×

A review of linear and nonlinear Cauchy singular integral and integro-differential equations arising in mechanics. (English) Zbl 1149.45002

The authors provide, numerical analysts and researchers interested in algorithm development, with ten model problems that arise in mechanics and involve linear (LSIDEs) and nonlinear singular integro-differential equations (NLSIDEs) with Cauchy kernels. Details of the origin of the problems are presented along with equations that arise (given in appropriate form), together with boundary conditions, typical parameter sizes (where available). The authors also present some existing analytical results such as asymptotic expansions and where appropriate, details of the techniques that have previously been employed for its numerical solution.
The problems are arranged in order of increasing numerical difficulty. Each of the problems considered possess distinctive characteristics and pose different numerical, asymptotic and analytical challenges. Any singular integro-differential equation solver that performs well on all of the test problems will indeed be a powerful tool. For the problem that deals with fluid suction equation, the authors discuss in detail the importance of the wellposedness of the problem. Under certain conditions, it is proved that the problem does not possess a Hölder continuous solution whereas the global collocation method produces a plausible looking solution.
A few other example problems are included in the appendix. This paper is a good contribution to the literature on singular integral and integro-differential equations. This will serve as a excellent source of test problems for those studying the numerical methods for solving LSIDEs and NLSIDEs.

MSC:

45E05 Integral equations with kernels of Cauchy type
45J05 Integro-ordinary differential equations
65R20 Numerical methods for integral equations
45G05 Singular nonlinear integral equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] S.M. Amer, On solution of nonlinear singular integral equations with shift in generalized Hölder Space , Chaos Solitons Fractals 12 (2001), 1323-1334. · Zbl 1017.45002 · doi:10.1016/S0960-0779(00)00066-7
[2] S.M. Amer and S. Dardery, On a class of nonlinear singular integral equations with shift on a closed contour , Appl. Math. Comput. 158 (2004), 781-791. · Zbl 1063.45002 · doi:10.1016/j.amc.2003.08.138
[3] R.S. Anderssen, F.R. de Hoog and M.A. Lukas, eds., The application and numerical solution of integral equations , Sijthoff & Noordhoff, Groningen, 198-0. · Zbl 0423.00019
[4] Y.A. Antipov and H. Gao, Exact solution of integro-differential equations of diffusion along a grain boundary , Quart. J. Mech. Appl. Math. 53 (2000), 645-674. · Zbl 0959.74006 · doi:10.1093/qjmam/53.4.645
[5] D. Berthold, W. Hoppe and B. Silbermann, A fast algorithm for solving the generalized airfoil equation , J. Comput. Appl. Math. 43 (1992), 185-219. · Zbl 0761.65102 · doi:10.1016/0377-0427(92)90266-Z
[6] E.J. Bissett and D.A. Spence, The line contact problem of elastohydrodynamic lubrication II. Numerical solutions of the integrodifferential equations in the transition and exit layers , Proc. Roy. Soc. London 424 (1989), 409-429. · Zbl 0674.73080 · doi:10.1098/rspa.1989.0092
[7] S.N. Brown, H.K. Cheng and F.T. Smith, Nonlinear instability and break-up of separated flow , J. Fluid Mech. 193 (1988), 191-216. · Zbl 0643.76058 · doi:10.1017/S0022112088002113
[8] M.R. Capobianco, The stability and the convergence of a collocation method for a class of Cauchy singular integral-equations , Math. Nachr. 162 (1993), 45-58. · Zbl 0801.65127 · doi:10.1002/mana.19931620105
[9] M.R. Capobianco, G. Criscuolo and P. Junghanns, A fast algorithm for Prandtl’s integro-differential equation , J. Comput. Appl. Math. 77 (1997), 103-128. · Zbl 0870.65135 · doi:10.1016/S0377-0427(96)00124-0
[10] M.R. Capobianco, G. Criscuolo, P. Junghanns and U. Luther, Uniform convergence of the collocation method for Prandtl’s integro-differential equation , Proc. of the David Elliott Conf., Hobart, 199-7. · Zbl 0966.65117
[11] A. Chakrabarti and G. Vanden Berghe, Approximate solutions of singular integral equations , Appl. Math. Lett. 17 (2004), 553-559. · Zbl 1061.65141 · doi:10.1016/S0893-9659(04)90125-5
[12] G.A. Chandler and I.G. Graham, The computation of water-waves modelled by Nekrasov’s equation , SIAM J. Numer. Anal. 30 (1993), 1041-1065. JSTOR: · Zbl 0780.76059 · doi:10.1137/0730054
[13] S. Childress, Solutions of Euler’s equations containing finite eddies , Phys. Fluids 9 (1966), 860-872. · Zbl 0148.20603 · doi:10.1063/1.1761786
[14] J.A. Cuminato, On the uniform-convergence of a collocation method for a class of singular integral-equations , BIT 27 (1987), 190-202. · Zbl 0629.65139 · doi:10.1007/BF01934184
[15] ——–, Numerical solution of Cauchy integral equations and applications , Ph.D. Thesis, Dept. of Mathematics, University of Oxford, 198-7.
