Research Projects

Fractional calculus emerged from discussions between mathematicians de l'Hospital and Leibnitz. Various definitions of fractional derivatives have emerged, often aimed at addressing the properties of classical ordinary derivatives or demonstrating physical processes. Mathematicians have proposed various definitions, including the Ψ-Hilfer derivative, K-Gamma function, k-Riemann-Liouville fractional integral, and (k, Ψ)-Hilfer derivative. The tempered Ψ-Hilfer derivative, defined by Kucche and Mali, is the most generalized fractional derivative and is used to analyze nonlinear fractional differential equations under initial, impulsive, and boundary conditions. The project aims to analyze these derivatives under initial, impulsive, and boundary conditions, examining new criteria for solution existence, uniqueness, extremal solutions, comparison results, data dependency results, stability results, and quasilinearization method application. The project also aims to merge various definitions of fractional derivatives into a single generalized fractional derivative operator, eliminating the need to study fractional differential equations with separate derivatives. This approach allows for the investigation of fractional differential equations with a single generalized fractional derivative operator, eliminating the need to study each derivative separately.