Advertisement

Direct Integrators for the General Third-Order Ordinary Differential Equations with an Application to the Korteweg–de Vries Equation

  • Samuel Jator
  • Temitayo Okunlola
  • Toheeb Biala
  • Raphael Adeniyi
Original Paper
  • 29 Downloads

Abstract

We construct a new class of implicit continuous linear multistep methods (LMMs) which are used as boundary value methods for the numerical integration of the general third order initial and boundary value problems in ordinary differential equations, including the Korteweg–de Vries equation. The boundary value methods obtained from these continuous LMMs are weighted the same and are used to simultaneously generate approximate solutions to the exact solutions in the entire interval of integration. We established the convergence analysis of the methods and several numerical examples are given to show the performance of the methods.

Keywords

Boundary value methods Third order problems Convergence Linear multistep methods Korteweg–de Vries equation 

Mathematics Subject Classification

65L05 65L06 65L10 65L12 

Notes

Acknowledgements

The authors are very grateful to the referee whose valuable suggestions greatly improved the manuscript.

References

  1. 1.
    Awoyemi, D.O.: A P-stable linear multistep methods for the general third order ordinary differential equations. Int. J. Comput. Math. 80, 985–991 (2003)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Salama, A.A., Mansour, A.A.: Fourth order finite difference method for third order boundary value problems. Numer. Heat Transf. Part B 47, 383–401 (2005)CrossRefGoogle Scholar
  3. 3.
    Li, Z., Wang, Y., Tan, F.: The solution of a class of third order boundary value problems by the reproducing kernel method. Abstr. Appl. Anal., Article ID: 195310 (2012)Google Scholar
  4. 4.
    Bhrawy, A.H., Abd-Elhameed, W.M.: New algorithm for the numerical solution of nonlinear third order differential equations using the Jacobi–Gauss collocation method. Math. Prob. Eng. Article ID: 837218 (2011)Google Scholar
  5. 5.
    Brugnano, L., Trigiante, D.: Higher order multistep methods for boundary value problems. Appl. Numer. Math. 18, 79–94 (1995)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brugnano, L., Trigiante, D.: Solving Differential Problems by Multistep Initial and Boundary Value Problems. Gordon and Breach Science Publishers, London (1998)zbMATHGoogle Scholar
  7. 7.
    Brugnano, L., Trigiante, D.: Stability properties of some BVM methods. Appl. Numer. Math. 13, 291–304 (1993)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Jator, S.N., Li, J.: A self starting linear multistep merhod for a direct solution of the general second order initial value problem. Int. J. Comput. Math. 86, 827–836 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Jator, S.N.: A continuous two-step method of order 8 with a block extension for \(y^{\prime \prime } = f(x, y, y^{\prime })\). Appl. Math Comput. 219, 781–791 (2012)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Hussain, K.A., Ismail, F., Senu, N., Rabiei, F.: Fourth-order improved Runge–Kutta method for directly solving special third-order ordinary differential equations. Iran. J. Sci. Technol. Trans. A Sci. 41, 429 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Jikantoro, Y., Ismail, F., Senu, N., Ibrahim, Z.: Hybrid methods for direct integration of special third order ordinary differential equations. Appl. Math. Comput. 320, 452 (2018)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ken, Y.L., Ismail, F.M., Suleiman, M., Amin, S.M.: Block methods based on Newton interpolations for solving special second order ordinary differential equations directly. J. Math. Stat. 4, 174 (2008)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Langkah, K.B.D.-T.E., Majid, Z.A., Azmi, N.A., Suleiman, M., Ibrahaim, Z.B.: Solving directly general third order ordinary differential equations using two-point four step block method. Sains Malays. 41, 623 (2012)zbMATHGoogle Scholar
  14. 14.
    Majid, Z.A., Suleiman, M., Azmi, N.A.: Variable step size block method for solving directly third order ordinary differential equations. Far East J. Math. Sci. 41, 632 (2010)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Mechee, M., Ismail, F., Hussain, Z., Siri, Z.: Direct numerical methods for solving a class of third-order partial differential equations. Appl. Math. Comput. 247, 663–674 (2014)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Mechee, M., Kadhim, M.: Explicit direct integrators of RK type for solving special fifth-order ordinary differential equations. Am. J. Appl. Sci. 13(12), 1452–1460 (2016).  https://doi.org/10.3844/ajassp.2016.1452.1460 CrossRefGoogle Scholar