[16] ——–, Uniform-convergence of a collocation method for the numerical-solution of Cauchy-type singular integral-equations-A generalization , IMA J. Numer. Anal. 12 (1992), 31-45. · Zbl 0747.65097 · doi:10.1093/imanum/12.1.31
[17] ——–, Numerical solution of Cauchy-type integral equations of index \(-1\) by collocation methods , Adv. Comput. Math. 6 (1996), 47-64. · Zbl 0870.65133 · doi:10.1007/BF02127695
[18] J.A. Cuminato, D. Butler and S. McKee, Sound scattering from an underwater finite flexible plate , Math. Engrg. Indust. 2 (1990), 233-251. · Zbl 0837.73048
[19] I.S. Duff, R.G. Grimes and J.G. Lewis, Sparse matrix test problems , ACM Trans. Math. Software 15 (1989), 1-14. · Zbl 0667.65040 · doi:10.1145/62038.62043
[20] D. Elliott, The classical collocation method for singular integral equations , SIAM J. Numer. Anal. 19 (1982), 816-832. JSTOR: · Zbl 0482.65069 · doi:10.1137/0719057
[21] ——–, The numerical treatment of singular integral equations-a review , in Treatment of integral equations by numerical methods (C.T.H. Baker and G.F. Miller, eds.), Academic Press, London, 198-2.
[22] ——–, A Galerkin-Petrov method for singular integral equations , J. Austral. Math. Soc. 25 (1983), 261-275. · Zbl 0517.65094 · doi:10.1017/S0334270000004057
[23] ——–, Rates of convergence for the method of classical collocation for solving singular integral equations , SIAM J. Numer. Anal. 21 (1984), 136-148. JSTOR: · Zbl 0549.65092 · doi:10.1137/0721009
[24] J. Elschner, Galerkin methods with splines for singular integral-equations over \((0,1)\) , Numer. Math. 43 (1984), 265-281. · Zbl 0549.65093 · doi:10.1007/BF01390127
[25] F. Erdogan, Approximate solutions of systems of singular integral equations , SIAM J. Appl. Math. 17 (1969), 1041-1059. JSTOR: · Zbl 0187.12404 · doi:10.1137/0117094
[26] F. Erdogan and G.D. Gupta, On the numerical solution of singular integral equations , Quart. Appl. Math. 29 (1972), 525-533. · Zbl 0236.65083
[27] A.D. Fitt, A.D. Kelly and C.P. Please, Crack-propagation models for rock fracture in a geothermal-energy reservoir , SIAM J. Appl. Math. 55 (1995), 1592-1608. JSTOR: · Zbl 0838.73062 · doi:10.1137/S0036139993260241
[28] A.D. Fitt, J.R. Ockendon and T.V. Jones, Aerodynamics of slot-film cooling : Theory and experiment , J. Fluid Mech. 160 (1985), 15-27.
[29] A.D. Fitt and M.P. Pope, The unsteady motion of two-dimensional flags with bending stiffness , J. Engrg. Math. 40 (2001), 227-248. · Zbl 0990.76025 · doi:10.1023/A:1017595632666
[30] A.D. Fitt and P. Wilmott, Slot film-cooling-the effect of separation angle , Acta Mech. 103 (1994), 79-88. · Zbl 0825.76073 · doi:10.1007/BF01180219
[31] J.I. Frankel, A Galerkin solution to a regularized Cauchy singular integrodifferential equation , Quart. Appl. Math. 53 (1995), 245-258. · Zbl 0823.65145
[32] K.O. Friedrichs and H. Lewy, The dock problem , Comm. Appl. Math. 1 (1951), 135-148. · Zbl 0030.37902 · doi:10.1002/cpa.3160010203
[33] F.D. Gakhov, Boundary value problems , Pergamon Press, Oxford, 196-6.
[34] D.M. Gay, Electronic mail distribution of linear programming test problems , Math. Programming Soc. COAL Newsletter 13 (1985), 10-12.
