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A Hybrid Method for All Types of Solutions of the System of Cauchy-Type Singular Integral Equations of the First Kind

Author

Listed:
  • H. X. Mamatova

    (Faculty of Ocean Engineering Technology and Informatics, University Malaysia Terengganu (UMT), Kuala Terengganu 21030, Malaysia)

  • Z. K. Eshkuvatov

    (Faculty of Ocean Engineering Technology and Informatics, University Malaysia Terengganu (UMT), Kuala Terengganu 21030, Malaysia
    Faculty of Applied Mathematics and Intellectual Technologies, National University of Uzbekistan (NUUz), Tashkent 100174, Uzbekistan)

  • Sh. Ismail

    (Faculty of Science and Technology, Universiti Sains Islam Malaysia (USIM), Bandar Baru Nilai 71800, Malaysia)

Abstract

In this note, the hybrid method (combination of the homotopy perturbation method (HPM) and the Gauss elimination method (GEM)) is developed as a semi-analytical solution for the first kind system of Cauchy-type singular integral equations (CSIEs) with constant coefficients. Before applying the HPM, we have to first reduce the system of CSIEs into a triangle system of algebraic equations using GEM, which is then carried out using the HPM. Using the theory of the bounded, unbounded and semi-bounded solutions of CSIEs, we are able to find inverse operators for the system of CSIEs of the first kind. A stability analysis and convergent of the proposed method has been conducted in the weighted L p space. Moreover, the proposed method is proven to be exact in the Holder class of functions for the system of characteristic SIEs for any type of initial guess. For each of the four cases, several examples are provided and examined to demonstrate the proposed method’s validity and accuracy. Obtained results are compared with the Chebyshev collocation method and modified HPM (MHPM). Example 3 reveals that the error term of the MHPM is slightly superior to that of the HPM. One of the features of the proposed method is that it can be solved as a complex-valued system of CSIEs. Numerical results revealed that the hybrid method dominates others.

Suggested Citation

  • H. X. Mamatova & Z. K. Eshkuvatov & Sh. Ismail, 2023. "A Hybrid Method for All Types of Solutions of the System of Cauchy-Type Singular Integral Equations of the First Kind," Mathematics, MDPI, vol. 11(20), pages 1-30, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:20:p:4404-:d:1265730
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    References listed on IDEAS

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