Abstract
A third derivative method (TDM) with continuous coefficients is derived and used to obtain a main and additional methods, which are simultaneously applied to provide all approximations on the entire interval for initial and boundary value problems of the form y′′ = f(x, y, y′). The convergence analysis of the method is discussed. An algorithm involving the TDMs is developed and equipped with an automatic error estimate based on the double mesh principle. Numerical experiments are performed to show efficiency and accuracy advantages.
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References
Amodio, P., Iavernaro, F.: Symmetric boundary methods for second initial and boundary value problems. Mediterr. J. Math. 3, 383–398 (2006)
Ananthakrishnaiah, U.: P-stable Obrechkoff methods with minimal phase-lag for periodic initial value problems. Math. Comput. 49, 553–559 (1987)
Ascher, U.M., Mattheij, R.M.M., Russell, R.D.: Numerical Solutions of Boundary Value Problems for Ordinary Differential Equations, 2nd edn. SIAM, Philadephia, PA (1995)
Awoyemi, D.O.: A new sixth-order algorithm for general second order ordinary differential equation. Int. J. Comput. Math. 77, 117–124 (2001)
Beentjes, P.A., Gerritsen, W.J.: High order Runge–Kutta methods for the numerical solution of second order differential equations without first derivative. Report NW 34/76. Center for Mathematics and Computer Science, Amsterdam (1976)
Brugnano, L., Trigiante, D.: Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon and Breach Science Publishers, Amsterdam (1998)
Chawla, M.M., Sharma, S.R.: Families of three-stage third order Runge–Kutta–Nystrom methods for y′′ = f(x, y, y′ ). J. Aust. Math. Soc. 26, 375–386 (1985)
Coleman, J.P., Gr. Ixaru, G.: P-stability and exponential-fitting methods for y′′ = f(x, y). IMA J. Numer. Anal. 16, 179–199 (1996)
Conte, S.D., de Boor, C.: Elementary numerical analysis. In: An Algorithmic Approach, 3rd edn. McGraw-Hill, Tokyo, Japan (1981)
Fang, Y., Song, Y., Wu, X.: A robust trigonometrically fitted embedded pair for perturbed oscillators. J. Comput. Appl. Math. 225, 347–355 (2009)
Franco, J.M.: Runge–Kutta–Nyström methods adapted to the numerical integration of perturbed oscillators. Comput. Phys. Commun. 147, 770–787 (2002)
Gladwell, I., Sayers, D.K.: Computational Techniques for Ordinary Differential Equations. Academic, London, New York (1976)
Hairer, E., Nörsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I. Springer, Berlin (1987)
Hairer, E.: Méthodes de Nyström pour l’équation différentielle y′′ = f(x, y). Numer. Math. 25, 283–300 (1977)
Jain, M.K., Aziz, T.: Cubic spline solution of two-point boundary value with signifigant first derivatives. Comput. Methods Appl. Mech. Eng. 39, 83–91 (1983)
Jator, S.N.: A sixth order linear multistep method for the direct solution of y′′ = f(x, y, y′). Int. J. Pure Appl. Math. 40, 457–472 (2007)
Jator, S.N., Li, J.: Boundary value methods via a multistep method with variable coefficients for second order initial and boundary value problems. Int. J. Pure Appl. Math. 50, 403–420 (2009)
Keller, H.B.: Numerical Methods for Two-Point Boundary Value Problems. Dover, New York (1992)
Lambert, J.D., Watson, A.: Symmetric multistep method for periodic initial value problem. J. Instrum. Math. Appl. 18, 189–202 (1976)
Mahmoud, S.M., Osman, M.S.: On a class of spline-collocation methods for solving second-order initial-value problems. Int. J. Comput. Math. 86, 613–630 (2009)
Onumanyi, P., Sirisena, U.W., Jator, S.N.: Continuous finite difference approximations for solving differential equations. Int. J. Comput. Math. 72, 15–27 (1999)
Ramadan, M.A., Lashien, I.F., Zahra, W.K.: Polynomial and nonpolynomial spline approaches to the numerical solution of second order boundary value problems. Appl. Math. Comput. 184, 476–484 (2007)
Simos, T.E.: Dissipative trigonometrically-fitted methods for second order IVPs with oscillating solution. Int. J. Mod. Phys. 13, 1333–1345 (2002)
Sommeijer, B.P.: Explicit, high-order Runge–Kutta–Nyström methods for parallel computers. Appl. Numer. Math. 13, 221–240 (1993)
Tsitouras, Ch.: Explicit eight order two-step methods with nine stages for integrating oscillatory problems. Int. J. Mod. Phys. 17, 861–876 (2006)
Twizell, E.H., Khaliq, A.Q.M.: Multiderivative methods for periodic IVPs. SIAM J. Numer. Anal. 21, 111–121 (1984)
Van der Houwen, P.J., Sommeijer, B.P.: Predictor-corrector methods for periodic second-order initial value problems. IMA J. Numer. Anal. 7, 407–422 (1987)
Vigo-Aguiar, J., Ramos, H.: Variable stepsize implementation of multistep methods for y′′ = f (x, y, y′). J. Comput. Appl. Math. 192, 114–131 (2006)
Yusuph, Y., Onumanyi, P.: New Multiple FDMs through multistep collocation for y′′ = f(x, y). In: Proceedings of the National Mathematical Center, Abuja Nigeria (2005)
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Jator, S.N., Li, J. An algorithm for second order initial and boundary value problems with an automatic error estimate based on a third derivative method. Numer Algor 59, 333–346 (2012). https://doi.org/10.1007/s11075-011-9492-3
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DOI: https://doi.org/10.1007/s11075-011-9492-3