×

Topics in singular perturbations. (English) Zbl 0203.40101


MSC:

34E15 Singular perturbations for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bellman, R., Perturbation Techniques in Mathematics, Physics, and Engineering (1964), Holt, Rinehart and Winston: Holt, Rinehart and Winston New York
[2] Birkhoff, G. D., On the asymptotic character of the solutions of certain linear differential equations containing a parameter, Trans. Am. Math. Soc., 9, 219-231 (1908) · JFM 39.0386.01
[3] Birkhoff, G. D., Quantum mechanics and asymptotic series, Bull. Am. Math. Soc., 39, 681-700 (1933) · Zbl 0008.08902
[4] Bobisud, L., On the single first-order partial differential equation with a small parameter, SIAM Rev., 8, 479-493 (1966) · Zbl 0143.32502
[5] Bobisud, L., Degeneration of the solutions of certain well-posed systems of partial differential equations depending on a small parameter, J. Math. Anal. Appl., 16, 419-454 (1966) · Zbl 0145.35901
[6] Bobisud, L., Second-order linear parabolic equations with a small parameter, Arch. Rat. Mech. Anal., 27, 385-397 (1968) · Zbl 0153.14203
[7] L. Bobisud; L. Bobisud · Zbl 0174.15501
[8] Boyce, W. E.; Handelman, G. H., Vibrations of rotating beams with tip mass, Z. Angew. Math. Phys., 12, 369-392 (1961) · Zbl 0105.38201
[9] Brock, P., The nature of solutions of a Rayleigh type forced vibration equation with a large coefficient of damping, J. Appl. Phys., 24, 1004-1007 (1953) · Zbl 0053.24506
[10] Carrier, G. F., Boundary Layer Problems in Applied Mechanics, (Advances in Applied Mechanics III (1953), Academic Press: Academic Press New York), 1-19 · Zbl 0051.42304
[11] Carrier, G. F., Boundary layer problems in applied mathematics, Commun. Pure Appl. Math., 7, 11-17 (1954) · Zbl 0058.33702
[12] Cartwright, M. L., Forced oscillations in nonlinear systems, Contrib. Theory Nonlinear Oscillations, 1, 149-241 (1950) · Zbl 0039.09901
[13] Cartwright, M. L., Van der Pol’s equation for relaxation oscillations, Contrib. Theory Nonlinear Oscillations, 2, 3-8 (1952) · Zbl 0048.06902
[14] Cochran, J. A., Problems in Singular Perturbation Theory (1962), Stanford University, unpublished Doctoral dissertation
[15] Cochran, J. A., On the uniqueness of solutions of linear differential equations, J. Math. Anal. Appl., 22, 418-426 (1968) · Zbl 0159.11701
[16] Coddington, E. A.; Levinson, N., A boundary value problem for a nonlinear differential equation with a small parameter, (Proc. Amer. Math. Soc., 3 (1952)), 73-81 · Zbl 0046.09503
[17] Coddington, E. A.; Levinson, N., Theory of Ordinary Differential Equations (1955), McGraw-Hill: McGraw-Hill New York · Zbl 0042.32602
[18] Cole, J. D.; Kevorkian, J., Uniformly Valid Asymptotic Expansions for Certain Nonlinear Differential Equations, (Nonlinear Differential Equations and Nonlinear Mechanics (1963), Academic Press: Academic Press New York), 113-120 · Zbl 0134.07502
[19] Copson, E. T., Asymptotic Expansions (1965), Cambridge University Press: Cambridge University Press London · Zbl 0123.26001
[20] van der Corput, J. G., Asymptotic developments I. Fundamental theorems of asymptotics, J. Anal. Math., 4, 341-418 (1956) · Zbl 0075.04901
[21] Courant, R.; Hilbert, D., Methods of Mathematical Physics II, Partial Differential Equations (1962), Interscience: Interscience New York · Zbl 0729.35001
[22] Davis, R. B., Asymptotic solutions of the first boundary value problem for a fourth-order elliptic partial differential equation, J. Ratl. Mech. Anal., 5, 605-620 (1956) · Zbl 0074.08401
