7 / 2019-05-29 17:33:32

FINITE DIFFERENCE SOLUTION OF (2+1) DIMENSIONAL SINE - GORDON EQUATION: A MATHEMATICAL MODEL FOR INVESTIGATING THE LONG JOSEPHSON JUNCTION

CONSISTENCY,STABILITY,SURFACE DAMPING PARAMETER, ADEI

The study considers the finite difference solution of (2+1) dimensional Sine-Gordon equation which is a mathematical model for investigating the long Josephson junction. Modeling of some physical phenomena and technological processes taking into account dissipation leads to use of Sine-Gordon equation with first time derivative. The effects of varying the junction penetration surface damping parameter on the current owing through the long Josephson junction have not yet been investigated using a model of (2+1) dimensional Sine-Gordon equation. This equation that governs the current flowing through long Josephson junction is solved numerically by Finite Difference Method using MATLAB computer software. The results obtained are used to investigate the effects of varying the junction surface damping parameter on the current flowing through the long Josephson junction. An Alternating Direction Implicit numerical scheme for the equation is developed and concepts of consistency and stability analysed and discussed. Taylor's series expansion is used to expand the finite difference approximations in the scheme for consistency while Matrix method is used to determine stability of the scheme. The scheme is found to consistent and unconditionally stable. The study also reveals that an increase in the surface damping parameter of the junction causes an increase in the current flowing through the Josephson junction. This study is a big contribution to mathematical knowledge, and the results obtained will find important applications in future electronic devices.