Numerical Method for Solving Quantum Fractional Differential Equations Using Quantum Fractional Taylor Functions

This paper presents a novel ‐Taylor operational method for solving linear and nonlinear ‐fractional differential equations (‐FDEs). The proposed approach transforms the original problem into an algebraic system using operational matrices constructed from ‐Taylor polynomials. A rigorous error analysis demonstrates the spectral convergence of the method, ensuring rapid error reduction with increasing collocation points. Numerical experiments on a wide range of test problems including linear equations, quadratic and cubic nonlinearities, and transcendental trigonometric models confirm the remarkable accuracy of the method, with errors as low as in certain cases. The results also highlight the robustness and stability of the approach across different ‐values, fractional orders, and extended time‐scale domains. Overall, the ‐Taylor operational method provides an efficient, accurate, and stable alternative to existing numerical techniques for solving ‐fractional differential equations.