[23] Am. Math. Soc. Transl., No. 88 (1953) · Zbl 0034.35602
[24] Eckhaus, W.; de Jager, E. M., Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type, Arch. Rational Mech. Anal., 23, 26-86 (1966) · Zbl 0151.15101
[25] Erdélyi, A., An expansion procedure for singular perturbations, Atti Acad. Sci. Torino, Cl. Sci. Fis. Mat. Natur., 95, 651-672 (1961) · Zbl 0107.31104
[26] Erdélyi, A., On a nonlinear boundary value problem involving a small parameter, J. Australian Math. Soc., 2, 425-439 (1962) · Zbl 0161.06201
[27] Erdélyi, A., Singular perturbations of boundary value problems involving ordinary differential equations, J. Soc. Indust. Appl. Math., 11, 105-116 (1963) · Zbl 0122.32501
[28] Erdélyi, A., The Integral Equations of Asymptotic Theory, (Asymptotic Solutions of Differential Equations and Their Applications (1964), Wiley: Wiley New York), 211-229
[29] Erdélyi, A., Two-variable expansions for singular perturbations, J. Inst. Math. Appl., 4, 113-119 (1968) · Zbl 0155.14402
[30] Flanders, D. A.; Stoker, J. J., The Limit Case of Relaxation Oscillations, (Studies in Nonlinear Vibration Theory (1946), New York University: New York University New York), 50-64 · Zbl 0061.19315
[31] L. E. Fraenkel; L. E. Fraenkel
[32] Friedman, B., Principles and Techniques of Applied Mathematics (1956), Wiley: Wiley New York · Zbl 0072.12806
[33] Friedman, B., Singular Pertubations of Ordinary Differential Equations, (Mathematical Note No. 245 (1961), Mathematics Research Laboratory, Boeing Scientific Research Laboratories: Mathematics Research Laboratory, Boeing Scientific Research Laboratories Seattle)
[34] Friedman, B., Singular Perturbations of Differential Equations II, (Mathematical Note No. 275 (1962), Mathematics Research Laboratory, Boeing Scientific Research Laboratories: Mathematics Research Laboratory, Boeing Scientific Research Laboratories Seattle)
[35] Friedrichs, K. O.; Wasow, W., Singular perturbations of nonlinear oscillations, Duke Math. J., 13, 367-381 (1946) · Zbl 0061.19509
[36] Friedrichs, K. O., Asymptotic phenomena in mathematical physics, Bull. Am. Math. Soc., 61, 485-504 (1955) · Zbl 0068.16406
[37] (Goldstein, S., Modern Developments in Fluid Dynamics, Vol. 1 (1938), Clarendon Press: Clarendon Press Oxford, England)
[38] Haag, J., Étude asymptotique des oscillations de relaxation, Ann. Sci. École Norm. Sup., 60, 65-111 (1943) · Zbl 0061.19107
[39] Haag, J., Exemples concrets d’étude asymptotique d’oscillations de relaxation, Ann. Sci. École Norm. Sup., 61, 73-117 (1944) · Zbl 0061.19108
[40] Hadamard, J., Lectures on Cauchy’s Problem in Linear Partial Differential Equations (1952), Dover: Dover New York, (original ed. 1923) · Zbl 0049.34805
[41] Handelman, G. H.; Keller, J. B., Small Vibrations of a Slightly Stiff Pendulum, (Proceedings of the 4th U.S. National Congress on Applied Mechanics, Vol. 1 (1962), American Society Mechanical Engineers: American Society Mechanical Engineers New York), 195-202
[42] Handelman, G. H.; Keller, J. B.; O’Malley, R. E., Loss of boundary conditions in the asymptotic solution of linear ordinary differential equations I. Eigenvalue problems, Commun. Pure Appl. Math., 21, 243-261 (1968) · Zbl 0198.42803
[43] Harris, W. A., Singular perturbations of two-point boundary problems for systems of ordinary differential equations, Arch. Ratl. Mech. Anal., 5, 212-225 (1960) · Zbl 0098.05801
