An efficient Hamiltonian numerical model for a fractional Klein–Gordon equation through weighted-shifted Grünwald differences

Abstract

In this work, we investigate numerically a nonlinear wave equation with fractional derivatives of the Riesz type in space. As opposed to previously published papers which employed fractional centered differences, the present approach is based on the use of weighted and shifted Grunwald difference operators. The mathematical model has an associated energy function which is preserved under suitable parameter conditions. In this manuscript, we propose a discrete energy function that estimates the continuous counterpart and which is preserved under the same conditions. As some of the main result of this work, we show that the method is stable and second-order convergent. Moreover, we establish that the technique is quadratically consistent, and we prove the existence and uniqueness of solutions of the numerical model for arbitrary initial conditions. Some numerical results are provided in order to confirm the quadratic order of convergence of the method.