The main aim of this paper is to introduce a new class of orthogonal polynomials that generalizes the class of Chebyshev polynomials of the first kind. Some basic properties of the generalized Chebyshev polynomials and their shifted ones are established. Additionally, some new formulas concerned with these generalized polynomials are established. These generalized orthogonal polynomials are employed to treat the multi-term linear fractional differential equations (FDEs) that include some specific problems that arise in many applications. The basic idea behind the derivation of our proposed algorithm is built on utilizing a new power form representation of the shifted generalized Chebyshev polynomials along with the application of the spectral Galerkin method to transform the FDE governed by its initial conditions into a system of linear equations that can be efficiently solved via a suitable numerical solver. Some illustrative examples accompanied by comparisons with some other methods are presented to show that the presented algorithm is useful and effective.

Fractional delay differential equations (FDDEs) and time-fractional delay partial differential equations (TFDPDEs) are the focus of the present research. The FDDEs is converted into a system of algebraic equations utilizing a novel numerical approach based on the spectral Galerkin (SG) technique. The suggested numerical technique is likewise utilized for TFDPDEs. In terms of shifted Jacobi polynomials, suitable trial functions are developed to fulfill the initial-boundary conditions of the main problems. According to the authors, this is the first time utilizing the SG technique to solve TFDPDEs. The approximate solution of five numerical examples is provided and compared with those of other approaches and with the analytic solutions to test the superiority of the proposed method.