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Applicable Analysis

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Volume 13, 1982 - Issue 1

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On two methods for elimination of non-unique solutions of an integral equation with logarithmic kernel/Footnote^†
A Very short version of part of this paper bas been presented on April 18, 1979 at Scientific Meeting of “Gesellschaft für Angewandte Mathematik und Mechanik” (GAMM) [6].

S⊘ren Christiansen Laboratory of Applied Mathematical Physics, The Technical University of Denmark, Lyngby, Denmark

Pages 1-18 | Received 01 Nov 1979, Published online: 02 May 2007

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https://doi.org/10.1080/00036818208839372

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Jeng-Tzong Chen, Ying-Te Lee, Jia-Wei Lee & Sheng-Kuang Chen. (2020) A study on the degenerate scale by using the fundamental solution with dimensionless argument for 2D elasticity problems. Journal of the Chinese Institute of Engineers 43:4, pages 373-385.
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Jeng-Tzong Chen, Wen-Sheng Huang, Ying-Te Lee, Shyh-Rong Kuo & Shing-Kai Kao. (2017) Revisit of degenerate scales in the BIEM/BEM for 2D elasticity problems. Mechanics of Advanced Materials and Structures 24:1, pages 1-15.
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Wolfgang L. Wendland & S⊘ren Christiansen. (1986) On the condition number of the influence matrix belonging to some first kind integral equations with logarithmic kernel. Applicable Analysis 21:3, pages 175-183.
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Jeng-Tzong Chen, Jeng-Hong Kao, Shing-Kai Kao, Ying-Te Lee & Shyh-Rong Kuo. (2022) Study on the double-degeneracy mechanism of BEM/BIEM for a plane elasticity problem with line segments. Engineering Analysis with Boundary Elements 136, pages 290-302.
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Hossein Hosseinzadeh, Mehdi Dehghan & Zeynab Sedaghatjoo. (2020) The stability study of numerical solution of Fredholm integral equations of the first kind with emphasis on its application in boundary elements method. Applied Numerical Mathematics 158, pages 134-151.
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A. Corfdir & G. Bonnet. (2018) Degenerate scale for 2D Laplace equation with mixed boundary condition and comparison with other conditions on the boundary. Engineering Analysis with Boundary Elements 88, pages 14-25.
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Zi-Cai Li, Ming-Gong Lee, Hung-Tsai Huang & John Y. Chiang. (2017) Neumann problems of 2D Laplace's equation by method of fundamental solutions. Applied Numerical Mathematics 119, pages 126-145.
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A. Corfdir & G. Bonnet. (2017) Degenerate scale for 2D Laplace equation with Robin boundary condition. Engineering Analysis with Boundary Elements 80, pages 49-57.
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Zeynab Sedaghatjoo, Mehdi Dehghan & Hossein Hosseinzadeh. (2017) On uniqueness of numerical solution of boundary integral equations with 3-times monotone radial kernels. Journal of Computational and Applied Mathematics 311, pages 664-681.
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A. Corfdir & G. Bonnet. (2016) Exact degenerate scales in plane elasticity using complex variable methods. International Journal of Solids and Structures 80, pages 430-444.
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Y.Z. Chen. (2015) Approximate solution for degenerate scale problem of two rigid lines in series in plane elasticity. International Journal of Solids and Structures 52, pages 205-208.
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Roman Vodička & Marek Petrík. (2015) Degenerate scales for boundary value problems in anisotropic elasticity. International Journal of Solids and Structures 52, pages 209-219.
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Roman Vodička. (2013) An asymptotic property of degenerate scales for multiple holes in plane elasticity. Applied Mathematics and Computation 220, pages 166-175.
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A. Corfdir & G. Bonnet. (2013) Degenerate scale for the Laplace problem in the half-plane; Approximate logarithmic capacity for two distant boundaries. Engineering Analysis with Boundary Elements 37:5, pages 836-841.
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Yi-zhou Chen. (2012) Degenerate scale problem in antiplane elasticity or Laplace equation for contour shapes of triangles or quadrilaterals. Applied Mathematics and Mechanics 33:4, pages 525-538.
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Y.Z. Chen. (2011) Numerical solution for degenerate scale problem arising from multiple rigid lines in plane elasticity. Applied Mathematics and Computation 218:1, pages 96-106.
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Y. Z. Chen, X. Y. Lin & Z. X. Wang. (2009) Degenerate scale problem for plane elasticity in a multiply connected region with outer elliptic boundary. Archive of Applied Mechanics 80:9, pages 1055-1067.
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G. Bonnet, A. Seghir & A. Corfdir. (2009) Coupling BEM with FEM by a direct computation of the boundary stiffness matrix. Computer Methods in Applied Mechanics and Engineering 198:30-32, pages 2439-2445.
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W. Dijkstra & R.M.M. Mattheij. (2008) Condition number of the BEM matrix arising from the Stokes equations in 2D. Engineering Analysis with Boundary Elements 32:9, pages 736-746.