[35] A. Gerasoulis, Piecewise-polynomial quadratures for Cauchy singular-integrals , SIAM J. Numer. Anal. 23 (1986), 891-902. JSTOR: · Zbl 0619.65010 · doi:10.1137/0723057
[36] A. Gerasoulis and R.P. Srivastav, A method for the numerical-solution of singular integral-equations with a principal value integral , Internat. J. Engrg. Sci. 19 (1981), 1293-1298. · Zbl 0476.73065 · doi:10.1016/0020-7225(81)90148-8
[37] M.A. Golberg, ed., Solution methods for integral equations : Theory and applications , Plenum Press, New York, 197-9.
[38] M.A. Golberg, Projections method for Cauchy singular integral equations with constant coefficients on [-1,1], in Treatment of integral equations by numerical methods (C.T.H. Baker and G.F. Miller, eds.), Academic Press, London, 198-2.
[39] ——–, Galerkin’s method for operator-equations with non negative index\emdash/With application to Cauchy singular integral-equations , J. Math. Anal. Appl. 91 (1983), 394-409. · Zbl 0513.65086 · doi:10.1016/0022-247X(83)90160-9
[40] ——–, A note on a superconvergence result for the generalized airfoil equation , Appl. Math. Comput. 26 (1988), 105-117. · Zbl 0656.76006 · doi:10.1016/0096-3003(88)90045-8
[41] ——–, The perturbed Galerkin method for Cauchy singular integral-equation with constant-coefficients , Appl. Math. Comput. 26 (1988), 1-33. · Zbl 0663.65139 · doi:10.1016/0096-3003(88)90084-7
[42] W. Hock and K. Schittkowski, Test examples for nonlinear programming codes , Lecture Notes in Econom. and Math. Systems, vol. 187 -1981. · Zbl 0452.90038
[43] T.E. Hull, W.E. Enright, B.M. Fellen and A.E. Sedgwick, Comparing numerical methods for ordinary differential equations , SIAM J. Numer. Anal. 9 (1972), 603-637. JSTOR: · Zbl 0221.65115 · doi:10.1137/0709052
[44] N.I. Ioakimidis, On the weighted Galerkin method of numerical-solution of Cauchy type singular integral-equations , SIAM J. Numer. Anal. 18 (1981), 1120-1127. JSTOR: · Zbl 0518.65098 · doi:10.1137/0718076
[45] ——–, A natural interpolation formula for Prandtl singular integrodifferential equation , Internat. J. Numer. Methods Fluids 4 (1984), 283-290. · Zbl 0553.65097 · doi:10.1002/fld.1650040306
[46] N.I. Ioakimidis and P.S. Theocaris, The numerical evaluation of a class of generalized stress intensity factors by use of the Lobatto-Jacobi numerical integration rule , Internat. J. Fracture 14 (1978), 469-484. · Zbl 0427.65087 · doi:10.1007/BF01390469
[47] D. Jinyuan, The collocation methods and singular integral equations with Cauchy kernels , Acta. Math. Sci. 20 (2000), 289-302. · Zbl 0967.65117
[48] A.F. Jones and S.D.R. Wilson, The film drainage problem in droplet coalescence , J. Fluid Mech. 87 (1978), 263-288. · Zbl 0378.76005 · doi:10.1017/S0022112078001585
[49] P. Junghanns, Numerical analysis of Newton projection methods for nonlinear singular integral equations , J. Comput. Appl. Math. 55 (1994), 145-163. · Zbl 0826.65119 · doi:10.1016/0377-0427(94)90017-5
[50] ——–, On the numerical solution of nonlinear singular integral equations , Z. Angew. Math. Mech. 76 (1996), 157-160. · Zbl 0888.76083
[51] ——–, Numerical solution of a free surface seepage problem from nonlinear channel , Appl. Anal. 63 (1996), 87-110. · Zbl 0865.65096 · doi:10.1080/00036819608840497
[52] ——–, Optimal control for a parametrized family of nonlinear Cauchy singular integral equations , J. Comput. Appl. Math. 164 (2004), 431-454. · Zbl 1063.49003 · doi:10.1016/j.cam.2003.11.009
[53] P. Junghanns and U. Luther, Uniform convergence of the quadrature method for Cauchy singular integral equations with weakly singular perturbation kernels , Rend. Circ. Mat. Palermo 2 (1998), 551-566. · Zbl 0905.65126
[54] ——–, Uniform convergence of a fast algorithm for Cauchy singular integral equations , Linear Algebra Appl. 275 - 276 (1998), 327-347. · Zbl 0939.65141 · doi:10.1016/S0024-3795(97)10025-8
