Skip to main content
SpringerLink
Log in

A Numerical Method for Solving Boundary Value Problem for Essentially Loaded Differential Equations

  • Published:
Lobachevskii Journal of Mathematics Aims and scope Submit manuscript

Abstract

A linear boundary value problem for essentially loaded differential equations is investigated. Using the properties of essentially loaded differential equation and assuming the invertibility of the matrix compiled through the coefficients at the values of the derivative of the desired function at load points, we reduce the considering problem to a two-point boundary value problem for loaded differential equations. The method of parameterization is used for solving the problem. The linear boundary value problem for loaded differential equations by introducing additional parameters at the loading points is reduced to equivalent boundary value problem with parameters. The equivalent boundary value problem with parameters consists of the Cauchy problem for the system of ordinary differential equations with parameters, boundary condition and continuity conditions. The solution of the Cauchy problem for the system of ordinary differential equations with parameters is constructed using the fundamental matrix of differential equation. The system of linear algebraic equations with respect to the parameters are composed by substituting the values of the corresponding points in the built solutions to the boundary condition and the continuity condition. Numerical method for finding solution of the problem is suggested, which based on the solving the constructed system and the Bulirsch–Stoer method for solving Cauchy problem on the subintervals.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. A. M. Nakhushev, Loaded Equations and Their Applications (Nauka, Moscow, 2012) [in Russian].

    MATH  Google Scholar 

  2. A. M. Nakhushev, Equations of Mathematical Biology (Vyssh. Shkola, Moscow, 1995) [in Russian].

    MATH  Google Scholar 

  3. A. M. Nakhushev, ‘‘An approximation method for solving boundary-value problems for differential equations with applications to the dynamics of soil moisture and groundwater,’’ Differ. Uravn. 18, 72–81 (1982).

    MathSciNet  Google Scholar 

  4. M. T. Dzhenaliev, ‘‘Loaded equations with periodic boundary conditions,’’ Differ. Equat. 37, 51–57 (2001).

    Article  MathSciNet  Google Scholar 

  5. V. M. Abdullaev and K. R. Aida-zade, ‘‘Numerical method of solution to loaded nonlocal boundary value problems for ordinary differential equations,’’ Comput. Math. Math. Phys. 54, 1096–1109 (2014).

    Article  MathSciNet  Google Scholar 

  6. V. M. Abdullaev and K. R. Aida-zade, ‘‘On the numerical solution to loaded systems of ordinary differential equations with non-separated multipoint and integral conditions,’’ Numer. Anal. Appl. 7, 1–14 (2014).

    Article  Google Scholar 

  7. A. A. Alikhanov, A. M. Berezkov, and M. Kh. Shkhanukhov-Lafishev, ‘‘Boundary value problems for certain classes of loaded differential equations and solving them by finite difference methods,’’ Comput. Math. Math. Phys. 48, 1581–1590 (2005).

    Article  Google Scholar 

  8. A. T. Assanova, A. E. Imanchiyev, and Zh. M. Kadirbayeva, ‘‘Numerical solution of systems of loaded ordinary differential equations with multipoint conditions,’’ Comput. Math. Math. Phys. 58, 508–516 (2018).

    Article  MathSciNet  Google Scholar 

  9. A. T. Assanova and Zh. M. Kadirbayeva, ‘‘On the numerical algorithms of parametrization method for solving a two-point boundary-value problem for impulsive systems of loaded differential equations,’’ Comput. Appl. Math. 37, 4966–4976 (2018).

    Article  MathSciNet  Google Scholar 

  10. E. A. Bakirova, A. B. Tleulesova, and Zh. M. Kadirbayeva, ‘‘On one algorithm for finding a solution to a two-point boundary value problem for loaded differential equations with impulse effect,’’ Bull. Karaganda Univ., Math. 87 (3), 43–50 (2017).

    Google Scholar 

  11. Zh. M. Kadirbayeva, ‘‘On the method for solving linear boundary-value problem for the system of loaded differential equations with multipoint integral condition,’’ Math. J. 17 (4), 50–61 (2017).

    Google Scholar 

  12. A. T. Assanova, E. A. Bakirova, and Zh. M. Kadirbayeva, ‘‘Numerical implementation of solving a boundary value problem for a system of loaded differential equations with parameter,’’ News NAS RK, Ser. Phys.-Math. 3 (325), 77–84 (2019).

    Google Scholar 

  13. T. K. Yuldashev, B. I. Islomov, and E. K. Alikulov, ‘‘Boundary-value problems for loaded third-order parabolic-hyperbolic equations in infinite three-dimensional domains,’’ Lobachevskii J. Math. 41 (5), 926–944 (2020).

    Article  MathSciNet  Google Scholar 

  14. M. T. Dzhenaliev, ‘‘Boundary value problems for loaded parabolic and hyperbolic equations with time derivatives in boundary conditions,’’ Differ. Equat. 28, 537–540 (1992).

    MathSciNet  Google Scholar 

  15. S. Z. Dzhamalov, S. R. Umarov, and R. R. Ashurov, ‘‘On unique solvability of a nonlocal boundary-value problem for a loaded multidimensional Chaplygin’s equation in the Sobolev space,’’ Lobachevskii J. Math. 41, 7–14 (2020).

    Article  MathSciNet  Google Scholar 

  16. K. U. Khubiev, ‘‘A maximum principle for a loaded hyperbolic-parabolic equation,’’ Vladikavk. Mat. Zh. 18 (4), 80–85 (2016).

    MathSciNet  MATH  Google Scholar 

  17. K. B. Sabitov, ‘‘Initial-boundary problem for parabolic-hyperbolic equation with loaded summands,’’ Russ. Math. 59 (6), 23–33 (2015).

    Article  MathSciNet  Google Scholar 

  18. U. I. Baltaeva, ‘‘On the solvability of boundary value problems with continuums and generalized gluing conditions for the equation of mixed type with loaded term,’’ Ukr. Math. J. 69, 1845–1854 (2018).

    Article  MathSciNet  Google Scholar 

  19. B. Islomov and U. I. Baltaeva, ‘‘Boundary-value problems for a third-order loaded parabolic-hyperbolic equation with variable coefficients,’’ Electron. J. Differ. Equat. 221, 1–10 (2015).

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. D. S. Dzhumabaev, ‘‘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 

  22. J. Stoer and R. Bulirsch, Introduction to Numerical Analysis (Springer Science, New York, USA, 2002).

    Book  Google Scholar 

  23. K. Atkinson, W. Han, and D. Stewart, Numerical Solution of Ordinary Differential Equations (Wiley, USA, 2009).

    Book  Google Scholar 

Download references

Funding

This research is funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (grant no. AP08955489).

Author information

Authors and Affiliations

  1. Institute of Mathematics and Mathematical Modeling, Department of Differential Equations, 050010/A26G7T4, Almaty, Kazakhstan

    Zh. M. Kadirbayeva

  2. International Information Technology University, Department of Mathematical and Computer Modeling, 050040, Almaty, Kazakhstan

    Zh. M. Kadirbayeva

Authors
  1. Zh. M. Kadirbayeva
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zh. M. Kadirbayeva.

Additional information

(Submitted by T. K. Yuldashev)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kadirbayeva, Z.M. A Numerical Method for Solving Boundary Value Problem for Essentially Loaded Differential Equations. Lobachevskii J Math 42, 551–559 (2021). https://doi.org/10.1134/S1995080221030112

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1995080221030112

Keywords:

Search

Navigation