Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-05T11:17:33.779Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact solution to a class of functional difference equations with application to a moving contact line flow

Published online by Cambridge University Press:  26 September 2008

J. B. Lawrie  and
A. C. King
J. B. Lawrie
Affiliation:
Department of Mathematics and Statistics, Brunel University, Uxbridge, Middlesex UB8 3PH, UK
A. C. King
Affiliation:
Department of Mathematics, Keele University, Keele, Staffs ST5 5BG, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

A new integral representation for the Barnes double gamma function is derived. This is canonical in the sense that solutions to a class of functional difference equations of first order with trigonometrical coefficients can be expressed in terms of the Barnes function. The integral representation given here makes these solutions very simple to compute. Several well-known difference equations are solved by this method, and a treatment of the linear theory for moving contact line flow in an inviscid fluid wedge is given.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 5 , Issue 2 , June 1994 , pp. 141 - 157
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramowitz, M. & Stegun, I. A. 1972 Handbook of Mathematical Functions. Dover.Google Scholar
Barnes, E. W. 1899 The genesis of the double gamma functions. Proc. Lond. Math. Soc. 31, 358–381.CrossRefGoogle Scholar
Brillouet, G. 1957 Publ. Sci. Tech. Ministeair, Paris, no. 329.Google Scholar
Ehrenmark, U. T. 1987 Far field asymptotics of the two-dimensional linearised sloping beach problem. SIAM J. Appl. Math. 47 (5), 965–981.CrossRefGoogle Scholar
Evans, D. V. 1984 The solution of a class of boundary-value problems with smoothly varying boundary conditions. Q. J. Mech. Appl. Math. 34 (4), 521–536.Google Scholar
Faulkner, T. R. 1965 Diffraction by a perfectly conducting wedge in an anisotropic plasma. Proc. Camb. Phil. Soc. 61, 767–776.CrossRefGoogle Scholar
Hongo, K. 1980 Polynomial approximation of Maliuzhinets' function. IEEE Trans. Ant. Prop. 34, 942–947.CrossRefGoogle Scholar
Isaccson, E. 1950 Water waves over a sloping bottom. Comm. Pure Appl. Math. 3, 11–31.CrossRefGoogle Scholar
Keller, J. B. & Miksis, M. J. 1983 Surface tension driven flows. SIAM J. Appl. Math. 43, 268–277.CrossRefGoogle Scholar
King, A. C. 1991 Moving contact lines in slender fluid wedges. Q. J. Appl. Math. 44 (2), 173–192.CrossRefGoogle Scholar
Koiter, W. T. 1955 On the diffusion of load from a stiffner into a sheet. Q. J. Mech. Appl. Math. 8(2), 164–178.CrossRefGoogle Scholar
Lauwerier, H. A. 1961 Solutions of the equation of Helmholtz in an angle. IV. Proc. Ned. Akacl. Wetensch. A64 (23), 348–359.CrossRefGoogle Scholar
Lawrie, J. B. 1990 Surface tension driven flow in a wedge. Q. J. Mech. Appl. Math. 43 (2), 251–273.CrossRefGoogle Scholar
Lewy, H. 1946 Water waves on sloping beaches. Bull. Amer. Math. Soc. 52, 737–775.CrossRefGoogle Scholar
Lipszyc, K. 1975 One-dimensional model of the rearrangement and dissociation processes and the Faddeev equations. II Phys. Rev. D11 (6), 1649–1661.Google Scholar
Lipszyc, K. 1980 On the application of the Sommerfeld-Maluzhinetz transformation to some one-dimensional three-particle problems. J. Math. Phys. 21 (5), 1092–1102.CrossRefGoogle Scholar
Maliuzhnets, G. D. 1958 Excitation, reflection and emission of surface waves from a wedge with given face impedances. Sov. Phys. Dokl. 3, 752–755.Google Scholar
Milne-Thomson, L. M. 1933 The Calculus of Finite Differences. Macmillan.Google Scholar
Noble, B. 1958 Methods Based on the Wiener-Hopf Technique. Pergamon.Google Scholar
Packham, B. A. 1989 A note on generalized edge waves on a sloping beach. Q. J. Mech. Appl. Math. 42 (3), 441–446.CrossRefGoogle Scholar
Peters, A. S. 1952 Water waves over sloping beaches and the solution of a mixed boundary value problem for δ2φ−k2φ = 0 in a sector. Comm. Pure Appl. Math. 5, 87–108.CrossRefGoogle Scholar
Roseau, M. 1958 Short waves parallel to the shore over a sloping beach. Comm. Pure Appl. Math. 11, 433–493.CrossRefGoogle Scholar
Roseau, M. 1976 Asymptotic Wave Theory. North-Holland.Google Scholar
Tayler, A. B. 1972 Singularities at flow separation. Quart. J. Mech. Appl. Math. 26, 153–164.CrossRefGoogle Scholar
Titchmarsh, E. C. 1948 Introduction to the Theory of Fourier Integrals, 2nd edition. Oxford University Press.Google Scholar
Tuzhilin, A. A. 1971 The theory of Malyuzhinets functional equations IV. Differential 'nye Uraveneniya 7, 968–976. (Translation.)Google Scholar
Van Dantzig, D. 1958 Solutions of the equation of Helmholtz in an angle with vanishing directional derivatives along each side. Kon. Ned. Akad. v. Wet. Proc. A61(4), 384–398.Google Scholar
Van Lennep, A. G. R. 1973 The Kernel of Sommerfield 's transform as solution to difference equations for a class of diffraction problems. J. Appl. Phys., 45, 4401–4405.CrossRefGoogle Scholar
Varley, E. & Walker, J. D. A. 1989 A method for solving Integrodifferential Equations. I.M.A. J. Appl. Maths. 43, 11–46.Google Scholar
Whittaker, E. T. & Watson, G. N. 1927 A Course of Modern Analysis, 4th EditionCambridge University Press.Google Scholar
Williams, W. E. 1959 Diffraction of an E-polarized plane wave by an imperfectly conducting wedge. Proc. R. Soc. Lond. A252, 376–393.Google Scholar