Numerical Treatments for Real-World Mathematical Models Governed by Non-Integer Order Differential Equations in Crossover Domains

Abstract

This thesis develops advanced numerical methods for solving dynamical systems governed by fractional differential equations within crossover domains. It incorporates Caputo, Atangana–Baleanu–Caputo, Caputo–Fabrizio, and Ψ-Caputo operators, including variable, fractal, and stochastic extensions. A unified framework based on Ψ-nonstandard finite difference, Toufik–Atangana schemes, and modified Euler–Maruyama methods ensures stability, accuracy, and positivity. A novel hybrid algorithm combining Caputo–Fabrizio discretization with stochastic techniques is proposed. Convergence and stability are rigorously proven. Applications to Monkeypox, breast cancer, and tumor growth models demonstrate strong agreement with real data, highlighting the effectiveness of crossover fractional modeling for complex systems.