Sadek, LakhlifaLdris, Sahar AhmedJarad, Fahd2025-05-132025-05-1320251110-01682090-2670https://doi.org/10.1016/j.aej.2025.02.065https://hdl.handle.net/20.500.12416/9991Sadek, Lakhlifa/0000-0001-9780-2592In this manuscript, we present the general fractional derivative (FD) along with its fractional integral (FI), specifically the psi-Caputo-Katugampola fractional derivative (psi-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 psi-Katugampola fractional integral (psi-KFI) and discuss several related theorems. An existence and uniqueness theorem for a psi-Caputo-Katugampola fractional Cauchy problem (psi-CKFCP) is established. Furthermore, we present an adaptive predictor-corrector algorithm for solving the psi-CKFCP. We include examples and applications to illustrate its effectiveness. The derivative used in our approach is significantly influenced by the parameters delta, gamma, and the function psi, which makes it a valuable tool for developing fractional calculus models.eninfo:eu-repo/semantics/closedAccessNumerical MethodsPsi-CfkdPsi-CfkiPsi-CkfcpArticle10.1016/j.aej.2025.02.0652-s2.0-86000149811