What is the difference between poisson and laplacian equations?

Question

I know that laplacian equation use when we didn't have a charge But i need the explanation between the poissons and laplacian equations physically

Answer 1

The only difference, as you say, is the right-hand side: $$\Delta u = f$$ where the special case $f=0$ is the Laplace equation.

The physical intuition is that $f$ is a "source term": if you think of the Poisson equation as describing the steady state temperature of a region, when $f=0$ you are assuming heat enters and leaves only through the boundary, whereas when $f\neq 0$ you are including some "hot plates" in the interior as well. The same intuition also applies when you interpret the Poisson equation as describing an electric potential: $f$ now is the charge density on the interior. And the same also applies for the gravitational potential and mass density.

The special case gets its own name since the solutions of the Laplace equation are harmonic and have very special properties, such as the maximum principle etc.