Consider the standard 1D non-dimensionalized heat equation, $$u_t = u_{xx}$$ subject to: $$u(x=0,t)=\delta(t), u(x=1,t)=1, u(x,0)=f(x)$$
The way I decided to break the problem down was by splitting it up into two parts. So, $$u = v+w$$ Where, $$v(x=0,t)=0, v(x=1,t)=1, v(x,0)=f(x)$$ and $$w(x=0,t)=\delta (t), w(x=1,t)=0, w(x,0) = 0$$
I can solve the PDE for $v$ using the steady-state solution and so on, but how do I resolve the dirac delta distribution showing up in my problem?