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Convergence of biorthogonal series of biharmonic eigenfunctions by the method of Titchmarsh. (English) Zbl 0486.35061


MSC:

35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
35J30 Higher-order elliptic equations
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[1] Benthem, J. P., A Laplace transform method for the solution of semi-infinite and finite strip problems in stress analysis. Quart. J. Mech. Appl. Math., 16, 413-429 (1963). · Zbl 0127.39904 · doi:10.1093/qjmam/16.4.413
[2] Buchwald, V. T., & Doran, H. E., Eigenfunctions of plane elastostatics. Proc. Roy. Soc., A, 284, 69-82 (1965). · doi:10.1098/rspa.1965.0052
[3] Dean, W. R., & Montagnon, P. E., On the steady motion of viscous liquid in a corner. Proc. Camb. Phil. Soc., 45, 389 (1949). · Zbl 0034.26802 · doi:10.1017/S0305004100025019
[4] de Socio, L., Slow M. H. D. flow in a rectangular cavity. Int. J. Eng. Sci., 17, 1237-1243 (1979). · Zbl 0428.76092 · doi:10.1016/0020-7225(79)90109-5
[5] de Socio, L. M., & Misici, L., Free convection in a trench with radiative wall conditions, ZAMM (in press). · Zbl 0447.76067
[6] de Socio, L., Convection driven by non-uniform surface tension. Letters in Heat and Mass Transfer (in press).
[7] de Socio, L., Misici, L., & Polzonetti, A., Natural convection in heat generating fluids in cavities. ASME paper, 18th Nat. Heat Transfer Conf., S. Diego (1979).
[8] Dixit, P., & Joseph, D. D., The shape of stress free surfaces on a sheared block (to be published in SIAM J. of Appl. Math.) (1982). · Zbl 0516.73040
[9] Dixit, P., Narain, A., & Joseph, D.D., Free surface problems induced by motions perturbing the natural state of simple solids, Arch. Rational Mech. Anal. 77, 199-261 (1981). · Zbl 0483.73001 · doi:10.1007/BF00279878
[10] Dussan, E. B., Immiscible displacement in a capillary tube: the moving contact angle. AIChE J., 23, 131 (1977). · doi:10.1002/aic.690230122
[11] Fadle, J., Die Selbstspannungs-Eigenwertfunktionen der quadratischen Scheibe. Ingenieur-Archiv, 11, 125 (1941). · JFM 66.1051.02 · doi:10.1007/BF02084699
[12] Gregory, R. D., Green’s functions, bi-linear forms, and completeness of the eigenfunctions for the elastostatic strip and wedge. J. Elasticity, 9, 1-27 (1979). · Zbl 0413.73026 · doi:10.1007/BF00041100
[13] Gregory, R. D., The semi-infinite strip x ?0, -1 ? y ? 1; completeness of the Papkovich-Fadle eigenfunctions when ?xx(0,y), ?yy(0,y) are prescribed. J. Elasticity, 10, 57-80 (1980A). · Zbl 0473.42022 · doi:10.1007/BF00043135
[14] Gregory, R. D., The traction boundary value problem for the elastostatic semiinfinite strip, existence of solution, and completeness of the Papkovich-Fadle eigenfunctions. J. Elasticity, 10, 295-307 (1980B). · Zbl 0473.73010 · doi:10.1007/BF00127452
[15] Hewitt, E., & Hewitt, R. E., The Gibbs-Wilbraham phenomenon: An episode in Fourier analysis. Arch. History of Exact Sciences, 21, 129-160 (1979). · Zbl 0424.42002 · doi:10.1007/BF00330404
[16] Johnson, M. W., & Little, R.W., The semi-infinite elastic strip. Quart. Appl. Math., 22, 335-344 (1965).
[17] Joseph, D. D., Slow motion and viscometric motion. Stability and bifurcation of the rest state of a simple fluid. Arch. Rational Mech. Anal., 56, 99 (1974). · Zbl 0295.76035 · doi:10.1007/BF00248137
[18] Joseph, D. D., The convergence of biorthogonal series for biharmonic and Stokes flow edge problems. Part I, SIAM J. Appl. Math., 33, 337-347 (1977). · Zbl 0373.31002 · doi:10.1137/0133021
[19] Joseph, D. D., A new separation of variables theory for problems of Stokes flow and elasticity, Proceedings of the Symposium ?Trends in Applications of Pure Mathematics to Mechanics? held in Kozubnik, Poland, in September 1977. Pitman, London (1979).
