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Discrete Splines and Its Applications

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Progress on Difference Equations and Discrete Dynamical Systems (ICDEA 2019)

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Abstract

In this paper, we survey the contributions made to discrete splines in the literature and present some applications of discrete splines in the numerical treatment of boundary value problems.

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References

  1. Agarwal, R.P., Wong, P.J.Y.: Error Inequalities in Polynomial Interpolation and Their Applications. Kluwer, Dordrecht (1993)

    Book  MATH  Google Scholar 

  2. Agarwal, R.P., Wong, P.J.Y.: Explicit error bounds for the derivatives of spline interpolation in \(L_{2}\) norm. Appl. Anal. 55, 189–205 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ahlberg, J.H., Nilson, E.N., Walsh, J.L.: The Theory of Splines and Applications. Academic Press, New York (1967)

    MATH  Google Scholar 

  4. Albasiny, E.L., Hoskins, W.D.: Cubic spline solutions to two-point boundary value problems. Comput. J. 12, 151–153 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  5. Al-Said, E.A.: Cubic spline method for solving two-point boundary-value problems. Korean J. Comput. Appl. Math. 5, 669–680 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Al-Said, E.A.: Quadratic spline solution of two point boundary value problems. J. Nat. Geom. 12, 125–134 (1997)

    MathSciNet  MATH  Google Scholar 

  7. Al-Said, E.A.: Spline solutions for system of second-order boundary-value problems. Int. J. Comput. Math. 62, 143–154 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  8. Al-Said, E.A.: Spline methods for solving system of second-order boundary-value problems. Int. J. Comput. Math. 70, 717–727 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. Al-Said, E.A.: The use of cubic splines in the numerical solution of a system of second-order boundary value problems. Comput. Math. Appl. 42, 861–869 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Al-Said, E.A., Noor, M.A.: Modified Numerov method for solving system of second-order boundary-value problems. Korean J. Comput. Appl. Math. 8, 129–136 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Al-Said, E.A., Noor, M.A., Al-Shejari, A.A.: Numerical solutions for system of second order boundary value problems. Korean J. Comput. Appl. Math. 5, 659–667 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  Google Scholar 

  13. Astor, P.H., Duris, C.S.: Discrete L-splines. Numer. Math. 22, 393–402 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  14. Aziz, T., Khan, A., Khan, I.: Quintic splines method for second-order boundary value problems. Int. J. Comput. Math. 85, 735–743 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Baiocchi, C., Capelo, A.: Variational and Quasi-Variational Inequalities. Wiley, New York (1984)

    MATH  Google Scholar 

  16. Bickley, W.G.: Piecewise cubic interpolation and two-point boundary problems. Comput. J. 11, 206–208 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chen, F., Wong, P.J.Y.: Error inequalities for quintic and biquintic discrete Hermite interpolation. J. Comput. Appl. Math. 235, 4589–4600 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Chen, F., Wong, P.J.Y.: Error estimates for discrete spline interpolation: quintic and biquintic splines. J. Comput. Appl. Math. 236, 3835–3854 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Chen, F., Wong, P.J.Y.: Solutions of Fredholm integral equations via discrete biquintic splines. Math. Comput. Model. 57, 551–563 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Chen, F., Wong, P.J.Y.: On periodic discrete spline interpolation: quintic and biquintic cases. J. Comput. Appl. Math. 255, 282–296 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Chen, F., Wong, P.J.Y.: Discrete cubic spline method for second order boundary value problems. Int. J. Comput. Math. 91, 1041–1053 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chen, F., Wong, P.J.Y.: Deficient discrete cubic spline solution for a system of second order boundary value problems. Numer. Algorithms 66, 793–809 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Chen, F., Wong, P.J.Y.: Numerical solutions of fourth order Lidstone boundary value problems using discrete quintic splines. J. Comput. Anal. Appl. 16, 540–551 (2014)

    MathSciNet  MATH  Google Scholar 

  24. Collatz, L.: The Numerical Treatment of Differential Equations. Springer, Berlin (1960)

    Book  MATH  Google Scholar 

  25. Cottle, R.W., Giannessi, F., Lions, J.L.: Variational Inequalities and Complementarity Problems: Theory and Applications. Wiley, Chichester (1980)

