A road map of methods for approximating solutions of two-point boundary-value problems

  • James W. Daniel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 76)


My aim in this paper has been to explain briefly what each of a variety of methods is, how methods relate to one another, and where are the difficulties today that stimulate interesting research problems. To describe what the methods are and how they relate we viewed each method as a Solution Technique for some Discrete Model of a Transformed Problem. Those areas which in my opinion deserve much more study and development include: numerical effects and efficiency of different methods of solving the linear algebraic systems that arise; methods for solving the special nonlinear algebraic systems that arise; comparative performance of codes implementing various methods on carefully chosen classes of test problems; and methods for estimating and controlling global errors by automatic selection of parameters of the method.


Finite Difference Discrete Model Richardson Extrapolation Linear Algebraic System Transform Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

9. References

  1. 1.
    Aktas, Z., and H. J. Stetter (1977), "A classification and survey of numerical methods for boundary-value problems in ordering differential equations," Int. J. Num. Meth. Eng., vol. 11, 771–796.Google Scholar
  2. 2.
    Frank, R. (1976), "The method of iterated defect correction and its application to two-point boundary-value problems, I," Numer. Math., vol. 25, 409–418.Google Scholar
  3. 3.
    Joyce, C. C. (1971), "Survey of extrapolation processes in numerical analysis," SIAM Rev., vol. 13, 435–590.Google Scholar
  4. 4.
    Keller, H. B. (1975), "Numerical methods for boundary-value problems in ordinary differential equations: Survey and some recent results on difference methods," in Numerical Solutions of Boundary-value Problems in Ordinary Differential Equations (A. K. Aziz, ed.), Academic Press, New York, 27–88.Google Scholar
  5. 5.
    Murray, W. (ed.) (1972), Numerical Methods for Unconstrained Optimization Problems, Academic Press, London.Google Scholar
  6. 6.
    Ortega, J. M., and W. C. Rheinboldt (1970), Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York.Google Scholar
  7. 7.
    Pereyra, V. (1967), "Iterated deferred corrections for nonlinear boundary-value problems," Numer. Math., vol. 10, 316–323.Google Scholar
  8. 8.
    Scott, M. R. (1973), Invariant Imbedding and Its Applications to Ordinary Differential Equations, Addison-Wesley, Reading, Mass.Google Scholar
  9. 9.
    Scott, M. R., and H. A. Watts (1977), "Computational solution of linear two-point boundary-value problems via orthonormalization," SIAM J. Numer. Anal., vol. 14, 40–70.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1979

Authors and Affiliations

  • James W. Daniel

There are no affiliations available

Personalised recommendations