# The method of fundamental solutions for elliptic boundary value problems

Article

- 2k Downloads
- 599 Citations

## Abstract

The aim of this paper is to describe the development of the method of fundamental solutions (MFS) and related methods over the last three decades. Several applications of MFS-type methods are presented. Techniques by which such methods are extended to certain classes of non-trivial problems and adapted for the solution of inhomogeneous problems are also outlined.

elliptic boundary value problems fundamental solutions nonlinear least squares boundary collocation 65N38 65N99

## References

- [1]M.A. Aleksidze, On approximate solutions of a certain mixed boundary value problem in the theory of harmonic functions, Differential Equations 2 (1966) 515-518.zbMATHGoogle Scholar
- [2]E. Almansi, Sull' integrazione dell' equazione differentiale Δ
^{2n}= 0, Annali di Mathematica Pura et Applicata, Series III, 2 (1897) 1-51.Google Scholar - [3]K.E. Atkinson, The numerical evaluation of particular solutions for Poisson's equation, IMA J. Numer. Anal. 5 (1985) 319-338.zbMATHMathSciNetGoogle Scholar
- [4]P.K. Banerjee and R. Butterfield,
*Boundary Element Methods in Engineering Science*(McGraw-Hill, Maidenhead, 1981).zbMATHGoogle Scholar - [5]J.R. Berger and A. Karageorghis, The method of fundamental solutions for heat conduction in layered materials, preprint.Google Scholar
- [6]J.R. Berger, J. Skilowitz and V.K. Tewary, Green's function for steady-state heat conduction in a bimaterial composite solid, preprint.Google Scholar
- [7]A. Boag, Y. Leviatan and A. Boag, Analysis of acoustic scattering from fluid cylinders using a multifilament source model, J. Acoust. Soc. Amer. 83 (1988) 1-8.CrossRefGoogle Scholar
- [8]A. Boag, Y. Leviatan and A. Boag, Analysis of acoustic scattering from fluid bodies using a multipoint source model, IEEE Trans. Ultrason. Ferroelectrics and Frequency Control 36 (1989) 119-128.CrossRefGoogle Scholar
- [9]A. Bogomolny, Fundamental solutions method for elliptic boundary value problems, SIAM J. Numer. Anal. 22 (1985) 644-669.zbMATHMathSciNetCrossRefGoogle Scholar
- [10]J.F. Brady and G. Bossis, Stokesian dynamics, Ann. Rev. Fluid Mech. 20 (1988) 111-157.CrossRefGoogle Scholar
- [11]G. Burgess and E. Mahajerin, Rotational fluid flow using a least squares collocation technique, Comput. & Fluids 12 (1984) 311-317.zbMATHCrossRefGoogle Scholar
- [12]G. Burgess and E. Mahajerin, A comparison of the boundary element and superposition methods, Comput. & Structures 19 (1984) 697-705.zbMATHCrossRefGoogle Scholar
- [13]G. Burgess and E. Mahajerin, An analytical contour integration method for handling body forces in elasticity, Appl. Math. Modelling 9 (1985) 27-32.zbMATHCrossRefGoogle Scholar
- [14]G. Burgess and E. Mahajerin, The fundamental collocation method applied to the nonlinear Poisson equation in two dimensions, Comput. & Structures 27(1987) 763-767.zbMATHMathSciNetCrossRefGoogle Scholar
- [15]Y. Cao, W.W. Schultz and R.F. Beck, Three-dimensional desingularized boundary integral methods for potential problems, Internat. J. Numer. Methods Fluids 12 (1991) 785-803.zbMATHMathSciNetCrossRefGoogle Scholar
- [16]B.C. Carlson, A table of elliptic integrals of the third kind, Math. Comp. 51 (1988) 267-280.zbMATHMathSciNetCrossRefGoogle Scholar
- [17]J.E. Caruthers, private communication.Google Scholar