[44] Harris, W. A., Singular perturbations of eigenvalue problems, Arch. Ratl. Mech. Anal., 7, 224-241 (1961) · Zbl 0098.05802
[45] Harris, W. A., Singular perturbations of a boundary value problem for a system of differential equations, Duke Math. J., 29, 429-445 (1962) · Zbl 0114.04204
[46] Harris, W. A., Singular perturbations of two-point boundary problems, J. Math. Mech., 11, 371-382 (1962) · Zbl 0121.07202
[47] Harris, W. A., Equivalent classes of singular perturbation problems, Rendi. Circ. Mat. Palermo, 14, 61-75 (1965) · Zbl 0139.04402
[48] Hoppensteadt, F. C., Singular perturbations on the infinite interval, Trans. Am. Math. Soc., 123, 521-535 (1966) · Zbl 0151.12502
[49] Hoppensteadt, F., Stability in Systems with Parameter, J. Math. Anal. Appl., 18, 129-134 (1967) · Zbl 0146.11702
[50] Huet, D., Phénomènes de perturbation singulière dans les problèmes aux limites, Ann. Inst. Fourier, Grenoble, 10, 1-96 (1960) · Zbl 0128.32904
[51] Huet, D., Perturbations singulières relatives au problème de Dirichlet dans un demi-espace, Ann. Scuola Norm. Sup. Pisa, 18, 427-448 (1964) · Zbl 0137.30001
[52] Huet, D., Sur quelques problèmes de perturbation singulière dans les espaces \(L_p\), Rev. Fac. Ci. Lisboa, 11, 137-164 (1965) · Zbl 0161.08801
[53] Huet, D., Remarque sur un théorème d’Agmon et applications à quelques problèmes de perturbation singulière, Boll. Un. Mat. Ital., 21, 219-227 (1966) · Zbl 0145.36802
[54] Kaplun, S., The role of coordinate systems in boundary layer theory, Z. Angew. Math. Phys., 5, 111-135 (1954) · Zbl 0055.19004
[55] Kaplun, S., Low Reynolds number flow past a circular cylinder, J. Math. Mech., 6, 595-603 (1957) · Zbl 0080.18502
[56] Kaplun, S., (Lagerstrom, P. A.; Howard, L. N.; Liu, C. S., Fluid Mechanics and Singular Perturbations (1967), Academic Press: Academic Press New York)
[57] Kaplun, S.; Lagerstrom, P. A., Asymptotic expansions of Navier-Stokes solutions for small Reynolds numbers, J. Math. Mech., 6, 585-593 (1957) · Zbl 0080.18501
[58] Kevorkian, J., The Two-Variable Expansion Procedure for the Approximate Solution of Certain Nonlinear Differential Equations, (Douglas Report SM-42620 (1962), Missile and Space Systems Division, Douglas Aircraft Company: Missile and Space Systems Division, Douglas Aircraft Company Santa Monica, California) · Zbl 0156.16502
[59] Kisyński, J., Sur les équations hyperboliques avec petit paramètre, Colloq. Math., 10, 331-343 (1963) · Zbl 0199.16203
[60] Kohn, J. J.; Nirenberg, L., Noncoercive boundary value problems, Commun. Pure Appl. Math., 18, 443-492 (1965) · Zbl 0125.33302
[61] Lagerstrom, P. A., Note on the preceding two papers, J. Math. Mech., 6, 605-606 (1957) · Zbl 0080.18503
[62] LaSalle, J. P.; Lefschetz, S., Stability by Liapunov’s Direct Method, with Applications (1961), Academic Press: Academic Press New York
[63] Latta, G. E., Singular Perturbation Problems (1951), California Institute of Technology: California Institute of Technology Pasadena, unpublished Doctoral dissertation
[64] Levin, J. J., First order partial differential equations containing a small parameter, J. Rational Mech. Anal., 4, 481-501 (1955) · Zbl 0067.07006
[65] Levinson, N., Perturbations of discontinuous solutions of nonlinear systems of differential equations, (Proc. Natl. Acad. Sci. U.S., 33 (1947)), 214-218
