On a Method of Solution of Systems of Fractional Pseudo-Differential Equations

Sabir Umarov; Ravshan Ashurov; YangQuan Chen

doi:10.1515/fca-2021-0011

On a Method of Solution of Systems of Fractional Pseudo-Differential Equations

Fract Calc Appl Anal. 2021;24(1):254-277. doi: 10.1515/fca-2021-0011. Epub 2021 Jan 29.

Authors

Sabir Umarov^{1

2}, Ravshan Ashurov², YangQuan Chen³

Affiliations

¹ Department of Mathematics, University of New Haven, 300 Boston Post Road, West Haven, CT 06516 USA.
² Institute of Mathematics, Academy of Sciences of Republic of Uzbekistan, Tashkent, Uzbekistan.
³ Department of Mechanical Engineering, California University, 5200 N. Lake Road, Merced, CA 95343 USA.

Abstract

This paper is devoted to the general theory of linear systems of fractional order pseudo-differential equations. Single fractional order differential and pseudo-differential equations are studied by many authors and several monographs and handbooks have been published devoted to its theory and applications. However, the state of systems of fractional order ordinary and partial or pseudo-differential equations is still far from completeness, even in the linear case. In this paper we develop a new method of solution of general systems of fractional order linear pseudo-differential equations and prove existence and uniqueness theorems in the special classes of distributions, as well as in the Sobolev spaces.

Keywords: 33E12; 35R11; Mittag-Leffler function; Primary 35E15; Secondary 35S10; fractional order differential equation; fractional system of differential equations; matrix symbol; pseudo-differential operator; solution operator; system of differential equations.