AccScience Publishing / IJOCTA / Online First / DOI: 10.36922/IJOCTA025160082
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RESEARCH ARTICLE

Solving parabolic differential equations via Haar wavelets: A focus on integral boundary conditions

Muhammad Nawaz Khan1 ,  Masood Ahmad2 ,  Rashid Jan3,4,5* ,  Imtiaz Ahmad6 ,  Mohamed Mousa7
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1 Institute of Engineering Mathematics, University Malaysia Perlis, Arau, Perlis, Malaysia
2 Department of Basic Sciences, University of Engineering and Technology, Peshawar, Pakistan
3 Department of Mathematics, Saveetha School of Engineering (SIMATS), Thandalam, Chennai, Tamil Nadu, India
4 Department of Mathematics, Khazar University, Baku, Azerbaijan
5 Institute of Energy Infrastructure (IEI), Department of Civil Engineering, College of Engineering, Universiti Tenaga Nasional (UNITEN), Putrajaya Campus, Jalan IKRAM-UNITEN, Kajang, Selangor, Malaysia
6 Institute of Informatics and Computing in Energy (IICE), Universiti Tenaga Nasional, Kajang, Selangor, Malaysia
7 Electrical Engineering Department, Future University in Egypt, Cairo, Egypt
IJOCTA, 025160082 https://doi.org/10.36922/IJOCTA025160082
Received: 17 April 2025 | Revised: 31 May 2025 | Accepted: 12 June 2025 | Published online: 7 July 2025
© 2025 by the Author(s). This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution -Noncommercial 4.0 International License (CC-by the license) ( https://creativecommons.org/licenses/by-nc/4.0/ )
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Abstract

The article addresses the solution of parabolic differential equations with integral boundary conditions using the Haar wavelet collocation method. This approach employs a linear combination of Haar wavelet functions to estimate the largest derivatives in the governing equation. The integral boundary conditions are incorporated by repeatedly integrating the highest derivative to formulate equations for the unknowns. Haar wavelets are particularly suitable for approximating solutions to differential equations due to their compact support and multiresolution properties. Numerical experiments on various test cases show that the proposed method yields accurate results, especially when the parameters of the integral boundary conditions are negative.

Keywords
Haar wavelets collocation method
Integral boundary conditions
Parabolic differential equations
Numerical analysis
Funding
None.
Conflict of interest
The authors declare there is no competing interest regarding this work.
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An International Journal of Optimization and Control: Theories & Applications, Electronic ISSN: 2146-5703 Print ISSN: 2146-0957, Published by AccScience Publishing
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