The general Caputo–Katugampola fractional derivative and numerical approach for solving the fractional differential equations

In 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.