[66] Levinson, N., The first boundary value problem for \(ϵΔu + A(x, y)u_x + B(x, y)u_y + C\)(x, y)u = D(x, y)\) for small ϵ, Ann. Math., 51, 428-445 (1950) · Zbl 0036.06801
[67] Levinson, N., Perturbations of Discontinuous Solutions of Nonlinear Systems of Differential Equations, Acta Math., 82, 71-106 (1951)
[68] Lewis, J. A.; Carrier, G. F., Some remarks on the flat plate boundary layer, Quart. Appl. Math., 7, 228-234 (1949) · Zbl 0035.25602
[69] Liénard, A., Étude des oscillations entretenues, Rev. Gen. Élec., 23, 946-954 (1928) · JFM 54.0806.05
[70] Lions, J. L., Singular Perturbations and some Nonlinear Boundary Value Problems, (Tech. Summary Rept. No. 421 (October 1963), Mathematics Research Center, U.S. Army, University of Wisconsin) · Zbl 0307.35029
[71] Lions, J. L., Sur certaines équations paraboliques non linéaires, Bull. Soc. Math. France, 93, 155-175 (1965) · Zbl 0132.10601
[72] Macki, J. W., Singular perturbations of a boundary value problem for a system of nonlinear ordinary differential equations, Arch. Ratl. Mech. Anal., 24, 219-232 (1967) · Zbl 0146.11601
[73] Miller, J. C.P, The Airy Integral (1946), Cambridge University Press: Cambridge University Press Cambridge, England, [British Association for Advancement of Science, Mathematical Tables, Part-Volume B] · Zbl 0061.30506
[74] Miranker, W. L., Singular perturbation eigenvalues by a method of undetermined coefficients, J. Math. Phys., 42, 47-58 (1963) · Zbl 0111.28204
[75] Am. Math. Soc. Transl., 18, 199-230 (1961), Ser. 2 · Zbl 0098.29002
[76] Mishchenko, E. F.; Pontryagin, L. S., Differential equations with a small parameter attached to the highest derivatives and some problems in the theory of oscillation, IRE Trans. Circuit Theory, 7, 527-535 (1960)
[77] Moser, J., Singular perturbation of eigenvalue problems for linear differential equations of even order, Commun. Pure Appl. Math., 8, 251-278 (1955) · Zbl 0064.33301
[78] Murphy, W. D., Numerical Analysis of Boundary Layer Problems (1966), Courant Institute of Mathematical Sciences, AEC Computing and Applied Mathematics Center, TID-4500, NYO-1480-63
[79] Noaillon, P., Développements asymptotiques dans les équations différentielles linéaires à paramètre variable, Mém. Soc. Sci. Liége, Ser. 3, 9, 197 (1912) · JFM 43.0375.01
[80] Oleinik, O. A., On equations of elliptic type with a small parameter in the highest derivatives, Mat. Sb., 31, 104-117 (1952) · Zbl 0049.07601
[81] O’Malley, R. E., Two-Parameter Singular Perturbation Problems (1965), Stanford University: Stanford University Stanford, California, unpublished Doctoral dissertation
[82] O’Malley, R. E., Two-parameter singular perturbation problems for second order equations, J. Math. Mech., 16, 1143-1164 (1967) · Zbl 0173.11102
[83] O’Malley, R. E., Singular perturbations of boundary value problems for linear ordinary differential equations involving two parameters, J. Math. Anal. Appl., 19, 291-308 (1967) · Zbl 0166.08001
[84] O’Malley, R. E., The first boundary value problem for certain linear elliptic differential equations involving two small parameters, Arch. Ratl. Mech. Anal., 26, 68-82 (1967) · Zbl 0157.18304
[85] O’Malley, R. E., A boundary value problem for certain nonlinear second order differential equations with a small parameter, Arch. Rational Mech. Anal., 29, 66-74 (1968) · Zbl 0157.14202
[86] R. E. O’Malley, Jr.; R. E. O’Malley, Jr.
[87] R. E. O’Malley, Jr.Arch. Rational Mech. Anal.; R. E. O’Malley, Jr.Arch. Rational Mech. Anal.
