Numerical solution of nonlinear Klein-Gordon equation using the element-free kp-Ritz method

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Original languageEnglish
Pages (from-to)2917-2928
Journal / PublicationApplied Mathematical Modelling
Issue number10-11
Publication statusPublished - 2015


This paper presents a numerical analysis of the one-dimensional Klein-Gordon equation with quadratic and cubic nonlinearity, using the element-free reproducing kernel particle Ritz method (kp-Ritz method). Approximation of the wave displacement is expressed according to a set of meshfree kernel particle functions. Based on the established functional corresponding to the nonlinear Klein-Gordon equation, a system of nonlinear discrete equations can be obtained by applying the Ritz minimization procedure. The Newmark integration scheme combined with an iterative technique is applied to the resulting nonlinear system equations. Numerical examples are tested to validate the proposition that the presented numerical solutions are in good agreement with analytical results available in extant literature.

Research Area(s)

  • Element-free kp-Ritz method, Meshless method, Newmark's integration scheme, Nonlinear Klein-Gordon equation