Fractional differential equation pertaining to an integral operator involving incomplete H-function in the kernel

Fractional differential equations (FDEs) involving a family of special functions and their solutions represent different physical phenomena. FDEs are characterizing and solving many problems of mathematical physics, chemistry, biology, and engineering. In this article, we establish an integral operator involving the family of incomplete H-function (IHF) in its kernel. First, we derive the solutions for FDEs involving the generalized composite fractional derivative (GCFD) and integral operator associated with the incomplete H-function. Several important special cases are revealed and analyzed. The main result derived in this study contains first-order Volterra-type integro-differential equation describing the unsaturated nature of the free electron laser as a special case. Further, we give the graphical interpretation of the solution of FDEs.