I'm having problems with solving a reaction-diffusion equation, since I barely have any experience with PDEs.
I know:
$u_t = Du_{xx}$ with $u(x,0)=u_0$ is solved by the so-called fundamental solution: $\frac{1}{2 \sqrt{\pi D t}}\int e^{- \frac{(x-y)^2}{4Dt}}u_0 \, dy$
But my question is, what if I have $u_t = Du_{xx}-\gamma u$. What is the solution of it then? $\gamma \in \mathbb{R}$.
I've tried using: $v := e^{\gamma t} u$
The derivative of $v$ w.r.t. $t$ is then: $$\frac{\partial}{\partial t}v = e^ {\gamma t}(Du_{xx}-\gamma u) + \gamma e^{\gamma t} u = e^{\gamma t}Du_{xx}$$ So if I solve $v_t = e^{\gamma t}Du_{xx}$, then $u$ would just be $u = e^{-\gamma t}v$. Unfortunately, I struggle with solving $v_t$. Does anyone know how I should proceed?
Use Fourier transform to get an algebraic equation of second order, then return by iversing the transform.