Book Review
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1567376
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Book Information:
Authors:
J. Kevorkian and
J. D. Cole
Title:
Perturbation methods in applied mathematics
Additional book information:
Applied Mathematical Sciences, vol. 34, Springer-Verlag, Berlin and New York, 1981, x + 558 pp., $42.00.
Author:
Ali Hasan Nayfeh
Title:
Introduction to perturbation techniques
Additional book information:
Wiley, New York, 1981, xiv + 519 pp., $29.95.
Carl M. Bender and Steven A. Orszag, Advanced mathematical methods for scientists and engineers, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1978. MR 538168
Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503330
George D. Birkhoff, On the asymptotic character of the solutions of certain linear differential equations containing a parameter, Trans. Amer. Math. Soc. 9 (1908), no. 2, 219–231. MR 1500810, DOI 10.1090/S0002-9947-1908-1500810-1
N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic methods in the theory of non-linear oscillations, Translated from the second revised Russian edition, International Monographs on Advanced Mathematics and Physics, Hindustan Publishing Corp., Delhi, Gordon and Breach Science Publishers, Inc., New York, 1961. MR 0141845
5. J. R. Bowen, A. Acrivos and A. K. Oppenheim [1963], Singular perturbation refinement to quasi-steady state approximations in chemical kinetics, Chem. Eng. Sci. 18, 177-188.
6. L. Brillouin [1926], Rémarques sur la méchaniques ondulatoire, J. Phys. Radium 7, 353-368.
J. D. Buckmaster and G. S. S. Ludford, Theory of laminar flames, Electronic & Electrical Engineering Research Studies: Pattern Recognition & Image Processing Series, vol. 2, Cambridge University Press, Cambridge-New York, 1982. MR 666866
G. F. Carrier, Boundary layer problems in applied mechanics, Advances in Applied Mechanics, vol. 3, Academic Press, Inc., New York, N.Y., 1953, pp. 1–19. MR 0062315
Carl E. Pearson (ed.), Handbook of applied mathematics, Van Nostrand Reinhold Co., New York-Toronto-London, 1974. Selected results and methods. MR 0345470
Julian D. Cole, Perturbation methods in applied mathematics, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1968. MR 0246537
Germund Dahlquist, A numerical method for some ordinary differential equations with large Lipschitz constants, Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968) North-Holland, Amsterdam, 1969, pp. 183–186. MR 0258290
Wiktor Eckhaus, Matched asymptotic expansions and singular perturbations, North-Holland Mathematics Studies, No. 6, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. MR 0670800
Wiktor Eckhaus, Asymptotic analysis of singular perturbations, Studies in Mathematics and its Applications, vol. 9, North-Holland Publishing Co., Amsterdam-New York, 1979. MR 553107
14. W. Eckhaus and E. M. deJager [1982] (Proc. Conf. Singular Perturbations and Appl., Oberwolfach).
Arthur Erdélyi, An expansion procedure for singular perturbations, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 95 (1960/61), 651–672. MR 137407
L. E. Fraenkel, On the method of matched asymptotic expansions. I. A matching principle, Proc. Cambridge Philos. Soc. 65 (1969), 209–231. MR 237898, DOI 10.1017/s0305004100044212
