## The numerical solution of boundary value problems for stiff differential equations

HTML articles powered by AMS MathViewer

- by Joseph E. Flaherty and R. E. O’Malley PDF
- Math. Comp.
**31**(1977), 66-93 Request permission

## Abstract:

The numerical solution of boundary value problems for certain stiff ordinary differential equations is studied. The methods developed use singular perturbation theory to construct approximate numerical solutions which are valid asymptotically; hence, they have the desirable feature of becoming more accurate as the equations become stiffer. Several numerical examples are presented which demonstrate the effectiveness of these methods.## References

- L. R. Abrahamsson, H. B. Keller, and H. O. Kreiss,
*Difference approximations for singular perturbations of systems of ordinary differential equations*, Numer. Math.**22**(1974), 367–391. MR**388784**, DOI 10.1007/BF01436920 - Richard C. Aiken and Leon Lapidus,
*An effective numerical integration method for typical stiff systems*, AIChE J.**20**(1974), no. 2, 368–375. MR**395228**, DOI 10.1002/aic.690200225 - Paul T. Boggs,
*An algorithm, based on singular perturbation theory, for ill-conditioned minimization problems*, SIAM J. Numer. Anal.**14**(1977), no. 5, 830–843. MR**519600**, DOI 10.1137/0714056 - Roland Bulirsch and Josef Stoer,
*Numerical treatment of ordinary differential equations by extrapolation methods*, Numer. Math.**8**(1966), 1–13. MR**191095**, DOI 10.1007/BF02165234 - Julian D. Cole,
*Perturbation methods in applied mathematics*, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1968. MR**0246537** - S. D. Conte,
*The numerical solution of linear boundary value problems*, SIAM Rev.**8**(1966), 309–321. MR**203945**, DOI 10.1137/1008063
S. D. CONTE & C. de BOOR, - Fred Dorr,
*The numerical solution of singular perturbations of boundary value problems*, SIAM J. Numer. Anal.**7**(1970), 281–313. MR**267781**, DOI 10.1137/0707021 - Fred W. Dorr,
*An example of ill-conditioning in the numerical solution of singular perturbation problems*, Math. Comp.**25**(1971), 271–283. MR**297142**, DOI 10.1090/S0025-5718-1971-0297142-0
W. E. FERGUSON, JR., "A singularly perturbed linear two-point boundary-value problem," Ph.D. Dissertation, California Inst. Tech., 1975.
- Paul C. Fife,
*Semilinear elliptic boundary value problems with small parameters*, Arch. Rational Mech. Anal.**52**(1973), 205–232. MR**374665**, DOI 10.1007/BF00247733 - Paul C. Fife,
*Transition layers in singular perturbation problems*, J. Differential Equations**15**(1974), 77–105. MR**330665**, DOI 10.1016/0022-0396(74)90088-6 - Nanny Fröman and Per Olof Fröman,
*JWKB approximation. Contributions to the theory*, North-Holland Publishing Co., Amsterdam, 1965. MR**0173481** - C. William Gear,
*Numerical initial value problems in ordinary differential equations*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR**0315898** - P. W. Hemker,
*A method of weighted one-sided differences for stiff boundary value problems with turning points*, Mathematisch Centrum, Afdeling Numerieke Wiskunde NW 9/74, Mathematisch Centrum, Amsterdam, 1974. MR**0351096** - F. A. Howes,
*The asymptotic solution of a class of singularly perturbed nonlinear boundary value problems via differential inequalities*, SIAM J. Math. Anal.**9**(1978), no. 2, 215–249. MR**477345**, DOI 10.1137/0509017 - Herbert B. Keller,
*Accurate difference methods for linear ordinary differential systems subject to linear constraints*, SIAM J. Numer. Anal.**6**(1969), 8–30. MR**253562**, DOI 10.1137/0706002 - 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 - Herbert B. Keller and Tuncer Cebeci,
*Accurate numerical methods for boundary-layer flows. II. Two-dimensional turbulent flows*, AIAA J.**10**(1972), 1193–1199. MR**311207**, DOI 10.2514/3.50349 - H. B. Keller and A. B. White Jr.,
*Difference methods for boundary value problems in ordinary differential equations*, SIAM J. Numer. Anal.**12**(1975), no. 5, 791–802. MR**413513**, DOI 10.1137/0712059 - M. Lentini and V. Pereyra,
*Boundary problem solvers for first order systems based on deferred corrections*, Numerical solutions of boundary value problems for ordinary differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1974) Academic Press, New York, 1975, pp. 293–315. MR**0488787** - Bengt Lindberg,
*On a dangerous property of methods for stiff differential equations*, Nordisk Tidskr. Informationsbehandling (BIT)**14**(1974), 430–436. MR**362909**, DOI 10.1007/bf01932539 - D. B. MacMillan,
