Advanced Spectral-Operational Frameworks for Fractional and Integer-Order Differential Equations: A Methodological Review

Abstract This work provides a methodological review and comparative analysis of several numerical frameworks for solving initial value problems (IVPs) and boundary value problems (BVPs) involving high-order ordinary differential equations (ODEs), multi-term fractional differential equations (MTFDEs), multiterm variable-order fractional differential equations (MTVOFDEs), and multi-term variable-order time-fractional diffusion-wave equations (MTVO-TFDWEs). The core of these approaches lies in the construction of operational matrices (OMs) for ordinary, fractional, and variable-order derivatives, as well as various integral operators (repeated integrals (RIs) and Riemann-Liouville fractional integral (RLFI)) based on generalized shifted Jacobi polynomials (GSJPs). These basis functions are specifically constructed to satisfy the given homogeneous initial conditions (ICs) or boundary conditions (BCs) a priori. The derived OMs are utilized within spectral collocation frameworks to easily transform the underlying differential equations (DEs) into solvable algebraic systems. The individual methodologies have been introduced in previous studies [8–10, 38], whereas this article combines them into a comprehensive spectral-operational framework with further insights into the integration of algorithms, convergence properties, and error analysis. The various numerical examples and methodical comparative tables are provided to prove that these schemes have accuracy, robustness, and a better accuracy-cost ratio to solve the complicated fractional differential equation models.