  17. 17.
    Mechee, M.S., Kadhim, M.A.: Direct explicit integrators of RK type for solving special fourth-order ordinary differential equations with an application. Glob. J. Pure Appl. Math. 12, 4687–4715 (2016)Google Scholar
  18. 18.
    Senu, N., Mechee, M., Ismail, F., Siri, Z.: Embedded explicit Runge–Kutta type methods for directly solving special third order differential equations. Appl. Math. Comput. 240, 281–293 (2014)MathSciNetzbMATHGoogle Scholar
  19. 19.
    You, X., Chen, Z.: Direct integrators of Runge–Kutta type for special third order ordinary differential equations. Appl. Numer. Math. 74, 128–150 (2013)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Amodio, P., Iavernaro, F.: Symmetric boundary value methods for second initial and boundary value problems. Medit. J. Math. 3, 383–398 (2006)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Amodio, P., Mazzia, F.: A boundary value approach to the numerical solution of initial value problems by multistep methods. J. Differ. Equ. Appl. 1, 353–367 (1995)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Amodio, P., Brugnano, L.: Parallel implementation of block boundary value methods for ODEs. J. Comput. Math. Appl. 78, 197–211 (1997)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Jator, S.N., Li, J.: An algorithm for second order initial and boundary value problems with an automatic error estimation based on a third derivative method. Numer. Algorithm 59, 333–346 (2012)CrossRefGoogle Scholar
  24. 24.
    Aceto, L., Ghelardoni, P., Magherini, C.: PGSCM: a family of P-stable Boundary value methods for second order initial value problems. J. Comput. Appl. Math. 236, 3857–3868 (2012)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Biala, T.A., Jator, S.N.: A family of boundary value methods for systems of second-order boundary value problems. Int. J. Differ. Equ. (2017).  https://doi.org/10.1155/2017/2464759 MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Vigo-Aguilar, J., Ramos, H.: Variable stepsize implementation of multistep methods for \(y^{\prime \prime } = f(x, y, y^{\prime })\). J. Comput. Appl. Math. 192, 114–131 (2006)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Onumanyi, P., Sirisena, U.W., Jator, S.N.: Continuous finite difference approximations for solving differential equations. Int. J. Comput. Math. 72, 15–27 (1999)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Jalilian, R.: Non-polynomial spline method for solving Bratu’s problem. Comput. Phys. Commun. 181(11), 1868–1872 (2010)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Jain, M.K., Aziz, T.: Cubic spline solution of two-point boundary value with signifigant first derivatives. Comput. Methods Appl. Mech. Eng. 39, 8391 (1983)CrossRefGoogle Scholar
  30. 30.
    Jator, S.N.: Novel finite difference scheme for third order boundary value problems. Int. J. Pure Appl. Math. 53(1), 37–54 (2009)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Sahi, R.K., Jator, S.N., Khan, N.A.: Continuous fourth derivative method for third order boundary value problems. Int. J. Pure Appl. Math. 85(5), 907–923 (2013)CrossRefGoogle Scholar
  32. 32.
    Jator, S.N.: On the numerical integration of third order boundary value problems by linear multistep methods. Int. J. Pure Appl. Math. 46(3), 375–388 (2008)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Hufford, C., Xing, Y.: Superconvergence of the local discontinuous Galerkin method for the linearized Korteweg–de Vries equation. J. Comput. Appl. Math. 255, 441–455 (2014)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Haq, S., Ul-Islam, S., Uddin, M.: A mesh-free method for the numerical solution of the KdV–Burgers equation. Appl. Math. Model. 33, 3442–3449 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature India Private Limited 2018

Authors and Affiliations

  • Samuel Jator
    • 1
  • Temitayo Okunlola
    • 2
  • Toheeb Biala
    • 3
  • Raphael Adeniyi
    • 4
  1. 1.Department of Mathematics and StatisticsAustin Peay State UniversityClarksvilleUSA
  2. 2.Department of Physical and Mathematical ScienceAfe Babalola UniversityAdo EkitiNigeria
  3. 3.Department of Mathematics and Computer ScienceJigawa State UniversityKafin HausaNigeria
  4. 4.Department of MathematicsUniversity of IlorinIlorinNigeria

Personalised recommendations