Numerical and Analytical Approaches to Solving Partial Differential Equations

Dr. Raman

Partial Differential Equations (PDEs), Finite Difference Method (FDM), Finite Element Method (FEM)

Partial Differential Equations (PDEs) are integral to modeling and solving problems involving continuous change across multiple variables. This paper delves into the theoretical foundations, analytical techniques, and diverse applications of PDEs. It begins with the classification of PDEs into elliptic, parabolic, and hyperbolic types, elucidating their unique characteristics and solution methods. We explore classical analytical methods, including separation of variables, Fourier and Laplace transforms, and Green’s functions, alongside modern numerical approaches such as the Finite Difference Method (FDM), Finite Element Method (FEM), and spectral methods. Through detailed case studies, we illustrate the application of PDEs in physics, engineering, biology, and economics, emphasizing their role in solving real-world problems. Additionally, the paper addresses advanced topics like nonlinear and stochastic PDEs, fractional calculus, and the burgeoning intersection of machine learning with PDEs. This comprehensive review aims to provide a thorough understanding of PDEs, highlighting current research trends and potential future directions in this pivotal area of mathematics.