https://doi.org/10.1140/epjp/s13360-023-04735-2
Regular Article
Numerical treatment of singular functional systems in quantum calculus: adaptive backpropagated Levenberg–Marquardt neural networks
1
Department of Mathematics, University of Gujrat, 50700, Gujrat, Pakistan
2
Yuan Ze University, AI Center, 320, Taoyuan, Taiwan
3
Future Technology Research Center, National Yunlin University of Science and Technology, 123 University Road, Section 3, 64002, Douliou, Yunlin, Taiwan, ROC
Received:
4
August
2023
Accepted:
20
November
2023
Published online:
2
January
2024
The purpose of this research is to investigate the numerical solutions of singular functional delay differential equations (SFDDEs) of third, fourth and fifth orders, which arises in quantum calculus models, by utilizing the applications of adaptive Levenberg–Marquardt backpropagation (LMB) neural networks. The presence of delay term as well as singularity causes difficulty in investigation of such types of functional differential equations, and these stiffness factors are overcome by employing the designed LMB technique. The stiff functional delay systems in quantum calculus are transformed into the iterative expressions, and synthetic datasets are generated for the SFDDEs by varying quantum and pantograph delay parameters, as well as the coefficients of delay and singular terms. The synthesize data is feed to the designed LMB neural networks for training, testing and validation processes to construct the approximate solution models of SFDDEs. The accuracy of proposed LMB technique is certified by comparison of the obtained numerical results with reference solutions through absolute error analysis, mean square error, regression index metric, convergence curves, histogram studies and training state parameters for each dynamical scenario of SFDDEs.
Supplementary Information The online version contains supplementary material available at https://doi.org/10.1140/epjp/s13360-023-04735-2.
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© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
