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.
Highlights
IntroductionThe paper is devoted to the method of separation of variables (the Fourier method)
This method, which is so widely used in solving boundary value problems for partial differential equations of integer order, until recently remained unsuitable for solving boundary value problems for differential equations with fractional derivatives
In this paper, a method of separation of variables is presented for solving boundary value problems for differential equations with fractional derivatives of the form
Summary
The paper is devoted to the method of separation of variables (the Fourier method) This method, which is so widely used in solving boundary value problems for partial differential equations of integer order, until recently remained unsuitable for solving boundary value problems for differential equations with fractional derivatives. In this paper, a method of separation of variables is presented for solving boundary value problems for differential equations with fractional derivatives of the form. The Green’s functions of boundary value problems for equations of the form (2) are considered in detail (it should be noted that these Green’s functions were first obtained by the author in his post-graduate student paper [2]), the study of which made it possible to approach problems of the distribution of zeros of a function of the Mittag type from completely new positions-Leffler and reveal the deeply hidden properties of these functions, which for many years have not been possible for specialists in the theory of functions.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
