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Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm–Volterra integrodifferential equations

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Abstract

In this article, we propose the reproducing kernel Hilbert space method to obtain the exact and the numerical solutions of fuzzy Fredholm–Volterra integrodifferential equations. The solution methodology is based on generating the orthogonal basis from the obtained kernel functions in which the constraint initial condition is satisfied, while the orthonormal basis is constructing in order to formulate and utilize the solutions with series form in terms of their r-cut representation form in the Hilbert space \( W_{2}^{2} \left( \varOmega \right) \oplus W_{2}^{2} \left( \varOmega \right) \). Several computational experiments are given to show the good performance and potentiality of the proposed procedure. Finally, the utilized results show that the present method and simulated annealing provide a good scheduling methodology to solve such fuzzy equations.

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Acknowledgments

The author would like to express his gratitude to the unknown referees for carefully reading the paper and their helpful comments.

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Correspondence to Omar Abu Arqub.

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Abu Arqub, O. Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm–Volterra integrodifferential equations. Neural Comput & Applic 28, 1591–1610 (2017). https://doi.org/10.1007/s00521-015-2110-x

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Keywords

  • Fuzzy integrodifferential equations
  • Reproducing kernel Hilbert space method
  • Strongly generalized differentiability
  • Gram–Schmidt process

Mathematics Subject Classification

  • 46S40
  • 47B32
  • 34K28