Analytical and Numerical Study of Fractional Logistic Equation With Variable Kernel in the Caputo Sense

Abstract

We consider a fractional logistic equation involving a Caputo-type fractional derivative of order (Formula presented.) with a variable kernel (Formula presented.), a formulation introduced for its versatility in modeling complex real-world phenomena through an appropriate selection of fractional derivatives. The equilibrium points are identified, and their stability is rigorously analyzed using the (Formula presented.) Laplace transform technique. The existence and uniqueness of the solution are established via the fixed-point theorem. Furthermore, we express the analytic solution as an infinite series by introducing the fractional (Formula presented.) series expansion, which has a positive radius of convergence. By truncating this series, we demonstrate its practical applicability for various kernel functions and different values of (Formula presented.). Additionally, we present an innovative adaptive predictor–corrector method for solving initial value problems (IVPs) that involve a Caputo-type fractional derivative with a variable kernel, taking graded meshes into account. We conducted extensive numerical simulations across various fractional orders and kernels, demonstrating that the obtained results closely align with exact solutions in the integer case, as well as with the truncated (Formula presented.) -series expansion when a large number of nodes are used. Moreover, our approach exhibits satisfactory numerical stability in fractional scenarios. © 2026 John Wiley & Sons Ltd.