Previous |  Up |  Next


boundary integral equation; interior Dirichlet boundary value problem; Laplace equation; collocation method
We present, in a uniform manner, several integral equations of the first kind for the solution of the two-dimensional interior Dirichlet boundary value problem. We apply a general numerical collocation method to the various equations, and thereby we compare the various integral equations, and recommend two of them. We give a survey of the various numerical methods, and present a simple method for the numerical solution of the recommended integral equations.
[A1] Abou El-Seoud M. Samir: Numerische Behandlung von schwach singulären Integralgleichungen 1. Art. Dissertation, Technische Hochschule Darmstadt, D17, 2. Februar 1979, 7+ 173pp.
[A2] Abou El-Seoud M. S.: Kollokationsmethode für schwach singuläre Integralgleichungen erster Art. Zeit. Angew. Math. Mech. 59 (1979) T45-T47. MR 0533975
[A3] Abou El-Seoud M. S.: Ein Vergleich von Kollokationsmethode und Galerkin-Verfahren für die numerische Losung von schwach singulären Integralgleichungen erster Art. Zeit. Angew. Math. Mech. 60 (1980) T278-T280. MR 0623868
[A4] Abou El-Seoud M. S.: Bemerkungen zur numerischen Behandlung einer Klasse von schwach singulären Integralgleichungen 1 . Art unter Verwendung von Kollokations- und Galerkin-Methoden. Zeit. Angew. Math. Mech. 65 (1985) 405-415. DOI 10.1002/zamm.19850650906 | MR 0814681 | Zbl 0598.65097
[A5] Arnold Douglas N.: A Spline-Trigonometric Galerkin Method and an Exponentially Convergent Boundary Integral Method. Math. Comput. 41 (1983) 383-397. DOI 10.1090/S0025-5718-1983-0717692-8 | MR 0717692 | Zbl 0543.65087
[A6] Arnold Douglas N., Wolfgang L. Wendland: On the Asymptotic Convergence of Collocation Methods. Math. Comput. 41 (1983) 349-381. DOI 10.1090/S0025-5718-1983-0717691-6 | MR 0717691
[A7] Arnold Douglas N., Wolfgang L. Wendland: The Convergence of Spline Collocation for Strongly Elliptic Equations on Curves. Numerische Mathematik 47 (1985) 317-341. DOI 10.1007/BF01389582 | MR 0808553
[B1] Bogomolny Alexander: Fundamental Solutions Method for Elliptic Boundary Value Problems. SIAM Jour. Numer. Analysis 22 (1985) 644-669. DOI 10.1137/0722040 | MR 0795946 | Zbl 0579.65121
[C1] Christiansen Søren: Numerical Solution of an Integral Equation with a Logarithmic Kernel. BIT, Nordisk Tidskrift Informations-Behandling 11 (1971) 276-287. MR 0300481
[C2] Christiansen S.: On Green's Third Identity as a Basis for Derivation of Integral Equations. Zeit. Angew. Math. Mech. 54 (1974) T185-T186. DOI 10.1002/zamm.197405412100
[C3] Christiansen S.: On Kupradze's Functional Equations for Plane Harmonic Problems. Zeit. Angew. Math. Mech. 55 (1975) T197-T199. MR 0408380 | Zbl 0303.31002
[C4] Christiansen Søren: Integral Equations without a Unique Solution can be made Useful for Solving some Plane Harmonic Problems. Jour. Inst. Math. Applications 16 (1975) 143-159. DOI 10.1093/imamat/16.2.143 | MR 0399482
[C5] Christiansen S.: On Kupradze's functional equations for plane harmonic problems. pp. 205-243 of: Gilbert, R. P. & R. J. Weinacht (Editors): Function theoretic methods in differential equations. Research Notes in Mathematics 8, Pitman Publishing; London, San Francisco, Melbourne, 1976, 10+ 309 pp. MR 0509079 | Zbl 0339.31002
[C6] Christiansen Søren: Numerical Investigation of Some Integral Equations Related to the Classical Saint-Venant Torsion Problem. pp. 87-104 of: Brebbia, C.A. (Editor): Recent Advances in Boundary Element Methods. Pentech Press; London, Plymouth, 1978, 6 + 424pp.
