Fractional differential equations related to an integral operator involving the incomplete I‐function as a kernel

Abstract

In this study, we present and examine a fractional integral operator with an ‐function in its kernel. This operator is used to solve several fractional differential equations (FDEs). FDE has a set of particular cases whose solutions represent different physical phenomena. Much mathematical physics, biology, engineering, and chemistry problems are identified and solved using FDE. We first solve the FDE and the integral operator for the incomplete ‐function (I F) for the generalized composite fractional derivative (GCFD). This is followed by the discovery and investigation of several important exceptional cases. The significant finding of this study is a first‐order integer‐differential equation of the Volterra type that clearly describes the unsaturated nature of free‐electron lasers.

New Trends on the Mathematical Models and Solitons Arising in Real-World Problems

“…Mathematical models in science and technology have recently attracted an increased amount of research attention with the aim to understand, describe, and predict the future behaviors of natural phenomena. Recent studies on fractional calculus have been particularly popular among researchers due to their favorable properties when analyzing real-world models associated with properties such as anomalous diffusion, non-Markovian processes, random walk, long range, and, most importantly, heterogeneous behaviors [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. The concept of local differential operators along with power law settings and non-local differential operators were suggested in order to accurately replicate the above-cited natural processes.…”