A Thesis Submitted In Partial Fulfillment Of The Requirements For The Degree Of Masters Of Science In Numerical Analysis.

Abstract

Partial differential equations are difficult to solve analytically in general but, there exist a variety of numerical schemes that convert impossible calculus into myriad simple calculations that a computer can readily perform. This study presents an explanation of a time dependent damped wave equation from a vibrating string considering the transverse displacement of a plucked string and the subsequent vibration of the string, and application of finite difference methods for solving damped wave equation. The study is planned to find numerical solution of damped wave equation using explicit and implicit finite difference methods. The finite difference method discretizations proceed by replacing the derivatives in the damped wave equations by finite difference approximations. These give a large number of algebraic systems of equations to be solved. The computational experiments are performed writing MATLAB program or using Thomas Algorithm for explicit and implicit finite difference methods to generate numerical solutions to the damped wave equation. Stability, consistency and convergence of the methods are considered based on and in the study. The methods approximate the solutions with consistency of the O {( )2 , ( )2}, for examining the accuracy of the results. Numerical examples are presented for each method. The results are compared with the analytic solution, obtained by the methods of separation of variable. Based on error analysis, we present results using tables and graphs.