Skip to main content
SpringerLink
Log in

Methods for solving singular boundary value problems using splines: a review

  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

This paper surveys and reviews papers of spline solution of singular boundary value problems. Among a number of numerical methods used to solve two-point singular boundary value problems, spline methods provide an efficient tool. Techniques collected in this paper include cubic splines, non-polynomial splines, parametric splines, B-splines and TAGE method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abu-Zaid, I.T., El-Gebeily, M.A.: A finite difference method for the spectral approximation of a class of singular two-point boundary value problems. IMA J. Numer. Anal. 14, 545–562 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  2. Abu-Zaid, I.T., El-Gebeily, M.A.: On a finite difference method for singular two-point boundary value problems. IMA J. Numer. Anal. 18(2), 179–190 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Agarwal, R.P.: Boundary Value Problems for High Ordinary Differential Equations. World Scientific, Singapore (1986)

    Google Scholar 

  4. Ahlberg, J.H.: The Theory of Splines and Their Applications. Academic Press, San Diego (1967)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Albasiny, E.L., Hoskins, W.D.: Increased accuracy cubic spline solutions to two-point boundary value problems. IMA J. Numer. Anal. 9(1), 47–55 (1972)

    MATH  MathSciNet  Google Scholar 

  7. Bellman, R., Kalaba, R.: Quasilinearization and Nonlinear Boundary Value Problems. American Elsevier, New York (1965)

    MATH  Google Scholar 

  8. Bickley, W.G.: Piecewise cubic interpolation and two point boundary problems. Comput. J. 11, 206–212 (1968)

    MATH  MathSciNet  Google Scholar 

  9. De Boor, C.: Practical Guide to Splines. Springer, Berlin (1978)

    MATH  Google Scholar 

  10. Brabston, D.C., Keller, H.B.: A numerical method for singular two point boundary value problems. SIAM J. Numer. Anal. 14, 779–791 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  11. Caglar, H., Caglar, N., Elfaituri, K.: B-spline interpolation compared with finite element and finite volume methods which applied to two-point boundary value problems. Appl. Math. Comput. 175, 72–79 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Caglar, H.N., Caglar, S.H., Twizell, E.H.: The numerical solution of third-order boundary-value problems with fourth-degree B-spline functions. Int. J. Comput. Math. 71, 373–381 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Caglar, N., Caglar, H.: B-spline solution of singular boundary value problems. Appl. Math. Comput. 182, 1509–1513 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Chawla, M.M.: A fourth-order finite difference method based on uniform mesh for singular two-point boundary-value problems. J. Comput. Appl. Math. 17, 359–364 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  15. Chawla, M.M., Katti, C.P.: A finite-difference method for a class of singular two-point boundary-value problems. IMA J. Numer. Anal. 4(4), 457–466 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  16. Chawla, M.M., Katti, C.P.: A uniform mesh finite difference method for a class of singular two-point boundary value problems. SIAM J. Numer. Anal. 22, 561–565 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  17. Chawla, M.M., Katti, C.P.: Finite difference methods and their convergence for a class of singular two-point boundary value problems. Numer. Math. 39, 341–350 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  18. Chawla, M.M., Mckee, S., Shaw, G.: Order h 2 method for a singular two-point boundary value problem. BIT 26, 326–381 (1986)

    MathSciNet  Google Scholar 

  19. Chawla, M.M., Subramanian, R.: A new fourth order cubic spline method for non-linear two-point boundary value problems. Int. J. Comput. Math. 22, 321–341 (1987)

    Article  MATH  Google Scholar 

  20. Chawla, M.M., Subramanian, R., Sathi, H.L.: A fourth-order spline method for singular two-point boundary-value problems. J. Comput. Appl. Math. 21, 189–202 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ciarlet, P.G., Natterer, F., Varga, R.S.: Numerical methods of high order accuracy for singular non-linear boundary value problems. Numer. Math. 15, 87–99 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  22. Cohen, A.M., Jones, D.E.: A note on the numerical solution of some singular second order differential equations. J. Inst. Math. Appl. 13, 379–384 (1974)

