A Discrete Model for a Caputo–Riesz Time-Space-Fractional Nonlinear Wave Equation: Numerical Efficiency and Energy Dissipation

The investigation of mathematical models with fractional derivatives has become a fruitful area of research in recent decades. For the first time, a new dissipationpreserving scheme is proposed and analyzed to solve a Caputo–Riesz time-spacefractional multidimensional nonlinear wave equation with generalized potential. Classically, Caputo-like fractional operators have found potential applications in studying systems with memory effects. In the present work, we consider a fractional extension of the nonlinear Klein–Gordon equation with damping, which involves Caputo temporal derivatives and Riesz spatial derivatives. We consider initial conditions and impose homogeneous Dirichlet data on the boundary of a bounded hypercube. We introduce an energy-type function and prove that the new mathematical model obeys a conservation law. Motivated by these facts, we propose a finite-difference scheme to approximate the solutions of the continuous model. A discrete form of the continuous energy is proposed and the discrete operator is shown to satisfy a conservation law, in agreement with its continuous counterpart. We employed a fixed-point theorem to establish theoretically the existence of solutions and study analytically the numerical properties of consistency, stability and convergence. We carried out a number of numerical simulations to verify the validity of our theoretical results.