An Entropy Paradox Free Fractional Diffusion Equation

Ortigueira, Manuel Duarte

doi:10.3390/fractalfract5040236

Cited by 6 publications

(1 citation statement)

References 42 publications

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“…It is important to point out that the theory of fractional calculus itself has benefited tremendously from these contributions [11,12]. On the one hand, various new fractional operators have been proposed and analyzed in the literature [13,14], and new analytical results have been derived in each of these cases [15,16].…”

Section: Introductionmentioning

confidence: 99%

An Efficient Dissipation-Preserving Numerical Scheme to Solve a Caputo–Riesz Time-Space-Fractional Nonlinear Wave Equation

Macías‐Díaz

Bountis

2022

Fractal Fract

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For the first time, a new dissipation-preserving scheme is proposed and analyzed to solve a Caputo–Riesz time-space-fractional multidimensional nonlinear wave equation with generalized potential. We consider initial conditions and impose homogeneous Dirichlet data on the boundary of a bounded hyper cube. We introduce an energy-type functional 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 employ a fixed-point theorem to establish theoretically the existence of solutions and study analytically the numerical properties of consistency, stability and convergence. We carry out a number of numerical simulations to verify the validity of our theoretical results.

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