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 Sine-Gordon equation with first time derivative. The effects of vary¬ing the junction penetration depth and surface damping parameter on the current flowing through the long Josephson junction have not yet been investigated using a model of (2+ 1) dimensional Sine-Gordon equation of the form Uxx + Uyy - _\ Utt - ~ Ut = ,\ sinu. C C "J This equation that governs the current flowing through long .Josephson junction is solved numerically by Finite Difference Method using MATLAB computer software programme. The results obtained are used to investigate the effects of varying the junction penetration depth and surface damping parameter on the current flowing through the long Josephson junction. An Alternating Direction Explicit and Alternating Direction Implicit numerical schemes for the equation are developed and concepts of consistency, stability and conver¬gence analysed and discussed. The two schemes are found to converge when Conditional Convergence theorem is used. Taylor's series expansion is used to expand the finite differ¬ence approximations in the two schemes and it is found out that both of the two schemes are consistent with the model equation. Matrix Method is used to analyse stability of the schemes developed and the two schemes are found to be unconditionally stable. The solution results for the model equation indicate that for a given value of x, the solution u values tends increase to infinity as t increases to infinity while for a given value of t, the solutions decrease to zero as x tends to increase to infinity. The study also reveals that an increase in the surface damping parameter of the junction causes an increases in the current flowing through Josephson Junction while an increase in the penetration depth of the junction causes a decrease in the current flowing through the junction. This study is a big contribution to mathematical knowledge, that the results obtained may find important applications in future electronic devices.