A FINITE DIFFERENCES SOLUTION TO THE VIBRATING MEMBRANE PROBLEM

Abstract

Abstract

A realistic approach to the solution of mechanical systems containing multiple parameters must take into account the fact that dependent variables depend not only on, but on more space variables. The modelling of such problems leads to partial differential equations (P.D.Es), rather than Ordinary Differential Equations (O.D.Es). Here, the wave equation, a P.D E that governs the vibrating membrane problem is considered. A finite difference method (F.D.M) is provided as an alternative to the analytic methods. F.D.Ms  basically involve three steps; dividing the solution into grids of nodes, approximating the given differential equation by finite difference equivalences that relate the solutions to grid points and solving the difference equations subject to the prescribed boundary and/or initial conditions. It is shown here that the error in the result is relatively negligible, and the conclusion made that the method developed can further be used to solve certain non-linear P.D.Es.

Key words; Wave Equation, Vibrating membrane, Fouling, Analytic solution, Numerical solution, Local truncation Error, Stability