article · Alexandria Engineering Journal
This research introduces the psi-Caputo-Katugampola fractional derivative and its corresponding integral, providing a generalised framework in fractional calculus. Existing forms such as the Caputo-Katugampola, Caputo, and Caputo-Hadamard fractional derivatives serve as special cases within this new formulation. Along with the psi-Katugampola fractional integral, the study presents related theoretical properties, establishing an existence and uniqueness theorem for the associated fractional Cauchy problem. To solve these equations, an adaptive predictor-corrector numerical algorithm is developed. The effectiveness of this computational method is demonstrated through specific examples and applications. Because the behaviour of the derivative depends heavily on chosen parameters and functions, it offers a flexible tool for formulating and analysing fractional calculus models across various technical contexts.
Fractional calculus provides mathematical tools to model complex systems with memory and hereditary properties. By unifying several existing fractional derivatives into a single, general framework and pairing it with a reliable numerical solver, this work simplifies the mathematical analysis and computational simulation of complex physical and engineering phenomena.
The abstract does not indicate an application pathway, representing early-stage theoretical and numerical mathematical research without specified industrial users or products.
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In this manuscript, we present the general fractional derivative (FD) along with its fractional integral (FI), specifically the ψ -Caputo–Katugampola fractional derivative ( ψ -CKFD). The Caputo–Katugampola (CKFD), the Caputo (CFD), and the Caputo–Hadamard FD (CHFD) are all special cases of this new fractional derivative. We also introduce the ψ -Katugampola fractional integral ( ψ -KFI) and discuss several related theorems. An existence and uniqueness theorem for a ψ -Caputo–Katugampola fractional Cauchy problem ( ψ -CKFCP) is established. Furthermore, we present an adaptive predictor–corrector algorithm for solving the ψ -CKFCP. We include examples and applications to illustrate its effectiveness. The derivative used in our approach is significantly influenced by the parameters δ , γ , and the function ψ , which makes it a valuable tool for developing fractional calculus models.
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DOI: 10.1016/j.aej.2025.02.065
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