- [18]J.E. Caruthers, J.C. French and G.K. Raviprakash, Green's function discretization for numerical solution of the Helmholtz equation, J. Sound Vibration 187 (1995) 553-568.MathSciNetCrossRefGoogle Scholar
- [19]J.E. Caruthers, J.C. French and G.K. Raviprakash, Recent developments concerning a new discretization method for the Helmholtz equation, in:
*Proceedings of the First CEAS/AIAA Aeroacoustics Conference*, June 1995, Munich, Germany, Vol. II (1995) pp. 819-826.Google Scholar - [20]C.Y. Chan and C.S. Chen, Method of fundamental solutions for multi-dimensional quenching problems, Proceedings of Dynamic Systems and Applications 2 (1996) 115-122.zbMATHMathSciNetGoogle Scholar
- [21]C.S. Chen, The method of fundamental solutions for nonlinear thermal explosion, Comm. Numer. Methods Engrg. 11 (1995) 675-681.zbMATHMathSciNetCrossRefGoogle Scholar
- [22]C.S. Chen, The method of fundamental solutions and the quasi-Monte Carlo method for Poisson's equation, in:
*Lecture Notes in Statistics*106, eds. H. Niederreiter and P. Shuie (Springer, New York, 1995) pp. 158-167.Google Scholar - [23]C.S. Chen and M.A. Golberg, A domain embedding method and quasi-Monte Carlo method for Poisson's equation, in:
*BEM 17*, eds. C.A. Brebbia, S. Kim, T.A. Osswald and H. Power (Computational Mechanics Publications, Southampton, 1995) pp. 115-122.Google Scholar - [24]C.S. Chen and M.A. Golberg, Las Vegas method for diffusion equations, in:
*Boundary Element Technology XII*, eds. J.I. Frankel, C.A. Brebbia and M.A.H. Aliabadi (Computational Mechanics Publications, Southampton, 1997) pp. 299-308.Google Scholar - [25]R.S. Cheng, Delta-trigonometric and spline-trigonometric methods using the simple-layer potential representation, Ph.D. thesis, Applied Mathematics Department, University of Maryland (1987).Google Scholar
- [26]S. Christiansen, On Kupradze's functional equations for plane harmonic problems, in:
*Function Theoretic Methods in Differential Equations*, eds. R.P. Gilbert and R.J. Weinacht (Pitman, London, 1976) pp. 205-243.Google Scholar - [27]D.L. Clements, Fundamental solutions for second order linear elliptic partial differential equations, in:
*Fundamental Solutions in Boundary Elements: Formulation and Integration*, ed. F.G. Benitez (University of Sevilla, Sevilla, 1997) pp. 1-12.Google Scholar - [28]T. D{ie91-01}broś, A singularity method for calculating hydrodynamic forces and particle velocities in low-Reynolds-number flows, J. Fluid. Mech. 156 (1986) 1-21.Google Scholar
- [29]G. De Mey, Integral equations for potential problems with the source function not located on the boundary, Comput. & Structures 8 (1978) 113-115.zbMATHCrossRefGoogle Scholar
- [30]G. Fairweather and R.L. Johnston, The method of fundamental solutions for problems in potential theory, in:
*Treatment of Integral Equations by Numerical Methods*, eds. C.T.H. Baker and G.F. Miller (Academic Press, London, 1982) pp. 349-359.Google Scholar - [31]G. Fairweather, F.J. Rizzo, D.J. Shippy and Y.S. Wu, On the numerical solution of two-dimensional potential problems by an improved boundary integral equation method, J. Comput. Phys. 31 (1979) 96-112.zbMATHMathSciNetCrossRefGoogle Scholar
- [32]G. Fichera, Linear elliptic equations of higher order in two independent variables and singular integral equations with applications to anisotropic inhomogeneous elasticity, in:
*Partial Differential Equations and Continuum Mechanics*, ed. R.E. Langer (University of Wisconsin Press, Madison, 1961) pp. 55-80.Google Scholar - [33]R.T. Fenner, Source field superposition analysis of two-dimensional potential problems, Internat. J. Numer. Methods Engrg. 32 (1991) 1079-1091.zbMATHCrossRefGoogle Scholar
