heat equation is: $\frac{\partial U}{\partial t} - \frac{1}{2}\sigma\frac{\partial^2U}{\partial z^2} = 0$ I need to solve for U through integration, but using the fact that the Normal density is given by the diffusion kernel of: $\frac{1}{\sqrt{2\pi \sigma^2t}}exp(\frac{-(z-\zeta)^2}{2\sigma^2 t})$ I have to give the final answer in CDF N.
I have been given that: I must split the integral into two integrals. Introduce a new variable of integration x into each integral. Where x: $(\frac{\zeta - z -\sigma^2 t}{\sigma\sqrt{t}})$ In the first integral, and let x : $\frac{\zeta - z}{\sigma\sqrt{t}}$ the second integral.
Given initial condition of: $U(z,0) = f(z)$ I dont really know how to approach this question. Can someone show me how I am meant to get two different integrals to substitute the values of x for?