STUDY OF SOME EVOLUTION EQUATIONS INVOLVING RIESZ FRACTIONAL DERIVATIVE WITH SINGULAR INITIAL

STUDY OF SOME EVOLUTION EQUATIONS INVOLVING RIESZ FRACTIONAL DERIVATIVE WITH SINGULAR INITIAL

M. Chaib , A. Taqbibt , M. Elomari , S. Melliani

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Abstract

The objective of this work is to study some evolution problems involving the Riesz fractional derivative with singular initial data which can be distributions. It is a question of proving the existence and uniqueness of the solutions of these problems in the extended Colombeau algebra GeR . It is established that the existence and uniqueness generalized solutions hold for both evolution problems associated to the Schr¨odinger equation and the heat equation involving the corresponding Riesz fractiononal operators derivatives.

Keywords

Colombeau algebra, Generalized functions, Evolution problem, Generalized semigroup, Riesz fractional derivative.