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.
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