Abstract—The nonlinear sine-Gordon equation is used to model many nonlinear phenomena. Numerical simulation of the solution to the one-dimensional generalized sine-Gordon equation is considered here. Two implicit three time-level difference schemes are developed, by using the exponential spline function approximation. We consider both Dirichlet and Neumann boundary conditions. The resulting spline difference schemes are analyzed for local truncation error, stability and convergence. It has been shown that by suitably choosing the parameters, we can obtain two schemes of    O   (  k   ² +k ²h  ²+h²) an d O(k²+k²h²+h⁴). In the end, some numerical examples are provided to demonstrate the effectiveness of the proposed schemes. 

Index Terms—Exponential spline, finite difference, generalizedsine-gordon equation, dirichlet and neumann boundary conditions, stability analysis, convergence.

The author is with the Department of Mathematics, University of Neyshabur, Postal code 91136-899 Neyshabur, Iran (email: rez.mohammadi@gmail.com).

Cite: Reza Mohammadi, "An Exponential Spline Approach to the Generalized Sine-Gordon Equation," International Journal of Applied Physics and Mathematics vol. 5, no. 4, pp. 259-266, 2015.