- [34]W. Freeden and H. Kersten, A constructive approximation theorem for the oblique derivative problem in potential theory, Math. Methods Appl. Sci. 3 (1981) 104-114.zbMATHMathSciNetGoogle Scholar
- [35]B.S. Garbow, K.E. Hillstrom and J.J. Moré,
*MINPACK Project*, Argonne National Laboratory (1980).Google Scholar - [36]M.A. Golberg, The method of fundamental solutions for Poisson's equation, Engrg. Anal. Boundary Elem. 16 (1995) 205-213.CrossRefGoogle Scholar
- [37]M.A. Golberg, Recent developments in the numerical evaluation of particular solutions in the boundary element method, Appl. Math. Comput. 75 (1996) 91-101.zbMATHMathSciNetCrossRefGoogle Scholar
- [38]M.A. Golberg and C.S. Chen, The theory of radial basis functions applied to the BEM for inhomogeneous partial differential equations, Boundary Elements Comm. 5 (1994) 57-61.Google Scholar
- [39]M.A. Golberg and C.S. Chen, On a method of Atkinson for evaluating domain integrals in the boundary element method, Appl. Math. Comput. 60 (1994) 125-138.zbMATHMathSciNetCrossRefGoogle Scholar
- [40]M.A. Golberg, C.S. Chen and S.R. Karur, Improved multiquadratic approximation for partial differential equations, Engrg. Anal. Boundary Elem. 18 (1996) 9-17.CrossRefGoogle Scholar
- [41]G. Gospodinov and D. Ljutskanov, The boundary element method applied to plates, Appl. Math. Modelling 6 (1982) 237-244.zbMATHCrossRefGoogle Scholar
- [42]A.K. Gupta, The boundary integral equation method for potential problems involving axisymmetric geometry and arbitrary boundary conditions, M.S. thesis, Department of Engineering Mechanics, University of Kentucky (1979).Google Scholar
- [43]P.S. Han and M.D. Olson, An adaptive boundary element method, Internat. J. Numer. Methods Engrg. 24 (1987) 1187-1202.zbMATHCrossRefGoogle Scholar
- [44]P.S. Han, M.D. Olson and R.L. Johnston, A Galerkin boundary element formulation with moving singularities, Engrg. Comput. 1 (1984) 232-236.Google Scholar
- [45]U. Heise, Numerical properties of integral equations in which the given boundary values and the sought solutions are defined on different curves, Comput. & Structures 8 (1978) 199-205.zbMATHMathSciNetCrossRefGoogle Scholar
- [46]U. Heise, Application of the singularity method for the formulation of plane elastostatical boundary value problems as integral equations, Acta Mechanica 31 (1978) 33-69.zbMATHMathSciNetCrossRefGoogle Scholar
- [47]I. Herrera,
*Boundary Methods: An Algebraic Theory*(Pitman, London, 1984).zbMATHGoogle Scholar - [48]M.J. Hopper, ed.,
*Harwell Subroutine Library Catalogue*, Theoretical Physics Division, AERE, Harwell, UK (1973).Google Scholar - [49]S. Ho-Tai, R.L. Johnston and R. Mathon, Software for solving boundary value problems for Laplace's equation using fundamental solutions, Technical Report 136/79, Department of Computer Science, University of Toronto (1979).Google Scholar
- [50]T. Inamuro, T. Saito and T. Adachi, A numerical analysis of unsteady separated flow by the discrete vortex method combined with the singularity method, Comput. & Structures 19 (1984) 75-84.zbMATHCrossRefGoogle Scholar
- [51]M.A. Jaswon and G.T. Symm,
*Integral Equation Methods in Potential Theory and Elastostatics*(Academic Press, New York, 1977).zbMATHGoogle Scholar - [52]D. Johnson, Plate bending by a boundary point method, Comput. & Structures 26 (1987) 673-680.zbMATHCrossRefGoogle Scholar
