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

Seham Al-Mekhlafi

doi:10.59628/jast.v4i2.2576

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

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Published 2026-02-28

DOI 10.59628/jast.v4i2.2576

Issue Vol. 4 No. 2 (2026): Sana'a University Journal of Applied Sciences and Technology

Section Article

Fractional partial differential equations Ψ- Caputo derivative Cancer modeling Stability analysis Numerical methods Tumor–immune– drug dynamics

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.

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Seham Al-Mekhlafi

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How to Cite

Efficient Numerical Method for Solving a Cancer Tumor Model Governed by Fractional Partial Differential Equations Involving the ψ-Caputo derivative. (2026). Sana’a University Journal of Applied Sciences and Technology, 4(2), 1682-1693. https://doi.org/10.59628/jast.v4i2.2576