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 Under the static assumptions, Maxwell's The steady-state heat diffusion equations reduce to are elliptic partial differential equations known as the Poisson and Laplace equations. These equations can be solved analytically only for a few canonical geometries with very simple boundary conditions. For most practical and realistic problems, you need to utilize a numerical technique and seek a computer solution. The Poisson and Laplace equations can be solved numerically using the finite difference (FD) method.