A Shifted One-Parameter Horadam Polynomial Framework for the Time-Fractional Supergeneralized Viscous Burgers Equation

A compact operational-matrix collocation algorithm is introduced and analyzed for the numerical treatment of the time-fractional supergeneralized viscous Burgers equation. The approximation space is generated by the shifted one-parameter Horadam polynomials Hjc, for a positive real number c, and the algebraic properties required by the algorithm are derived explicitly. In particular, we obtain a power form representation, a monomial inversion relation, and closed-form operational matrices for integer and Caputo fractional derivatives. These relations allow the nonlinear fractional equation to be discretized directly in a tensor-product polynomial space and converted into a finite nonlinear system for the expansion coefficients. The resulting system is treated by the standard Newton iteration method. The convergence discussion and the error estimates support the proposed approximation, while the numerical experiments show that accurate solutions are obtained with a relatively small number of collocation points. Comparisons with existing methods indicate the efficiency of the present approach for Burgers-type fractional models.