Algorithmic Program for Solving Fractional Differential Equations

This paper presents a novel predictor-corrector method specifically designed for solving nonlinear equations, with particular focus on fractional differential equations. We detail the algorithmic workflow involving iterative prediction and correction phases, where the predictor step generates an initial approximation using Grünwald-Letnikov discretization, followed by a corrector step refining the solution through weighted integration. The implementation utilizes MATLAB's vectorization capabilities for efficient memory handling of fractional-order operators. Additionally, we provide complete computational code demonstrating adaptive step-size control and error estimation mechanisms. The discussion covers comparative analysis of computational complexity (O(N²) for N discretization points) and convergence properties, along with practical applications in viscoelastic modeling and anomalous diffusion phenomena. Numerical examples include Caputo-type fractional derivatives implementation using Simpson's rule integration and Mittag-Leffler function approximations. This resource offers valuable insights and reusable code templates for researchers interested in nonlinear systems and fractional calculus applications.