Approximation of functions using the Haar basis with scale 2 and 3

Voronova, Y. N.; Arora, G.; Kravchenko, O. V.

doi:10.47974/JIM-2323

Open Access Research Article

Approximation of functions using the Haar basis with scale 2 and 3

Y. N. Voronovajulija.n.voronova@bmstu.ruDepartment of Higher Mathematics Bauman Moscow State Technical UniversityMoscow, 105005, Russian FederationView full profile → , ^*G. AroraCorresponding authorgeetadma@gmail.comDepartment of Mathematics Lovely Professional UniversityPhagwara, Punjab, 144411, IndiaView full profile → , O. V. Kravchenkookravchenko@frcccsc.ruFederal Research Center Computer Science and Control of Russian Academy of SciencesMoscow, 119333, Russian FederationView full profile →

* Corresponding author · click or hover a name for details

pp. 1897–1908Vol. 29Issue 6June 2026DOI: 10.47974/JIM-2323XML

Received:: 01 Mar 2025
Published Online:: 30 Jun 2026
Article type:: Research Article
Language:: EN
Article no.:: JIM-2323
Pages:: 1897–1908

Abstract

The orthonormal basis of Haar wavelets with both scale 2 and 3 is a useful tool in various scientific and engineering problems. Simultaneously, applications often require solutions that involve functions with extrema. In this study, a comparative analysis of approximation by Haar wavelets with scale 2 and 3 of functions with one and multiple extrema is considered. The properties and approximation capabilities of Haar wavelets at different levels of decomposition are analyzed, using the example of approximation of one-dimensional functions. Examples are also provided for the approximation of specific functions to illustrate the effectiveness of the approach. Also, the numerical analysis of errors of the considered approximations is carried out in the tables. Finally, numerical calculations are presented which are supplemented by graphical results.

Keywords

Haar wavelets Haar basis with scale 2 Haar basis with scale 3 Functions approximation

Subject Classifications

26A0633F0565D20

References

[1] A. Haar, “Zur Theorie der orthogonalen Funktionensysteme: Erste Mitteilung,” Mathematische Annalen, vol. 69, no. 3, pp. 331-371 (1910), doi: 10.1007/bf01456326.
[2] C. F. Chen and C. H. Hsiao, “Haar wavelet method for solving lumped and distributed-parameter systems,” IEE Proceedings - Control Theory and Applications, vol. 144, no. 1, pp. 87-94 (1997), doi: 10.1049/ip-cta:19970702.
[3] C. F. Chen and C. H. Hsiao, “Wavelet approach to optimising dynamic systems,” IEE Proceedings - Control Theory and Applications, vol. 146, no. 2, pp. 213-219 (1999), doi: 10.1049/ip-cta:19990516.
[4] C. H. Hsiao, “Haar wavelet approach to linear stiff systems,” Mathematics and Computers in Simulation, vol. 64, no. 5, pp. 561-567 (2004), doi: 10.1016/j.matcom.2003.11.011.
[5] Ü. Lepik, “Numerical solution of differential equations using Haar wavelets,” Mathematics and Computers in Simulation, vol. 68, no. 2, pp. 127-143 (2005), doi: 10.1016/j.matcom.2004.10.005.
[6] Ü. Lepik, “Solving PDEs with the aid of two-dimensional Haar wavelets,” Computers & Mathematics with Applications, vol. 61, no. 7, pp. 1873-1879 (2011), doi: 10.1016/j.camwa.2011.02.016.
[7] G. Hariharan and K. Kannan, “Review of wavelet methods for the solution of reaction-diffusion problems in science and engineering,” Applied Mathematical Modelling, vol. 38, no. 3, pp. 799-813 (2014), doi: 10.1016/j.apm.2013.08.003.
[8] U. Lepik and H. Hein, Haar Wavelets: With Applications. Cham, Switzerland: Springer International Publishing (2014), doi: 10.1007/978-3-319-04295-4.
[9] S. C. Shiralashetti and L. Lamani, “Numerical solution of stochastic ordinary differential equations using HAAR wavelet collocation method,” Journal of Interdisciplinary Mathematics, vol. 25, no. 2, pp. 195-211 (2022), doi: 10.1080/09720502.2021.1874085.
[10] R. C. Mittal and S. Pandit, “New scale-3 Haar wavelets algorithm for numerical simulation of second order ordinary differential equations,” Proceedings of the National Academy of Sciences, India Section A: Physical Sciences, vol. 89, no. 4, pp. 799-808 (2019), doi: 10.1007/s40010-018-0538-y.
[11] R. Kumar and J. Gupta, “A comparative study using scale-2 and scale-3 Haar wavelet for the solution of higher order differential equation,” International Journal of Mathematical, Engineering and Management Sciences, vol. 8, no. 5, pp. 966-978 (2023), doi: 10.33889/ijmems.2023.8.5.055.
[12] S. S. Zaman, R. Amin, N. Haider, A. Aloqaily, and N. Mlaiki, “Haar wavelet collocation technique for numerical solution of porous media equations,” Partial Differential Equations in Applied Mathematics, vol. 10, Art. no. 100728 (2024), doi: 10.1016/j.padiff.2024.100728.
[13] V. B. Awati, A. Goravar, and M. Kumar, “Spectral and Haar wavelet collocation method for the solution of heat generation and viscous dissipation in micro-polar nanofluid for MHD stagnation point flow,” Mathematics and Computers in Simulation, vol. 215, pp. 158-183 (2024), doi: 10.1016/j.matcom.2023.07.031.
[14] Siraj-ul-Islam, I. Aziz, and B. Šarler, “The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets,” Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1577-1590 (2010), doi: 10.1016/j.mcm.2010.06.023.
[15] R. Singh, H. Garg, and V. Guleria, “Haar wavelet collocation method for Lane-Emden equations with Dirichlet, Neumann and Neumann-Robin boundary conditions,” Journal of Computational and Applied Mathematics, vol. 346, pp. 150-161 (2019), doi: 10.1016/j.cam.2018.07.004.
[16] L. Bouzid, N. Lahmar-Ablaoui, and M. Hamou Maamar, “New 2D numerical integration formula based on the Legendre wavelets,” Journal of Interdisciplinary Mathematics, vol. 26, no. 5, pp. 835-847 (2023), doi: 10.47974/JIM-1509.
[17] M. Ahsan, W. Lei, A. A. Khan, A. Ullah, S. Ahmad, S. U. Arifeen, Z. Uddin, and H. Qu, “A high-order reliable and efficient Haar wavelet collocation method for nonlinear problems with two point-integral boundary conditions,” Alexandria Engineering Journal, vol. 71, pp. 185-200 (2023), doi: 10.1016/j.aej.2023.03.011.
[18] I. Aziz and Siraj-ul-Islam, “An efficient modified Haar wavelet collocation method for numerical solution of two-dimensional elliptic PDEs,” Differential Equations and Dynamical Systems, vol. 25, no. 2, pp. 347-360 (2017), doi: 10.1007/s12591-015-0262-x.
[19] L. Sadek and A. S. Bataineh, “The general Bernstein function: application to χ-fractional differential equations,” Mathematical Methods in the Applied Sciences, vol. 47, no. 7, pp. 6117-6142 (2024), doi: 10.1002/mma.9910.

Views: 34Downloads: 18Citations: 0