Eur. Phys. J. Plus (2020) 135: 881
https://doi.org/10.1140/epjp/s13360-020-00865-z

Regular Article

Normalized Lucas wavelets: an application to Lane–Emden and pantograph differential equations

¹ School of Mathematics, Computer and Information Sciences, Central University of Himachal Pradesh, Dharamshala, India
² Department of Mathematics, Cankaya University, Eskisehir Yolu29.km, 06810, Ankara, Turkey
³ Institute of Space Sciences, Magurele-Bucharest, Romania

^* e-mail: rakesh@cuhimachal.ac.in

Received: 8 June 2020
Accepted: 16 October 2020
Published online: 3 November 2020

Abstract

In this paper, a novel normalized Lucas wavelet scheme based on tau approach is proposed for the two classes of second-order differential equations, namely Lane–Emden and pantograph equations. The introduced scheme depends on shifted Lucas polynomials (SLPs) and their operational matrix of derivative (which are developed here). The weight function for the orthogonality of Lucas polynomials, and Rodrigues formula are proposed for the first time, which form the basis for the construction of SLPs. Normalized Lucas wavelets are constructed by utilizing SLPs and their novel properties. Literally, the present scheme transforms the given method to a set of nonlinear algebraic equations with undetermined coefficients which are here tackled by tau method. Meanwhile, new treatment of convergence and error analysis is provided for the established approach. Finally, the accuracy and applicability of present scheme is ensured by considering several examples.