- [53]R.L. Johnston and G. Fairweather, The method of fundamental solutions for problems in potential flow, Appl. Math. Modelling 8 (1984) 265-270.zbMATHCrossRefGoogle Scholar
- [54]R.L. Johnston, G. Fairweather and P.S. Han, The method of fundamental solutions, an adaptive boundary element method, for problems in potential flow and solid mechanics, in:
*Proceedings of the 5th ASCE Specialty Conference*(Engineering Mechanics Division, Laramie, WY, 1984) pp. 140-143.Google Scholar - [55]R.L. Johnston, G. Fairweather and A. Karageorghis, An adaptive indirect boundary element methodt with applications, in:
*Boundary Elements VIII, Proceedings of the 8th International Conference*, Tokyo, Japan, September 1986, Vol. II, eds. M. Tanaka and C. Brebbia (Springer, New York, 1987) pp. 587-598.Google Scholar - [56]R.L. Johnston and R. Mathon, The computation of electric dipole fields in conducting media, Internat. J. Numer. Methods Engrg. 14 (1979) 1739-1760.CrossRefGoogle Scholar
- [57]A. Karageorghis, Modified methods of fundamental solutions for harmonic and biharmonic problems with boundary singularities, Numer. Methods Partial Differential Equations 8 (1992) 1-19.zbMATHMathSciNetCrossRefGoogle Scholar
- [58]A. Karageorghis, The method of fundamental solutions for the solution of steady-state free boundary problems, J. Comput. Phys. 98 (1992) 119-128.zbMATHCrossRefGoogle Scholar
- [59]A. Karageorghis and G. Fairweather, The method of fundamental solutions for the numerical solution of the biharmonic equation, J. Comput. Phys. 69 (1987) 434-459.zbMATHMathSciNetCrossRefGoogle Scholar
- [60]A. Karageorghis and G. Fairweather, The Almansi method of fundamental solutions for solving biharmonic problems, Internat. J. Numer. Methods Engrg. 26 (1988) 1668-1682.Google Scholar
- [61]A. Karageorghis and G. Fairweather, The simple layer potential method of fundamental solutions for certain biharmonic problems, Internat. J. Numer. Methods Fluids 9 (1989) 1221-1234.zbMATHMathSciNetCrossRefGoogle Scholar
- [62]A. Karageorghis and G. Fairweather, The method of fundamental solutions for the solution of nonlinear plane potential problems, IMA J. Numer. Anal. 9 (1989) 231-242.zbMATHMathSciNetGoogle Scholar
- [63]A. Karageorghis and G. Fairweather, The method of fundamental solutions for axisymmetric potential problems, Technical Report 01/98, Department of Mathematics and Statistics, University of Cyprus (1998).Google Scholar
- [64]A. Karageorghis and G. Fairweather, The method of fundamental solutions for axisymmetric acoustic scattering and radiation problems, Technical Report 02/98, Department of Mathematics and Statistics, University of Cyprus (1998).Google Scholar
- [65]M. Katsurada, A mathematical study of the charge simulation method II, J. Fac. Sci., Univ. of Tokyo, Sect. 1A, Math. 36 (1989) 135-162.zbMATHMathSciNetGoogle Scholar
- [66]M. Katsurada, Asymptotic error analysis of the charge simulation method in a Jordan region with an analytic boundary, J. Fac. Sci., Univ. of Tokyo, Sect. 1A, Math. 37 (1990) 635-657.zbMATHMathSciNetGoogle Scholar
- [67]M. Katsurada, Charge simulation method using exterior mapping functions, Japan J. Indust. Appl. Math. 11 (1994) 47-61.zbMATHMathSciNetCrossRefGoogle Scholar
- [68]M. Katsurada and H. Okamoto, A mathematical study of the charge simulation method I, J. Fac. Sci., Univ. of Tokyo, Sect. 1A, Math. 35 (1988) 507-518.zbMATHMathSciNetGoogle Scholar