17. K. O. Friedrichs [1953], Special topics in analysis, lecture notes, New York University.
K. O. Friedrichs, Asymptotic phenomena in mathematical physics, Bull. Amer. Math. Soc. 61 (1955), 485–504. MR 74614, DOI 10.1090/S0002-9904-1955-09976-2
K. O. Friedrichs and J. J. Stoker, The non-linear boundary value problem of the buckled plate, Amer. J. Math. 63 (1941), 839–888. MR 5866, DOI 10.2307/2371625
K. O. Friedrichs and W. R. Wasow, Singular perturbations of non-linear oscillations, Duke Math. J. 13 (1946), 367–381. MR 18308
A. L. Gol′denveĭzer, Theory of elastic thin shells, International Series of Monographs in Aeronautics and Astronautics, Published for the American Society of Mechanical Engineers by Pergamon Press, Oxford-London-New York-Paris, 1961. Translation from the Russian edited by G. Herrmann. MR 0135763
W. M. Greenlee and R. E. Snow, Two-timing on the half line for damped oscillation equations, J. Math. Anal. Appl. 51 (1975), no. 2, 394–428. MR 382798, DOI 10.1016/0022-247X(75)90129-8
F. A. Howes, Boundary-interior layer interactions in nonlinear singular perturbation theory, Mem. Amer. Math. Soc. 15 (1978), no. 203, iv+108. MR 499407, DOI 10.1090/memo/0203
Saul Kaplun, Low Reynolds number flow past a circular cylinder, J. Math. Mech. 6 (1957), 595–603. MR 0091694, DOI 10.1512/iumj.1957.6.56029
Paco A. Lagerstrom and Louis N. Howard (eds.), Fluid mechanics and singular perturbations: A collection of papers by Saul Kaplun, Academic Press, New York-London, 1967. MR 0214326
Joseph B. Keller, A geometrical theory of diffraction, Calculus of variations and its applications. Proceedings of Symposia in Applied Mathematics, Vol. VIII, McGraw-Hill Book Co., Inc., New York-Toronto-London, for the American Mathematical Society, Providence, R.I., 1958, pp. 27–52. MR 0094120
Joseph B. Keller, Rays, waves and asymptotics, Bull. Amer. Math. Soc. 84 (1978), no. 5, 727–750. MR 499726, DOI 10.1090/S0002-9904-1978-14505-4
J. Kevorkian, The two variable expansion procedure of the approximate solution of certain nonlinear differential equations, Space Mathematics (Proc. Summer Seminar, Ithaca, N.Y., 1963) Amer. Math. Soc., Providence, R.I., 1966, pp. 206–275. MR 0205468
29. H. A. Kramers [1926], Wellenmechanik and halbzahlige Quantisierung, Z. Physik 39, 828-840.
Paco A. Lagerstrom, Forms of singular asymptotic expansions in layer-type problems, Rocky Mountain J. Math. 6 (1976), no. 4, 609–635. MR 430442, DOI 10.1216/RMJ-1976-6-4-609
Rudolph E. Langer, On the asymptotic solutions of ordinary differential equations, with an application to the Bessel functions of large order, Trans. Amer. Math. Soc. 33 (1931), no. 1, 23–64. MR 1501574, DOI 10.1090/S0002-9947-1931-1501574-0
32. N. Levinson [1950a], The first boundary value problem for ε∆u + A(x, y)u, Ann. of Math. (2) 51, 428-445.
Norman Levinson, Perturbations of discontinuous solutions of non-linear systems of differential equations, Acta Math. 82 (1950), 71–106. MR 35356, DOI 10.1007/BF02398275
J.-L. Lions, Perturbations singulières dans les problèmes aux limites et en contrôle optimal, Lecture Notes in Mathematics, Vol. 323, Springer-Verlag, Berlin-New York, 1973 (French). MR 0600331
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
Richard E. Meyer and Seymour V. Parter (eds.), Singular perturbations and asymptotics, Publication of the Mathematics Research Center, University of Wisconsin, vol. 45, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 606032
J. J. H. Miller (ed.), Boundary and interior layers—computational and asymptotic methods, Boole Press, Dún Laoghaire, 1980. MR 589347