[C7] Christiansen S.: An Investigation of an Integral Method for Determination of the Shearing Stress in Saint-Venant Torsion. Zeit. Angew. Math. Mech. 59 (1979) T173-T175. MR 0543467
[C8] Christiansen Søren: Numerical Treatment of an Integral Equation Originating from a Two-Dimjnsional Dirichlet Boundary Value Problem. pp. 73-91 of: Albrecht, J. & L. Collatz (Editors): Numerical Treatment of Integral Equations. (Proceedings, Workshop on Numerical Treatment of Integral Equations, Mathematical Research Institute, Oberwolfach, Fed. Rep. Germany. November 18-24, 1979.) International Series of Numerical Mathematics, ISNM 55. Birkhäuser Verlag; Basel, Boston, Stuttgart, 1980, 275 pp. MR 0590442
[C9] Christiansen S.: How to Suppress Non-unique Solutions to Integral Equations of the First Kind. Zeit. Angew. Math. Mech. 60 (1980) T244-T245. MR 0623855 | Zbl 0459.65095
[C10] Christiansen S.: Condition Number of Matrices Derived from Two Classes of Integral Equations. Math. Methods Appl. Sci. 3 (1981) 364-392. DOI 10.1002/mma.1670030126 | MR 0657303 | Zbl 0485.65089
[C11] Christiansen Søren: On Two Methods for Elimination of Non-unique Solutions of an Integral Equation with Logarithmic Kernel. Applicable Analysis 13 (1982) 1- 18. MR 0647662
[C12] Christiansen Søren: A comparison of various integral equations for treating the Dirichlet problem. pp. 13-24 of: Baker, Christopher T. H. & Geoffrey F. Miller (Editors): Treatment of Integral Equations by Numerical Methods. (Proceedings of a Symposium held in Durham, 19-29 July 1982.) Academic Press; London, et al., 1982, 16 -f 493 pp. MR 0755338
[C13] Christiansen S.: Numerical Investigation of an Integral Equation of Hsiao and MacCamy. Zeit. Angew. Math. Mech. 63 (1983) T341-T343. MR 0711928 | Zbl 0531.65076
[C14] Christiansen S.: Integral Equations: An Outline. pp. Q1-Q27 of: Bock, R. K., K. Bos, S. Brandt, J. Myrheim & M. Regler: Formulae and Methods in Experimental Data Evaluation with Emphasis on High Energy Physics. VoL 3: Articles on Statistical and Numerical Methods. European Physical Society (Computational Physics Group), CERN, Geneve, January 1984, 337 pp.
[C15] Christiansen S.: Modifications of Some First Kind Integral Equations with Logarithmic Kernel to Improve Numerical Conditioning. Computing 34 (1985) 221-242. DOI 10.1007/BF02253319 | MR 0799824 | Zbl 0551.65093
[C16] Christiansen S. E. B. Hansen: Numerical Solution of Boundary Value Problems through Integral Equations. Zeit. Angew. Math. Mech. 58 (1978) T14-T25. MR 0502074
[C17] Christiansen S. H. Rasmussen: Numerical Solutions for Two-dimensional Annular Electrochemical Machining Problems. Jour. Inst. Math. Applications 18 (1976) 295-307. DOI 10.1093/imamat/18.3.295
[D1] De Mey G.: Integral Equations for Potential Problems with the Source Function not located on the Boundary. Computers and Structures 8 (1978) 113-115. DOI 10.1016/0045-7949(78)90166-9 | Zbl 0369.35019
[H1] Hansen Per Christian, Søren Christiansen: An SVD analysis of linear algebraic equations derived from first kind integral equations. Jour. Computational Appl. Math. 12 & 13 (1985) 341-357. MR 0793966
[H2] Hsiao George C.: On the Stability of Integral Equations of the First Kind with Logarithmic Kernels. Archive Rational Mech. Anal. 94 (1986) 179-192. DOI 10.1007/BF00280433 | MR 0832291 | Zbl 0606.65089
[H3] Hsiao George R. C. MacCamy: Solution of Boundary Value Problems by Integral Equations of the First Kind. SIAM Review 15 (1973) 687-705. DOI 10.1137/1015093 | MR 0324242
[H4] Hsiao George C., Wolfgang L. Wendland: A Finite Element Method for Some Integral Equations of the First Kind. Jour. Math. Analysis Applications 58 (1977) 449-481. DOI 10.1016/0022-247X(77)90186-X | MR 0461963
[H5] Hsiao G. C. P. Kopp W. L. Wendland: A Galerkin Collocation Method for Some Integral Equations of the First Kind. Computing 25 (1980) 89-130. DOI 10.1007/BF02259638 | MR 0620387
[H6] Hsiao G. C. P. Kopp W. L. Wendland: Some Applications of a Galerkin-Collocation Method for Boundary Integral Equations of the First Kind. Math. Methods Appl. Sci. 6 (1984) 280-325. MR 0751746
[J1] Jaswon M. A. G. T. Symm: Integral Equation Methods in Potential Theory and Elastostatics. Academic Press; London, New York, San Francisco, 1977, 14 + 287 pp. MR 0499236
[K1] Купрадзе В. Д. Т. Г. Гегелна M. О. Башелейшвили T. В. Бурчуладзе: Трехмерные Задачи Математической Теории Упругости. Издательство Тбилисского Университета; Тбилиси, 1968, 627 стр. Zbl 1230.78011
