I was wondering how I can determine an integral expression that accounts for $x<0$ for the diffusion equation on the half-plane governed by the PDE
$\frac{\partial C}{\partial t} = D \frac{\partial^2 c}{\partial x^2}$ over $0<x<\infty$ for $t>0$.
My initial condition is $C(x,0)=0$ for all $x\ge0$. I have another condition of $C(0,t)=C_0$. My thoughts on approaching this is to take an even extension, however in lectures an even extension was only shown when we have $C(x,0)=f(x)$. What should I do for when my condition equals $0$? Thanks.