[55] P. Junghanns and G. Mastroianni, On the stability of collocation for Cauchy singular integral equations on an interval , Oper. Theory Adv. Appl., vol. 121, Birkhauser-Verlag, New York, 2001, pp. 261-277. · Zbl 1004.65143
[56] P. Junghanns and K. Müller, A collocation method for nonlinear Cauchy singular integral equations , J. Comput. Appl. Math. 115 (2000), 283-300. · Zbl 0947.65141 · doi:10.1016/S0377-0427(99)00117-X
[57] P. Junghanns, K. Müller and K. Rost, On collocation methods for nonlinear Cauchy singular integral operators , Oper. Theory Adv. Appl., vol. 135, Birkhäuser Verlag, Basel, 2002, pp. 209-233. · Zbl 1026.65136
[58] P. Junghanns, G. Semmler, U. Weber and E. Wegert, Nonlinear singular integral equations on a finite interval , Math. Meth. Appl. Sci. 24 (2001), 1275-1288. · Zbl 1003.45001 · doi:10.1002/mma.272
[59] P. Junghanns and B. Silbermann, Numerical analysis for one-dimensional Cauchy singular integral equations , J. Comput. Appl. Math. 125 (2000), 395-421. · Zbl 0978.65125 · doi:10.1016/S0377-0427(00)00482-9
[60] P. Junghanns, B. Silbermann and S. Roch, Collocation methods for systems of Cauchy singular integral equations on an interval , Comp. Tech. 6 (2001), 88-126. · Zbl 0969.65118
[61] A.I. Kalandiya, Mathematical methods of two-dimensional elasticity , Mir Publ., Moscow, 197-5.
[62] A.C. King and E.O. Tuck, Thin liquid layers supported by steady air-flow surface traction , J. Fluid Mech. 251 (1993), 709-718. · Zbl 0783.76031 · doi:10.1017/S0022112093003581
[63] V.G. Kravchenko and G.S. Litvinchuk, Introduction to the theory of singular integral operators with shift , Kluwer, Amsterdam, 199-4. · Zbl 0811.47049
[64] S. Krenk, On quadrature formulae for singular integral equations of the first and the second kind , Quart. Appl. Math. 33 (1975), 225-232. · Zbl 0322.45022
[65] F.T. Krogh, On testing a subroutine for the numerical integration of ordinary differential equations , J. Assoc. Comput. Mach. 20 (1973), 545-562. · Zbl 0292.65039 · doi:10.1145/321784.321786
[66] E.G. Ladopoulos, Collocation-approximation methods for nonlinear singular integrodifferential equations in Banach spaces , J. Comp. Appl. Math. 79 (1997), 289-297. · Zbl 0878.65123 · doi:10.1016/S0377-0427(96)00174-4
[67] ——–, Singular integral equations : Linear and nonlinear theory and its applications in science and engineering , Springer, Berlin, 200-0.
[68] J. Lam, Resistivity and specific heat of a dilute magnetic alloy I: Electron self energy , J. Phys. F: Metal Phys. 3 (1973), 1197-1206.
[69] M.J. Lighthill, Proof of a conjecture of Spence , Rep. Aero. Res. Coun. 20 (1959), 793-801.
[70] M.A.D. Madurasinghe, Splashless ship bows with stagnant attachment , J. Ship. Res. 32 (1988), 194-202.
[71] S.R. Manam, A note on a singular integral equation arising in water wave scattering , IMA J. Appl. Math. 69 (2004), 483-491. · Zbl 1083.45001 · doi:10.1093/imamat/69.5.483
[72] I.N. Meleshko, Approximate solution of a singular integrodifferential equation , Differential Equations 25 (1989), 644-652. · Zbl 0698.65091
[73] G.R. Miller and L.M. Keer, A numerical technique for the solution of singular integral-equations of the \(2\)nd kind , Quart. Appl. Math. 42 (1985), 455-465. · Zbl 0593.65094
[74] J.J. Moré, A collection of nonlinear problems , in Computational solution of nonlinear systems of equations (E.L. Allgower and K. Georg, eds.), Lectures in Appl. Math., vol. 26, Amer. Math. Soc., Providence, 1990, pp. 723-762.
[75] J.J. Moré, B.S. Garbow and K.E. Hillstrom, Testing unconstrained optimization software , ACM Trans. Math. Software 7 (1981), 17-41. · Zbl 0454.65049 · doi:10.1145/355934.355936
[76] M.J.S. Mphaka, Partial singular integro-differential equation models for dryout in boilers , Ph.D. Thesis, Dept. of Mathematics, University of Southampton, 2000.
