Abstract
In this chapter, the methods developed previously are applied to partial differential equations. The plan is the same as for the cases of ordinary differential equations discussed earlier. First, the very simplest case is discussed, in which a singular perturbation problem arises. This is a second-order equation which becomes a first-order one in the limit ε → 0. Following this, various more complicated physical examples of singular perturbations and boundary-layer theory are discussed. Next, the ideas of matching and inner and outer expansions are applied in some problems that are analogous to the singular boundary problems of Section 2.7. The final section deals with multiple variable expansions for partial differential equations, and several applications dealing with different aspects of the procedure are discussed.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1, Interscience Publishers, Inc., New York, 1953.
Cole, J. D., On a quasilinear parabolic equation occurring in aerodynamics, Q. Appl. Math. 9, 1951, pp. 225–236.
Hopf, E., The partial differential equation u t + uu_, = µu_ß-_r, Comm. Pure Appl. Math. 3, 1950, pp. 201–230.
Lax, P. D., Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, S.I.A.M. Regional Conference Series in Applied Mathematics, Vol. 11, 1973.
Whitham, G. B., Linear and Nonlinear Waves, John Wiley and Sons, New York, 1974.
S. Kaplun, The role of coordinate systems in boundary layer theory, Z.A.M.P. V, 2 (1954) 111–135.
P. A. Lagerstrom, Laminar flow theory, High Speed Aerodynamics and Jet Propulsion, IV, Princeton Univ. Press, 1964, 20–282.
H. Weyl, On the simplest differential equations of boundary layer theory. Annals of Math. 43, 2, 381–407.
P. A. Lagerstrom, and J. D. Cole, Examples illustrating expansion procedures for the Navier-Stokes equations, Journal of Rational Mechanics and Analysis 4, 6 (Nov. 1955), 817–882.
M. J. Lighthill, Mathematical Biofluiddynamics, Society for Industrial and Applied Mathematics, Philadelphia, Penn. 19103, 1975.
S. Kaplun, Low Reynolds number flow past a circular cylinder, Journal of Math. und Mech. 6, 5 (1957), 595–603.
S. Kaplun and P. A. Lagerstrom, Asymptotic expansions of Navier—Stokes solutions for small Reynolds numbers, Journal of Math. and Mech. 6, 5 (1957), 515–593.
P. A. Lagerstrom, Note on the Preceding Two Papers, Journal of Math. and Mech. 6, 5 (1957), 605–606.
I. Proudman and J. R. A. Pearson, Expansions at small Reynolds number for the flow past a sphere and a circular cylinder, Journal of Fluid Mechanics 2, Part 3 (1957).
A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity. 4th ed., England: Cambridge University Press, 1927.
V. Z. Vlasov, Allgemeine Schalentheorie und ihre Anwendung in der Technik (translated from Russian), Berlin: Akademie Verlag, 1958.
S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells. 2nd ed., New York: McGraw-Hill Book Co., 1959.
A. Peskoff, R. S. Eisenberg, and J. D. Cole, Potential Induced by a Point Source of Current in the Interior of a Spherical Cell, University of California at Los Angeles, Rept. U.C.L.A.-ENG-7259, December 1972.
A. Peskoff and R. S. Eisenberg, The time-dependent potential in a spherical cell using matched asymptotic expansions, J. Math. Bio. 2, 1975, pp. 277–300.
A. Peskoff, R. S. Eisenberg, and J. D. Cole. Matched asymptotic expansions of the Green’s function for the electric potential in an infinite cylindrical cell, S.I.A.M. Jnl. Appl. Math. 30, No. 2. March 1976, pp. 222–239.
A. Peskoff, Green’s function for Laplace’s equation in an infinite cylindrical cell, J. Math. Phys. 15 (1974), pp. 2112–2120.
R. D. Taylor, Cable theory, Physical Techniques in Biological Research. W. L. Nastuk ed. Vol. VIB. Academic Press, N.Y. 1963, pp. 219–262.
V. Barcilon, J. D. Cole, and R. S. Eisenberg. A singular perturbation analysis of induced electric fields in cells, S.I.A.M. J. Appl. Math. 21 (1971), pp. 339–353.
C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Method for solving the Korteweg-de Vries equation, Physics Review Letters 19, 1967, pp. 1095–1097.
G. B. Whitham, Linear and Nonlinear Waves, John Wiley and Sons, New York, 1974.
G. F. Carrier, M. Krook, and C. E. Pearson, Functions of a Complex Variable, Theory and Technique, McGraw-Hill Book Company, New York, 1966.
4] S. C. Chikwendu, Nonlinear wave propagation solution by Fourier transform perturbation, International Journal of Nonlinear Mechanics,to appear.
G. B. Whitham, A general approach to linear and nonlinear dispersive waves using a Lagrangian, Journal of Fluid Mechanics 22, 1965, pp. 273–283.
J. C. Luke, A perturbation method for nonlinear dispersive wave problems, Proceedings of the Royal Society, Series A, 292, 1966, pp. 403–412.
J. B. Keller and S. Kogelman, Asymptotic solutions of initial value problems for nonlinear partial differential equations, S.I.A.M. Journal on Applied Mathematics 18, 1970, pp. 748–758.
S. C. Chikwendu and J. Kevorkian, A perturbation method for hyperbolic equations with small nonlinearities, S.I A.M. Journal on Applied Mathematics 22, 1972, pp. 235–258.
W. Eckhaus, New approach to the asymptotic theory of nonlinear oscillations and wave-propagation, Journal of Mathematical Analysis and Applications 49, 1975, pp. 575–611.
R. W. Lardner, Asymptotic solutions of nonlinear wave equations using the methods of averaging and two-timing, Quarterly of Applied Mathematics 35, 1977, pp. 225–238.
S. C. Chikwendu, Asymptotic solutions of some weakly nonlinear elliptic equations, S.IA.M. Journal on Applied Mathematics 31, 1976, pp. 286–303.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1981 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Kevorkian, J., Cole, J.D. (1981). Applications to Partial Differential Equations. In: Perturbation Methods in Applied Mathematics. Applied Mathematical Sciences, vol 34. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-4213-8_4
Download citation
DOI: https://doi.org/10.1007/978-1-4757-4213-8_4
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-2812-2
Online ISBN: 978-1-4757-4213-8
eBook Packages: Springer Book Archive