Example of 1D and 2D Poisson Equation

Question

I need some example of 1D and 2D Poisson Equation in Engineering, with Dirichlet Boundaries, that difficult enough to find the exact solution, but it can be solved in numerical ways. Difficult for me to find the example. I will try to find the numerical solution from that equation(with fast fourier transform). So, can you help me please to give the example of that equation? Thank you so much.

Answer 1

The Poisson equation is encountered in electrostatics:

$$\nabla^2\phi = -\frac{\rho_{\text{el}}}{\varepsilon}.$$

Poisson's equation is also used for grid generation in numerical schemes as an extension to the Laplace equation method.