Page 52 - IJOCTA-15-4

An International Journal of Optimization and Control: Theories & Applications
ISSN: 2146-0957 eISSN: 2146-5703
Vol.15, No.4, pp.594-609 (2025)
https://doi.org/10.36922/IJOCTA025160082

RESEARCH ARTICLE

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

6
2
1
Muhammad Nawaz Khan , Masood Ahmad , Rashid Jan 3,4,5* , Imtiaz Ahmad , and
Mohamed Mousa 7
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
mnawaz77@gmail.com, masood.suf@gmail.com, rashid.jan@uniten.edu.my, imtiazkakakhil@gmail.com,
mohamed.Mossa@fue.edu.eg

ARTICLE INFO ABSTRACT

Article History:
The article addresses the solution of parabolic differential equations with in-
Received: April 17, 2025
tegral boundary conditions using the Haar wavelet collocation method. This
Revised: May 31, 2025
approach employs a linear combination of Haar wavelet functions to estimate
Accepted: June 12, 2025
the largest derivatives in the governing equation. The integral boundary con-
Published Online: July 7, 2025
ditions are incorporated by repeatedly integrating the highest derivative to
Keywords: formulate equations for the unknowns. Haar wavelets are particularly suitable
Haar wavelets collocation method for approximating solutions to differential equations due to their compact sup-
Integral boundary conditions port and multiresolution properties. Numerical experiments on various test
Parabolic differential equations cases show that the proposed method yields accurate results, especially when
Numerical analysis the parameters of the integral boundary conditions are negative.
AMS Classification 2010:
26A33; 34A08; 35H15; 34K50
47H10; 60H10

1. Introduction conduction problem with integral boundary con-
ditions is notably significant due to its non-self-
adjoint nature, which poses challenges for thor-
ough investigation. 4
Integral boundary conditions play a crucial role
in numerous mathematical and physical prob- Certain chemical diffusion and heat conduc-
lems, such as heat conduction and fluid mechan- tion processes are modeled by the nonclassical
ics. These conditions incorporate integrals of the parabolic initial-boundary value problem :
5
solution over the spatial domain, adding complex- 2
1
ity to the use of standard techniques. Research ∂s = ∂ s 2 + a ∂s + cs + f(κ, t),
has delved into optimal control problems that in- ∂t ∂κ ∂κ
(κ, t) ∈ (0, 1) × (0, T],
volve integral boundary conditions in fields, like
physics, engineering, and mechanics. 2,3 The heat s(κ, 0) = g 1 (κ), κ ∈ (0, 1),
*Corresponding Author
594