Previous |  Up |  Next

Article

Keywords:
non-linear singular integral equations; existence and uniqueness theorems; Banach spaces; Hölder conditions; fluid mechanics; equations over finite set of contours; steady incompressible motion; turbomachines
Summary:
Non-linear singular integral equations are investigated in connection with some basic applications in two-dimensional fluid mechanics. A general existence and uniqueness analysis is proposed for non-linear singular integral equations defined on a Banach space. Therefore, the non-linear equations are defined over a finite set of contours and the existence of solutions is investigated for two different kinds of equations, the first and the second kind. Moreover, the existence of solutions is further studied for non-linear singular integral equations over a finite number of arbitrarily ordered arcs. An application to fluid mechanics theory is finally given for the determination of the form of the profiles of a turbomachine in two-dimensional flow of an incompressible fluid.
References:
[1] E. G. Ladopoulos: On the numerical solution of the finite-part singular integral equations of the first and the second kind used in fracture mechanics. Comp. Meth. Appl. Mech. Engng 65 (1987), 253–266. DOI 10.1016/0045-7825(87)90159-9 | MR 0919245
[2] E. G. Ladopoulos: On a new integration rule with the Gegenbauer polynomials for singular integral equations, used in the theory of elasticity. Ing. Archiv 58 (1988), 35–46. DOI 10.1007/BF00537198 | Zbl 0627.73018
[3] E. G. Ladopoulos: On the numerical evaluation of the general type of finite-part singular integrals and integral equations used in fracture mechanics. J. Engrg. Fract. Mech. 31 (1988), 315–337. DOI 10.1016/0013-7944(88)90075-6
[4] E. G. Ladopoulos: The general type of finite-part singular integrals and integral equations with logarithmic singularities used in fracture mechanics. Acta Mech. 75 (1988), 275–285. DOI 10.1007/BF01174641 | Zbl 0667.73072
[5] E. G. Ladopoulos: On the solution of the two-dimensional problem of a plane crack of arbitrary shape in an anisotropic material. J. Engrg Fract. Mech. 28 (1987), 187–195. DOI 10.1016/0013-7944(87)90212-8
[6] E. G. Ladopoulos: On the numerical evaluation of the singular integral equations used in two and three-dimensional plasticity problems. Mech. Res. Commun. 14 (1987), 263–274. DOI 10.1016/0093-6413(87)90039-5 | Zbl 0635.73044
[7] E. G. Ladopoulos: Singular integral representation of three-dimensional plasticity fracture problem. Theor. Appl. Fract. Mech. 8 (1987), 205–211. DOI 10.1016/0167-8442(87)90047-4
[8] E. G. Ladopoulos: On the numerical solution of the multidimensional singular integrals and integral equations used in the theory of linear viscoelasticity. Internat. J. Math. Math. Scien. 11 (1988), 561–574. DOI 10.1155/S0161171288000675 | MR 0947288 | Zbl 0665.65097
[9] E. G. Ladopoulos: Relativistic elastic stress analysis for moving frames. Rev. Roum. Sci. Tech., Méc. Appl. 36 (1991), 195–209. MR 1171626
[10] J. Andrews and J. M. Ball: Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity. J. Diff. Eqns 44 (1982), 306–341. DOI 10.1016/0022-0396(82)90019-5 | MR 0657784
[11] S. S. Antman: Ordinary differential equations of nonlinear elasticity I: Foundations of the theories of nonlinearly elastic rods and shells. Arch. Ration. Mech. Anal. 61 (1976), 307–351. MR 0418580
[12] S. S. Antman: Ordinary differential equations of nonlinear elasticity II: Existence and regularity for conservative boundary value problems. Arch. Ration. Mech. Anal. 61 (1976), 352–393.
