doi: 10.17706/ijapm.2015.5.4.259-266
An Exponential Spline Approach to the Generalized Sine-Gordon Equation
Reza Mohammadi
- the Department of Mathematics, University of Neyshabur, Postal code 91136-899 Neyshabur, Iran
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 <span id="cke_bm_90S" style="display: none;"> </span><span id="cke_bm_86S" style="display: none;"> </span><span id="cke_bm_84S" style="display: none;"> </span><em>O</em><span id="cke_bm_90E" style="display: none;"> </span><span id="cke_bm_86E" style="display: none;"> </span><span id="cke_bm_84E" style="display: none;"> </span>(<span id="cke_bm_91S" style="display: none;"> </span><span id="cke_bm_87S" style="display: none;"> </span><em>k</em><span id="cke_bm_91E" style="display: none;"> </span><span id="cke_bm_87E" style="display: none;"> </span><span id="cke_bm_88S" style="display: none;"> </span><sup>2</sup><span id="cke_bm_92S" style="display: none;"> </span><em>+k</em><span id="cke_bm_92E" style="display: none;"> </span><sup>2</sup><em>h</em><span id="cke_bm_93E" style="display: none;"> </span><span id="cke_bm_88E" style="display: none;"> </span><sup>2</sup>+<em>h</em><sup>2</sup>) an<span id="cke_bm_89E" style="display: none;"> </span>d <em>O</em>(<em>k</em><sup>2</sup><em>+k</em><sup>2</sup><em>h</em><sup>2</sup>+<em>h</em><sup>4</sup>). In the end, some numerical examples are provided to demonstrate the effectiveness of the proposed schemes.<span id="cke_bm_85E" style="display: none;"> </span>
Keywords
- Exponential spline
- finite difference
- generalizedsine-gordon equation
- dirichlet and neumann boundary conditions
- stability analysis
- convergence
How to 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. https://doi.org/ 10.17706/ijapm.2015.5.4.259-266
Copyright & License
Copyright © 2015 by the authors. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).