38. W. L. Miranker [1981], Numerical methods for stiff equations, Reidel, Dordrecht.
E. F. Mishchenko and N. Kh. Rozov, Differential equations with small parameters and relaxation oscillations, Mathematical Concepts and Methods in Science and Engineering, vol. 13, Plenum Press, New York, 1980. Translated from the Russian by F. M. C. Goodspeed. MR 750298, DOI 10.1007/978-1-4615-9047-7
Mitio Nagumo, Über das Verhalten der Integrale von $\lambda y''+f(x,y,y’,\lambda )=0$ für $\lambda \to 0$, Proc. Phys.-Math. Soc. Japan (3) 21 (1939), 529–534 (German). MR 1085
Ali Hasan Nayfeh, Perturbation methods, Pure and Applied Mathematics, John Wiley & Sons, New York-London-Sydney, 1973. MR 0404788
Ali Hasan Nayfeh and Dean T. Mook, Nonlinear oscillations, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1979. MR 549322
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., Singular perturbations and optimal control, Mathematical control theory (Proc. Conf., Australian Nat. Univ., Canberra, 1977) Lecture Notes in Math., vol. 680, Springer, Berlin, 1978, pp. 170–218. MR 515718
L. S. Pontryagin, Asymptotic behavior of the solutions of systems of differential equations with a small parameter in the higher derivatives, Amer. Math. Soc. Transl. (2) 18 (1961), 295–319. MR 0124591, DOI 10.1090/trans2/018/18
46. L. Prandtl [1905], Über Flüssigkeits-bewegung bei kleiner Reibung, Verh. III. Int. Math.-Kongresses, Tuebner, Leipzig, pp. 484-491.
Ian Proudman and J. R. A. Pearson, Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder, J. Fluid Mech. 2 (1957), 237–262. MR 86545, DOI 10.1017/S0022112057000105
Erich Rothe, Asymptotic solution of a boundary value problem, Iowa State Coll. J. Sci. 13 (1939), 369–372. MR 327
Jan A. Sanders, Second quantization and averaging: Fermi resonance, J. Chem. Phys. 74 (1981), no. 10, 5733–5736. MR 613274, DOI 10.1063/1.440938
Zeev Schuss, Theory and applications of stochastic differential equations, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., New York, 1980. MR 595164
J. J. Stoker, Mathematical problems connected with the bending and buckling of elastic plates, Bull. Amer. Math. Soc. 48 (1942), 247–261. MR 6324, DOI 10.1090/S0002-9904-1942-07646-4
A. Tihonov, On the dependence of the solutions of differential equations on a small parameter, Mat. Sbornik N.S. 22(64) (1948), 193–204 (Russian). MR 0025047
Yü-Why Tschen, Über das Verhalten der Lösungen einer Folge von Differentialgleichungsproblemen, welche im Limes ausarten, Compositio Math. 2 (1935), 378–401 (German). MR 1556923
Milton Van Dyke, Perturbation methods in fluid mechanics, Annotated edition, Parabolic Press, Stanford, Calif., 1975. MR 0416240
A. B. Vasil′eva, Asymptotic behaviour of solutions of certain problems for ordinary non-linear differential equations with a small parameter multiplying the highest derivatives, Uspehi Mat. Nauk 18 (1963), no. 3 (111), 15–86 (Russian). MR 0158137
A. B. Vasil′eva and V. F. Butuzov, Asimptoticheskie razlozheniya resheniĭ singulyarno- vozmushchennykh uravneniĭ, Izdat. “Nauka”, Moscow, 1973 (Russian). MR 0477344
A. B. Vasil′eva and V. M. Volosov, Works of A. N. Tihonov and his students in differential equations containing a small parameter, Uspehi Mat. Nauk 22 (1967), no. 2 (134), 149–168 (Russian). MR 0205800
M. I. Višik and L. A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 5(77), 3–122 (Russian). MR 0096041
V. M. Volosov, Averaging in systems of ordinary differential equations, Uspehi Mat. Nauk 17 (1962), no. 6 (108), 3–126 (Russian). MR 0146454
60. W. Wasow [1941], On boundary layer problems in the theory of ordinary differential equations, doctoral dissertation, New York Univ., New York.
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
62. W. Wasow [1944b], Asymptotic solution of boundary value problems for the differential equation $\Delta U+łambda\partial U/\partial x=łambda f(x,y)$, Duke Math. J. 11 (1944), 405-415.