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W. Dijkstra & R. M. M. Mattheij. (2006) A relation between the logarithmic capacity and the condition number of the BEM‐matrices. Communications in Numerical Methods in Engineering 23:7, pages 665-680.
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J. T. Chen, S. R. Lin & K. H. Chen. (2004) Degenerate scale problem when solving Laplace's equation by BEM and its treatment. International Journal for Numerical Methods in Engineering 62:2, pages 233-261.
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S?ren Christiansen. (2001) Detecting non-uniqueness of solutions to biharmonic integral equations through SVD. Journal of Computational and Applied Mathematics 134:1-2, pages 23-35.
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S?ren Christiansen. (2001) Derivation and investigation of fifth order quadrature formulas for biharmonic boundary integral operators. Applied Numerical Mathematics 37:1-2, pages 145-159.
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S?ren Christiansen. (1998) Derivation and analytical investigation of three direct boundary integral equations for the fundamental biharmonic problem. Journal of Computational and Applied Mathematics 91:2, pages 231-247.
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Wen-jun He, Hao-jiang Ding & Hai-chang Hu. (1996) Non-equivalence of the conventional boundary integral formulation and its elimination for two-dimensional mixed potential problems. Computers & Structures 60:6, pages 1029-1035.
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He Wenjun, Ding Haojiang & Hu Haichang. (1993) Numerical comparison of two boundary element methods for plane harmonic functions. Acta Mechanica Sinica 9:4, pages 312-319.
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R.S.-C. Cheng. (1993) On Using a Modified Nyström Method to Solve the 2-D Potential Problem. Journal of Integral Equations and Applications 5:2.
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I. G. Graham & Y. Yan. (2009) Piecewise-constant collocation for first-kind boundary integral equations. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics 33:1, pages 39-64.
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S?ren Christiansen. (1990) A stability analysis of a Eulerian solution method for moving boundary problems in electrochemical machining. Journal of Computational and Applied Mathematics 33:3, pages 269-296.
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Yi Yan. (1990) The collocation method for first-kind boundary integral equations on polygonal regions. Mathematics of Computation 54:189, pages 139-154.
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G. Athanasiadis. (1989) Direkte und indirekte Randelementmethoden zur Bestimmung der Lage des Schubmittelpunktes beliebig geformter Stabquerschnitte. Forschung im Ingenieurwesen 55:2, pages 39-50.
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U. Heise. (1989) Condition of boundary integral equations in which the sought-for function and the given right-hand side are defined on different domains; round-off errors in the numerical solutions. Computational Mechanics 4:4, pages 245-259.
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Ulrich Heise. (1987) Dependence of the round-off error in the solution of boundary integral equations on a geometrical scale factor. Computer Methods in Applied Mechanics and Engineering 62:2, pages 115-126.
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Jean‐Louis Migeot. (2005) BEM—condition number of systems obtained when solving mixed boundary value potential problems. Communications in Applied Numerical Methods 2:4, pages 429-435.
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W. McLean. (1986) A spectral Galerkin method for a boundary integral equation. Mathematics of Computation 47:176, pages 597-607.
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S. Christiansen. (1985) Modifications of some first kind integral equations with logarithmic kernel to improve numerical conditioningModifikation einiger Integralgleichungen erster Art mit logarithmischem Kern zur Verbesserung ihrer numerischen Kondition. Computing 34:3, pages 221-242.
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G. Athanasiadis. (1985) Numerical investigations of direct and indirect integral equations for solving the heat conduction problem. Computer Methods in Applied Mechanics and Engineering 49:2, pages 203-220.
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Per Christian Hansen & S?ren Christiansen. (1985) An SVD analysis of linear algebraic equations derived from first kind integral equations. Journal of Computational and Applied Mathematics 12-13, pages 341-357.
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On two methods for elimination of non-unique solutions of an integral equation with logarithmic kernel/Footnote^†
A Very short version of part of this paper bas been presented on April 18, 1979 at Scientific Meeting of “Gesellschaft für Angewandte Mathematik und Mechanik” (GAMM) [6].

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On two methods for elimination of non-unique solutions of an integral equation with logarithmic kernel/Footnote†A Very short version of part of this paper bas been presented on April 18, 1979 at Scientific Meeting of “Gesellschaft für Angewandte Mathematik und Mechanik” (GAMM) [6].

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On two methods for elimination of non-unique solutions of an integral equation with logarithmic kernel/Footnote^†
A Very short version of part of this paper bas been presented on April 18, 1979 at Scientific Meeting of “Gesellschaft für Angewandte Mathematik und Mechanik” (GAMM) [6].