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Error Estimation for Approximate Solutions of Delay Volterra Integral Equations

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Frontiers in Functional Equations and Analytic Inequalities

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Abstract

This work is related to inequalities in the approximation theory. Mainly, we study numerical solutions of delay Volterra integral equations by using a collocation method based on sigmoidal function approximation. Error estimation and convergence analysis are provided. At the end of the paper we display numerical simulations verifying our results.

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References

  1. G.A. Anastassiou, in Intelligent Systems: Approximation by Artificial Neural Networks, Intelligent Systems Reference Library, vol. 19 (Springer-Verlag, Berlin, 2011)

    MATH  Google Scholar 

  2. G.A. Anastassiou, Multivariate sigmoidal neural network approximation. Neural Netw. 24(4), 378–386 (2011)

    Article  Google Scholar 

  3. G.A. Anastassiou, in Intelligent Systems II: Complete Approximation by Neural Network Operators, Studies in Computational Intelligence, vol. 608 (Springer, Cham, 2016)

    Google Scholar 

  4. G.A. Anastassiou, L. Coroianu, S.G. Gal, Approximation by a nonlinear Cardaliaguet-Euvrard neural network operator of max-product kind. J. Comput. Anal. Appl. 12(2), 396–406 (2010)

    MathSciNet  MATH  Google Scholar 

  5. K.E. Atkinson, in The Numerical Solution of Integral Equations of the Second Kind, Cambridge Monographs on Applied and Computational Mathematics, vol. 4 (Cambridge University Press, Cambridge, 1997)

    Book  Google Scholar 

  6. H. Brunner, Iterated collocation methods for Volterra integral equations with delay arguments. Math. Comp. 62, 581–599 (1994)

    Article  MathSciNet  Google Scholar 

  7. H. Brunner, in Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge Monographs on Applied and Computational Mathematics, vol. 15 (Cambridge University Press, Cambridge 2004)

    Google Scholar 

  8. K.L. Cooke, An epidemic equation with immigration. Math. Biosci. 29(1–2), 135–158 (1976)

    Article  MathSciNet  Google Scholar 

  9. D. Costarelli, R. Spigler, Approximation results for neural network operators activated by sigmoidal functions. Neural Netw. 44, 101–106 (2013)

    Article  Google Scholar 

  10. D. Costarelli, R. Spigler, Constructive approximation by superposition of sigmoidal functions. Anal. Theory Appl. 29(2), 169–196 (2013)

    Article  MathSciNet  Google Scholar 

  11. D. Costarelli, R. Spigler, Solving Volterra integral equations of the second kind by sigmoidal functions approximation. J. Integral Equations Appl. 25(2), 193–222 (2013)

    Article  MathSciNet  Google Scholar 

  12. D. Costarelli, R. Spigler, A collocation method for solving nonlinear Volterra integro-differential equations of neutral type by sigmoidal functions. J. Integral Equations Appl. 26(1), 15–52 (2014)

    Article  MathSciNet  Google Scholar 

  13. D. Costarelli, G. Vinti, Pointwise and uniform approximation by multivariate neural network operators of the max-product type. Neural Netw. 81, 81–90 (2016)

    Article  Google Scholar 

  14. J.K. Hale, S.M.V. Lunel, in Introduction to Functional-Differential Equations, Applied Mathematical Sciences, vol. 99 (Springer-Verlag, New York, 1993)

    Book  Google Scholar 

  15. W. Ming, C. Huang, Collocation methods for Volterra functional integral equations with non-vanishing delays. Appl. Math. Comput. 296, 198–214 (2017)

    MathSciNet  MATH  Google Scholar 

  16. W. Ming, C. Huang, M. Li, Superconvergence in collocation methods for Volterra integral equations with vanishing delays. J. Comput. Appl. Math. 308, 361–378 (2016)

    Article  MathSciNet  Google Scholar 

  17. S. Nemati, Numerical solution of Volterra-Fredholm integral equations using Legendre collocation method. J. Comput. Appl. Math. 278, 29–36 (2015)

    Article  MathSciNet  Google Scholar 

  18. K. Yang, Zhang, R.: Analysis of continuous collocation solutions for a kind of Volterra functional integral equations with proportional delay. J. Comput. Appl. Math. 236(5), 743–752 (2011)

    Google Scholar 

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Authors and Affiliations

  1. TOBB University of Economics and Technology, Department of Mathematics, Söğütözü, Ankara, Turkey

    Oktay Duman

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  1. Oktay Duman
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Correspondence to Oktay Duman .

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Editors and Affiliations

  1. Department of Mathematical Sciences, University of Memphis, Memphis, TN, USA

    George A. Anastassiou

  2. Pedagogical Department E. E., Section of Mathematics and Informatics, National and Kapodistrian University of Athens, Athens, Greece

    John Michael Rassias

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Duman, O. (2019). Error Estimation for Approximate Solutions of Delay Volterra Integral Equations. In: Anastassiou, G., Rassias, J. (eds) Frontiers in Functional Equations and Analytic Inequalities. Springer, Cham. https://doi.org/10.1007/978-3-030-28950-8_29

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