Wolfgang Wasow, Asymptotic expansions for ordinary differential equations, Pure and Applied Mathematics, Vol. XIV, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0203188
64. G. Wentzel [1926], Eine Verallgemeinerun der Quantenbedingung für die Zwecke der Wellenmechanik, Z. Physik 38, 518-529.
- 1.
- C. M. Bender and S. A. Orszag [1978], Advanced mathematical methods for scientists and engineers, McGraw-Hill, New York. MR 0538168
- 2.
- A. Bensoussan, J.-L. Lions and G. Papanicolaou [1978], Asymptotic analysis for periodic structures, North-Holland, Amsterdam. MR 503330
- 3.
- G. D. Birkhoff [1908], On the asymptotic character of the solutions of certain linear differential equations containing a parameter, Trans. Amer. Math. Soc. 9, 219-231. MR 1500810
- 4.
- N. N. Bogoliubov and Y. A. Mitropolsky [1961], Asymptotic methods in the theory of non-linear oscillations, 2nd ed., Hindustan Publishing, Delhi. MR 141845
- 5.
- J. R. Bowen, A. Acrivos and A. K. Oppenheim [1963], Singular perturbation refinement to quasi-steady state approximations in chemical kinetics, Chem. Eng. Sci. 18, 177-188.
- 6.
- L. Brillouin [1926], Rémarques sur la méchaniques ondulatoire, J. Phys. Radium 7, 353-368.
- 7.
- J. D. Buckmaster and G. S. S. Ludford [1982], Theory of laminar flames, Cambridge Univ. Press, Cambridge and New York. MR 666866
- 8.
- G. F. Carrier [1953], Boundary layer problems in applied mechanics, Advances in Applied Mechanics. III, Academic Press, New York, pp. 1-19. MR 62315
- 9.
- G. F. Carrier [1974], Perturbation methods, Handbook of Applied Mathematics (C. E. Pearson, ed. ), Van Nostrand-Reinhold, New York, pp. 761-828. MR 345470
- 10.
- J. D. Cole [1968], Perturbation methods in applied mathematics, Blaisdell, Waltham, Mass. MR 246537
- 11.
- G. Dahlquist [1969], A numerical method for some ordinary differential equations with large Lipschitz constants, Information Processing, vol. 68 (A. J. H. Morrell, ed. ), North-Holland, Amsterdam, pp. 183-186. MR 258290
- 12.
- W. Eckhaus [1973], Matched asymptotic expansions and singular perturbations, North-Holland, Amsterdam. (See review in SIAM Rev. 16 (1974), 564-565.) MR 670800
- 13.
- W. Eckhaus[1979], Asymptotic analysis of singular perturbations, North-Holland, Amsterdam. MR 553107
- 14.
- W. Eckhaus and E. M. deJager [1982] (Proc. Conf. Singular Perturbations and Appl., Oberwolfach).
- 15.
- A. Erdélyi [1961], An expansion procedure for singular perturbations, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 95, 651-672. MR 137407
- 16.
- L. E. Fraenkel [1969], On the method of matched asymptotic expansions, Proc. Cambridge Philos. Soc. 65, 209-284. MR 237898
- 17.
- K. O. Friedrichs [1953], Special topics in analysis, lecture notes, New York University.
- 18.
- K. O. Friedrichs [1955], Asymptotic phenomena in mathematical physics, Bull. Amer. Math. Soc. 61, 367-381. MR 74614
- 19.
- K. O. Friedrichs and J. J. Stoker [1941], The nonlinear boundary value problem of the buckled plate, Amer. J. Math. 63, 839-888. MR 5866
- 20.
- K. O. Friedrichs and W. Wasow [1946], Singular perturbations of nonlinear oscillations, Duke Math. J. 13, 367-381. MR 18308
- 21.
- A. L. Gol'denveizer [1961], Theory of elastic thin shells, Pergamon Press, New York and Oxford. MR 135763
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- W. M. Greenlee and R. E. Snow [1975], Two-timing on the half line for damped oscillator equations, J. Math. Anal. Appl. 51, 394-428. MR 382798
- 23.