[88] O’Malley, R. E.; Keller, J. B., Loss of boundary conditions in the asymptotic solution of linear ordinary differential equations, II, boundary value problems Commun, Pure Appl. Math., 21, 263-270 (1968) · Zbl 0198.42804
[89] Am. Math. Soc. Transl., 18, 295-319 (1961), Ser. 2 · Zbl 0098.29003
[90] Ponzo, P. J.; Wax, N., On certain relaxation oscillations: asymptotic solutions, J. Soc. Indust. Appl. Math., 13, 740-766 (1965) · Zbl 0136.08201
[91] Prandtl, L., Über Flüssigkeits-bewegung bei kleiner Reibung, Verhandlungen des III. Internationalen Mathematiker-Kongresses, ((1905), Tuebner: Tuebner Leipzig), 484-491
[92] Proudman, I.; Pearson, J. R.A, Expansions at small reynolds numbers for the flow past a sphere and a circular cylinder, J. Fluid Mech., 2, 237-262 (1957) · Zbl 0077.39103
[93] Rayleigh, Lord; Stutt, J. W., (The Theory of Sound, Vol. 1 (1945), Dover: Dover New York), (original ed. 1894)
[94] Rellich, F., Störungstheorie der Spectralzerlegung, Article in five parts appearing, Math. Ann., 118, 462-484 (1937-1942), Part V · JFM 68.0243.02
[95] Segel, L. A., The importance of asymptotic analysis in applied mathematics, Am. Math. Monthly, 73, 7-14 (1966) · Zbl 0134.13103
[96] Sibuya, Y., Sur réduction analytique d’un système d’équations différentielles ordinaires linéaires contenant un paramètre, J. Fac. Sci. Univ. Tokyo, Sec. I, 7, 527-540 (1958) · Zbl 0081.08103
[97] Smoller, J. A., Singular perturbations of Cauchy’s problem, Commun. Pure Appl. Math., 18, 665-677 (1965) · Zbl 0151.20302
[98] Stoker, J. J., Nonlinear Vibrations in Mechanical and Electrical Systems (1950), Interscience: Interscience New York · Zbl 0035.39603
[99] Tikhonov, A. N., Systems of differential equations containing small parameters in the derivatives, Mat. Sb., 31, 575-586 (1952)
[100] Tricomi, F. G., Integral Equations (1957), Interscience: Interscience New York · Zbl 0078.09404
[101] Turrittin, H. L., Asymptotic solutions of certain ordinary differential equations associated with multiple roots of the characteristic equation, Am. J. Math., 58, 364-378 (1936) · Zbl 0013.40005
[102] Turrittin, H. L., Asymptotic expansions of solutions of systems of ordinary linear differential equations containing a parameter, Contrib. Theory Nonlinear Oscillations, 2, 81-116 (1952) · Zbl 0047.08602
[103] Urabe, M., Periodic solutions of van der Pol’s equation with damping coefficient λ = 0 ∼ 10, IRE, Trans. Circuit Theory, 7, 382-386 (1960)
[104] Urabe, M., Numerical Study of Periodic Solutions of the van der Pol Equation, (International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics (1963), Academic Press: Academic Press New York), 184-192
[105] Van Dyke, M., Perturbation Methods in Fluid Dynamics (1964), Academic Press: Academic Press New York · Zbl 0136.45001
[106] Russian Math. Surveys, 18, 13-84 (1963) · Zbl 0135.14001
[107] Am. Math. Soc. Transl., Ser. 2, 20, 239-364 (1961) · Zbl 0122.32402
[108] Russian Math. Survey, 15, 1-73 (1960) · Zbl 0096.08702
[109] Wasow, W., On the asymptotic solution of boundary value problems for ordinary differential equations containing a parameter, J. Math. Phys., 23, 173-183 (1944) · Zbl 0061.18202
[110] Wasow, W., Asymptotic solution of boundary value problems for the differential equation \(ΔU + λ(∂U∂x) = λf(x, y)\), Duke Math. J., 11, 405-415 (1944) · Zbl 0061.23708
[111] Wasow, W., On the construction of periodic solutions of singular perturbation problems, Contrib. Theory Nonlinear Oscillations, 1, 313-350 (1950)
