New Operational Matrix Via Gnocchi Polynomial for Solving Non-Linear Fractional Differential Equations.

Fractional differential equations (FDEs) have emerged as essential tools in modeling complex dynamical systems exhibiting memory and hereditary properties. Traditional operational matrices arising from Legendre, Chebyshev, and Jacobi polynomials are generally known to be numerically unstable, computationally expensive, and inefficient in approximating fractional operators. In this study an operational matrix based on Gnocchi polynomial is introduced for solving non linear fractional differential equations (NFDE) with better sparsity, stability and computational efficiency. The proposed method transforms NFDEs into tractable algebraic systems by constructing a fractional differentiation operational matrix using Gnocchi polynomials. The method is validated by theoretical formulations, spectral convergence analysis, error estimation proofs. It is also compared with existing polynomial based approaches to demonstrate better performance in function approximation and numerical stability. The Gnocchi operational matrix is based on Gnocchi, and it achieves exponential convergence, reduced computational complexity and increased numerical robustness compared to classical techniques. It is effective in fractional modeling because it can accurately approximate non-linear fractional operators. The author develops a mathematically rigorous, computationally efficient framework to solve NFDEs. Further improvements will be done by other researchers in the future for higher dimension applications, for adaptive techniques in the spectral method and for hybrid AI assisted optimization.