[20] Joseph, D. D., & Fosdick, R., The free surface on a liquid between cylinders rotating at different speeds. Part I, Arch. Rational Mech. Anal., 49, 321-380 (1973). · Zbl 0265.76113
[21] Joseph, D. D., & Sturges, L., The free surface on a liquid filling a trench heated from its side. J. Fluid Mech., 69, 565 (1975). · Zbl 0325.76130 · doi:10.1017/S0022112075001565
[22] Joseph, D. D., & Sturges, L., The convergence of biorthogonal series for biharmonic and Stokes flow edge problems. Part 2, SIAM J. Appl. Math., 34, 7-26 (1978). · Zbl 0379.31001 · doi:10.1137/0134002
[23] Liu, C. H., & Joseph, D. D., Stokes flow in wedge-shaped trenches. J. Fluid Mech., 80, 443 (1977). · Zbl 0355.76027 · doi:10.1017/S0022112077001785
[24] Liu, C. H., & Joseph, D. D., Stokes flow in conical cavities. SIAM J. Appl. Math., 34, 286-296 (1978). · Zbl 0376.76018 · doi:10.1137/0134023
[25] Lugt, H. J., & Schwiderski, E. W., Flows around dihedral angles I. Eigenmotion analysis. Proc. Roy. Soc. A, 285, 382-389 (1965). · Zbl 0133.20701 · doi:10.1098/rspa.1965.0111
[26] May, P. J. D., Some mixed boundary value problems in elasticity and Stokes flow. Master thesis, Dept. of Engineering Science, Oxford Univ. (1980).
[27] Moffatt, H. K., Viscous and resistive eddies near a sharp corner. J. Fluid Mech., 18, 1-18 (1964). · Zbl 0118.20501 · doi:10.1017/S0022112064000015
[28] Papkovich, P. V., Über eine Form der Lösung des biharmonischen Problems für das Rechteck, Comptes Rendus (Doklady) de l’Académie des Sciences de l’U.R.S.S., 27, 335-338 (1940).
[29] Richardson, S., A ?stick-slip? problem related to the motion of free jet at low Reynolds Number. Proc. Camb. Phil. Soc., 67, 477-489 (1970). · Zbl 0198.30202 · doi:10.1017/S0305004100045758
[30] Sanders, J., O’Brien, V., & Joseph, D. D., Stokes flow in a driven sector by two different methods. J. Appl. Mech., 47, 482-484 (1980). · doi:10.1115/1.3153717
[31] Smith, R. C. T., The bending of a semi-infinite strip. Aust. J. Sci. Res., 5, 227-237 (1952).
[32] Spence, D. A., Mixed boundary value problems for the elastic strip: the eigenfunction expansion, University of Wisconsin Mathematics Research Center Technical Summary Report 1863 (1978).
[33] Spence, D. A., A note on the eigenfunction expansion for the elastic strip (to be published, SIAM J. Appl. Math.) (1981).
[34] Sturges, L., & Joseph, D. D., The free surface on a simple fluid between cylinders undergoing torsional oscillations. Part III. Oscillating planes. Arch. Rational Mech. Anal., 69, 245 (1977). · Zbl 0369.76009 · doi:10.1007/BF00280149
[35] Sturges, L. D., A theoretical study of extrudate swell (Preprint, 1981). · Zbl 0476.76007
[36] Tamarkin, J., Some general problems of the theory of ordinary linear differential equations and expansion of an arbitrary function in series of fundamental functions. Math. Zeit, 27, 1-54 (1928). · JFM 53.0419.02 · doi:10.1007/BF01171084
[37] Teodorescu, P. P., Sur le problème de la demibande élastique. Archiwum Mechaniki Stosowanej, 12, 313-331 (1960). · Zbl 0104.18801
[38] Titchmarsh, E. C., Eigenfunction expansions associated with second-order differential equations, Oxford University Press (1946). · Zbl 0061.13505
[39] Trogdon, S. A., & Joseph, D. D., The stick-slip problem for a round jet. Part 1. Large surface tension. Rheol. Acta, 19, 404-420 (1980). · Zbl 0457.76032 · doi:10.1007/BF01524013
[40] Trogdon, S. A., & Joseph, D. D., Matched eigenfunction expansions for slow flow over a slot (preprint, 1980). · Zbl 0504.76006
[41] Williams, M. L., Stress singularities resulting from various boundary conditions in angular corners of plates in extension. J. Appl. Mech., 19, 526-528 (1952).
[42] Yoo, J. Y., & Joseph, D. D., Stokes flow in a trench between concentric cylinders. SIAM J. Appl. Math., 34, 247-285 (1978). · Zbl 0376.76019 · doi:10.1137/0134022
[43] Yoo, J. Y., Joseph, D. D., & Beavers, G. S., Higher order theory of the Weissenberg effect. J. Fluid Mech., 92, 529-590 (1979). · Zbl 0411.76005 · doi:10.1017/S0022112079000768
[44] Zidan, M., Zur Rheologie des Spinnprozesses. Rheol. Acta, 8, 89-123 (1969). · doi:10.1007/BF02321359
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