[112] Wasow, W., Singular Perturbation Methods for Nonlinear Oscillations, (Proceedings of the Symposium on Nonlinear Circuit Analysis (1953), Edwards Brothers: Edwards Brothers Ann Arbor, Michigan), 75-98
[113] Wasow, W., Singular perturbation of boundary value problems for nonlinear differential equations of the second order, Commun. Pure Appl. Math., 9, 93-113 (1956) · Zbl 0074.30502
[114] Wasow, W., Asymptotic Expansions for Ordinary Differential Equations: Trends and Problems, (Asymptotic Solutions of Differential Equations and Their Applications (1964), Wiley: Wiley New York), 3-26 · Zbl 0136.08101
[115] Wasow, W., Asymptotic Decomposition of Systems of Linear Differential Equations Having Poles with respect to a Parameter, Rend. Circ. Mat. Palermo, 13, 329-344 (1964) · Zbl 0139.04401
[116] Wasow, W., Asymptotic Expansions for Ordinary Differential Equations (1965), Interscience: Interscience New York · Zbl 0169.10903
[117] Wendell, J. G., Singular perturbations of a van der Pol equation, Contrib. Theory Nonlinear Oscillations, 1, 243-290 (1950)
[118] Willett, D., On a nonlinear boundary value problem with a small parameter multiplying the highest derivative, Arch. Ratl. Mech. Anal., 23, 276-287 (1966) · Zbl 0152.28402
[119] Zlámal, M., The parabolic equation as a limiting case of a certain elliptic equation, Ann. Mat. Pure Appl., 4, 143-150 (1962) · Zbl 0143.14002
[120] Zlámal, M., On a Singular Perturbation Problem Concerning Hyperbolic Equations, (Lecture Series No. 45 (1964), Institute for Fluid Dynamics and Applied Mathematics, University of Maryland: Institute for Fluid Dynamics and Applied Mathematics, University of Maryland College Park, Maryland) · Zbl 0161.30303
[121] Bellman, R.; Cooke, K. L., On the limit of solutions of differential-difference equations as the retardation approaches zero, (Proc. Nat. Acad. Sci. U.S., 45 (1959)), 1026-1028 · Zbl 0088.29803
[122] Bobisud, L., On the behavior of the solution of the telegraphist’s equation for large velocities, Pacific J. Math., 22, 213-219 (1967) · Zbl 0171.07204
[123] Carrier, G. F.; Pearson, C. E., Ordinary Differential Equations (1968), Ginn-Blaisdell: Ginn-Blaisdell Waltham · Zbl 0165.40601
[124] Chang, K. W., Almost periodic solutions of singularly perturbed systems of differential equations, J. Diff. Eq., 4, 300-307 (1968) · Zbl 0157.15303
[125] Cole, J. D., Perturbation Methods in Applied Mathematics (1968), Ginn-Blaisdell: Ginn-Blaisdell Waltham · Zbl 0162.12602
[126] Comstock, C., Boundary layers for almost characteristic boundaries, J. Math. Anal. Appl., 22, 54-61 (1968) · Zbl 0159.40102
[127] Cooke, K. L., The condition of regular degeneration for singularly perturbed differential-difference equations, J. Diff. Eq., 1, 39-94 (1965) · Zbl 0151.10303
[128] Cooke, K. L.; Meyer, K. R., The condition of regular degeneration of singularly perturbed systems of linear differential-difference equations, J. Math. Anal. Appl., 14, 83-106 (1966) · Zbl 0142.05802
[129] Eckhaus, W., On the Foundations of the Method of Matched Asymptotic Approximations (July 1968), Mathematisch Instituut der Technische Hogeschool Delft: Mathematisch Instituut der Technische Hogeschool Delft Nederland
[130] Erdélyi, A., Approximate solutions of a nonlinear boundary value problem, Arch. Rational Mech. Anal., 29, 1-17 (1968) · Zbl 0155.13203
[131] Fife, P. C., Nonlinear deflection of thin elastic plates under tension, Commun. Pure Appl. Math., 14, 81-112 (1961) · Zbl 0099.40802
[132] Fife, P. C., Considerations regarding the mathematical basis for Prandtl’s boundary layer theory, Arch. Rational Mech. Anal., 28, 184-216 (1968) · Zbl 0172.53801
