187
Views
6
CrossRef citations to date
0
Altmetric
Section B

Discrete cubic spline method for second-order boundary value problems

&
Pages 1041-1053 | Received 14 Dec 2012, Accepted 19 Jun 2013, Published online: 05 Aug 2013

References

  • J. H. Ahlberg, E. N. Nilson and J. L. Walsh, The Theory of Splines and Applications, Academic Press, New York, 1967
  • E. L. Albasiny and W. D. Hoskins, Cubic spline solutions to two-point boundary value problems, Comput. J.12 (1969), pp. 151–153 doi: 10.1093/comjnl/12.2.151
  • E. A. Al-Said, Spline solutions for system of second-order boundary-value problems, Int. J. Comput. Math.62 (1996), pp. 143–154 doi: 10.1080/00207169608804531
  • E. A. Al-Said, Quadratic spline solution of two point boundary value problems, J. Natur. Geom.12 (1997), pp. 125–134
  • E. A. Al-Said, Cubic spline method for solving two-point boundary-value problems, Korean J. Comput. Appl. Math.5 (1998), pp. 669–680
  • E. A. Al-Said, Spline methods for solving system of second-order boundary-value problems, Int. J. Comput. Math.70 (1999), pp. 717–727 doi: 10.1080/00207169908804784
  • E. A. Al-Said, The use of cubic splines in the numerical solution of a system of second-order boundary value problems, Comput. Math. Appl.42 (2001), pp. 861–869 doi: 10.1016/S0898-1221(01)00204-8
  • E. A. Al-Said and M. A. Noor, Modified Numerov method for solving system of second-order boundary-value problems, Korean J. Comput. Appl. Math.8 (2001), pp. 129–136
  • E. A. Al-Said, M. A. Noor and A. A. Al-Shejari, Numerical solutions for system of second order boundary value problems, Korean J. Comput. Appl. Math.5 (1998), pp. 659–667
  • U. M. Ascher, R. M. Mettheij and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, SIAM, Philadelphia, 1995
  • P. H. Astor and C. S. Duris, Discrete L-splines, Numer. Math.22 (1974), pp. 393–402 doi: 10.1007/BF01436921
  • T. Aziz, A. Khan and I. Khan, Quintic splines method for second-order boundary value problems, Int. J. Comput. Math.85 (2008), pp. 735–743 doi: 10.1080/00207160701219803
  • C. Baiocchi and A. Capelo, Variational and Quasi-Variational Inequalities, John Wiley & Sons, New York, 1984
  • W. G. Bickley, Piecewise cubic interpolation and two-point boundary problems, Comput. J.11 (1968), pp. 206–208 doi: 10.1093/comjnl/11.2.206
  • F. Chen and P. J.Y. Wong, Error estimates for discrete spline interpolation: Quintic and biquintic splines, J. Comput. Appl. Math.236 (2012), pp. 3835–3854 doi: 10.1016/j.cam.2012.01.011
  • F. Chen and P. J.Y. Wong, On periodic discrete spline interpolation: Quintic and biquintic cases, J. Comput. Appl. Math.255 (2014), pp. 282–296 doi: 10.1016/j.cam.2013.05.005
  • F. Chen and P.J.Y. Wong, Deficient discrete cubic spline solution for a system of second order boundary value problems, under peer review.
  • L. Collatz, The Numerical Treatment of Differential Equations, Springer-Verlag, Berlin, 1960
  • R. W. Cottle, F. Giannessi and J. L. Lions, Variational Inequalities and Complementarity Problems: Theory and Applications, John Wiley & Sons, New York, 1980
  • J. Crank, Free and Moving Boundary Problems, Clarendon Press, Oxford, 1984
  • C. Fröberg, Numerical Mathematics, Theory and Computer Applications, Benjamin/Cummings, Reading, MA, 1985
  • D. J. Fyfe, The use of cubic splines in the solution of two-point boundary value problems, Comput. J.12 (1969), pp. 188–192 doi: 10.1093/comjnl/12.2.188
  • P. Henrici, Discrete Variable Methods in Ordinary Differential Equations, John Wiley, New York, 1961
  • A. K. Khalifa and J. C. Eilbeck, Collocation with quadratic and cubic splines, IMA J. Numer. Anal.2 (1982), pp. 111–121 doi: 10.1093/imanum/2.1.111
  • A. Khan, Parametric cubic spline solution of two point boundary value problems, Appl. Math. Comput.154 (2004), pp. 175–182 doi: 10.1016/S0096-3003(03)00701-X
  • N. Kikuchi and J. T. Oden, Contact Problems in Elasticity, SIAM, Philadelphia, 1988
  • T. Lyche, Discrete polynomial spline approximation methods, in Lecture Notes in Mathematics 501: Spline Functions, Springer-Verlag, New York, 1976
  • T. Lyche, Discrete cubic spline interpolation, BIT16 (1976), pp. 281–290 doi: 10.1007/BF01932270
  • O. L. Mangasarian and L. L. Schumaker, Discrete splines via mathematical programming, SIAM J. Control9 (1971), pp. 174–183 doi: 10.1137/0309015
  • M. A. Noor, Some recent advances in variational inequalities, Part I, Basic concepts, N Z J. Math.26 (1997), pp. 53–80
  • M. A. Noor, Some recent advances in variational inequalities, Part II, Other concepts, N Z J. Math.26 (1997), pp. 229–255
  • M. A. Noor and A. K. Khalifa, Cubic splines collocation methods for unilateral problems, Int. J. Eng. Sci.25 (1987), pp. 1527–1530
  • M. A. Noor and S. I.A. Tirmizi, Finite difference technique for solving obstacle problems, Appl. Math. Lett.1 (1988), pp. 267–271 doi: 10.1016/0893-9659(88)90090-0
  • J. F. Rodrigues, Obstacle Problems in Mathematical Physics, North-Holland, Amsterdam, 1987
  • M. Sakai and R. A. Usmani, Quadratic spline and two-point boundary value problem, Publ. RIMS, Kyoto Univ.19 (1983), pp. 7–13 doi: 10.2977/prims/1195182972
  • L. L. Schumaker, Constructive aspects of discrete polynomial spline functions, Approximation Theory, G. G. Lorentz, ed., Academic Press, New York, 1973, pp. 469–476
  • R. A. Usmani, Bounds for the solution of a second order differential equation with mixed boundary conditions, J. Eng. Math.9 (1975), pp. 159–164 doi: 10.1007/BF01535397
  • R. A. Usmani and M. Sakai, A connection between quartic spline and Numerov solution of a boundary value problems, Int. J. Comput. Math.26 (1989), pp. 263–273 doi: 10.1080/00207168908803700
  • P. J.Y. Wong and R. P. Agarwal, Error inequalities for discrete Hermite and spline interpolation, Recent Progress in Inequalities, G. V. Milovanovic, ed., Kluwer, Dordrecht, 1998, pp. 397–422

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.