| Abstract: |
| We consider a Caputo-type fractional derivative of order $q\in (0,1]$ with a variable kernel $\psi$, which has been introduced in the literature for its efficacy in analyzing real-world models through appropriate selection of fractional derivatives. This inspired us to incorporate this generalized fractional operator in the logistic differential equation, pivotal in studying population dynamics. We identify the equilibrium points and evaluate their stability using the $\psi-$Laplace transform technique. The proof of the solution`s existence and uniqueness is achieved through employing the fixed-point theorem. Additionally, we derive the representation for the analytic solution as an infinite series by introducing the fractional $\psi-$series expansion, which has a positive radius of convergence. To conclude, by considering various kernels, we demonstrate the utility of the truncated series in closely approximating the analytical solution for different values of $q$. |
|