Approximation and moduli of continuity for a function belonging to Holder’s class Hα [0, 1) and solving Lane-Emden differential equation by Boubaker wavelet technique

Authors

  • Swatantra Yadav Banaras Hindu University-221005, Varanasi
  • Shyam Lal Banaras Hindu University image/svg+xml

DOI:

https://doi.org/10.23755/rm.v48i0.1092

Keywords:

Boubaker wavelet, Boubaker polynomial, Boubaker wavelet approximation, Moduli of continuity, Collocation method, Lane-Emden differential equations

Abstract

In this paper, Boubaker wavelet is considered. The Boubaker wavelets are orthonormal. The series of this wavelet is verified for the function f(t) = t. The convergence analysis of solution function of Lane-Emden differential equation has been studied. New Boubaker wavelet estimator E2 k,M(f) for the approximation of solution function f belong to H¨older’s class Hα[0, 1) of order 0 < α ≤ 1, has been devloped. Furthermore, the moduli of continuity of solution function of Lane-Emden differential equation has been introduced and it has been estimated for solution function f∈ Hα[0, 1) class. These estimator and moduli of continuity are new and best possible in wavelet analysis. Boubaker wavelet collocation method has been proposed to solve Lane-Emden differential equations with unkown Boubaker coefficients. In this process, Lane-Emden differential equations are reduced into a system of algebraic equations and these equations are solved by collocation method. Three Lane-Emden type equations are solved to demonstrate the applicability of the proposed method. The solutions obtained by the proposed method are compared with their exact solutions. The absolute errors are negligible. Thus,this shows that the method described in this paper is applicable and accurate.

References

C.K.Chui, Wavelets: A mathematical tool for signal analysis, SIAM, Philadelphia PA, (1997).

Devore, R.A.: Nonlinear approximation, Acta Numerica, Vol.7, pp. 51-150, Cambridge University Press, Cambridge (1998).

Debnath, L.: Wavelet Transform and Their Applications, Birkhauser Boston, Massachusetts (2002).

Meyer, Y.: Wavelets; their past and their future, Progress in Wavelet Analysis and (Applications) (Toulouse, 1992) (Meyer, Y. and Roques, S. eds) Frontieres, Gif-sur-Yvette, pp. 9-18 (1998).

Morlet, J.; Arens, G; Fourgeau, E.; and Giard, D.: Wave propagation and sampling theory, part II. Sampling theory and complex waves, Geophysics 47(2), 222-236 (1982).

Daubechies, I.: Ten Lectures on Wavelets, SIAM, Philadelphia, PA (1992).

Wazwaz, A.-M.: A new algorithm for solving differential equations of Lane–Emden type. Applied Mathematics and Computation, 118(2), 287–310 (2001).

Shiralashetti, Siddu & Lamani, Lata. (2020). Boubaker wavelet based numerical method for the solution of Abel’S integral equations. 28. 114-124.

Das, G.; Ghosh, T.; Ray, B.K.: Degree of approximation of functions by their Fourier series in the generalized H¨older metric. Proc. Indian Acad. Sci. Math. Sci. Vol.106, no.2, pp. 139–153 (1996).

Zygmund A.: Trigonometric Series, vol.I. Cambridge University Press, Cambridge (1959).

Lal, Shyam, & Satish Kumar. ”CAS wavelet approximation of functions of Holder class and solutions of Fredholm integral equation.” Ratio Mathematica [Online], 39 (2020): 187-212.

Downloads

Published

2023-12-30

Issue

Vol. 48 (2023)

Section

Volume 48-2023

License

This journal provides immediate open access to its content on the principle that making research freely available to the public supports a greater global exchange of knowledge. The journal is committed to real and immediate open access for academic work. All of the journal articles are free to access immediately from the date of publication. There are no charge for readers to download articles and reviews for their own scholarly use. Benefits of open access for authors, include:

  • Free access for all users worldwide
  • Increased visibility and readership
  • Rapid publication
  • No spatial constraints
  • Authors can share their research papers on their personal web-pages (i.e. ResearchGate, etc.)

Ratio Mathematica publishes open access articles under the terms of the Creative Commons Attribution (CC BY) License. The Creative Commons Attribution License (CC-BY) allows users to copy, distribute and transmit an article, adapt the article and make commercial use of the article. The CC BY license permits commercial and non-commercial re-use of an open access article, as long as the author is properly attributed. Copyright on any research article published by Ratio Mathematica is retained by the author(s). Authors grant Ratio Mathematica a license to publish the article and identify itself as the original publisher. Authors also grant any third party the right to use the article freely as long as its original authors, citation details and publisher are identified. 
For further information visit: https://creativecommons.org/licenses/by/3.0/. The following logo will appear in any paper that will be published from the volume number 31/2016 onwards.