Efficient Numerical Method for Solving a Cancer Tumor Model Governed by Fractional Partial Differential Equations Involving the ψ-Caputo derivative

Abstract

This paper introduces a novel fractional-order framework for modeling tumor-immune-drug interactions using Ψ-Caputo derivatives with general memory kernels. Unlike classical approaches, our model captures hereditary effects and anomalous transport inherent in cancer biology through flexible time-scaling laws. We establish rigorous theoretical guarantees, including existence, uniqueness, non-negativity, and boundedness of solutions, ensuring biological feasibility. Numerically, we develop a fully implicit θ-method with Ψ-Caputo discretization, proving unconditional stability and demonstrating its efficacy through comprehensive simulations. Our results reveal how the fractional order ξ and kernel selection critically regulate tumor persistence, immune activation, and treatment outcomes, uncovering memory-driven dynamics inaccessible to integer-order models. This work demonstrates that Ψ-Caputo operators are a powerful tool for predictive oncology, bridging fractional partial differential equation theory with cancer systems biology.