[77] N.I. Muskhelishvili, Singular integral equations , Noordhoff, Groningen, 195-3.
[78] J.N. Newman, Marine hydrodynamics , MIT Press, Boston, 197-7.
[79] K. O’Malley, A.D. Fitt, T.V. Jones, J.R. Ockendon and P. Wilmott, Models for high-Reynolds-number flow down a step , J. Fluid Mech. 222 (1991), 139-155. · Zbl 0717.76041 · doi:10.1017/S0022112091001039
[80] A.S. Peters, A note on the integral equation of the first kind with a Cauchy kernel , Comm. Pure Appl. Math. 41 (1963), 57-61. · Zbl 0119.30903 · doi:10.1002/cpa.3160160108
[81] ——–, Abel’s equation and the Cauchy integral equation of the second kind , Comm. Pure Appl. Math. 21 (1968), 51-65. · Zbl 0177.39001 · doi:10.1002/cpa.3160210105
[82] M.P. Pope, Mathematical modelling of unsteady problems in thin aerofoil theory , Ph.D. Thesis, Dept. of Mathematics, University of Southampton, 199-9.
[83] M.J.D. Powell, A hybrid method for nonlinear algebraic equations , in Numerical methods for nonlinear algebraic equations (P. Rabinowitz, ed.), Gordon & Breach, London, 197-0.
[84] A. Robinson and J.A. Laurmann, Wing theory , Cambridge Univ. Press, Cambridge, 195-6. · Zbl 0073.41901
[85] G. Schmidt, Spline collocation for singular integro-differential equations over \((0,1)\) , Numer. Math. 50 (1987), 337-352. · Zbl 0624.65141 · doi:10.1007/BF01390710
[86] D.A. Spence and P. Sharp, Self-similar solutions for elastohydrodynamic cavity flow , Proc. Roy. Soc. London 400 (1985), 289-313. · Zbl 0581.76007 · doi:10.1098/rspa.1985.0081
[87] D.A. Spence and P.W. Sharp, Distortion and necking of a viscous inclusion in Stokes flow , Proc. Roy. Soc. London 422 (1989), 173-192. · Zbl 0698.76039 · doi:10.1098/rspa.1989.0024
[88] D.A. Spence, P.W. Sharp and D.L. Turcotte, Buoyancy-driven crack propagation : A mechanism for magma migration , J. Fluid. Mech. 174 (1987), 135-153. · Zbl 0602.73112 · doi:10.1017/S0022112087000077
[89] K. Stewartson, A note on lifting line theory , Quart. J. Mech. Appl. Math. 13 (1960), 49-56. · Zbl 0093.18102 · doi:10.1093/qjmam/13.1.49
[90] P.S. Theocaris and N.I. Ioakimidis, Numerical integration methods for the solution of singular integral equations , Quart. Appl. Math. 35 (1977), 173-183. · Zbl 0353.45016
[91] B. Thwaites, The aerodynamic theory of sails I. Two-dimensional sails , Proc. Roy. Soc. London 261 (1961), 402-422. · Zbl 0096.40803 · doi:10.1098/rspa.1961.0086
[92] E.O. Tuck, Ship-hydrodynamic free-surface problems without waves , J. Ship. Res. 35 (1991), 277-287.
[93] E.O. Tuck and J.-M. Vanden-Broeck, Ploughing flows , European J. Appl. Math. 9 (1998), 463-483. · Zbl 0932.76010 · doi:10.1017/S0956792598003519
[94] M. Van Dyke, Perturbation methods in fluid mechanics , Parabolic Press, New York, 197-5.
[95] E. Varley and J.D.A. Walker, A method for solving singular integro-differential equations , IMA J. Appl. Math. 43 (1989), 11-45. · Zbl 0687.45006 · doi:10.1093/imamat/43.1.11
[96] K. Voelz, Profil und Auftrieb eines Segels , Z. Angew. Math. Mech. 30 (1950), 301-317. · Zbl 0039.21003
[97] J.V. Wehausen and E.V. Laitone, Surface waves , in Handbuch der Physik (S. Flugge, ed.) 9 Springer-Verlag, Berlin, 1960, pp. 446-778. · Zbl 1339.76009
[98] L.V. Wolfersdorf, On a nonlinear singular integro-differential equation , Z. Angew. Math. Mech. 67 (1987), 333-334. · Zbl 0624.45007 · doi:10.1002/zamm.19870670711
[99] E. Zachariou, P. Wilmott and A.D. Fitt, A cavitating aerofoil with a Prandtl-Batchelor eddy , Aeronaut. J. 98 (1994), 171-176.
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.