[133] Flatto, L.; Levinson, N., Periodic solutions of singularly perturbed systems, J. Rational Mech. Anal., 4, 943-950 (1955) · Zbl 0066.07302
[134] Fowkes, N. D., A singular perturbation method, Quart. Appl. Math., 26, 71-85 (1968) · Zbl 0162.16203
[135] Friedman, A., Singular perturbations for partial differential equations, Arch. Rational Mech. Anal., 29, 289-303 (1968) · Zbl 0164.41701
[136] Greenlee, W. M., Rate of Convergence in Singular Perturbations, University of Kansas, Department of Mathematics, Technical Report 12 (June 1967)
[137] Haber, S.; Levinson, N., A boundary value problem for a singularly perturbed differential equation, (Proc. Amer. Math. Soc., 6 (1955)), 866-872 · Zbl 0066.06503
[138] Hale, J. K.; Seifert, G., Bounded and almost periodic solutions of singularly perturbed equations, J. Math. Anal. Appl., 3, 18-24 (1961) · Zbl 0099.29401
[139] Heineken, F. G.; Tsuchiya, H. M.; Aris, R., On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics, Math. Biosciences, 1, 95-113 (1967)
[140] F. Hoppensteadt; F. Hoppensteadt · Zbl 0167.08204
[141] F. Hoppensteadt; F. Hoppensteadt · Zbl 0264.35044
[142] Hurd, C. C., Asymptotic theory of linear differential equations containing two parameters, Tohoku Math. J., 45, 58-68 (1939) · Zbl 0021.02701
[143] Iwano, M., Asymptotic solutions of Whittaker’s equation as the moduli of the independent variable and two parameters tend to infinity, Japanese J. Math., 33, 1-92 (1963) · Zbl 0142.03704
[144] Iwano, M.; Sibuya, Y., Reduction of order of a linear ordinary differential equation containing a small parameter, Kodai Math. Sem. Rep., 15, 1-28 (1963) · Zbl 0115.07001
[145] Kazarinoff, W. D., Asymptotic theory of second order differential equations with two simple turning points, Arch. Rational Mech. Anal., 2, 129-150 (1958) · Zbl 0082.07102
[146] Keller, J. B., Perturbation Theory, (lecture notes (1968), Michigan State University: Michigan State University East Lansing) · Zbl 0169.12702
[147] Knowles, J. K., The Dirichlet problem for a thin rectangle, (Proc. Edinburgh Math. Soc., 15 (1967)), 315-320 · Zbl 0164.13901
[148] Knowles, J. K.; Messick, R. E., On a class of singular perturbation problems, J. Math. Anal. Appl., 9, 42-58 (1964) · Zbl 0138.35802
[149] Levinson, N., A boundary value problem for a singularly perturbed differential equation, Duke Math. J., 25, 331-342 (1958) · Zbl 0173.35301
[150] Lin, C. C.; Rabinstein, A. L., On the asymptotic solutions of a class of ordinary differential equations of the fourth order, Trans. Amer. Math. Soc., 94, 24-57 (1960) · Zbl 0092.07804
[151] MacMillan, D. B., Asymptotic methods for systems of differential equations in which some variables have very short response times, SIAM J. Appl. Math., 16, 704-722 (1968) · Zbl 0176.14603
[152] Markus, L.; Amundson, N. R., Nonlinear boundary value problems arising in chemical reaction theory, J. Diff. Eq., 4, 102-113 (1968) · Zbl 0155.13301
[153] McKelvey, R. W., The solution of second order linear ordinary differential equations about a turning point of order two, Trans. Amer. Math. Soc., 79, 103-123 (1955) · Zbl 0065.31801
[154] Meyer, R. E., On the approximation of double limits by single limits and the Kaplun extension theorem, J. Inst. Math. Appl., 3, 245-249 (1967) · Zbl 0162.07703
[155] Oleinik, O. A., On the second boundary-value problem for elliptic equations with a small parameter before the highest derivative, Dokl. Akad. Nauk SSSR, 79, 735-737 (1951)
[156] R. E. O’MalleyJ. Math. Mech.; R. E. O’MalleyJ. Math. Mech.
