Hybrid Shifted Gegenbauer Integral-Pseudospectral Method for Solving Time-Fractional Benjamin-Bona-Mahony-Burgers Equation

This paper presents a high-order hybrid shifted Gegenbauer integral-pseudospectral (HSG-IPS) method for solving the time-fractional Benjamin-Bona-Mahony-Burgers (FBBMB) equation. A key innovation of our approach is the transformation of the original equation into a fractional partial-integro differential form that contains only a first-order derivative, which can be accurately approximated using a first-order shifted Gegenbauer differentiation matrix (SGDM), while all other terms in the transformed equation are resolved using highly accurate quadrature rules. The method combines several advanced numerical techniques including the shifted Gegenbauer pseudospectral (SGPS) method, Gegenbauer-based fractional approximation (GBFA), shifted Gegenbauer integration matrix (SGIM), shifted Gegenbauer integration row vector (SGIRV), and SGDM to achieve spectral accuracy. Numerical experiments demonstrate that the HSG-IPS method outperforms existing numerical approaches, achieving significantly lower average absolute errors (AAEs) with computational times as low as 0.04-0.05 seconds. The method's robustness is validated across various fractional orders, showing excellent agreement with analytical solutions. The transformation strategy effectively circumvents the numerical instability associated with direct approximation of high-order derivatives in the original equation, while the use of shifted Gegenbauer (SG) polynomials and barycentric representations ensures numerical stability and efficiency. This work provides a powerful computational framework for modeling wave propagation, dispersion, and nonlinearity in fractional calculus applications.