[K2] Kupradze V. D. (Editor): V. D. Kupradze T. G. Gegelia M. O. Basheleishvili T. V. Burchuladze: Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. North-Holland Publishing Company; Amsterdam, New York, Oxford, 1979, 19 + 929 pp. (North-Holland series in applied mathematics and mechanics, Vol. 25). (Translated from the Russian second revised and supplemented edition. "Nauka" state publishing house for physics and mathematics literature, Moscow, 1976.) MR 0530377
[L1] Lamp U. K.-T. Schleicher W. L. Wendland: The Fast Fourier Transform and the Numerical Solution of One-Dimensional Boundary Integral Equations. Numerische Mathematik 47 (1985) 15-38. DOI 10.1007/BF01389873 | MR 0797875
[L2] Landkof N. S.: Foundations of Modern Potential Theory. Springer-Verlag; Berlin, Heidelberg, New York, 1972, 10 + 424 pp. (Die Grundlehren der mathematischen Wissenschaften, Band 180). (Translated from Основы современной теории потенциала Nauka, Moscow, 1966.) MR 0350027 | Zbl 0253.31001
[M1] Mathon Rudolf R. L. Johnston: The Approximate Solution of Elliptic Boundary-Value Problems by Fundamental Solutions. SIAM Jour. Numer. Analysis 14 (1977) 638-650. DOI 10.1137/0714043 | MR 0520337
[M2] McLean W.: A Spectral Galerkin Method for a Boundary Integral Equation. Math. Comput. 47 (1986) 597-607. DOI 10.1090/S0025-5718-1986-0856705-2 | MR 0856705 | Zbl 0614.65124
[M3] Mehlhorn G.: Anwendung eines potentialtheoretischen Analogieverfahrens in der Elastomechanik. Zeit. Angew. Math. Mech. 54 (1974) T92-T93. DOI 10.1002/zamm.19740541238 | Zbl 0305.73016
[M4] Mokin Yu. I.: Methods of solving the integral equations of potential theory. U.S.S.R. Computational Math. and Math. Physics 26, 4 (1986) 206. [Ž. Vyčisl. Mat. i Mat. Fiz. 26, 8 (1986) 1272-1273.] Zbl 0629.65142
[M5] Mokin Yu. I.: Numerical Methods for the Solution of Integral Equations in Potential Theory. Differential Equations 23 (1987) 843-852. [Differentsiaľnye Uravneniya 23 (1987) 1250-1262.] MR 0903980 | Zbl 0643.65096
[R1] Reichel Lothar: A Method for Preconditioning Matrices Arising from Linear Integral Equations for Elliptic Boundary Value Problems. Computing 37 (1986) 125-136. DOI 10.1007/BF02253186 | MR 0854581 | Zbl 0589.65089
[R2] Ruotsalainen K. J. Saranen: Some Boundary Element Methods Using Dirac's Distributions as Trial Functions. SIAM Jour. Numer. Analysis 24 (1987) 816-827. DOI 10.1137/0724052 | MR 0899705
[S1] Saranen J. W. L. Wendland: On the Asymptotic Convergence of Collocation Methods With Spline Functions of Even Degree. Math. Comput. 45 (1985) 91-108. DOI 10.1090/S0025-5718-1985-0790646-3 | MR 0790646
[S2] Sloan I. H. A. Spence: The Galerkin Method for Integral Equations of the First Kind with Logarithmic Kernel: Theory. IMA Jour. Numer. Analysis 8 (1988) 105- 122. DOI 10.1093/imanum/8.1.105 | MR 0967846
[S3] Sloan I. H. A. Spence: The Galerkin Method for Integral Equations of the First Kind with Logarithmic Kernel: Applications. IMA Jour. Numer. Analysis 8 (1988) 123-140. DOI 10.1093/imanum/8.1.123 | MR 0967847
[S4] Symm George T.: Numerical Mapping of Exterior Domains. Numerische Mathematik 10 (1967) 437-445. DOI 10.1007/BF02162876 | MR 0220465
[V1] Voronin V. V. V. A. Cecoho: An interpolation method for solving an integral equation of the first kind with a logarithmic singularity. Soviet Mathematics, Doklady 15 (1974) 949-952. [Dokl. Akad. Nauk SSSR 216 (1974) 1209-1211.] MR 0487361
[V2] Voronin V. V. V. A. Tsetsokho: Numerical solution, by interpolation and collocation, of integral equations of the 1st kind with logarithmic singularity. U.S.S.R. Computational Math. and Math. Physics 27, 1 (1981) 38-52. [Ž. Vyčisl. Mat. i Mat. Fiz. 27, 1 (1981) 40-53.] DOI 10.1016/0041-5553(81)90131-2 | MR 0608618
[W1] Wendland W. L.: Strongly Elliptic Boundary Integral Equations. pp. 511-562 of: Iserles, A. & M. J. D. Powell (Editors): The State of the Art in Numerical Analysis. (Proceedings of the joint IMA/SIAM conference held at the University of Birmingham, 14-18 April 1986.) Clarendon Press; Oxford, 1987, 14 + 719 pp. MR 0921677
[W2] Wendland Wolfgang L., Søren Christiansen: On the condition number of the influence matrix belonging to some first kind integral equations with logarithmic kernel. Applicable Analysis 27 (1986) 175-183. DOI 10.1080/00036818608839589 | MR 0840310
[Y1] Young David M., Robert Todd Gregory: A Survey of Numerical Mathematics. I + II. Addison-Wesley Publishing Company; Reading, Massachusetts, 1973, 51+ 63 + 1099 pp. MR 0408189
Partner of
EuDML logo