Finite Difference Solution of (2+1)-Dimensional Sine-Gordon Equation: A Model for Investigating the Effects of Varying Surface Damping Parameter on Josephson Current Flowing through the Long Josephson Junction

Modeling of some physical phenomena and technological processes taking into account dissipation leads to the Sine-Gordon equation with the first time derivative. The (2+1) Sine-Gordon equation with the first time derivative is used in explaining a number of physical phenomena including the propagation of fluxons in Josephson junctions. This study uses Finite Difference Method to solve (2+1) dimensional Sine-Gordon equation with the first time derivative that models dissipation of the current flow through the long Josephson junction. An Alternating Direction Implicit numerical scheme for the equation is developed with concepts of stability tested using Matrix Method. The value of surface damping parameter used are =1.1, =3.7, =7.2 and =9.3 for Alluminium (Al), Tin (Sn), Lead (Pb), and Niobium (Nb) respectively. The numerical results obtained are presented in tables and graphs. The computational results obtained indicate that as the length of long Josephson junction increases, the current flowing through the long Josephson junction decreases to zero. The results also indicate that when the surface damping parameter increases, the current flowing through the long Josephson junction also increases.