An Operational Matrix Approach Using Chebyshev Polynomials for Solving Generalized Caputo Fractal-Fractional Differential Equations

This study introduces an approach relying on the application of an operational matrix based on shifted Chebyshev polynomials for numerically addressing fractal-fractional (FF) linear and nonlinear differential equations, as well as systems of equations, utilizing the generalized fractional derivative of Caputo type. The proposed method transforms the generalized Caputo-type FF derivatives transformed into a system of algebraic equations, enabling the determination of unknown solutions. Theoretical analysis was conducted to establish convergence criteria and derive error bounds for the method. The approach was validated through quantitative analysis across various scenarios and benchmarked against established techniques to affirm its precision and computational effectiveness. Additionally, the method was applied to the SIRD (Susceptible-Infected-Recovered-Deceased) mathematical model with a FF operator, employing the spectral collocation method to demonstrate the effectiveness of the proposed numerical approach in handling FF derivatives.