Analytical and Numerical Solutions of Partial Differential Equations with Emphasis on Finite Difference Methods
Abstract:
This study explores the analytical and numerical solutions of partial differential equations (PDEs), focusing on hyperbolic the first part presents their analytical solutions using initial and boundary conditions and delves into the finite difference method (FDM), discussing forward, backward, and central difference schemes. These methods are applied to numerically solve one- and two-dimensional heat. The Crank-Nicolson method, recognized for its unconditional stability, is employed to improve the accuracy of wave equation solutions, overcoming limitations of explicit and implicit schemes. We then analyze the performance, strengths, and weaknesses of FDM through nu merical simulations of one-dimensional heat. Due to computational constraints, Crank-Nicolson for 1Dsimulation, was not executed. Results indicate that the implicit backward difference method demonstrates superior stability by allowing unrestricted step sizes compared to the explicit forward difference method. These findings contribute to a deeper understanding of numerical PDE solutions and stability considerations in computational mathematics.
