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Abstract

The well-known Laplace equation, having an elliptical character, is popularly known as the Poisson partial differential equation. It occupies a central place in mathematical physics and engineering due to its ability to describe steady-state diffusion, electrostatics, gravitational potential, and heat conduction problems. The Poisson partial differential equation in three variables is solved by employing the triple Laplace transform in Laplace domain. In original domain, the solutions are obtained numerically using the Brancik technique. The numerous illustrative examples are presented to demonstrate the behavior and sensitivity of the solutions. This dual approach analytical simplification via the triple Laplace transform and accurate numerical inversion via Brancik’s method enables us to effectively solve the Poisson equation in three variables with improved precision and computational efficiency. Graphical representations are illustrated using MATLAB software.

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