SOLVING BOUNDARY VALUE PROBLEMS USING POTENTIAL THEORY. THEORETICAL FOUNDATIONS AND COMPUTATIONAL IMPLEMENTATIONS

Abstract

Boundary Value Problems (BVPs) are central to modeling physical phenomena such as electrostatic fields, steady-state heat conduction, fluid dynamics, and linear elasticity. While classical analytical methods like the separation of variables (Fourier method) are highly effective for simple geometries (e.g., spheres, cylinders, or rectangles), they fail when applied to domains with complex or arbitrary boundaries. To overcome this limitation, Potential Theory offers a powerful alternative. Instead of solving a differential equation inside a domain , the problem is reformulated as an integral equation over the boundary . This method possesses two extraordinary advantages.