Solving the Boundary Value Problems for Differential Equations with Fractional Derivatives by the Method of Separation of Variables

Abstract

This paper is devoted to solving boundary value problems for differential equations with fractional derivatives by the Fourier method. The necessary information is given (in particular, theorems on the completeness of the eigenfunctions and associated functions, multiplicity of eigenvalues, and questions of the localization of root functions and eigenvalues are discussed) from the spectral theory of non-self-adjoint operators generated by differential equations with fractional derivatives and boundary conditions of the Sturm–Liouville type, obtained by the author during implementation of the method of separation of variables (Fourier). Solutions of boundary value problems for a fractional diffusion equation and wave equation with a fractional derivative are presented with respect to a spatial variable.

Keywords

• eigenvalue
• eigenfunction
• function of Mittag–Leffler
• fractional derivative
• Fourier method
• method of separation of variables