Development of numerical methods to solve partial differential equations is extremely important as most of the real life physical systems can be formulated in the form of these. By solving partial differential equations, the characteristics of a physical system can be simulated even without performing laboratory experiments. This book discusses the development of few efficient numerical tools to solve diffusion and wave equation. Methods for treating diffusion of atoms, molecules, ions and charge carriers in solid state and optoelectronic structures are presented. Additionally, split-step based numerical method for bi-directional wave propagation is presented. This method requires only matrix multiplications at each step and no inversion or diagonalization is required.The numerical tools presented in the book are applied to analyze numerous diffusion and wave propagation based processes.