Authors: Ravindra K. Joshi, Meena L. Patwardhan
Abstract: Partial Differential Equations (PDEs) play a central role in modeling a wide variety of physical, biological, and engineering phenomena such as heat conduction, fluid flow, wave propagation, electromagnetics, and population dynamics. Analytical solutions to PDEs are available only for a limited class of problems with simplified geometries and boundary conditions. As a result, numerical methods have become an essential tool for obtaining approximate solutions to PDEs arising in real-world applications. This paper presents a comprehensive review of numerical methods for solving PDEs, with emphasis on classical and modern approaches. Finite difference methods, finite element methods, and finite volume methods are discussed in detail, along with spectral methods and meshless techniques. Issues related to stability, convergence, accuracy, and computational efficiency are also examined. The paper further highlights recent developments in adaptive schemes and high-performance computing for PDE solvers. The aim of this review is to provide a structured understanding of numerical PDE methods for students, researchers, and practitioners in engineering mathematics and applied sciences.
Keywords: Partial differential equations; numerical methods; finite difference method; finite element method; finite volume method; stability and convergence
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