    MATH  Google Scholar 

  23. Doede, E.J., Reddien, G.W.: Finite difference methods for singular two-point boundary value problems. SIAM J. Numer. Anal. 21, 300–313 (1984)

    Article  MathSciNet  Google Scholar 

  24. Doolan, E.P., Miller, J.J.H., Schilders, W.H.A.: Uniform Numerical Methods for Problems with Initial and Boundary Layers. Boole, Dublin (1980)

    MATH  Google Scholar 

  25. El-Gebeily, M.A., Boumunir, A., Elgindi, M.B.: Existence and uniqueness of solutions of a class of two-point boundary value problems in one dimension. J. Comput. Appl. Math. 46, 345–355 (1992)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  27. Golub, G.H., Ortega, J.M.: Scientific Computing and Differential Equations. Academic Press, San Diego (1992)

    MATH  Google Scholar 

  28. Guoqiang, H.: Correct solution of finite difference and spline approximation for a class of singular two-point boundary value problems. Math. Numer. Sin. 13, 187–192 (1991) (in Chinese)

    MATH  Google Scholar 

  29. Guoqiang, H., : J. Comput. Appl. Math. 126, 145–157 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  30. Guoqiang, H., Jiong, W., Hayami, K., Yuesheng, X.: Correction method and extrapolation method for singular two-point boundary value problems. J. Comput. Math. 126, 145–157 (2000)

    Article  MATH  Google Scholar 

  31. Guoqiang, H.: Spline finite difference methods and their extrapolation for singular two-point boundary value problems. J. Comput. Math. 11, 289–296 (1993)

    MATH  MathSciNet  Google Scholar 

  32. Gustafsson, B.: A numerical method for solving singular boundary value problems. Numer. Math. 21, 328–344 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  33. De Hoog, F.R., Weiss, R.: Difference methods for boundary value problems with a singularity of the first kind. SIAM J. Numer. Anal. 13, 775–813 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  34. Iyengar, S.R.K., Jain, P.: Spline finite difference methods for singular two point boundary value problems. Numer. Math. 50, 363–376 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  35. Jamet, P.: On the convergence of finite difference approximations to one dimensional singular boundary value problems. Numer. Math. 14, 355–378 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  36. Kadalbajoo Mohan, K., Aggarwal Vivek, K.: Cubic spline for solving singular two-point boundary value problems. Appl. Math. Comput. 156, 249–259 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  37. Kadalbajoo Mohan, K., Aggarwal Vivek, K.: Numerical solution of singular boundary value problems via Chebychev polynomial and B-spline. Appl. Math. Comput. 160, 851–863 (2005)

    MATH  MathSciNet  Google Scholar 

  38. Kadalbajoo Mohan, K., Kumar, V.: B-spline method for a class of singular two-point boundary value problems using optimal grid. Appl. Math. Comput. 188, 1856–1869 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  39. Kadalbajoo Mohan, K., Raman, K.S.: Cubic spline and invariant imbedding for solving singular two-point boundary value problems. J. Math. Anal. Appl. 112, 22–35 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  40. Kadalbajoo Mohan, K., Raman, K.S.: Numerical solution of singular boundary value problems by invariant imbedding. J. Comput. Phys. 55, 268–277 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  41. Keller, H.B.: Numerical Methods for Two Point Boundary Value Problems. Blaisdell, London (1992)

    Google Scholar 

  42. Kumar, M.: Higher order method for singular boundary value problems by using spline function. Appl. Math. Comput. 192(1), 175–179 (2007)

    Article  MathSciNet  Google Scholar 

  43. Kumar, M.: A fourth-order spline finite difference method for singular two-point boundary value problems. Int. J. Comput. Math. 80(12), 1499–1504 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  44. Kumar, M.: A second order spline finite difference method for singular two-point boundary value problems. Appl. Math. Comput. 142, 283–290 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  45. Micula, G., Micula, S.: Handbook of Splines. Kluwer Academic, Dordrecht (1999)

