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

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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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ascher, U.M., Mattheij, R., Russell, R.D.: Numerical Solution of Boundary Value Problems for Ordinary Differential Equations. SIAM, Philadelphia (1995)

    Book  MATH  Google Scholar 

  4. Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Courier Corporation, Chelmsford (2001)

    MATH  Google Scholar 

  5. Butcher, J.C.: The Numerical Analysis of Ordinary Differential Equations: Runge–Kutta and General Linear Methods. Wiley-Interscience, Hoboken (1987)

    MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  7. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods in Fluid Dynamics. Springer, New York (1988)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  12. Conte, D., DAmbrosio, R., Jackiewicz, Z.: Two-step Runge–Kutta methods with quadratic stability functions. J. Sci. Comput. 44(2), 191–218 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  15. Costabile, F., Napoli, A.: A collocation method for global approximation of general second order BVPs. Comput. Lett. 3(1), 23–34 (2007)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  20. Gottlieb, D., Orszag, S.A.: Numerical Analysis of Spectral Methods: Theory and Applications. SIAM, Philadelphia (1977)

    Book  MATH  Google Scholar 

  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)

  22. Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I: Nonstiff Problems. Springer Series in Computational Mathematics. Springer, Berlin (1993)

    MATH  Google Scholar 

  23. Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. Wiley, Interscience Publishers Inc., Hoboken (1962)

    MATH  Google Scholar 

  24. Jackiewicz, Z.: General Linear Methods for Ordinary Differential Equations. Wiley, Hoboken (2009)

    Book  MATH  Google Scholar 

  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. Numerov, B.: Note on the numerical integration of \(d^2x/dt^2=f (x,t)\). Astronom. Nachr. 230(19), 359–364 (1927)

    MATH  Google Scholar 

  27. Pereyra, V.: High Order Finite Difference Solution of Differential Equations. Stanford University, Stanford (1973)

    Google Scholar 

  28. Phillips, G.G.M.A., Taylor, P.J.: Theory and Applications of Numerical Analysis. Academic Press Incorporated, Boca Raton (1973)

    MATH  Google Scholar 

  29. Powers, D.L.: Boundary Value Problems: and Partial Differential Equations. Academic Press, Boca Raton (2009)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  31. Tsitouras, C.: Explicit two-step methods for second-order linear IVPs. Comput. Math. Appl. 43(8–9), 943–949 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

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Correspondence to Maria Italia Gualtieri.

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Costabile, F.A., Caira, R. & Gualtieri, M.I. A Block Hybrid Method for Non-linear Second Order Boundary Value Problems. Mediterr. J. Math. 16, 17 (2019). https://doi.org/10.1007/s00009-018-1286-6

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  • DOI: https://doi.org/10.1007/s00009-018-1286-6

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