Abstract
This paper is concerned with establishing novel expressions that express the derivative of any order of the orthogonal polynomials, namely, Chebyshev polynomials of the sixth kind in terms of Chebyshev polynomials themselves. We will prove that these expressions involve certain terminating hypergeometric functions of the type 4F3(1) that can be reduced in some specific cases. The derived expressions along with the linearization formula of Chebyshev polynomials of the sixth kind serve in obtaining a numerical solution of the non-linear one-dimensional Burgers’ equation based on the application of the spectral tau method. Convergence analysis of the proposed double shifted Chebyshev expansion of the sixth kind is investigated. Numerical results are displayed aiming to show the efficiency and applicability of the proposed algorithm.
Highlights
Non-Linear One-DimensionalOrthogonal polynomials are successfully employed for the numerical solutions of various differential equations
In order to proceed in our proposed algorithm, the following linearization formula is essential in the sequel
We developed new expressions for the high-order derivatives of Chebyshev polynomials of the sixth kind in terms of their original ones
Summary
Orthogonal polynomials are successfully employed for the numerical solutions of various differential equations. The authors in [36] established expressions of the derivatives of the fifth kind Chebyshev polynomials and they proved that these expressions involve hypergeometric functions of the type 4 F3 (1). In the two papers [18,37], the authors developed new formulas for the high-order derivatives and repeated integrals of Chebyshev polynomials of the third and fourth kinds and utilized the developed formulas for treating some special types of differential equations. It will be shown that these derivatives can be expressed in terms of hypergeometric functions of the type 4 F3 (1) that can be reduced in some specific cases Another principal goal of this article is to utilize these formulas for obtaining numerical solutions of the non-linear one dimensional Burgers’.
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