*Asymptotic methods for systems of differential equations in which some variables have very short response times*, SIAM J. Appl. Math.**16**(1968), 704–722. MR**234084**, DOI 10.1137/0116058 - James A. M. McHugh,
*An historical survey of ordinary linear differential equations with a large parameter and turning points*, Arch. History Exact Sci.**7**(1971), no. 4, 277–324. MR**1554147**, DOI 10.1007/BF00328046 - W. L. Miranker,
*Numerical methods of boundary layer type for stiff systems of differential equations*, Computing (Arch. Elektron. Rechnen)**11**(1973), no. 3, 221–234 (English, with German summary). MR**386276**, DOI 10.1007/bf02252912 - W. L. Miranker and J. P. Morreeuw,
*Semianalytic numerical studies of turning points arising in stiff boundary value problems*, Math. Comput.**28**(1974), 1017–1034. MR**0381329**, DOI 10.1090/S0025-5718-1974-0381329-5 - David E. Muller,
*A method for solving algebraic equations using an automatic computer*, Math. Tables Aids Comput.**10**(1956), 208–215. MR**83822**, DOI 10.1090/S0025-5718-1956-0083822-0 - W. D. Murphy,
*Numerical analysis of boundary-layer problems in ordinary differential equations*, Math. Comp.**21**(1967), 583–596. MR**225496**, DOI 10.1090/S0025-5718-1967-0225496-9 - Robert E. O’Malley Jr.,
*Introduction to singular perturbations*, Applied Mathematics and Mechanics, Vol. 14, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR**0402217** - R. E. O’Malley Jr.,
*On multiple solutions of a singular perturbation problem*, Arch. Rational Mech. Anal.**49**(1972/73), 89–98. MR**335985**, DOI 10.1007/BF00281412 - R. E. O’Malley Jr.,
*Phase-plane solutions to some singular perturbation problems*, J. Math. Anal. Appl.**54**(1976), no. 2, 449–466. MR**450722**, DOI 10.1016/0022-247X(76)90214-6 - R. E. O’Malley Jr.,
*Boundary layer methods for ordinary differential equations with small coefficients multiplying the highest derivatives*, Constructive and computational methods for differential and integral equations (Sympos., Indiana Univ., Bloomington, Ind., 1974) Lecture Notes in Math., Vol. 430, Springer, Berlin, 1974, pp. 363–389. MR**0486872** - Robert E. O’Malley Jr. and Joseph B. Keller,
*Loss of boundary conditions in the asymptotic solution of linear ordinary differential equations. II. Boundary value problems*, Comm. Pure Appl. Math.**21**(1968), 263–270. MR**224929**, DOI 10.1002/cpa.3160210305 - F. W. J. Olver,
*Asymptotics and special functions*, Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR**0435697**
S. V. PARTER, "Singular perturbations of second order differential equations." (Unpublished.)
- Carl E. Pearson,
*On a differential equation of boundary layer type*, J. Math. and Phys.**47**(1968), 134–154. MR**228189** - Carl E. Pearson,
*On non-linear ordinary differential equations of boundary layer type*, J. Math. and Phys.**47**(1968), 351–358. MR**237107**
M. R. SCOTT & H. A. WATTS, - M. I. Višik and L. A. Lyusternik,
*Initial jump for nonlinear differential equations containing a small parameter*, Soviet Math. Dokl.**1**(1960), 749–752. MR**0120427** - Wolfgang Wasow,
*On the asymptotic solution of boundary value problems for ordinary differential equations containing a parameter*, J. Math. Phys. Mass. Inst. Tech.**23**(1944), 173–183. MR**10907**, DOI 10.1002/sapm1944231173 - Wolfgang Wasow,
*Singular perturbations of boundary value problems for nonlinear differential equations of the second order*, Comm. Pure Appl. Math.**9**(1956), 93–113. MR**79161**, DOI 10.1002/cpa.3160090107 - Wolfgang Wasow,
*Connection problems for asymptotic series*, Bull. Amer. Math. Soc.**74**(1968), 831–853. MR**228757**, DOI 10.1090/S0002-9904-1968-12055-5 - Ralph A. Willoughby (ed.),
*Stiff differential systems*, The IBM Research Symposia Series, Plenum Press, New York-London, 1974. MR**0343619**
J. YARMISH, - Joshua Yarmish,
*Newton’s method techniques for singular perturbations*, SIAM J. Math. Anal.**6**(1975), 661–680. MR**426448**, DOI 10.1137/0506058

*Elementary Numerical Analysis*, 2nd ed., McGraw-Hill, New York, 1972, Chapter 5.

*Support-A Computer Code for Two-Point Boundary-Value Problems via Orthonormalization*, Sandia Laboratories Report SAND 75-0198, June 1975. S. TIMOSHENKO,

*Strength of Materials, Part*II,

*Advanced Theory and Problems*, 3rd ed., Van Nostrand, Princeton, N. J., 1956.

*Aspects of the Numerical and Theoretical Treatment of Singular Perturbation*, Doctoral Dissertation, New York Univ., 1972.

## Additional Information

- © Copyright 1977 American Mathematical Society
- Journal: Math. Comp.
**31**(1977), 66-93 - MSC: Primary 65L10
- DOI: https://doi.org/10.1090/S0025-5718-1977-0657396-0
- MathSciNet review: 0657396