Piecewise Cancer Tumor Disease of Partial Differential Equations Based on Exponential and Non-Singular Kernel; Numerical Treatments

Abstract

This work presents a novel fractional-stochastic reaction–diffusion model for simulating cancer dynamics, incorporating the Caputo–Fabrizio derivative and additive Gaussian noise within a temporally piecewise framework. The model captures the nonlinear spatiotemporal interactions among normal cells, tumor cells, immune responses, and chemotherapeutic agents. By employing fractional-order derivatives with exponential kernels, the model accounts for biological memory and nonlocal effects, while the stochastic component reflects environmental and treatment-related uncertainties. In the early phase, memory-driven deterministic dynamics dominate, transitioning to stochastic behavior in later stages to simulate real-world perturbations. A hybrid numerical scheme, combining finite difference discretization of the Caputo–Fabrizio operator with the Euler–Maruyama method, is developed to simulate system behavior under varying fractional orders and noise intensities. Simulation results reveal that lower fractional orders correspond to delayed immune responses, persistent tumor presence, and slower drug clearance, highlighting the biological impact of memory. Conversely, higher fractional orders accelerate therapeutic effects. Stochastic noise introduces fluctuations that can destabilize outcomes, emphasizing the need for robust modeling in clinical contexts. Sensitivity analyses confirm the dominant influence of fractional parameters on system dynamics. The proposed framework offers a biologically informed, predictive platform for optimizing cancer treatment strategies and exploring the role of memory and randomness in tumor evolution.