Approximation of functions belonging to Hölder’s class and solution of Lane-Emden differential equation using Gegenbauer wavelets

Abstract

In this paper, a very new technique based on the Gegenbauer wavelet series is introduced to solve the Lane-Emden differential equation. The Gegenbauer wavelets are derived by dilation and translation of an orthogonal Gegenbauer polynomial. The orthonormality of Gegenbauer wavelets is verified by the orthogonality of classical Gegenbauer polynomials. The convergence analysis of Gegenbauer wavelet series is studied in Hölder’s class. Hölder’s class Hα[0, 1) and Hϕ[0, 1) of functions are considered, Hϕ[0,1) class consides with classical Hölder’s class Hα[0, 1) if ϕ(t) = tα, 0 < α ≤ 1. The Gegenbauer wavelet approximations of solution functions of the Lane-Emden differential equation in these classes are determined by partial sums of their wavelet series. In briefly, four approximations E(1) 2k−1,0, E(1) 2k−1,M, E(2) 2k−1,0, E(2) 2k−1,M of solution functions of classes Hα[0, 1), Hϕ[0, 1) by (2k−1, 0)th and (2k−1, M)th partial sums of their Gegenbauer wavelet expansions have been estimated. The solution of the Lane-Emden differential equation obtained by the Gegenbauer wavelets is compared to its solution derived by using Legendre wavelets and Chebyshev wavelets. It is observed that the solutions obtained by Gegenbauer wavelets are better than those obtained by using Legendre wavelets and Chebyshev wavelets, and they coincide almost exactly with their exact solutions. This is an accomplishment of this research paper in wavelet analysis. © Faculty of Sciences and Mathematics, University of Niš, Serbia.