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New sinusoidal basis functions and a neural network approach to solve nonlinear Volterra–Fredholm integral equations

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Abstract

In this paper, we present and investigate the analytical properties of a new set of orthogonal basis functions derived from the block-pulse functions. Also, we present a numerical method based on this new class of functions to solve nonlinear Volterra–Fredholm integral equations. In particular, an alternative and efficient method based on the formalism of artificial neural networks is discussed. The efficiency of the mentioned approach is theoretically justified and illustrated through several qualitative and quantitative examples.

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Acknowledgements

The authors wishes to thank the anonymous reviewers and the editor in charge of handling this paper for all their criticisms and comments. All of their suggestions contributed significantly to improve the quality of this work.

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Authors and Affiliations

  1. Consorzio di Ricerca Sistemi ad Agenti (CORISA), Università degli Studi di Salerno, Fisciano, SA, Italy

    Stefania Tomasiello

  2. Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Aguascalientes, Ags., Mexico

    Jorge E. Macías-Díaz

  3. Department of Mathematics, Institute for Advanced Studies, Basic Sciences, Zanjan, Iran

    Alireza Khastan

  4. Department of Mathematics and Statistics, University of Tartu, Tartu, Estonia

    Zahra Alijani

Authors
  1. Stefania Tomasiello
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  2. Jorge E. Macías-Díaz
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  3. Alireza Khastan
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  4. Zahra Alijani
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Correspondence to Stefania Tomasiello.

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Tomasiello, S., Macías-Díaz, J.E., Khastan, A. et al. New sinusoidal basis functions and a neural network approach to solve nonlinear Volterra–Fredholm integral equations. Neural Comput & Applic 31, 4865–4878 (2019). https://doi.org/10.1007/s00521-018-03984-y

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  • DOI: https://doi.org/10.1007/s00521-018-03984-y

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