Skip to main content
SpringerLink
Log in

Solvability of Multipoint-Integral Boundary Value Problem for a Third-Order Differential Equation and Parametrization Method

  • Conference paper
  • First Online:
Functional Analysis in Interdisciplinary Applications (FAIA 2017)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 216))

Included in the following conference series:

Abstract

A multipoint-integral boundary value problem for a third order differential equation with variable coefficients is considered. The questions of the existence of a unique solution of the considered problem and ways of its construction are investigated. The multipoint-integral boundary value problem for the differential equation of third order with variable coefficients is reduced to a multipoint-integral boundary value problem for a system of three differential equations by introducing new functions. To solve the resulting multipoint-integral boundary value problem, a parametrization method is applied. Algorithms of finding the approximate solution to the multipoint-integral boundary value problem for the system of three differential equations are constructed and their convergence is proved. The conditions of the unique solvability of the multipoint-integral boundary value problem for the system of three differential equations are established in the terms of initial data. The results are also formulated relative to the original of the multipoint-integral boundary value problem for the differential equation of third order with variable coefficients. The obtained results are applied to a two-point boundary value problem for the third order ordinary differential equation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

References

  1. Agarwal, R.P.: Existence-uniqueness and iterative methods for third-order boundary value problems. J. Comput. Appl. Math. 17, 271–289 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  2. Asanova, A.T.: About of the solvability of family multi-point boundary value problems for system of differential equations and their applications to nonlocal boundary value problems. Math. J. 13(3), 38–42 (2013) (in Russian)

    Google Scholar 

  3. Asanova, A.T.: Multipoint problem for a system of hyperbolic equations with mixed derivative. J. Math. Sci. 212(3), 213–233 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Assanova, A.T., Imanchiev, A.E.: On the unique solvability of a multi-point boundary value problem for a third-order differential equation. News of the NAS of the RK. Phys.-mathem ser. 5(303), 124–132 (2015)

    Google Scholar 

  5. Brown, R.C., Krall, A.M.: Adjoints of multipoint-integral boundary value problems. Proc. Am. Math. Soc. 37, 213–216 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brown, R.C., Krall, A.M.: Ordinary differential operators under Stiltjes boundary conditions. Trans. Am. Math. Soc. 198, 73–92 (1974)

    Article  MATH  Google Scholar 

  7. Cabada, A., Heikkila, S.: Uniqueness, comparison, and existence results for third-order initial-boundary value problems. Comput. Math. Appl. 41, 607–618 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cabada, A., Minhos, F., Santos, A.I.: Solvability for a third-order discontinuous fully equation with nonlinear functional boundary conditions. J. Math. Anal. Appl. 322, 735–748 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Du, Z., Cai, G., Ge, W.: A class a third-order multi-point boundary value problem. Taiwanese J. Math. 9, 81–94 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Du, Z., Ge, W., Zhou, M.: Singular perturbations for third-order nonlinear multi-point boundary value problem. J. Differ. Equ. 218, 69–90 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dzhumabayev, D.S.: Criteria for the unique solvability of a linear boundary-value problem for an ordinary differential equation. USSR Comput. Math. Math. Phys. 29, 34–46 (1989)

    Article  Google Scholar 

  12. Dzhumabaev, D.S., Imanchiev, A.E.: Well-posedness of linear multi-point boundary value problem. Math. J. 5, 30–38 (2005) (in Russian)

    Google Scholar 

  13. Dzhumabaev, D.S., Imanchiev, A.E.: Criteria of existence izolated solution of multi-point boundary value problem for system of ordinary differential equations. Izvestiya NAN RK. Ser. phyz.-matem. 3, 117–121 (2010) (in Russian)

    Google Scholar 

  14. Henderson, J., Tisdale, C.C.: Five-point boundary value problems for third-order differential equations by solution matching. Math. Comput. Model. 42, 133–137 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kiguradze, I.T.: Boundary value problems for systems of ordinary differential equations. In: Modern Problems of Mathematics. Advancement Achievements, vol. 30, pp. 3–103. Nauka, Moscow (1987) (in Russian)

