DIRECT AND INVERSE SOURCE PROBLEMS FOR GENERALIZED FRACTIONAL DIFFERENTIAL EQUATIONS

Abstract

In this paper, we investigate the question of solution existence for higher order fractional differential equations that involve both Riemann-Liouville and Caputo type fractional derivatives. To prove our main results, we use the Laplace transform method, which provides a powerful tool for dealing with fractional operators and finding explicit formulas for solutions. Moreover, we investigate some inverse source problems for the class of higher order fractional differential equations under consideration. We study the problem of determining unknown sources in the equations under some additional conditions imposed on the solutions, called the over-determination condition y(T) = h. The results obtained in this study contribute to the development of the theory of fractional calculus and its applications in various fields of mathematical physics and engineering sciences, in which fractional differential equations arise in a natural way to describe memory and hereditary properties of various phenomena.