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Solving parabolic differential equations via Haar wavelets: A focus on integral boundary conditions
decomposition or adaptive techniques. These en- Availability of data
hancements are suggested as promising avenues
for future research to broaden the method’s ap- Not applicable.
plicability to real-world problems. Numerical
tests have confirmed the method’s effectiveness
in terms of accuracy, integration, and computa- AI tools statement
tional efficiency, highlighting its practical applica-
bility to real-world problems. The Haar wavelets All authors confirm that no AI tools were used in
collocation method has presented a promising ap- the preparation of this manuscript.
proach for numerically treating PDEs with inte-
gral boundary conditions, providing a valuable
tool for researchers and practitioners across vari-
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Author contributions
order parabolic equation subject to nonlocal spec-
Conceptualization: Muhammad Nawaz Khan, ifications. Appl Numer Math. 2005;52(1):39-62.
Imtiaz Ahmad, Mohamed Mousa 11. Day W. Extensions of a property of the heat equa-
Formal analysis: Muhammad Nawaz Khan, Ma- tion to linear thermoelasticity and other theories.
sood Ahmad, Rashid Jan Qtly Appl Math. 1982;40(3):319-330.
12. Kumar A, Kumar M, Goswami P. Numerical so-
Investigation: Mohamed Mousa, Rashid Jan, Im-
lution of coupled system of Emden–Fowler equa-
tiaz Ahmad
tions using artificial neural network technique.
Methodology: Muhammad Nawaz Khan, Masood
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Ahmad, Mohamed Mousa
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Writing-original draft: Muhammad Nawaz Khan, 13. Arslan D, C¸elik E. An approximate solution
Imtiaz Ahmad of singularly perturbed problem on uniform
Writing-review & editing: Rashid Jan, Mohamed mesh. Int J Optimiz Control Theor Appl. 2024;
Mousa 14(1):74-80.
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