Analyzing two-dimensional sine–Gordon equation with the mesh-free reproducing kernel particle Ritz method

We consider an analysis of the non-linear two-dimensional sine–Gordon (SG) equation. The analysis is performed using the mesh-free reproducing kernel particle Ritz method (kp-Ritz method). The mesh-free kernel particle estimate is employed to approximate the 2D displacement field. A system of discrete equations is obtained through application of the Ritz minimization procedure to the energy expressions. To validate the accuracy of the results and stability of the present method, convergence studies were carried out based on influences of support size and number of nodes. The present results were compared with results reported in extant literature and were found to be in good agreement with the literature.

► Nonlinear sine-Gordon equation arises in many engineering and scientific fields.
► Nonlinear sine–Gordon equation is a basic equation of nonlinear wave theory.
► Reproducing kernel particle estimation is used to study the sine–Gordon equation.
► The kp-Ritz method overcomes the disadvantages of the conventional Ritz method.