IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i20p4404-d1265730.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/20/4404/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/20/4404/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Odibat, Zaid & Momani, Shaher, 2008. "Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 36(1), pages 167-174.
    2. Abbasbandy, S., 2007. "Application of He’s homotopy perturbation method to functional integral equations," Chaos, Solitons & Fractals, Elsevier, vol. 31(5), pages 1243-1247.
    3. Siddiqui, A.M. & Mahmood, R. & Ghori, Q.K., 2008. "Homotopy perturbation method for thin film flow of a third grade fluid down an inclined plane," Chaos, Solitons & Fractals, Elsevier, vol. 35(1), pages 140-147.
    4. Cveticanin, L., 2006. "Homotopy–perturbation method for pure nonlinear differential equation," Chaos, Solitons & Fractals, Elsevier, vol. 30(5), pages 1221-1230.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Golbabai, A. & Keramati, B., 2008. "Modified homotopy perturbation method for solving Fredholm integral equations," Chaos, Solitons & Fractals, Elsevier, vol. 37(5), pages 1528-1537.
    2. Cveticanin, L., 2009. "Application of homotopy-perturbation to non-linear partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 40(1), pages 221-228.
    3. Biazar, J. & Eslami, M. & Aminikhah, H., 2009. "Application of homotopy perturbation method for systems of Volterra integral equations of the first kind," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 3020-3026.
    4. Biazar, J. & Ghazvini, H., 2009. "He’s homotopy perturbation method for solving systems of Volterra integral equations of the second kind," Chaos, Solitons & Fractals, Elsevier, vol. 39(2), pages 770-777.
    5. Golbabai, A. & Keramati, B., 2009. "Solution of non-linear Fredholm integral equations of the first kind using modified homotopy perturbation method," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2316-2321.
    6. Biazar, J. & Ghazvini, H. & Eslami, M., 2009. "He’s homotopy perturbation method for systems of integro-differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1253-1258.
    7. Beléndez, A. & Beléndez, T. & Neipp, C. & Hernández, A. & Álvarez, M.L., 2009. "Approximate solutions of a nonlinear oscillator typified as a mass attached to a stretched elastic wire by the homotopy perturbation method," Chaos, Solitons & Fractals, Elsevier, vol. 39(2), pages 746-764.
    8. Golbabai, A. & Keramati, B., 2008. "Easy computational approach to solution of system of linear Fredholm integral equations," Chaos, Solitons & Fractals, Elsevier, vol. 38(2), pages 568-574.
    9. Yusufoğlu (Agadjanov), Elcin, 2009. "Improved homotopy perturbation method for solving Fredholm type integro-differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 41(1), pages 28-37.
    10. Moosaei, H. & Ketabchi, S. & Noor, M.A. & Iqbal, J. & Hooshyarbakhsh, V., 2015. "Some techniques for solving absolute value equations," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 696-705.
    11. Jim Gatheral & Radoš Radoičić, 2019. "Rational Approximation Of The Rough Heston Solution," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(03), pages 1-19, May.
    12. Fathy, Mohamed & Abdelgaber, K.M., 2022. "Approximate solutions for the fractional order quadratic Riccati and Bagley-Torvik differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 162(C).
    13. S. Balaji, 2014. "Legendre Wavelet Operational Matrix Method for Solution of Riccati Differential Equation," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2014, pages 1-10, June.
    14. Shloof, A.M. & Senu, N. & Ahmadian, A. & Salahshour, Soheil, 2021. "An efficient operation matrix method for solving fractal–fractional differential equations with generalized Caputo-type fractional–fractal derivative," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 188(C), pages 415-435.
    15. Wang, Shu-Qiang & He, Ji-Huan, 2008. "Nonlinear oscillator with discontinuity by parameter-expansion method," Chaos, Solitons & Fractals, Elsevier, vol. 35(4), pages 688-691.
    16. Abolvafaei, Mahnaz & Ganjefar, Soheil, 2020. "Maximum power extraction from wind energy system using homotopy singular perturbation and fast terminal sliding mode method," Renewable Energy, Elsevier, vol. 148(C), pages 611-626.
    17. M. Motawi Khashan & Rohul Amin & Muhammed I. Syam, 2019. "A New Algorithm for Fractional Riccati Type Differential Equations by Using Haar Wavelet," Mathematics, MDPI, vol. 7(6), pages 1-12, June.
    18. Adesanya, Samuel O. & Makinde, Oluwole D., 2015. "Irreversibility analysis in a couple stress film flow along an inclined heated plate with adiabatic free surface," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 432(C), pages 222-229.
    19. Ghorbani, Asghar, 2009. "Beyond Adomian polynomials: He polynomials," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1486-1492.
    20. S M, Sivalingam & Kumar, Pushpendra & Govindaraj, V., 2023. "A novel numerical scheme for fractional differential equations using extreme learning machine," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 622(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:11:y:2023:i:20:p:4404-:d:1265730. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.