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.
References
  1. Bouziani A. On the solvability of parabolic and hyperbolic problems with a boundary integral condition. Int J Math Math Sci. 2002;31(4):201-

 

  1. Barbu T,  Miranville  A,  Moro¸sanu    On  a local  and  nonlocal  second-order  boundary value  problem  with  in-homogeneous  cauchy– neumann  boundary  conditions—applications in  engineering  and  industry.  Mathematics. 2024;12(13):2050.

 

  1. Molaei H. Optimal control and problem with in- tegral boundary conditions. Int J Contemp Math Sci. 2011;6(48):2385-2390.

 

  1. Almomani R, Almefleh H. On heat conduction problem with integral boundary condition. J Emerg Trends Eng Appl Sci. 2012;3(6):977-979.

 

  1. Bouziani A. On a class of parabolic equations with a nonlocal boundary condition. Bulll’Acad R Belg. 1999;10(1):61-77.

 

  1. Yin H-M. On a class of parabolic equations with nonlocal boundary conditions. J Math Anal Appl. 2004;294(2):712-728.

 

  1. Bouziani A. Strong solution for a mixed problem with nonlocal condition for certain pluriparabolic equations. Hiroshima Math J. 1997;27(3):373-390.

 

  1. Day W. A decreasing property of solutions of parabolic equations with applications to thermoelas Qtly Appl Math. 1983;40(4):468-475.

 

  1. Ang W. A method of solution for the one- dimensional heat  equation  subject  to  nonlocal    Southeast  Asian  Bull  Math. 2003;26:185-191.

 

  1. Dehghan M. Efficient techniques for the second- order parabolic equation subject to nonlocal spec- ifications. Appl Numer Math. 2005;52(1):39-62.

 

  1. Day W. Extensions of a property of the heat equation to linear thermoelasticity and other theories. Qtly Appl Math. 1982;40(3):319-330.

 

  1. Kumar A, Kumar M, Goswami P. Numerical solution of coupled system of Emden–Fowler equations using artificial neural network technique. Int J Optimiz Control Theor Appl. 2024;14(1):62-

 

  1. Arslan D, C¸ elik E. An approximate solution of singularly perturbed problem on uniform mesh. Int J Optimiz Control Theor Appl. 2024;14(1):74-80.

 

  1. Baleanu D, Hajipour M, Jajarmi A. An accurate finite difference formula for the numerical solution of delay-dependent fractional optimal control problems. Int J Optimiz Control Theor Appl. 2024;14(3):183-192.

 

  1. Abbas WS, El-wakad MT, Darwish RR. Finite element modeling for two-dimensional wireless capsule endoscope manipulation system. Trends Adv Sci Technol. 2025;2(1): 2.

 

  1. Masti I, Sayevand K, Jafari H. On analyzing two- dimensional fractional order brain tumor model based on orthonormal Bernoulli polynomials and Newton’s method. Int J Optimiz Control Theor Appl. 2024;14(1):12-19.

 

  1. Malagi NS, Veeresha P, Prasanna GD, Prasan- nakumara BC, Prakasha DG. Novel approach for nonlinear time-fractional Sharma–Tasso–Olver equation using Elzaki transform. Int J Optimiz Control Theor Appl. 2023;13(1):46-58.

 

  1. Erdogan U, Ozis T. A smart nonstandard finite difference scheme for second order nonlinear boundary value problems. J Comput Phys. 2011;230(17):6464-6474.

 

  1. Mukhtarov O, C¸ AVUS¸OG˘ LU S, OLG˘ AR H. Numerical solution of one boundary value problem using finite difference method. Turk J Math Com- put Sci. 2019;11:85-89.

 

  1. Cheng F, Li W, Zhou Y, et al. admetSAR: a com- prehensive source and free tool for assessment of chemical ADMET properties. J Chem Inf Model. 2012;52(11):3099-3105.

 

  1. Hoppe RH, Kieweg M. Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems. Comput Optimiz Appl. 2010;46:511-

 

  1. Hesameddini E, Riahi M. Hybrid legendre block-pulse functions method for solving partial differential equations with non-local integral boundary conditions. J Inform Optimiz Sci. 2019;40(7):1391-1403.

