Highly Accurate Galerkin Approach for Modified Anomalous Time-Fractional Sub-Diffusion Equations Based on Fibonacci Coefficient Polynomials

A spectral Galerkin algorithm is developed for the numerical treatment of modified anomalous time-fractional sub-diffusion equations (MATFSDEs). The algorithm is constructed using Fibonacci coefficient polynomials, from which two space–time trial families are formed. These families are selected so that the homogeneous initial and boundary conditions are automatically incorporated after a suitable transformation of the original problem. The Galerkin formulation then reduces the model to a finite algebraic matrix system whose entries can be explicitly evaluated. To support the construction, inversion, moment, and linearization identities for the selected polynomials are obtained and then used to express the required matrices in closed form. The convergence of the expansion is investigated, and explicit error bounds are derived. Numerical tests are reported to demonstrate the accuracy, applicability, and competitiveness of the proposed scheme in comparison with some existing methods in the literature.