The analytical solution for advection-diffusion equation with source term.

We have: $$\frac{\partial w}{\partial t} + a(x) \frac{\partial w}{\partial x} - v \frac{\partial^2 w}{\partial x^2} = f(t)$$ within a domain $x \in [0,1]$

Simplest Sample is $a(x) = 1$ (constant) and $v = 0.01$ (constant) and $f(t) = 1$

with the initial condition

$w(x,0) = 0$ and Boundary condition

$w(0,t)=w(1,t) = 0$

I am stuck in solving this Advection-Diffusion equation with a constant source term. Hopf–Cole transformation could be used to solve the one without the source term, if it has the source term how can I solve that.

Could anyone show the paper or the method how to solve it? Thanks very much.

Consider the special case you mention, constant $a$ and $v$ and $f=1$.In view of the condition $w(0,t)=w(1,t)=0$ you can make an expansion in a Fourier series $$w(x,t)=∑_{n}d_{n}(t)exp[i2πnx]=∑_{n}d_{n}(t)exp[ik_{n}x]$$. Then you can obtain the values of the differential operators acting on the basis functions and the equations for the time-dependent Fourier coefficients.