Main Article Content
This paper is concerned with introducing an explicit expression for orthogonal Boubaker polynomial functions with some important properties. Taking advantage of the interesting properties of Boubaker polynomials, the definition of Boubaker wavelets on interval [0,1) is achieved. These basic functions are orthonormal and have compact support. Wavelets have many advantages and applications in the theoretical and applied fields, and they are applied with the orthogonal polynomials to propose a new method for treating several problems in sciences, and engineering that is wavelet method, which is computationally more attractive in the various fields. A novel property of Boubaker wavelet function derivative in terms of Boubaker wavelet themselves is also obtained. This Boubaker wavelet is utilized along with a collocation method to obtain an approximate numerical solution of singular linear type of Lane-Emden equations. Lane-Emden equations describe several important phenomena in mathematical science and astrophysics such as thermal explosions and stellar structure. It is one of the cases of singular initial value problem in the form of second order nonlinear ordinary differential equation. The suggested method converts Lane-Emden equation into a system of linear differential equations, which can be performed easily on computer. Consequently, the numerical solution concurs with the exact solution even with a small number of Boubaker wavelets used in estimation. An estimation of error bound for the present method is also proved in this work. Three examples of Lane-Emden type equations are included to demonstrate the applicability of the proposed method. The exact known solutions against the obtained approximate results are illustrated in figures for comparison
Published Online First 30/4/2021
This work is licensed under a Creative Commons Attribution 4.0 International License.
Miguel G F, Patrick K P. Marcelo G V. Electromagnetic device modeling using a new adaptive wavelet finite element method. Math Comput Simul. In press. 2020
Nighila V P, Johney I, Jacob P. Application of Wavelet Transform to the study of Lattice Dynamics of two-dimensional Nanostructures. Mater Today PR. 2019; 18(3): 1524-1531.
Dilshad A H. A well-balanced adaptive Haar wavelet finite volume scheme for 1D free surface water flows. Ain Shams Eng. J. 2019; 10(4): 891-895.
Zhaoming X, Hao Z, Kongchao S, Yongquan Q, Zheng J. Wavelet analysis of extended X-ray absorption fine structure data: Theory, application J. Phys. Condens. Matter. 2018; 542(1): 12-19.
Seyed A T, Seyed M T. Wavelet based damage identification and dynamic pull-in instability analysis of electrostatically actuated coupled domain microsystems using generalized differential quadrature method. Mech Syst Signal Process. 2019; 133(1): 106256.
Muhammad A I, Umar K, Ayyaz A, Syed T M D. Modified Chebyshev Wavelet-Picard Technique for Thin Film Flow of Non-Newtonian Fluid Down an Inclined Plane. Proc. Nat. Acad. Sci. A., 2018, 89(3): 533-538.
Hariharan G. An efficient wavelet based approximation method to water quality assessment model in a uniform channel. Ain Shams Eng. J. 2014; 5(2): 525-532.
Keshavarz E, Ordokhani Y. A fast numerical algorithm based on the Taylor wavelets for solving the fractional integro-differential equations with weakly singular kernels. Math. Methods Appl. Sci. 2019; 42(13): 4427–4443.
Jing G, Yao L. Trigonometric Hermite wavelet approximation for the integral equations of second kind with weakly singular kernel. J. Comput. Appl. Math. 2008; 215(1): 242-259.
Saeed U. Hermite wavelet method for fractional delay differential equations. J. Differ. Equ. 2014.
Mohsen R, Sohrabali Y. Legendre wavelets method for constrained optimal control problems. Math. Methods Appl. Sci.2002; 25(7): 529-539.
Mujeebur R, Rahmat A K. The Legendre wavelet method for solving fractional differential equations. Commun Nonlinear Sci Numer Simul. 2011; 16(11): 4163-4173.
Nanshan L E, Bing L. Legendre wavelet method for numerical solutions of partial differential equations. Numer. Methods Partial Differential Equations. 2010; 26(1): 81-94.
Sohrab A Y. Legendre wavelets method for solving differential equations of Lane–Emden type. Appl. Math. Comput. 2006; 181(2): 1417-1422.
Moustafa A S. Approximate solution for the Lane-Emden equation of the second kind in a spherical annulus. J. King Saud Univ. Sci. 2019; 31(1): 1-5.
Joachim M, Herbert S. Statistical mechanics of the isothermal lane-emden equation. J. Stat. Phys. 1982; 29(3): 561-578.
Asghar G, Mojtaba B. A variational iteration method for solving nonlinear Lane–Emden problems. New Astron. 2017; 54: 1-6.
Boubaker K, Chaouachi A, Amlouk M, Bouzouita H. Enhancement of pyrolysis spray disposal performance using thermal time-response to precursor uniform deposition. Eur. Phys. J. Appl. Phys. 2007; 37(1): 105-109.
Boubaker K. On modified Boubaker polynomials: some differential and analytical properties of the new polynomials issued from an attempt for solving bi-varied heat equation. Trends Appl. Sci. Res. 2007: 2(6): 540-544.
Tuğçe A, Salih Y. Boubaker polynomial approach for solving high-order linear differential-difference equations. AIP Conf. Proc. 2012; 26: 1493.