New Fractional Derivative Expression of the Shifted Third-Kind Chebyshev Polynomials: Application to a Type of Nonlinear Fractional Pantograph Differential Equations

Abstract

The main goal of this paper is to develop a new formula of the fractional derivatives of the shifted Chebyshev polynomials of the third kind. This new formula expresses approximately the fractional derivatives of these polynomials in the Caputo sense in terms of their original ones. The linking coefficients are given in terms of a certain ${}_4{F}_3\left(1\right)$ terminating hypergeometric function. The integer derivatives of the shifted third-kind Chebyshev polynomials can be calculated using this formula after performing some reductions. To solve a nonlinear fractional pantograph differential equation with quadratic nonlinearity, the fractional derivative formula is used in conjunction with the tau technique. The role of the tau method is to convert the pantograph differential equation with its governing initial/boundary conditions into a nonlinear system of algebraic equations that can be treated with the aid of Newton’s iterative scheme. To test the method’s convergence, certain estimations are included. The proposed numerical method is demonstrated by numerical results to ensure its applicability and efficiency.