- F. A. Howes [1978], Boundary and interior layer behavior and their interaction, Mem. Amer. Math. Soc. No. 203. MR 499407
- 24.
- S. Kaplun [1957], Low Reynolds number flow past a circular cylinder, J. Math. Mech. 6, 595-603. MR 91694
- 25.
- S. Kaplun [1967], Fluid mechanics and singular perturbations (P. A. Lagerstrom, L. N. Howard and C. S. Liu, eds. ), Academic Press, New York. MR 214326
- 26.
- J. B. Keller [1958], A geometrical theory of diffraction, Calculus of Variations and its Applications (L. M. Graves, ed. ), McGraw-Hill, New York. MR 94120
- 27.
- J. B. Keller[1978], Rays, waves, and asymptotics, Bull. Amer. Math. Soc. 84, 727-750. MR 499726
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- J. Kevorkian [1962], The two-variable expansion procedure for the approximate solution of certain nonlinear differential equations, Report SM-42620, Douglas Aircraft, Santa Monica, Calif.; also in Space Mathematics, Part III (J. B. Rosser, ed. ), Lectures in Appl. Math., vol. 7, Amer. Math. Soc., Providence, R. I., 1966, pp. 206-275. MR 205468
- 29.
- H. A. Kramers [1926], Wellenmechanik and halbzahlige Quantisierung, Z. Physik 39, 828-840.
- 30.
- P. A. Lagerstrom [1976], Forms of singular asymptotic expansions in layer-type problems, Rocky Mountain J. Math. 6, 609-635. MR 430442
- 31.
- R. E. Langer [1931], On the asymptotic solution of ordinary differential equations with an application to the Bessel functions of large order, Trans. Amer. Math. Soc. 33, 23-64. MR 1501574
- 32.
- N. Levinson [1950a], The first boundary value problem for ε∆u + A(x, y)u, Ann. of Math. (2) 51, 428-445.
- 33.
- N. Levinson [1950b], Perturbations of discontinuous solutions of non-linear systems of differential equations, Acta Math. 82, 71-106. MR 35356
- 34.
- J. -L. Lions [1973], Perturbation singulières dans les problèmes aux limites et en contrôle optimal, Lecture Notes in Math., vol. 323, Springer-Verlag, Berlin and New York. MR 600331
- 35.
- J. A. M. McHugh [1971], An historical survey of ordinary differential equations with a large parameter and turning points, Arch. Hist. Exact Sci. 7, 277-324. MR 1554147
- 36.
- R. E. Meyer and S. V. Parter (eds.) [1980], Singular perturbations and asymptotics, Academic Press, New York. MR 606032
- 37.
- J. J. H. Miller (ed.) [1980], Boundary and interior layers, computational and asymptotic methods, Boole Press, Dublin. MR 589347
- 38.
- W. L. Miranker [1981], Numerical methods for stiff equations, Reidel, Dordrecht.
- 39.
- E. F. Mishchenko and N. Kh. Rozov [1980], Differential equations with small parameters and relaxation oscillations, Plenum Press, New York. MR 750298
- 40.
- M. Nagumo [1939], Über das Verhalten der Integrale von λy" + f(x, y, y', λ) = 0 für λ → 0, Proc. Phys. Math. Soc. Japan 21, 529-534. MR 1085
- 41.
- A. H. Nayfeh [1973], Perturbation methods, Wiley, New York. (See review in J. Fluid Mech. 63 (1974), 623.) MR 404788
- 42.
- A. H. Nayfeh and D. T. Mook [1979], Nonlinear oscillations, Wiley, New York. MR 549322
- 43.
- R. E. O'Malley, Jr. [1974], Introduction to singular perturbations, Academic Press, New York. MR 402217
- 44.
- R. E. O'Malley, Jr. [1978], Singular perturbations and optimal control, Lecture Notes in Math., vol. 680 Springer-Verlag, Heidelberg, pp. 170-218. MR 515718
- 45.
