Numerical investigation of fractional nonlinear sine-Gordon and Klein-Gordon models arising in relativistic quantum mechanics

Abstract

This paper presents a method for the approximate solution of the time-fractional nonlinear sine-Gordon and the Klein-Gordon models described in Caputo sense and with the order 1 < γ < 2. This method discretizes the unknown solution in two main steps. First, the temporal discretization of the governing problems is obtained by means of the finite difference scheme. Second, the spatial terms are expanded using local radial basis functions, where each basis function is approximated by a weighted linear summation of function values. The spatial discretization achieved through the local radial basis functions and finite difference (LRBF-FD) is highly accurate, since in local collocation techniques we only consider the nodes located in the subdomain around the collocation node surrounding the local collocation point. In fact, the new approach avoids the ill-conditioning problem resulting from the adoption of the global collocation and leads to sparse systems. The numerical stability and convergence are examined and confirmed numerically to support the theoretical formulation. Numerical experiments assess the effectiveness and capability of the algorithm and demonstrate its good computational performance on irregular domains.