Finite difference solution of (1+1) sinegordon equation: A mathematical model for the rigid pendula attached to a stretched wire

Abstract

The nonlinear (1+1) Sine-Gordon equation that governs the vibrations of the rigid pendula attached to a stretched wire is solved. The equation is discretized and solved by Finite Difference Method with specific initial and boundary conditions. A Crank Nicolson numerical scheme is developed with concepts of stability of the scheme analysed using matrix method. The resulting systems of linear algebraic equations are solved using Mathematica software. The solutions are presented graphically in three dimensions and interpreted. The numerical results obtained indicate that the amplitudes of the rigid pendula attached to a stretched wire vary inversely as the position of the travelling waves produced on the stretched wire. The efficacy of the proposed approach and the results obtained are acceptable and in good agreement with earlier studies on the rigid pendula attached to a stretched wire.