Dérivations fractionnaires discrètes et applications numériques.

Abstract

This thesis focuses on the construction of discrete fractional operators as fractional powers of sectorial operators. This approach requires the construction of resolvents and control of their norms in various Banach spaces. This original approach in the discrete case, allowed us to find some operators and to build a new one. Following this, essential convergence results are proved. We show the uniform convergence of the operator nabla h-sum to the Riemann-Liouville integral in the spaces of the continuous functions and then in weighted continuous spaces. Also, the strong convergence of fractional nabla operators to the fractional derivative operator in Hölder spaces is proved. As applications the resolution of some problems of fractional Cauchy is proposed.