- L. S. Pontryagin [1961], Asymptotic behavior of the solutions of systems of differential equations with a small parameter in the higher derivatives, Amer. Math. Soc. Transl. (2) 18, 295-319. MR 124591
- 46.
- L. Prandtl [1905], Über Flüssigkeits-bewegung bei kleiner Reibung, Verh. III. Int. Math.-Kongresses, Tuebner, Leipzig, pp. 484-491.
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- I. Proudman and J. R. A. Pearson [1957], Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder, J. Fluid Mech. 2, 237-262. MR 86545
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- E. Rothe [1939], Asymptotic solution of a boundary value poblem, Iowa State College J. Sci. 13. MR 327
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- J. Sanders and F. Verhuist [1981], Two chapters in the theory of averaging, Preprint No. 201, Dept. of Math., University of Utrecht. MR 613274
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- Z. Schuss [1980], Theory and applications of stochastic differential equations, Wiley, New York. (See review in Phys. Today 34 (1981), 95-97.) MR 595164
- 51.
- J. J. Stoker [1942], Mathematical problems connected with the bending and buckling of elastic plates, Bull. Amer. Math. Soc. 48, 247-261. MR 6324
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- A. Tikhonov [1948], On the dependence of the solutions of differential equations on a small parameter, Mat. Sb. 22, 193-204. MR 25047
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- Y. Tschen [1935], Über das Verhalten der Lösungen einer Folge von Differential gleichungen, welche im Limes ausarten, Comp. Math. 2, 378-401. MR 1556923
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- M. Van Dyke [1964], Perturbation methods in fluid dynamics, Academic Press, New York. (Annotated edition, Parabolic Press, Stanford, Calif., 1975.) MR 416240
- 55.
- A. B. Vasil'eva [1963], Asymptotic behavior of solutions to certain problems involving nonlinear differential equations containing a small parameter multiplying the highest derivatives, Russian Math. Surveys 18, 13-84. MR 158137
- 56.
- A. B. Vasil'eva and V. F. Butuzov [1973], Asymptotic expansions of solutions of singularly perturbed equations, "Nauka", Moscow. (Russian) MR 477344
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- A. B. Vasil'eva and V. M. Volosov [1967], The work of Tikhonov and his pupils on ordinary differential equations containing a small parameter, Russian Math. Surveys 22, 124-142. MR 205800
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- M. I. Vishik and L. A. Lyusternik [1957], Regular degeneration and boundary layer for linear differential equations with a small parameter, Uspehi Mat. Nauk 12, 3-122. (Also Amer. Math. Soc. Transl. (2) 20 (1961), 239-364.) MR 96041
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- V. M. Volosov [1962], Averaging in systems of ordinary differential equations, Russian Math. Surveys 17, 1-126. MR 146454
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- W. Wasow [1941], On boundary layer problems in the theory of ordinary differential equations, doctoral dissertation, New York Univ., New York.
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- W. Wasow [1944a], On the asymptotic solution of boundary value problems for ordinary differential equations containing a parameter, J. Math. and Phys. 23, 173-183. MR 10907
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- W. Wasow [1944b], Asymptotic solution of boundary value problems for the differential equation $\Delta U+łambda\partial U/\partial x=łambda f(x,y)$, Duke Math. J. 11 (1944), 405-415.
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- W. Wasow [1965], Asymptotic expansions for ordinary differential equations, Interscience, New York. (Reprinted by Kreiger, Huntington, 1976.) MR 203188
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- G. Wentzel [1926], Eine Verallgemeinerun der Quantenbedingung für die Zwecke der Wellenmechanik, Z. Physik 38, 518-529.
Review Information:
Reviewer:
R. E. O'Malley, Jr.
Journal:
Bull. Amer. Math. Soc.
7 (1982), 414-420
DOI:
https://doi.org/10.1090/S0273-0979-1982-15053-4