Advertisement

A Block Hybrid Method for Non-linear Second Order Boundary Value Problems

  • Francesco Aldo Costabile
  • Rosanna Caira
  • Maria Italia GualtieriEmail author
Article
  • 43 Downloads

Abstract

In this paper, we propose a new method of hybrid type, for non-linear second order BVPs, to overcome some drawbacks of classic difference finite methods. An estimation of the local and global error is also given. Numerical tests on BVPs which is well known in the literature confirm the accuracy of the theoretical results.

Keywords

Boundary value problem difference method sparse matrix 

Mathematics Subject Classification

65L10 65L12 65F50 

Notes

Acknowledgements

Funding was provided by Ministero dell’Istruzione, dell’Università e della Ricerca.

References

  1. 1.
    Amodio, P., Settanni, G.: A finite differences MATLAB code for the numerical solution of second order singular perturbation problems. J. Comput. Appl. Math. 236(16), 3869–3879 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Amodio, P., Sgura, I.: High-order finite difference schemes for the solution of second-order BVPs. J. Comput. Appl. Math. 176(1), 59–76 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ascher, U.M., Mattheij, R., Russell, R.D.: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. SIAM, Philadelphia (1995)zbMATHCrossRefGoogle Scholar
  4. 4.
    Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Courier Corporation, Chelmsford (2001)zbMATHGoogle Scholar
  5. 5.
    Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations: Runge–Kutta and General Linear Methods. Wiley-Interscience, Hoboken (1987)zbMATHGoogle Scholar
  6. 6.
    Caira, R., Costabile, F.: Two steps methods of Runge-Kutta type for initial value problem \(y^{\prime \prime }=f(x, y)\). Rendiconti di Matematica e delle sue Applicazioni VII 6(4), 441–465 (1986)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, New York (1988)zbMATHCrossRefGoogle Scholar
  8. 8.
    Cash, J.R., Hollevoet, D., Mazzia, F., Nagy, A.M.: Algorithm 927: the MATLAB code bvptwp.m for the numerical solution of two point boundary value problems. ACM Trans. Math. Softw. (TOMS) 39(2), 15 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Chawla, M.M., Katti, C.P.: Finite difference methods for two-point boundary value problems involving high order differential equations. BIT Numer. Math. 19(1), 27–33 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Coleman, J.P.: Numerical methods for \(y^{\prime \prime }=f(x, y)\) via rational approximations for the cosine. IMA J. Numer. Anal. 9(2), 145–165 (1989)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Coleman, J.P.: Order conditions for a class of two-step methods for \(y^{\prime \prime } = f(x, y)\). IMA J. Numer. Anal. 23(2), 197–220 (2003)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Conte, D., DAmbrosio, R., Jackiewicz, Z.: Two-step Runge–Kutta methods with quadratic stability functions. J. Sci. Comput. 44(2), 191–218 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Costabile, F., Luceri, R.: On the order of a Runge–Kutta method for initial value problem \(y^{\prime \prime }= f(y)\). Rendiconti di Matematica VII 6(4), 547–553 (1986)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Costabile, F., Napoli, A.: A method for polynomial approximation of the solution of general second order BVPs Far East. J. Appl. Math. 25(3), 289–305 (2006)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Costabile, F., Napoli, A.: A collocation method for global approximation of general second order BVPs. Comput. Lett. 3(1), 23–34 (2007)CrossRefGoogle Scholar
  16. 16.
    Costabile, F., Napoli, A.: A new spectral method for a class of linear boundary value problems. J. Comput. Appl. Math. 292, 329–341 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Costabile, F., Varano, A.: Convergence, stability and truncation error estimation of a method for the numerical integration of the initial value problem \(Y^{\prime \prime }=F(X, Y)\). CALCOLO 18(4), 371–382 (1981)MathSciNetzbMATHGoogle Scholar
  18. 18.
    D’Ambrosio, R., Ferro, M., Jackiewicz, Z., Paternoster, B.: Two-step almost collocation methods for ordinary differential equations. Numer. Algorithms 53(2–3), 195–217 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    D’Ambrosio, R., Ferro, M., Paternoster, B.: Two-step hybrid collocation methods for \(y^{\prime \prime }=f(x, y)\). Appl. Math. Lett. 22(7), 1076–1080 (2009)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. SIAM, Philadelphia (1977)zbMATHCrossRefGoogle Scholar
  21. 21.
    Groza, G., Jianu, M.: Polynomial approximations of solutions of boundary value problems for ODEs which arise from engineering. In Proceedings of RIGA 2014, Riemannian Geometry and Applications to Engineering and Economics, Bucharest, Romania (2014)Google Scholar
  22. 22.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer Series in Computational Mathematics. Springer, Berlin (1993)zbMATHGoogle Scholar
  23. 23.
    Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. Wiley, Interscience Publishers Inc., Hoboken (1962)zbMATHGoogle Scholar
  24. 24.
    Jackiewicz, Z.: General Linear Methods for Ordinary Differential Equations. Wiley, Hoboken (2009)zbMATHCrossRefGoogle Scholar
  25. 25.
    Lakshmi, R., Muthuselvi, M.: Numerical solution for boundary value problem using difference method. Int. J. Innov. Res. Sci. Eng. Technol. 2(10), 5305–5313 (2013)Google Scholar
  26. 26.
    Numerov, B.: Note on the numerical integration of \(d^2x/dt^2=f (x,t)\). Astronom. Nachr. 230(19), 359–364 (1927)zbMATHGoogle Scholar
  27. 27.
    Pereyra, V.: High Order Finite Difference Solution of Differential Equations. Stanford University, Stanford (1973)Google Scholar
  28. 28.
    Phillips, G.G.M.A., Taylor, P.J.: Theory and Applications of Numerical Analysis. Academic Press Incorporated, Boca Raton (1973)zbMATHGoogle Scholar
  29. 29.
    Powers, D.L.: Boundary Value Problems: and Partial Differential Equations. Academic Press, Boca Raton (2009)zbMATHGoogle Scholar
  30. 30.
    Simos, T.E.: Explicit two-step methods with minimal phase-lag for the numerical integration of special second-order initial-value problems and their application to the one-dimensional Schrödinger equation. J. Comput. Appl. Math. 39(1), 89–94 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Tsitouras, C.: Explicit two-step methods for second-order linear IVPs. Comput. Math. Appl. 43(8–9), 943–949 (2002)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Francesco Aldo Costabile
    • 1
  • Rosanna Caira
    • 1
  • Maria Italia Gualtieri
    • 1
    Email author
  1. 1.Department of Mathematics and Computer ScienceUniversity of CalabriaRendeItaly

Personalised recommendations