Numerical analysis of a space-time ψ-Caputo fractional reaction-diffusion system with applications to tumor-immune dynamics

Fractional calculus has become an essential tool for describing memory and nonlocal effects in complex biological systems, particularly in tumor-immune-drug interactions. In this work, we propose a space-time ψ-Caputo fractional reaction-diffusion model to describe the coupled dynamics of normal cells, tumor cells, immune response and chemotherapy drug distribution within a heterogeneous tumor microenvironment. The ψ-Caputo framework enables flexible incorporation of general kernel functions, unifying several classical fractional operators and allowing independent characterization of temporal memory and spatial nonlocality. A θ-weighted finite difference scheme is developed to approximate the model, and rigorous von Neumann stability and convergence analyzes are established. Numerical simulations demonstrate that stronger temporal memory suppresses tumor progression, while spatial nonlocality enhances immune infiltration and drug penetration into tumor regions. Moreover, the choice of kernel function significantly influences system dynamics, providing a mechanism for tailoring model behavior to patient-specific characteristics. Overall, the proposed framework offers a robust and general modeling platform for analyzing nonlinear fractional diffusion processes in cancer dynamics with potential applications in personalized therapeutic design. These results highlight the importance of fractional operators in capturing realistic biological heterogeneity and support the use of generalized kernels for improved predictive accuracy in biomedical modeling frameworks and clinical applications.