Hybrid Laplace and Poisson Solvers I: Dirichlet Boundary Conditions

Abstract

Herein we numerically solve Laplace and Poisson problems using a “hybrid” version of DFI (Direct Formal Integration) that starts from an IDE (Integro-Differential Equation) formulation of the problem. DFI [1] has been intensively implemented with analytic and numeric success for two decades across physics (e.g., aerodynamics, electromagnetism, heat transfer, solid state physics, “chaos,” population dynamics, and orbital mechanics). DFI has three phases, namely, 1) formally integrate any DE system yielding Volterra-type integral or integrodifferential equations; 2) study both IE/IDE and DE forms for multiple new insights; 3) integrate by hand a few times, near the IP, for further insights and, lastly, code for a digital computer to any desired accuracy, subject only to machine limitations. This method is conceptionally so simple that even Sophomores can and do solve nonlinear DEs using it. A major goal of this chapter is to encourage a wider use of DFI, believed to be a simpler and more accurate DE solver.