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On an Approximate Solution of a Class of Systems of Curvilinear Integral Equations

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Abstract

A substantiation of the collocation method for the system of integral equations in the problems of shielding electromagnetic fields for cylindrical bodies is given. The system of integral equations is approximated at specially chosen points by a system of algebraic equations, for which the existence and uniqueness of a solution are established. The convergence of the solution of the resulting system of algebraic equations to the exact solution of the system of integral equations is proved, and the rate of convergence of the method is indicated.

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REFERENCES

  1. Kravchenko, V.F. and Erofeenko, V.T., Diffraction of electromagnetic waves by superconducting thin cylindrical shells, Dokl. Ross. Akad. Nauk, 1994, vol. 337, no. 1, pp. 25–27.

    MATH  Google Scholar 

  2. Erofeenko, V.T. and Kozlovskaya, I.S., Integral equations in problems of shielding electromagnetic fields for cylindrical bodies, Differ. Equations, 1992, vol. 28, no. 2, pp. 209–213.

    MATH  Google Scholar 

  3. Boikov, I.V. and Tynda, A.N., Approximate solution of nonlinear integral equations of the theory of developing systems, Differ. Equations, 2003, vol. 39, no. 9, pp. 1277–1288.

    Article  MathSciNet  Google Scholar 

  4. Bulatov, M.V., Numerical solution of systems of Volterra integral equations of the first kind, Comput. Math. Math. Phys., 1998, vol. 38, no. 4, pp. 585–589.

    MathSciNet  MATH  Google Scholar 

  5. Ghiat, M., Kamouche, S., Khellaf, A., and Merchela, W., On a system of Volterra integral equations with a weakly singular kernel, Itogi Nauki Tekh. Ser. Sovrem. Mat. Pril. Temat. Obz., 2021, vol. 193, pp. 33–44.

    Article  Google Scholar 

  6. Stavtsev, S.L., An iterative approach to the numerical solution of the system of integral equations for boundary value problems for the scalar Helmholtz equation, Differ. Equations, 2006, vol. 42, no. 9, pp. 1352–1360.

    Article  MathSciNet  Google Scholar 

  7. Tynda, A.N., Sidorov, D.N., and Muftakhov, I.R., A numerical method for solving systems of nonlinear Volterra integral equations of the first kind with discontinuous kernels, Zh. Sredne-Volzhsk. Mat. O-va, 2018, vol. 20, no. 1, pp. 55–63.

    MATH  Google Scholar 

  8. Khalilov, E.H., Justification of the collocation method for one class of systems of integral equations, Ukr. Mat. Zh., 2017, vol. 69, no. 6, pp. 823–835.

    Article  Google Scholar 

  9. Khalilov, E.H. and Bakhshaliyeva, M.N., Quadrature formulas for simple and double layer logarithmic potentials, Proc. IMM NAS Azerb., 2019, vol. 45, no. 1, pp. 155–162.

    MathSciNet  MATH  Google Scholar 

  10. Kress, R., Boundary integral equations in time-harmonic acoustic scattering, Math. Comput. Model., 1991, vol. 15, no. 3–5, pp. 229–243.

    Article  MathSciNet  Google Scholar 

  11. Muskheleshvili, N.I., Singulyarnye integral’nye uravneniya (Singular Integral Equations), Moscow: Fizmatgiz, 1962.

    Google Scholar 

  12. Vladimirov, V.S., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Nauka, 1976.

    Google Scholar 

  13. Khalilov, E.H., Justification of the collocation method for a class of surface integral equations, Math. Notes, 2020, vol. 107, no. 4, pp. 663–678.

    Article  MathSciNet  Google Scholar 

  14. Vainikko, G.M., Regular convergence of operators and approximate solution of equations, J. Math. Sci., 1981, vol. 15, no. 6, pp. 675–705.

    Article  Google Scholar 

  15. Bakhshalyeva, M.N. and Khalilov, E.H., Justification of the collocation method for an integral equation of the exterior Dirichlet problem for the Laplace equation, Comput. Math. Math. Phys., 2021, vol. 61, no. 6, pp. 923–937.

    Article  MathSciNet  Google Scholar 

  16. Khalilov, E.H. and Bakhshaliyeva, M.N., Study of approximate solution to integral equation associated with mixed boundary value problem for Laplace equation, Ufa Math. J., 2021, vol. 13, no. 1, pp. 85–97.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Azerbaijan State Oil and Industry University, Baku, AZ1010, Azerbaijan

    E. H. Khalilov

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  1. E. H. Khalilov
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Correspondence to E. H. Khalilov.

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Translated by V. Potapchouck

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Khalilov, E.H. On an Approximate Solution of a Class of Systems of Curvilinear Integral Equations. Diff Equat 58, 556–570 (2022). https://doi.org/10.1134/S0012266122040115

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  • DOI: https://doi.org/10.1134/S0012266122040115

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