    Google Scholar 

  26. Crank, J.: Free and Moving Boundary Problems. Clarendon Press, Oxford, UK (1984)

    MATH  Google Scholar 

  27. Danumjaya, P., Pani, A.K.: Orthogonal cubic spline collocation method for the extended Fisher-Kolmogorov equation. J. Comput. Appl. Math. 174, 101–117 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Dikshit, H.P., Powar, P.L.: Discrete cubic spline interpolation. Numer. Math. 40, 71–78 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  29. Dikshit, H.P., Rana, S.S.: Discrete cubic spline interpolation over a nonuniform mesh. Rocky Mt. J. Math. 17, 709–718 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ding, Q., Wong, P.J.Y.: Mid-knot cubic non-polynomial spline for a system of second-order boundary value problems. Boundary Value Problems 2018, paper no. 156, 16 pp (2018)

    Google Scholar 

  31. Fröberg, C.: Numerical Mathematics. Theory and Computer Applications. Benjamin/Cummings, Reading, MA (1985)

    MATH  Google Scholar 

  32. Fyfe, D.J.: The use of cubic splines in the solution of two-point boundary value problems. Comput. J. 12, 188–192 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  33. Han, X.: A degree by degree recursive construction of Hermite spline interpolants. J. Comput. Appl. Math. 225, 113–123 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  34. Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. Wiley, New York (1961)

    MATH  Google Scholar 

  35. Khalifa, A.K., Eilbeck, J.C.: Collocation with quadratic and cubic splines. IMA J. Numer. Anal. 2, 111–121 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  36. Khan, A.: Parametric cubic spline solution of two point boundary value problems. Appl. Math. Comput. 154, 175–182 (2004)

    MathSciNet  MATH  Google Scholar 

  37. Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity. SIAM Publishing Co., Philadelphia (1988)

    Book  MATH  Google Scholar 

  38. Kouibia, A., Pasadas, M.: Approximation by interpolating variational splines. J. Comput. Appl. Math. 218, 342–349 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  39. Lewy, H., Stampacchia, G.: On the regularity of the solution of a variational inequality. Comm. Pure Appl. Math. 22, 153–188 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  40. Lyche, T.: Discrete cubic spline interpolation. Report RRI 5, University of Oslo (1975)

    Google Scholar 

  41. Lyche, T.: Discrete polynomial spline approximation methods. Spline Functions. Lecture Notes in Mathematics, vol. 501, pp. 144–176. Springer, Berlin (1976)

    Google Scholar 

  42. Lyche, T.: Discrete cubic spline interpolation. BIT 16, 281–290 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  43. Mangasarian, O.L., Schumaker, L.L.: Discrete splines via mathematical programming. SIAM J. Control 9, 174–183 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  44. Mangasarian, O.L., Schumaker, L.L.: Best summation formulae and discrete splines. SIAM J. Numer. Anal. 10, 448–459 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  45. Momani, S.: Solving a system of second order obstacle problems by a modified decomposition method. Appl. Math. E-Notes 6, 141–147 (2006)

    MathSciNet  MATH  Google Scholar 

  46. Noor, M.A.: Some recent advances in variational inequalities, Part I. Basic concepts. New Zealand J. Math. 26, 53–80 (1997)

    MathSciNet  MATH  Google Scholar 

  47. Noor, M.A.: Some recent advances in variational inequalities, Part II. Other concepts. New Zealand J. Math. 26, 229–255 (1997)

    MathSciNet  MATH  Google Scholar 

  48. Noor, M.A., Khalifa, A.K.: Cubic splines collocation methods for unilateral problems. Internat. J. Engrg. Sci. 25, 1527–1530 (1987)

    MathSciNet  MATH  Google Scholar 

  49. Noor, M.A., Noor, K.I., Rafiq, M., Al-Said, E.A.: Variational iteration method for solving a system of second-order boundary value problems. Int. J. Nonlinear Sci. Numer. Simul. 11, 1109–1120 (2010)

    Article  Google Scholar 

  50. Noor, M.A., Noor, K.I., Rassias, Th: Some aspects of variational inequalities. J. Comput. Appl. Math. 47, 285–312 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  51. Noor, M.A., Tirmizi, S.I.A.: Finite difference technique for solving obstacle problems. Appl. Math. Lett. 1, 267–271 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  52. Rana, S.S.: Local behaviour of the first difference of a discrete cubic spline interpolator. Approx. Theory Appl. 6, 58–64 (1990)