- [69]M. Katsurada and H. Okamoto, The collocation points of the fundamental solution method for the potential problem, Comput. Math. Appl. 31 (1996) 123-137.zbMATHMathSciNetCrossRefGoogle Scholar
- [70]M. Keshavarzi, A modified integral equation applied to problems of elastostatics, Comput. Methods Appl. Mech. Engrg. 16 (1978) 1-9.zbMATHCrossRefGoogle Scholar
- [71]S. Kim and S.J. Karrila,
*Microhydrodynamics: Principles and Selected Applications*(Butterworth-Heinemann, Stoneham, 1991).Google Scholar - [72]T. Kitagawa, On the numerical stability of the method of fundamental solution applied to the Dirichlet problem, Japan J. Appl. Math. 5 (1988) 123-133.zbMATHMathSciNetGoogle Scholar
- [73]T. Kitagawa, Asymptotic stability of the fundamental solution method, J. Comput. Appl. Math. 38 (1991) 263-269.zbMATHMathSciNetCrossRefGoogle Scholar
- [74]J.A. Kolodziej, Review of application of boundary collocation methods in mechanics of continuous media, Solid Mech. Arch. 12 (1987) 187-231.zbMATHGoogle Scholar
- [75]P.S. Kondapalli, Time-harmonic solutions in acoustics and elastodynamics by the method of fundamental solutions, Ph.D. thesis, Department of Engineering Mechanics, University of Kentucky (1991).Google Scholar
- [76]P.S. Kondapalli, D.J. Shippy and G. Fairweather, Analysis of acoustic scattering in fluids and solids by the method of fundamental solutions, J. Acoust. Soc. Amer. 91 (1992) 1844-1854.CrossRefGoogle Scholar
- [77]P.S. Kondapalli, D.J. Shippy and G. Fairweather, The method of fundamental solutions for transmission and scattering of elastic waves, Comput. Methods Appl. Mech. Engrg. 96 (1992) 255-269.zbMATHCrossRefGoogle Scholar
- [78]G.H. Koopman, L. Song and J.B. Fahnline, A method for computing acoustic fields based on the principle of wave superposition, J. Acoust. Soc. Amer. 86 (1989) 2433-2438.CrossRefGoogle Scholar
- [79]R. Kress and A. Mohsen, On the simulation source technique for exterior problems in acoustics, Math. Methods Appl. Sci. 8 (1986) 585-597.zbMATHMathSciNetGoogle Scholar
- [80]V.D. Kupradze, A method for the approximate solution of limiting problems in mathematical physics, Comput. Math. Math. Phys. 4 (1964) 199-205.zbMATHCrossRefGoogle Scholar
- [81]V.D. Kupradze,
*Potential Methods in the Theory of Elasticity*(Israel Program for Scientific Translations, Jerusalem, 1965).zbMATHGoogle Scholar - [82]V.D. Kupradze, On the approximate solution of problems in mathematical physics, Russian Math. Surveys 22 (1967) 58-108.MathSciNetCrossRefGoogle Scholar
- [83]V.D. Kupradze and M.A. Aleksidze, The method of functional equations for the approximate solution of certain boundary value problems, Comput. Math. Math. Phys. 4 (1964) 82-126.zbMATHMathSciNetCrossRefGoogle Scholar
- [84]D. Levin and A. Tal, A boundary collocation method for the solution of a flow problem in a complex three-dimensional porous medium, Internat. J. Numer. Methods Fluids 6 (1986) 611-622.zbMATHCrossRefGoogle Scholar
- [85]S.A. Lifits and S. Yu. Reutsky, The method of fundamental solutions and singular expansions for the numerical solution of the elliptic boundary-value problems with singularities, Prépublication 41, Institut de Mathématiques de Jussieu, Unité Mixte de Recherche 9994, Universités Paris VI et Paris VII/CNRS (October 1995).Google Scholar
- [86]M. MacDonell, A boundary method applied to the modified Helmholtz equation in three dimensions and its application to a waste disposal problem in the deep ocean, M.S. thesis, Department of Computer Science, University of Toronto (1985).Google Scholar
