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Tridiagonal fourth order approximations to general two-point nonlinear boundary value problems with mixed boundary conditions


Author: Robert S. Stepleman
Journal: Math. Comp. 30 (1976), 92-103
MSC: Primary 65L10
MathSciNet review: 0408259
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Abstract: This paper develops fourth order discretizations to the two-point boundary value problem

\begin{displaymath}\begin{array}{*{20}{c}} {{y^{(2)}}(t) = f(t,y(t),{y^{(1)}}(t)... ...1}y(1) + {\beta _1}{y^{(1)}}(1) = {\delta _1}.} \\ \end{array} \end{displaymath}

These discretizations have the desirable properties that they are tridiagonal and of "positive type".

References [Enhancements On Off] (What's this?)

  • [1] B. T. Allen, A new method of solving second-order differential equations when the first derivative is present, Comput. J. 8 (1965/1966), 392–394. MR 0189248
  • [2] A. K. Aziz and B. E. Hubbard, Bounds for the solution of the Sturm-Liouville problem with application to finite difference methods, J. Soc. Indust. Appl. Math. 12 (1964), 163–178. MR 0165701
  • [3] L. COLLATZ, The Numerical Treatment of Differential Equations, Springer, Berlin, 1966.
  • [4] J. DANIEL & B. SWARTZ, Extrapolated Collocation for Two-Point Boundary-Value Problems Using Cubic Splines, Technical Report LA-DC-72-1520, Los Alamos Scientific Laboratory, Los Alamos, 1972.
  • [5] Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons, Inc., New York-London, 1962. MR 0135729
  • [6] Herbert B. Keller, Numerical methods for two-point boundary-value problems, Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London, 1968. MR 0230476
  • [7] Herbert B. Keller, Accurate difference methods for nonlinear two-point boundary value problems, SIAM J. Numer. Anal. 11 (1974), 305–320. MR 0351098
  • [8] Milton Lees, Discrete methods for nonlinear two-point boundary value problems, Numerical Solution of Partial Differential Equations (Proc. Sympos. Univ. Maryland, 1965) Academic Press, New York, 1966, pp. 59–72. MR 0202323
  • [9] V. PEREYRA, High Order Finite Difference Solution of Differential Equations, Technical Report STAN-CS-73-348, Computer Science Dept., Stanford University, 1973.
  • [10] R. D. Russell and L. F. Shampine, A collocation method for boundary value problems, Numer. Math. 19 (1972), 1–28. MR 0305607
  • [11] J. SHOOSMITH, A Study of Monotone Matrices With an Application to the High-Order, Finite-Difference Solution of a Linear Two-Point Boundary-Value Problem, Dissertation, Department of Applied Mathematics and Computer Science, University of Virginia, Charlottesville, 1973.
  • [12] R. STEPLEMAN, "High order solution of mildly nonlinear elliptic boundary value problems," Proceedings of the AICA International Symposium on Computer Methods for Partial Differential Equations, Lehigh University, 1975.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1976-0408259-6
Keywords: Boundary value problems, mixed boundary conditions, fourth order discretization
Article copyright: © Copyright 1976 American Mathematical Society