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.
Author(s) Details:
Jorge E. Macías-Díaz,
Department of Mathematics and Didactics of Mathematics, School of
Digital Technologies, Tallinn University, Narva Rd. 25, 10120 Tallinn, Estonia
and Departamento de Matemáticas y Física, Universidad Autónoma de
Aguascalientes, Avenida Universidad 940, Ciudad Universitaria, Aguascalientes
20100, Mexico.
Tassos
Bountis,
Department
of Mathematics, University of Patras, 26500 Patras, Greece and Center for
Integrable Systems, P. G. Demidov Yaroslavl State University, 150003 Yaroslavl,
Russia.
Please see the link here: https://stm.bookpi.org/RUMCS-V5/article/view/14345
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