Tridiagonal fourth order approximations to general two-point nonlinear boundary value problems with mixed boundary conditions
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- by Robert S. Stepleman PDF
- Math. Comp. 30 (1976), 92-103 Request permission
Abstract:
This paper develops fourth order discretizations to the two-point boundary value problem \[ \begin {array}{*{20}{c}} {{y^{(2)}}(t) = f(t,y(t),{y^{(1)}}(t)),} \\ {{\alpha _0}y(0) - {\beta _0}{y^{(1)}}(0) = {\delta _0},\quad {\alpha _1}y(1) + {\beta _1}{y^{(1)}}(1) = {\delta _1}.} \\ \end {array} \] These discretizations have the desirable properties that they are tridiagonal and of "positive type".References
- B. T. Allen, A new method of solving second-order differential equations when the first derivative is present, Comput. J. 8 (1965/66), 392–394. MR 189248, DOI 10.1093/comjnl/8.4.392
- 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 165701 L. COLLATZ, The Numerical Treatment of Differential Equations, Springer, Berlin, 1966. 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.
- Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons, Inc., New York-London, 1962. MR 0135729
- 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
- Herbert B. Keller, Accurate difference methods for nonlinear two-point boundary value problems, SIAM J. Numer. Anal. 11 (1974), 305–320. MR 351098, DOI 10.1137/0711028
- 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 V. PEREYRA, High Order Finite Difference Solution of Differential Equations, Technical Report STAN-CS-73-348, Computer Science Dept., Stanford University, 1973.
- R. D. Russell and L. F. Shampine, A collocation method for boundary value problems, Numer. Math. 19 (1972), 1–28. MR 305607, DOI 10.1007/BF01395926 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. 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.
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Math. Comp. 30 (1976), 92-103
- MSC: Primary 65L10
- DOI: https://doi.org/10.1090/S0025-5718-1976-0408259-6
- MathSciNet review: 0408259