AN INHOMOGENEOUS COMPLETE PDE SOLUTION AND THE IMPLEMENTATION OF BOUNDARY CONDITIONS THROUGH RANDOM WALKS

Abstract

In a Partial Differential Equation (PDE), the inhomogeneous term describes different physical systems that contain sources or sinks such as: charge, matter or energy, while the term that does not have a derivative (zero order term) is related to diverse physical processes such as: Newton cooling, Lambert absorption or radioactive disintegration among others. There are stochastic methods such as random walks to solve equations, for example, Suxo (2011) formulated a theorem applied exclusively to homogeneous PDEs, without taking into account the zero order term. Therefore, in order to extend the study to several physical phenomena, in this work we reformulate that theorem taking into account the zero order term within an inhomogeneous PDE. In addition, we implement the study of the Dirichlet, Neumann and Mixed boundary conditions. Finally, we verify the effectiveness of the study by comparing the results obtained with analytical solutions of the Poisson, Fick and Fourier equations.