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Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method

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  • Al-Smadi, Mohammed
  • Arqub, Omar Abu
  • Shawagfeh, Nabil
  • Momani, Shaher

Abstract

The reproducing kernel method is a numerical as well as analytical technique for solving a large variety of ordinary and partial differential equations associated to different kind of boundary conditions, and usually provides the solutions in term of rapidly convergent series in the appropriate Hilbert spaces with components that can be elegantly computed. The aim of the present analysis is to implement a relatively recent computational method, reproducing kernel Hilbert space, for obtaining the solutions for systems of second-order differential equations with periodic boundary conditions. A reproducing kernel space is constructed in which the periodic conditions of the systems are satisfied, whilst, the smooth kernel functions are used throughout the evolution of the method to obtain the required grid points. An efficient construction is given to obtain the approximate solutions for the systems together with an existence proof of the exact solutions is proposed based upon the reproducing kernel theory. Convergence analysis and error behavior of the presented method are also discussed. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed.

Suggested Citation

  • Al-Smadi, Mohammed & Arqub, Omar Abu & Shawagfeh, Nabil & Momani, Shaher, 2016. "Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method," Applied Mathematics and Computation, Elsevier, vol. 291(C), pages 137-148.
    • Handle: RePEc:eee:apmaco:v:291:y:2016:i:c:p:137-148
    DOI: 10.1016/j.amc.2016.06.002
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    References listed on IDEAS

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    1. Geng, F.Z. & Qian, S.P. & Cui, M.G., 2015. "Improved reproducing kernel method for singularly perturbed differential-difference equations with boundary layer behavior," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 58-63.
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    Cited by:

    1. Hasan, Shatha & El-Ajou, Ahmad & Hadid, Samir & Al-Smadi, Mohammed & Momani, Shaher, 2020. "Atangana-Baleanu fractional framework of reproducing kernel technique in solving fractional population dynamics system," Chaos, Solitons & Fractals, Elsevier, vol. 133(C).
    2. Akgül, Ali, 2018. "A novel method for a fractional derivative with non-local and non-singular kernel," Chaos, Solitons & Fractals, Elsevier, vol. 114(C), pages 478-482.
    3. Hasan, Shatha & Al-Smadi, Mohammed & El-Ajou, Ahmad & Momani, Shaher & Hadid, Samir & Al-Zhour, Zeyad, 2021. "Numerical approach in the Hilbert space to solve a fuzzy Atangana-Baleanu fractional hybrid system," Chaos, Solitons & Fractals, Elsevier, vol. 143(C).

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