Numerical Solution and Stability for Model of Extensible Beam

Abstract

In this paper, numerical methods (finite differences methods for explicit and implicit) has been applied, to solve nonlinear partial differential equations. In methodology, the beam was divided into very smaller squares, then the study discussed three partial differential equations generating from model. The first equation called longitudinal vibrations of a beam, second equation known as transverse vibrations of a beam and then the third equation considered the extensible beam. The equation of extensible beam was defined by Woiniwsky- Krieger as a model for transverse deflection of an extensible beam of natural length. The study discussed the stability of these models (longitudinal vibrations, transverse vibrations and extensible beams). The stability solution has been counted and considered unconditionally for implicit method, but it's conditional for an explicit method. Obtaining the stability and convergent solution for longitudinal vibrations of a beam if width divisions is less than length divisions (R<2), and for transverse vibrations of a beam if width divisions less than the square length divisions (R<0.25), as well as for extensible beam if width divisions less than the square length divisions, the study recommended to use an implicit method. But in case of using an explicit method, the divisions must be adhered for a stable and convergent solution