    MathSciNet  MATH  Google Scholar 

  53. Rana, S.S., Dubey, Y.P.: Local behaviour of the deficient discrete cubic spline interpolator. J. Approx. Theory 86, 120–127 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  54. Rashidinia, J., Jalilian, R., Mohammadi, R.: Non-polynomial spline methods for the solution of a system of obstacle problems. Appl. Math. Comput. 188, 1984–1990 (2007)

    MathSciNet  MATH  Google Scholar 

  55. Rodrigues, J.F.: Obstacle Problems in Mathematical Physics. North-Holland, Amsterdam (1987)

    MATH  Google Scholar 

  56. Sakai, M., Usmani, R.A.: Quadratic spline and two-point boundary value problem. Publ. Res. Inst. Math. Sci. 19, 7–13 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  57. Sakai, M., Usmani, R.A.: On recursion relations for splines. J. Approx. Theory 65, 200–206 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  58. Schultz, M.H.: Spline Analysis. Prentice-Hall, Englewood Cliffs (1973)

    MATH  Google Scholar 

  59. Schumaker, L.L.: Constructive aspects of discrete polynomial spline functions. In: Lorentz, G.G. (ed.) Approximation Theory, pp. 469–476. Academic Press, New York (1973)

    Google Scholar 

  60. Siewer, R.: A constructive approach to nodal splines. J. Comput. Appl. Math. 203, 289–308 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  61. Siraj-ul-Islam, Noor, M.A., Tirmizi, I.A., Khan, M.A.: Quadratic non-polynomial spline approach to the solution of a system of second-order boundary-value problems. Appl. Math. Comput. 179, 153–160 (2006)

    Google Scholar 

  62. Siraj-ul-Islam, Tirmizi, I.A.: Nonpolynomial spline approach to the solution of a system of second-order boundary-value problems. Appl. Math. Comput. 173, 1208–1218 (2006)

    Google Scholar 

  63. Usmani, R.A.: Bounds for the solution of a second order differential equation with mixed boundary conditions. J. Engrg. Math. 9, 159–164 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  64. Usmani, R.A.: On quadratic spline interpolation. BIT 27, 615–622 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  65. Usmani, R.A., Sakai, M.: A connection between quartic spline and Numerov solution of a boundary value problems. Int. J. Comput. Math. 26, 263–273 (1989)

    Article  MATH  Google Scholar 

  66. Usmani, R.A., Warsi, S.A.: Quintic spline solutions of boundary value problems. Comput. Math. Appl. 6, 197–203 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  67. Wong, P.J.Y., Agarwal, R.P.: Explicit error estimates for quintic and biquintic spline interpolation. Comput. Math. Appl. 18, 701–722 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  68. Wong, P.J.Y., Agarwal, R.P.: Quintic spline solutions of Fredholm integral equations of the second kind. Int. J. Comput. Math. 33, 237–249 (1990)

    Article  MATH  Google Scholar 

  69. Wong, P.J.Y., Agarwal, R.P.: Sharp error bounds for the derivatives of Lidstone-spline interpolation. Comput. Math. Appl. 28, 23–53 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  70. Wong, P.J.Y., Agarwal, R.P.: Error inequalities for discrete Hermite and spline interpolation. In: Milovanovic, G.V. (ed.) Recent Progress in Inequalities, pp. 397–422. Kluwer, Dordrecht (1998)

    Chapter  Google Scholar 

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Authors and Affiliations

  1. School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore, 639798, Singapore

    Patricia J. Y. Wong

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  1. Patricia J. Y. Wong
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Correspondence to Patricia J. Y. Wong .

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Editors and Affiliations

  1. Department of Mathematics, University College London, London, UK

    Steve Baigent

  2. Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO, USA

    Martin Bohner

  3. Department of Mathematics, Trinity University, San Antonio, TX, USA

    Saber Elaydi

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Wong, P.J.Y. (2020). Discrete Splines and Its Applications. In: Baigent, S., Bohner, M., Elaydi, S. (eds) Progress on Difference Equations and Discrete Dynamical Systems. ICDEA 2019. Springer Proceedings in Mathematics & Statistics, vol 341. Springer, Cham. https://doi.org/10.1007/978-3-030-60107-2_5

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