Numerical analysis of boundary-layer problems in ordinary differential equations

Author:
W. D. Murphy

Journal:
Math. Comp. **21** (1967), 583-596

MSC:
Primary 65.61

DOI:
https://doi.org/10.1090/S0025-5718-1967-0225496-9

MathSciNet review:
0225496

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Abstract | References | Similar Articles | Additional Information

Abstract: We categorize some of the finite-difference methods that can be used to treat the initial-value problem for the boundary-layer differential equation

The idea of relating to allows us to study the nature of the difference equation for very small . We can, however, look at this in another way. Given a differential equation in the form of Eq. (1) we ask how can we choose so that the associated difference equation will give an accurate approximation. If is sufficiently small, choose by the formula where .

**[1]**Germund Dahlquist,*Convergence and stability in the numerical integration of ordinary differential equations*, Math. Scand.**4**(1956), 33–53. MR**0080998**, https://doi.org/10.7146/math.scand.a-10454**[2]**Peter Henrici,*Discrete variable methods in ordinary differential equations*, John Wiley & Sons, Inc., New York-London, 1962. MR**0135729****[3]**T. E. Hull and W. A. J. Luxemburg,*Numerical methods and existence theorems for ordinary differential equations*, Numer. Math.**2**(1960), 30–41. MR**0114017**, https://doi.org/10.1007/BF01386206**[4]**T. E. Hull and A. C. R. Newbery,*Error bounds for a family of three-point integration procedures*, J. Soc. Indust. Appl. Math.**7**(1959), 402–412. MR**0136079****[5]**T. E. Hull and A. C. R. Newbery,*Integration procedures which minimize propagated errors*, J. Soc. Indust. Appl. Math.**9**(1961), 31–47. MR**0120770****[6]**W. D. Murphy, "Numerical analysis of boundary layer problems," AEC Research and Development Report NYO-1480-63, New York University.**[7]**A. B. Vasiĺeva, "Asymptotic behavior of solutions to certain problems involving nonlinear differential equations containing a small parameter multiplying the highest derivatives,"*Russian Math. Surveys*, v. 18, 1963, no. 3, pp. 13-84.

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DOI:
https://doi.org/10.1090/S0025-5718-1967-0225496-9

Article copyright:
© Copyright 1967
American Mathematical Society