Algorithms to Approximate Solutions of Poisson\u27s Equation in Two and Three Dimensions

The focus of this research was to develop numerical algorithms to approximate solutions to Poisson\u27s equation in two and three dimensions. Numerical analysis of partial differential equations is vital to understanding and modeling these complex problems. A finite difference approximation of Poisson\u27s equation can be used to form a system of linear equations of solutions through a region. A computer program was developed to solve this system with inputs such as boundary conditions and a nonhomogenous source function. Approximate solutions were compared with exact solutions to prove their accuracy. The program was tested with an increasing number of subintervals to ensure that the approximations got closer to the actual solution. Then, an experiment was performed to find the temperatures through a heated piece of aluminum foil to show how this approximation can predict real world phenomenon