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A Solution to a Boundary-Value Problem for Integro-Differential Equations with Weakly Singular Kernels

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Abstract

A linear boundary-value problem for a system of integro-differential equations with weakly singular kernels is considered. Questions of the unique solvability and the construction of algorithms for finding solution to the considered problem are studied. Conditions for the solvability of the boundary-value problem for a system of integro-differential equations with weakly singular kernels are established using the Dzhumabaev parametrization method based on splitting the interval and introducing additional parameters. Necessary and sufficient conditions for the solvability of the two-point problem for integro-differential equations with weakly singular kernels are obtained.

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REFERENCES

  1. Brunner H. Collocation methods for Volterra integral and related functional equations (Cambridge University Press, Cambridge, 2004).

    Book  Google Scholar 

  2. Dzhumabaev D.S. "A method for solving the linear boundary value problem for an integro-differential equation", Comput. Math. Math. Phys. 50 (7), 1150-1161 (2010).

    Article  MathSciNet  Google Scholar 

  3. Dzhumabaev D.S., Bakirova E.A. "Criteria for the well-posedness of a linear two-point boundary value problem for systems of integro-differential equations", Differ. Equ. 46 (4), 553-567 (2010).

    Article  MathSciNet  Google Scholar 

  4. Dzhumabaev D.S., Bakirova E.A. "Criteria for the unique solvability of a linear two-point boundary value problem for systems of integro-differential equations", Differ. Equ. 49 (9), 1087-1102 (2013).

    Article  MathSciNet  Google Scholar 

  5. Dzhumabaev D.S., Bakirova E.A. "On unique solvability of a boundary-value problem for Fredholm intergo-differential equations with degenerate kernel", J. Math. Sci. 220 (3), 440-460 (2017).

    Article  MathSciNet  Google Scholar 

  6. Dzhumabaev D.S. "An algorithm for solving a linear two-point boundary value problem for an integrodifferential equation", Comput. Math. Math. Phys. 53 (6), 736-758 (2013).

    Article  MathSciNet  Google Scholar 

  7. Dzhumabaev D.S. "Necessary and sufficient conditions for the solvability of linear boundary-value problems for the Fredholm integrodifferential equations", Ukr. Math. J. 66 (8), 1200-1219 (2014).

    Article  MathSciNet  Google Scholar 

  8. Dzhumabaev D.S. "Solvability of a linear boundary value problem for a fredholm integro-differential equation with impulsive inputs", Differ. Equ. 51 (9), 1180-1196 (2015).

    Article  MathSciNet  Google Scholar 

  9. Dzhumabaev D.S. "On one approach to solve the linear boundary value problems for Fredholm integro-differential equations", J. Comput. Appl. Math. 294 (2), 342-357 (2016).

    Article  MathSciNet  Google Scholar 

  10. Parts I., Pedas A., Tamme E. "Piecewise polynomial collocation for Fredholm integro-differential equations with weakly singular kernels", SIAM J. Numer. Anal. 43 (9), 1897-1911 (2005).

    Article  MathSciNet  Google Scholar 

  11. Pedas A., Tamme E. "Spline collocation method for integro-differential equations with weakly singular kernels", J. Comput. Appl. Math. 197 (2), 253-269 (2006).

    Article  MathSciNet  Google Scholar 

  12. Pedas A., Tamme E. "Discrete Galerkin method for Fredholm integro-differential equations with weakly singular kernels", J. Comput. Appl. Math. 213 (1), 111-126 (2008).

    Article  MathSciNet  Google Scholar 

  13. Kolk M., Pedas A., Vainikko G. "High-order methods for Volterra integral equations with general weak singularities", Numer. Funct. Anal. Optim. 30 (6), 1002-1024 (2009).

    Article  MathSciNet  Google Scholar 

  14. Kangro R., Tamme E. "On fully discrete collocation methods for solving weakly singular integro-differential equations", Math. Model. Anal. 15 (1), 69-82 (2010).

    Article  MathSciNet  Google Scholar 

  15. Orav-Puurand K., Pedas A., Vainikko G. "Nyström type methods for Fredholm integral equations with weak singularities", J. Comput. Appl. Math. 234 (10), 2848-2858 (2010).

    Article  MathSciNet  Google Scholar 

  16. Pedas A., Tamme E. "A discrete collocation method for Fredholm integro-differential equations with weakly singular kernels", Appl. Numer. Math. 61 (4), 738-751 (2011).

    Article  MathSciNet  Google Scholar 

  17. Pedas A., Tamme E. "Product Integration for Weakly Singular Integro-Differential Equations", Math. Model. Anal. 16 (1), 153-172 (2011).

    Article  MathSciNet  Google Scholar 

  18. Pedas A., Tamme E. "On the convergence of spline collocation methods for solving fractional differential equations", J. Comput. Appl. Math. 235 (12), 3502-3514 (2011).

    Article  MathSciNet  Google Scholar 

  19. Pedas A., Tamme E. "Piecewise polynomial collocation for linear boundary value problems of fractional differential equations", J. Comput. Appl. Math. 236 (12), 3349-3359 (2012).

    Article  MathSciNet  Google Scholar 

  20. Pedas A., Tamme E. "Numerical solution of nonlinear fractional differential equations by spline collocation methods", J. Comput. Appl. Math. 255 (1), 216-230 (2014).

    Article  MathSciNet  Google Scholar 

  21. Dzhumabaev D.S. "Criteria for the unique solvability of a linear boundary-value problem for an ordinary differential equation", U.S.S.R. Comput. Math. Math. Phys. 29 (1), 34-46 (1989).

    Article  MathSciNet  Google Scholar 

  22. Muskhelishvili N.I. Singular integral equations (Nauka, Moscow, 1968) [in Russian].

    MATH  Google Scholar 

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ACKNOWLEDGMENTS

The authors are sincerely grateful to the reviewer for worthy advises and helpful remarks which helped to improve the results of this work significantly.

Funding

This research is funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP 09258829).

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Authors and Affiliations

  1. Institute of Mathematics and Mathematical Modeling, 125 Pushkin str., 050010, Almaty, Republic of Kazakhstan

    A. T. Assanova & Sh. N. Nurmukanbet

  2. al-Farabi Kazakh National University, 71 Al-Farabi ave., 050040, Almaty, Republic of Kazakhstan

    Sh. N. Nurmukanbet

Authors
  1. A. T. Assanova
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  2. Sh. N. Nurmukanbet
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Corresponding authors

Correspondence to A. T. Assanova or Sh. N. Nurmukanbet.

Additional information

Russian Text © The Author(s), 2021, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2021, No. 11, pp. 3–15.

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Assanova, A.T., Nurmukanbet, S.N. A Solution to a Boundary-Value Problem for Integro-Differential Equations with Weakly Singular Kernels. Russ Math. 65, 1–13 (2021). https://doi.org/10.3103/S1066369X21110013

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  • DOI: https://doi.org/10.3103/S1066369X21110013

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