I am wondering whether the following differential equation can be solved.
$$\frac{\partial^{2}f}{\partial x^2}+ \frac{\partial^{2}f}{\partial y^2}+ \frac{\partial^{2}f}{\partial z^2}+ \alpha \frac{\partial f}{\partial t} = g,$$
where $f= f(x,y,z,t)$, $\alpha$ is a constant and $g=g(x,y,z,t)$ is a given continuous function. This equation is similar to the heat equation except for function $g$.
Is there any analytic solution for this kind of equation? If not, what is the appropriate numerical method to solve it?
Thanks for any hint!
The equation is called inhomogeneous heat equation.
If and how a analytical solution can be obtained is depending on the domain $(x,y,z)\in\Omega$, the boundary conditions the initial data and $g$. Especially the following cases are noteworthy
$\Omega=\mathbb{R}^3$: Use the fourier transform in the space domain.
$\Omega$ is a cuboid: Use a separation ansatz for the space domain which (for suitable boundary conditions) results in a sum over sine and cosine functions.
Additionally, it is possible to solve via fundamental solution but depending on the domain it might be hard to achieve.
Hope that helps.