    MATH  Google Scholar 

  46. Mohanty, R.K., Evans, D.J.: A fourth order accurate cubic spline alternating group explicit method for non linear singular two-point boundary value problems. Int. J. Comput. Math. 80, 479–492 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  47. Mohanty, R.K., Evans, D.J., Khosla, N.: An O(h 3 k ) non-uniform mesh cubic spline TAGE method for non-linear singular two-point boundary value problems. Int. J. Comput. Math. 82(9), 1125–1139 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  48. Mohanty, R.K., Sachdev, P.L., Jha, N.: An O(h 4) accurate cubic spline TAGE method for nonlinear singular two point boundary value problem. Appl. Math. Comput. 158, 853–868 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  49. Natterer, F.: A generalized spline method for singular boundary value problems of ordinary differential equations. Linear Algebra Appl. 7, 189–216 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  50. Nelson, P.: Invariant embedding applied to homogeneous two point boundary value problems with a singularity of first kind. Appl. Math. Comput. 9, 93–110 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  51. Pandey, R.K.: On the convergence of a spline method for singular two point boundary value problems arising in physiology. Int. J. Comput. Math. 79(3), 357–366 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  52. Prenter, P.M.: Splines and Variational Methods. Wiley, New York (1975)

    MATH  Google Scholar 

  53. Rashidinia, J.: Applications of Spline to Numerical Solution of Differential Equations. M. Phil Dissertation, AMU, India (1990)

  54. Rashidinia, J., Mahmoodi, Z., Ghasemi, M.: Parametric spline method for a class of singular two-point boundary value problems. Appl. Math. Comput. 188, 58–63 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  55. Rashidinia, J., Mohammadi, R., Jalilian, R.: The numerical solution of non-linear singular boundary value problems arising in physiology. Appl. Math. Comput. 185, 360–367 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  56. Ravi Kanth, A.S.V., Bhattacharya, V.: Cubic spline for a class of non-linear singular boundary value problems arising in physiology. Appl. Math. Comput. 74, 768–774 (2006)

    Article  MathSciNet  Google Scholar 

  57. Ravi Kanth, A.S.V.: Cubic spline polynomial for non-linear singular two-point boundary value problems. Appl. Math. Comput. 189(2), 2017–2022 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  58. Ravi Kanth, A.S.V., Reddy, Y.N.: Cubic spline for a class of singular two-point boundary value problems. Appl. Math. Comput. 170, 733–740 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  59. Reddien, G.W.: Projection methods and singular two point boundary value problems. Numer. Math. 21, 193–205 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  60. Reddien, G.W., Schumaker, L.L.: On a collocation method for singular two-point boundary value problems. Numer. Math. 21, 193–205 (1976)

    Article  Google Scholar 

  61. Reddien, G.W., Schumaker, L.L.: On the collocation method for singular two point boundary value problems. Numer. Math. 25, 427–432 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  62. Regan, D.O.: Theory of Singular Boundary Value Problems. World Scientific, Singapore (1994)

    MATH  Google Scholar 

  63. Rentrop, P.: A Taylor series method for the numerical solution of two point boundary value problems. Numer. Math. 31, 359–375 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  64. Russell, R.D., Shampine, L.F.: Numerical methods for singular boundary value problems. SIAM J. Numer. Anal. 12, 13–36 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  65. Schreiber, R.: Finite element methods of high order accuracy for singular two-point boundary value problems with non smooth solutions. SIAM J. Numer. Anal. 17, 547–566 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  66. Shampine, L.F., Reichelt, M.W., Kierzenka, J.: Solving boundary value problems for ODE in matlab using bvp4c. Available on ftp://ftp.mathworks.com/pub/doc/papers/bvp

  67. Shampine, L.F.: Singular boundary value problems for odes. Available on ftp://ftp.mathworks.com/pub/doc/papers/bvp

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad, 211004, (UP), India

    Manoj Kumar & Yogesh Gupta

Authors
  1. Manoj Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yogesh Gupta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Manoj Kumar.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kumar, M., Gupta, Y. Methods for solving singular boundary value problems using splines: a review. J. Appl. Math. Comput. 32, 265–278 (2010). https://doi.org/10.1007/s12190-009-0249-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-009-0249-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation