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Finite difference solution of a nonlinear Klein-Gordon equation with an external source

Authors: G. Berikelashvili, O. Jokhadze, S. Kharibegashvili and B. Midodashvili
Journal: Math. Comp. 80 (2011), 847-862
MSC (2010): Primary 65M06, 35L70
Published electronically: August 25, 2010
MathSciNet review: 2772098
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Abstract: In this paper, we consider the Darboux problem for a (1+1)-dimensional cubic nonlinear Klein-Gordon equation with an external source. Stable finite difference scheme is constructed on a four-point stencil, which does not require additional iterations for passing from one level to another. It is proved, that the finite difference scheme converges with the rate $ O(h^2)$, when the exact solution belongs to the Sobolev space $ W_2^2$.

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Additional Information

G. Berikelashvili
Affiliation: A. Razmadze Mathematical Institute, 1 M. Aleksidze str., 0193, Tbilisi, Georgia

O. Jokhadze
Affiliation: A. Razmadze Mathematical Institute, 1 M. Aleksidze str., 0193, Tbilisi, Georgia

S. Kharibegashvili
Affiliation: A. Razmadze Mathematical Institute, 1 M. Aleksidze str., 0193, Tbilisi, Georgia

B. Midodashvili
Affiliation: I. Javakhishvili Tbilisi State University, 2, University str., 0186, Tbilisi, Georgia

Keywords: Nonlinear Klein-Gordon equation, Darboux problem, finite difference solution
Received by editor(s): May 13, 2008
Received by editor(s) in revised form: February 4, 2010
Published electronically: August 25, 2010
Article copyright: © Copyright 2010 American Mathematical Society