 

  1. Siraj-ul-Isalm, Aziz I, Ahmad M. Numerical solution of two-dimensional elliptic PDEs with non-local boundary conditions. Comput Math Appl.2015;69(3):180-205.

 

  1. Ooi E H, Popov V. A simplified approach for imposing the boundary conditions in the local boundary integral equation method. Comput 2013;51(5):717-729.

 

  1. Kai Y, Yin Z. On the gaussian traveling wave solution to a special kind of schr¨odinger equation with logarithmic nonlinearity. Mod Phys Lett B., 2022;36(02):2150543.

 

  1. Yang Y, Li H. Neural ordinary differential equations for robust parameter estimation in dynamic systems with physical priors. Appl Soft Comput. 2025;169:112649.

 

  1. Chen Z, Wu J, Xu Y. Higher-order finite volume methods for elliptic boundary value problems. Adv Comput Math. 2012;37(2):191- 253.

 

  1. Jang G-W, Kim Y Y, Choi K K. Remesh-free shape optimization using the wavelet-Galerkin method. Int J Solids Struct. 2004;41(22-23):6465-

 

  1. Liu Y, Cen Z. Daubechies wavelet meshless method for 2-D elastic problems. Tsinghua Sci 2008;13(5):605-608.

 

  1. Lepik U¨ . Solving PDEs with the aid of two-dimensional haar wavelets. Comput Math Appl. 2011;61(7):1873-1879.

 

  1. D´ıaz L A, Mart´ın M T, Vampa V. Daubechies wavelet beam and plate finite elements. Finite El- ements Anal Des. 2009;45(3):200-209.

 

  1. Khan A A, Ahsan M, Ahmad I, Alwuthay- nani M. Enhanced resolution in solving first-order nonlinear differential equations with integral condition: a high-order wavelet approach. Eur Phys J Spec Top. 2024;2024:1-14.

 

  1. Ahsan M, Khan A A, Dinibutun S, et al. The haar wavelets based numerical solution of Reccati equation with integral boundary condition. Thermal Sci. 2023;27(1):93-100.

 

  1. Shah K, Amin R, Abdeljawad T. Utilization of Haar wavelet collocation technique for fractal-fractional order problem. Heliyon. 2023;9(6):e17123.

 

  1. Amin R, Shah K, Awais M, Mahariq I, Nisar KS, Sumelka W. Existence and solution of third-order integro-differential equations via Haar wavelet Fractals 2023;31(02):2340037.

 

  1. Liu X, Ahsan M, Ahmad M, et al. Applications of haar wavelet-finite difference hybrid method and its convergence for hyperbolic nonlinear Schr¨o dinger equation with energy and mass conver Energies 2021;14(23):7831.

 

  1. Ahsan M, Lin S, Ahmad M, et al. A haar wavelet-based scheme for finding the control parameter in nonlinear inverse heat conduction equation. Open Phys. 2021;19(1): 722-734.

 

  1. Zhou S, He Z, Chen X, Chang W. An anomaly detection method for uav based on wavelet de- composition and stacked denoising autoencoder. 2024;11(5):393.

 

  1. Aziz I, Sˇarler B. The numerical solution of second-order boundary-value problems by collocation method with the haar wavelets. Math Com- put Model. 2010;52(9-10):1577-1590.

 

  1. Tatari M, Dehghan M. On the solution of the non-local parabolic partial differential equations via radial basis functions. Appl Math Model., 2009;33(3):1729-1738.

 

  1. Ivanauskas F, Meˇskauskas T, Sapagovas M. Stability of difference schemes for two-dimensional parabolic equations with non-local boundary con- ditions. Appl Math Comput. 2009;215(7):2716-

 

  1. Sajaviˇcius S. Stability of the weighted splitting finite-difference scheme for a two-dimensional parabolic equation with two nonlocal integral conditions. Comput Math Appl. 2012;64(11):3485-3499.
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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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