[157] O’Malley, R. E., On a Boundary Value Problem for a Nonlinear Differential Equation with a Small Parameter (December 1967), Mathematics Research Center, U. S. Army, The University of Wisconsin, TSR 831
[158] O’Malley, R. E., On Singular Perturbation Problems with Interior Non-uniformities (April 1968), Mathematics Research Center, U. S. Army, The University of Wisconsin, TSR 864
[159] O’Malley, R. E., On the Asymptotic Solution of Boundary Value Problems for Nonhomogeneous Ordinary Differential Equations Containing a Parameter (May 1968), Mathematics Research Center, U. S. Army, The University of Wisconsin, TSR 872
[160] O’Malley, R. E., On the Asymptotic Solution of Multi-Point Boundary Value Problems (July 1968), Mathematics Research Center, U. S. Army, The University of Wisconsin, TSR 885
[161] S. V. Parter; S. V. Parter
[162] Payne, L. E.; Sather, D., On singular perturbation in non-well posed problems, Ann. Mat. Pura Appl., 75, 4, 219-230 (1967) · Zbl 0146.34503
[163] Perko, L. M., A method of error estimation in singular perturbation problems with application to the restricted three body problem, SIAM J. Appl. Math., 15, 738-753 (1967) · Zbl 0155.51702
[164] Ponzo, P. J., Forced oscillations of the generalized Liénard equation, SIAM J. Appl. Math., 15, 75-87 (1967) · Zbl 0146.32804
[165] Rang, E. R., Periodic Solutions of Singular Perturbation Problems, (Nonlinear Differential Equations and Nonlinear Mechanics (1963), Academic Press: Academic Press New York), 377-383 · Zbl 0132.32003
[166] Sibuya, Y., Asymptotic solutions of a system of linear ordinary differential equations containing a parameter, Funkcial. Ekvac., 4, 83-113 (1962) · Zbl 0123.04902
[167] Sibuya, Y., On the Convergence of Formal Solutions of Linear Ordinary Differential Equations Containing a Parameter (September 1964), Mathematics Research Center, U. S. Army, The University of Wisconsin, TSR 511
[168] Steele, C. R., On the asymptotic solution of nonhomogeneous ordinary differential equations with a large parameter, Quart. Appl. Math., 23, 193-201 (1965) · Zbl 0132.31602
[169] Stenger, F., Error bounds for asymptotic solutions of differential equations, J. Res. Natl. Bur. Stds., 70B, 167-210 (1966)
[170] G. Stengle; G. Stengle
[171] Sugiyama, S., Continuity properties of the retardation in the theory of difference-differential equations, (Proc. Japan Acad., 37 (1961)), 179-182 · Zbl 0102.30001
[172] Takahashi, K., Über eine Asymptotische Darstellung der Lösung eines Systems von Differentialgleichungen welche von zwei Parameter abhängen…, Tôhoku Math. J., 13, 1-17 (1961) · Zbl 0101.30201
[173] Ting, L.; Chen, S., Perturbation solutions and asymptotic solutions in boundary layer theory, J. Eng. Math., 1, 327-340 (1967) · Zbl 0173.53104
[174] Vasil’eva, A. B.; Imanaliev, M., The asymptotic form of the solution to the Cauchy problem for an integro-differential equation with a small parameter, Sibirskii Mat. Z., 7, 61-69 (1966)
[175] Russian Math. Surveys, 22, No. 2, 124-142 (1967) · Zbl 0158.24401
[176] Vasil’eva, A. B.; Tupciev, V. A., Periodic nearly-discontinuous solutions of systems of differential equations with a small parameter in the derivatives, Soviet Math. Dokl., 9, No. 1, 179-183 (1968) · Zbl 0172.12003
[177] Visik, M. I.; Lyusternik, L. A., On the asymptotic behavior of the solutions of boundary problems for quasi-linear differential equations, Dokl. Akad. Nauk SSSR, 121, 778-781 (1958) · Zbl 0094.06203
[178] Visik, M. I.; Lyusternik, L. A., Initial jump for nonlinear differential equations containing a small parameter, Dokl. Akad. Nauk SSSR, 132, 1242-1245 (1960)
[179] Wasow, W., On Boundary Layer Problems in the Theory of Ordinary Differential Equations (December 1941), New York University, unpublished thesis
[180] Wasow, W., Singular perturbation problems of systems of two ordinary analytic differential equations, Arch. Rational Mech. Anal., 14, 61-80 (1963) · Zbl 0113.29801
[181] Wasow, W., Asymptotic simplification of self-adjoint differential equations with a parameter, J. Diff. Eq., 2, 378-390 (1966) · Zbl 0148.06503
[182] Wasow, W., On turning point problems for systems with almost diagonal coefficient matrix, Funkcial. Ekvac., 8, 143-171 (1966) · Zbl 0151.12602
[183] W. Wasow; W. Wasow · Zbl 0174.39601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.