J. Nonl. Evol. Equ. Appl. 2023 (2), pp. 19-33, published on June 5, 2023:

Stability and convergence of a numerical scheme for advection-diffusion equations involving a fractional Laplace operator

Martin Nitiema1, Somdouda Sawadogo2,

1 Université Joseph KI-ZERBO, Ouagadougou, Burkina Faso, and Laboratoire LAMIA, Université des Antilles, Pointe-à-Pitre, Guadeloupe
2 Ecole Normale Supérieure, Laboratoire LANIBIO, Ouagadougou, Burkina Faso

Received on November 28, 2022, revised version on January 16, 2023
Accepted on January 18, 2023

Communicated by Mahamadi Warma

Abstract.  We consider an advection-diffusion equation involving a Laplace fractional operator of order 1/2 < s < 1. We first assume that the solution of this equation is regular. Then we use a combination of left and right fractional Riemann-Liouville derivatives of order 2s to approximate the fractional Laplace operator. This allows us to obtain a numerical scheme of Euler explicit type which is proven to be conditionally stable, first order in time an space accurate. Numerical results are given to illustrate the results.
Keywords: C0 fractional Laplace operator, fractional Riemann-Liouville derivatives, Euler explicit scheme, stability and convergence.
2010 Mathematics Subject Classification:   35R11; 35S15; 65M12.

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