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Book Review

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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.

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

  • 1. C. M. Bender and S. A. Orszag [1978], Advanced mathematical methods for scientists and engineers, McGraw-Hill, New York. MR 538168
  • 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
  • 22. 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
  • 28. 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.
  • 47. 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
  • 48. E. Rothe [1939], Asymptotic solution of a boundary value poblem, Iowa State College J. Sci. 13. MR 327
  • 49. J. Sanders and F. Verhuist [1981], Two chapters in the theory of averaging, Preprint No. 201, Dept. of Math., University of Utrecht. MR 613274
  • 50. 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
  • 52. A. Tikhonov [1948], On the dependence of the solutions of differential equations on a small parameter, Mat. Sb. 22, 193-204. MR 25047
  • 53. Y. Tschen [1935], Über das Verhalten der Lösungen einer Folge von Differential gleichungen, welche im Limes ausarten, Comp. Math. 2, 378-401. MR 1556923
  • 54. 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
  • 57. 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
  • 58. 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
  • 59. V. M. Volosov [1962], Averaging in systems of ordinary differential equations, Russian Math. Surveys 17, 1-126. MR 146454
  • 60. W. Wasow [1941], On boundary layer problems in the theory of ordinary differential equations, doctoral dissertation, New York Univ., New York.
  • 61. 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
  • 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.
  • 63. W. Wasow [1965], Asymptotic expansions for ordinary differential equations, Interscience, New York. (Reprinted by Kreiger, Huntington, 1976.) MR 203188
  • 64. 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
American Mathematical Society