- [87]E. Mahajerin, An extension of the superposition method for plane anisotropic elastic bodies, Comput. & Structures 21 (1985) 953-958.zbMATHCrossRefGoogle Scholar
- [88]M. Maiti and S.K. Chakrabarty, Integral equations solutions for simply supported polygonal plates, Internat. J. Engrg. Sci. 12 (1974) 793-806.zbMATHCrossRefGoogle Scholar
- [89]R. Mathon and R.L. Johnston, The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM J. Numer. Anal. 14 (1977) 638-650.zbMATHMathSciNetCrossRefGoogle Scholar
- [90]S. Murashima and H. Kuahara, An approximate method to solve two-dimensional Laplace's equation by means of superposition of Green's functions on a Riemann surface, J. Inform. Process. 3 (1980) 127-139.zbMATHMathSciNetGoogle Scholar
- [91]S. Murashima and Y. Nonaka, Interactive Laplace's equation analyzing system ILAS, in:
*Boundary Elements VIII, Proceedings of the 8th International Conference*, Tokyo, Japan, 1986, eds. M. Tanaka and C.A. Brebbia (Springer, Berlin, 1987) pp. 621-630.Google Scholar - [92]S. Murashima, Y. Nonaka and H. Nieda, The charge simulation method and its applications to two-dimensional elasticity, in:
*Boundary Elements V, Proceedings of the 5th International Conference*, Hiroshima, Japan, 1983, eds. C.A. Brebbia, T. Fugatami and M. Tanaka (Springer, Berlin, 1983) pp. 75-80.Google Scholar - [93]Y. Niwa, S. Kobayashi and T. Fukui, An application of the integral equation method to plate-bending problems, Mem. Fac. Eng. Kyoto Univ. (Japan) 36 (1974) 140-158.Google Scholar
- [94]Numerical Algorithms Group Library, NAG(UK) Ltd, Oxford, UK.Google Scholar
- [95]E.R. Oliveira, Plane stress analysis by a general integral method, J. Engrg. Mech. Div. ASCE 94 (1968) 79-101.Google Scholar
- [96]Y.H. Pao and V. Varathatajulu, Huygens' principle, radiation conditions and integral formulae for the scattering of elastic waves, Department of Theoretical and Applied Mechanics, Cornell University (1975).Google Scholar
- [97]P.W. Partridge, C.A. Brebbia and L.C. Wrobel,
*The Dual Reciprocity Boundary Element Method*(Computational Mechanics Publications, Southampton, 1992).zbMATHGoogle Scholar - [98]C. Patterson and M.A. Sheikh, Regular boundary integral equations for stress analysis, in:
*Boundary Element Methods, Proceedings of the 3rd International Seminar on Boundary Element Methods*, Irvine, CA, 1981, ed. C.A. Brebbia (Springer, New York, 1981) pp. 85-104.Google Scholar - [99]C. Patterson and M.A. Sheikh, On the use of fundamental solutions in Trefftz method for potential and elasticity problems, in:
*Boundary Element Methods in Engineering, Proceedings of the Fourth International Seminar on Boundary Element Methods*, Southampton, 1982, ed. C.A. Brebbia (Springer, New York, 1982) pp. 43-54.Google Scholar - [100]A. Poullikkas, The method of fundamental solutions for the solution of elliptic boundary value problems, Ph.D. thesis, Department of Mechanical Engineering, Loughborough University (1998).Google Scholar
- [101]A. Poullikkas, A. Karageorghis and G. Georgiou, Methods of fundamental solutions for harmonic and biharmonic boundary value problems, Comput. Mech. 21 (1998) 416-423.zbMATHMathSciNetCrossRefGoogle Scholar
- [102]A. Poullikkas, A. Karageorghis and G. Georgiou, The method of fundamental solutions for Signorini problems, IMA J. Numer. Anal. 18 (1998) 273-285.zbMATHMathSciNetCrossRefGoogle Scholar