    Google Scholar 

  16. Krall, A.M.: Differential operators and their adjoints under integral and multiple point boundary conditions. J. Differ. Equ. 4, 327–336 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lan, C.: Boundary value problems for second and third order differential equations. J. Differ. Equ. 18, 258–274 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, S.: Positive solutions of nonlinear singular third-order two-point boundary value problem. J. Math. Anal. Appl. 323, 413–425 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lin, X., Du, Z.: Uniqueness and existence results for a third-order nonlinear multi-point boundary value problem. Appl. Math. Comput. 205, 187–196 (2008)

    MathSciNet  MATH  Google Scholar 

  20. Liu, B., Yu, J.: Solvability of multi-point boundary value problems at resonance (II). Appl. Math. Comput. 129, 119–143 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Minghe, P., Chang, S.K.: Existence and uniqueness of solutions for third-order nonlinear boundary value problems. J. Math. Anal. Appl. 327, 23–35 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Palamides, A.P., Smyrlis, G.: Positive solutions to a singular third-order three-point boundary value problem with an indefinitely signed Green’s functions. Nonlinear Anal. 68, 2104–2118 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  23. Samoilenko, A.M., Ronto, N.I.: Numerical-Analytic Methods of Investigation of Solutions of Boundary Value Problems. Naukova Dumka, Kyiv (1985) (in Russian)

    Google Scholar 

  24. Samoilenko, A.M., Laptinskii, V.N., Kenzhebayev, K.K.: Constructive methods of investigation of periodic and multipoint boundary value problems. In: Proceedings of Institute of Mathematics of NAS of Ukraine, vol. 29, pp. 3-220. Institute of Mathematics of NAS of Ukraine, Kiev (1999) (in Russian)

    Google Scholar 

  25. Sun, B., Zhao, J., Yang, P., Ge, W.: Successive iteration and positive solutions for a third-order multipoint generalized right-focal boundary value problem with p-Laplacian. Nonlinear Anal. 70, 220–230 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Valarmathi, N., Ramanujam, N.: An asymptotic numerical method for singularly perturbed third-order ordinary differential equations of convection-diffusion type. Comput. Math. Appl. 44, 693–710 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Xie, S., Li, P., Gao, Z., Wang, H.: High order compact finite difference schemes for a system of third order boundary value problem. Appl. Math. Comput. 219, 2564–2573 (2012)

    MathSciNet  MATH  Google Scholar 

  28. Yao, Q.: Solution and positive solution for a semilinear third-order two-point boundary value problem. Appl. Math. Lett. 17, 1171–1175 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Yao, Q.: Successive iteration of positive solution for a discontinuous third-order boundary value problem. Comput. Math. Appl. 53, 741–749 (2007)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This research is financially supported by a grant from the Ministry of Science and Education of the Republic of Kazakhstan (Grant No. 0822/\(\varGamma \varPhi \)4). This publication is supported by the target program 0085/PTSF-14 from the Ministry of Science and Education of the Republic of Kazakhstan.

Author information

Authors and Affiliations

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

    Anar T. Assanova

  2. Zhubanov Aktobe Regional State University, 34 A.Moldagulova str., Aktobe, 030000, Kazakhstan

    Askarbek E. Imanchiev

Authors
  1. Anar T. Assanova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Askarbek E. Imanchiev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anar T. Assanova .

Editor information

Editors and Affiliations

  1. Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

    Tynysbek Sh. Kalmenov

  2. Department of Mechanics and Mathematics, Kazakhstan Branch of Lomonosov Moscow State University, Astana, Kazakhstan

    Erlan D. Nursultanov

  3. Department of Mathematics, Imperial College London, London, United Kingdom

    Michael V. Ruzhansky

  4. Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

    Makhmud A. Sadybekov

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Assanova, A.T., Imanchiev, A.E. (2017). Solvability of Multipoint-Integral Boundary Value Problem for a Third-Order Differential Equation and Parametrization Method. In: Kalmenov, T., Nursultanov, E., Ruzhansky, M., Sadybekov, M. (eds) Functional Analysis in Interdisciplinary Applications. FAIA 2017. Springer Proceedings in Mathematics & Statistics, vol 216. Springer, Cham. https://doi.org/10.1007/978-3-319-67053-9_11

Download citation

Publish with us

Policies and ethics

Search

Navigation