[13] S. S. Antman, E. R. Carbone: Shear and necking instabilities in nonlinear elasticity. J. Elasticity 7 (1977), 125–151. DOI 10.1007/BF00041087 | MR 0451990
[14] J. M. Ball: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal. 63 (1977), 337-403. MR 0475169 | Zbl 0368.73040
[15] J. M. Ball: Discontinuous equilibrium solutions and cavitation in nonlinear elasticity. Phil. Trans. R. Soc. Lond. A 306 (1982), 557–611. DOI 10.1098/rsta.1982.0095 | MR 0703623 | Zbl 0513.73020
[16] J. M. Ball: Remarques sur l’existence et la régularité des solutions d’elastostatique nonlinéaire, in: Recent Contributions to Nonlinear Partial Differential Equations. Pitman, Boston, 1981, pp. 50–62. MR 0639745
[17] H. Brezis: Equations et inéquations non lineaires dans les éspaces vectoriels en dualite. Ann. Inst. Fourier 18 (1968), 115–175. DOI 10.5802/aif.280 | MR 0270222 | Zbl 0169.18602
[18] P. G. Ciarlet, P. Destuynder: A justification of a nonlinear model in plate theory. Comp. Meth. Appl. Mech. Engng 17 (1979), 227–258. MR 0533827
[19] P. G. Ciarlet, J. Nečas: Injectivité presque partout, autocontact, et noninterpénétrabilité en élasticité non linéaire tridimensionnelle. C. R. Akad. Sci. Paris, Sér I 301 (1985), 621–624. MR 0816644
[20] P. G. Ciarlet, J. Nečas: Injectivity and self-contact in non-linear elasticity. Arch. Ration. Mech. Anal. 97 (1987), 171–188. DOI 10.1007/BF00250807 | MR 0862546
[21] C. M. Dafermos: The mixed initial-boundary value problem for the equations of nonlinear one dimensional viscoelasticity. J. Diff. Eqns 6 (1969), 71–86. DOI 10.1016/0022-0396(69)90118-1 | MR 0241831 | Zbl 0218.73054
[22] C. M. Dafermos: Development of singularities in the motion of materials with fading memory. Arch. Ration. Mech. Anal. 91 (1985), 193–205. MR 0806001
[23] C. M. Dafermos, L. Hsiao: Development of singularities in solutions of the equations of nonlinear thermoelasticity. Q. Appl. Math. 44 (1986), 463–474. MR 0860899
[24] J. E. Dendy: Galerkin’s method for some highly nonlinear problems. SIAM J. Num. Anal. 14 (1977), 327–347. DOI 10.1137/0714021 | MR 0433914 | Zbl 0365.65065
[25] Guo Zhong-Heng: The unified theory of variational principles in nonlinear elasticity. Arch. Mech. 32 (1980), 577–596. MR 0619303
[26] H. Hattori: Breakdown of smooth solutions in dissipative nonlinear hyperbolic equations. Q. Appl. Math. 40 (1982), 113–127. MR 0666668 | Zbl 0505.76008
[27] D. Hoff, J. Smoller: Solutions in the large for certain nonlinear parabolic systems. Anal. Non Lin. 2 (1985), 213–235. MR 0797271
[28] W. J. Hrusa: A nonlinear functional differential equation in Banach space with applications to materials with fading memory. Arch. Ration. Mech. Anal. 84 (1983), 99–137. DOI 10.1007/BF00252129 | MR 0713121 | Zbl 0544.73056
[29] R. C. MacCamy: Nonlinear Volterra equations on a Hilbert space. J. Diff. Eqns 16 (1974), 373–393. DOI 10.1016/0022-0396(74)90021-7 | MR 0377605 | Zbl 0263.45010
[30] R. C. MacCamy: Stability theorems for a class of functional differential equations. SIAM J. Appl. Math. 30 (1976), 557–576. DOI 10.1137/0130050 | MR 0404818 | Zbl 0346.34059
[31] R. C. MacCamy: A model for non-dimensional, nonlinear viscoelasticity. Q. Appl. Math. 35 (1977), 21–33. MR 0478939
[32] B. Neta: Finite element approximation of a nonlinear parabolic problem. Comput. Math Appl. 4 (1987), 247–255. MR 0518696
[33] B. Neta: Numerical solution of a nonlinear integro-differential equation. J. Math. Anal. Appl. 89 (1982), 598–611. DOI 10.1016/0022-247X(82)90119-6 | MR 0677747 | Zbl 0488.65074
[34] R. W. Ogden: Principal stress and strain trajectories in nonlinear elastostatics. Q. Appl. Math. 44 (1986), 255–264. MR 0856179 | Zbl 0608.73022
[35] R. L. Pego: Phase transitions in one-dimensional nonlinear viscoelasticity: admissibility and stability. Arch. Ration. Mech. Anal. 97 (1987), 353–394. DOI 10.1007/BF00280411 | MR 0865845 | Zbl 0656.73023
[36] M. Slemrod: Global existence, uniqueness and asymptotic stability of classical smooth solutions in one-dimensional, nonlinear thermoelasticity. Arch. Ration. Mech. Anal. 76 (1981), 97–133. MR 0629700 | Zbl 0481.73009
[37] O. J. Staffans: On a nonlinear hyperbolic Volterra equation. SIAM J. Math. Anal. 11 (1980), 793–812. DOI 10.1137/0511071 | MR 0586908 | Zbl 0464.45010
[38] J. Schauder: Der Fixpunktsatz in Funktionalräumen. Studia Math. 2 (1930), 171–180.
[39] I. Privalov: On a boundary problem in the theory of analytic functions. Math. Sb. 41 (1934), 519–526.
[40] S. Banach: Sur les opérations dans les ensembles abstraits et leurs applications aux équations intégrales. Fundam. Math. 3 (1922), 133–181.
[41] V. Cacciopoli: Un teorema generale sull’asistenza di elementi uniti in una transformazione functionale. Rend. Accad. Lincei 2 (1930).
[42] M. I. Zhykovskiy: Calculation of the Flow in Lattices of Profiles of Turbomachines. Mashgiz, Moscow, 1960. (Russian)
[43] N. I. Muskhelishvili: Some Basic Problems of the Mathematical Theory of Elasticity. Noordhoff, Groningen, Netherlands, 1953. MR 0058417 | Zbl 0052.41402
[44] N. I. Muskhelishvili: Singular Integral Equations. Noordhoff, Groningen, Netherlands, 1972. MR 0355494
[45] V. V. Ivanov: The Theory of Approximate Methods and their Application to the Numerical Solution of Singular Integral Equations. Noordhoff, Leyden, Netherlands, 1976. MR 0405045 | Zbl 0346.65065
Partner of
EuDML logo