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Analysis and Spectral Element Solution of Nonlinear Integral Equations of Hammerstein Type

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Topics in Integral and Integro-Differential Equations

Part of the book series: Studies in Systems, Decision and Control ((SSDC,volume 340))

Abstract

We employ the spectral element method with Gauss-Lobatto-Legendre collocation points to approximate nonlinear integral equations of Hammerstein type. Using the Banach Fixed Point Theorem, we establish sufficient conditions for the existence and uniqueness of solutions in the \(L^2\) norm, as well as the convergence of the proposed method, under a different aspect of the existing works in the literature, indicating that the numerical error decays exponentially provided that the kernel function be smooth enough. The iterative Picard process was used to approximate the nonlinear problem. Numerical experiments involving one- and two-dimensional nonlinear equations illustrate the effectiveness of this approach.

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Acknowledgements

S. P. Oliveira is supported by CNPq under the grant 313100/2017-9.

Author information

Authors and Affiliations

  1. ICTI-UFBA Centro, Camaçari, BA, 42809-000, Brazil

    Juarez S. Azevedo

  2. DMAT-UFPR and INCT-GP, Curitiba, PR, 81531-980, Brazil

    Saulo P. Oliveira

  3. Universidade Estadual Paulista (UNESP) IGCE, Rio Claro, SP, 13506-900, Brazil

    Suzete M. Afonso

  4. CETEC-UFRB Centro, Cruz das Almas, BA, 44380-000, Brazil

    Mariana P. G. da Silva

Authors
  1. Juarez S. Azevedo
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  2. Saulo P. Oliveira
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  3. Suzete M. Afonso
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  4. Mariana P. G. da Silva
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Corresponding author

Correspondence to Saulo P. Oliveira .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Post Graduate College, Ghazipur, Uttar Pradesh, India

    Harendra Singh

  2. Department of Mathematics, Gauhati University, Guwahati, Assam, India

    Hemen Dutta

  3. Departamento de Matemática, Universidade Estadual de Maringá, Maringá, Brazil

    Marcelo M. Cavalcanti

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Azevedo, J.S., Oliveira, S.P., Afonso, S.M., da Silva, M.P.G. (2021). Analysis and Spectral Element Solution of Nonlinear Integral Equations of Hammerstein Type. In: Singh, H., Dutta, H., Cavalcanti, M.M. (eds) Topics in Integral and Integro-Differential Equations. Studies in Systems, Decision and Control, vol 340. Springer, Cham. https://doi.org/10.1007/978-3-030-65509-9_2

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