Lachache, QamarToufik, HERAIZ: Supervisor2024-07-172024-07-172024-06-10https://dspace.univ-msila.dz/handle/123456789/43886Fractional analysis is a branch of mathematical analysis which studies the possibility of defining noninteger powers of derivative and integrating operators. we define some useful functions such as Gamma function, Beta function and MittagLeffler function. These functions play a very important role in the theory of fractional order differential calculus. we will present the definition of a fractional derivative and then we study the three most popular approaches to fractional derivatives: the Grünwald-Letnikov, Riemann-Liouville and Caputo approaches as well as their properties. Finally, we study some examples of fractional derivatives. Keywords: Fractional derivation, Grünwald-Letnikov, Riemann-Liouville, Caputo. and operator theory to study the spectral properties (such as eigenvalues and eigenfunctions) of fractional differential operators.enClosed OperatorspectrumDifferential of fractional orderThesis