How to solve this coupled linear differential equations?

Question

$\partial_t f(x,t)= \alpha \partial_x^2f+\beta f + \gamma g \\ \partial_t g(x,t)= \alpha \partial_x^2g -\beta f - \gamma g$

With everything real.

I tried to take the first equation and express g(x,t) as function of f(x,t)

$g=\frac{1}{\gamma}\partial_t f-\alpha \partial_x^2f-\beta f$

and substitute g in the second expression. Then trying to solve with Fourier transform doesn't seem to work.

Or adding both equation leads to

$\partial_t(f+g)=\alpha\partial_x^2(f+g)$.

Assuming variables can be separated for (f+g) this can be solved for (f+g) like a wave propagation equation. But anyway this does not solve the problem.

Thanks for indications.

Answer 1

Introduce the new unknown functions $$u:=\beta f+\gamma g,\qquad s:=f+g\ .$$ Then the system gets decoupled: $$u_t=\alpha u_{xx}+(\beta-\gamma) u,\qquad s_t=\alpha s_{xx}\ .$$ Maybe there will be difficulties when $\beta=\gamma$; see Juan Ospina's answer.

Answer 2

Using the Fourier transform and Maple I am obtaining

I am using the initial conditions

$$f \left( x,0 \right) =a{\it Dirac} \left( x \right)$$

$$g \left( x,0 \right) =b{\it Dirac} \left( x \right)$$