- [103]A. Poullikkas, A. Karageorghis, G. Georgiou and J. Ascough, The method of fundamental solutions for Stokes flows with a free surface, Numer. Methods Partial Differential Equations, to appear.Google Scholar
- [104]A. Poullikkas, A. Karageorghis and G. Georgiou, The method of fundamental solutions for inhomogeneous elliptic problems, Comput. Mech., to appear.Google Scholar
- [105]C. Pozrikidis,
*Boundary Integral and Singularity Methods for Linearized Viscous Flow*(Cambridge University Press, Cambridge, 1992).zbMATHGoogle Scholar - [106]J. Raamachandran and C. Rajamohan, Analysis of composite plates using charge simulation method, Engrg. Anal. Boundary Elem. 18 (1996) 131-135.CrossRefGoogle Scholar
- [107]D. Redekop, Fundamental solutions for the collocation method in planar elastostatics, Appl. Math. Modelling 6 (1982) 390-393.zbMATHCrossRefGoogle Scholar
- [108]D. Redekop and R.S.W. Cheung, Fundamental solutions for the collocation method in three-dimensional elastostatics, Comput. & Structures 26 (1987) 703-707.zbMATHCrossRefGoogle Scholar
- [109]D. Redekop and J.C. Thompson, Use of fundamental solutions in the collocation method in axisymmetric elastostatics, Comput. & Structures 17 (1983) 485-490.CrossRefGoogle Scholar
- [110]F.J. Rizzo and D.J. Shippy, A boundary integral approach to potential and elasticity problems for axisymmetric bodies with arbitrary boundary conditions, Mech. Res. Comm. 6 (1979) 99-103.zbMATHCrossRefGoogle Scholar
- [111]A.F. Seybert, B. Soenarko, F.J. Rizzo and D.J. Shippy, A special integral equation formulation for acoustic radiation and scattering for axisymmetric bodies and boundary conditions, J. Acoust. Soc. Amer. 80 (1986) 1241-1247.CrossRefGoogle Scholar
- [112]D.J. Shippy, P.S. Kondapalli and G. Fairweather, Analysis of acoustic scattering in fluids and solids by the method of fundamental solutions, Math. Comput. Modelling 14 (1990) 74-79.zbMATHCrossRefGoogle Scholar
- [113]N. Simos and A.M. Sadegh, An indirect BIM for static analysis of spherical shells using auxiliary boundaries, Internat. J. Numer. Methods Engrg. 32 (1991) 313-325.zbMATHCrossRefGoogle Scholar
- [114]L. Song, G.H. Koopman and J.B. Fahnline, Numerical errors associated with the method of superposition for computing acoustic fields, J. Acoust. Soc. Amer. 89 (1991) 2625-2633.CrossRefGoogle Scholar
- [115]L. Song, G.H. Koopman and J.B. Fahnline, Active control of the acoustic radiation of a vibrating structure using a superposition formulation, J. Acoust. Soc. Amer. 89 (1991) 2786-2792.CrossRefGoogle Scholar
- [116]S.P. Walker, Diffusion problems using transient discrete source superposition, Internat. J. Numer. Methods Engrg. 35 (1992) 165-178.zbMATHCrossRefGoogle Scholar
- [117]J.L. Wearing and O. Bettahar, The analysis of plate bending problems using the regular direct boundary element method, Engrg. Anal. Boundary Elem. 16 (1995) 261-271.CrossRefGoogle Scholar
- [118]W.C. Webster, The flow about arbitrary, three-dimensional smooth bodies, J. Ship Res. 19 (1975) 206-218.Google Scholar
- [119]S. Weinbaum, P. Ganatos and Z.-Y. Yan, Numerical multipole and boundary integral equation techniques in Stokes flow, Ann. Rev. Fluid Mech. 22 (1990) 275-316.zbMATHMathSciNetCrossRefGoogle Scholar
- [120]T. Westphal, C.S. de Barcellos and J. Tomás Pereira, On general fundamental solutions of some linear elliptic differential operators, Engrg. Anal. Boundary Elem. 17 (1996) 279-285.CrossRefGoogle Scholar
- [121]B.C. Wu and N.J. Altiero, A boundary integral method applied to plates of arbitrary plan form and arbitrary boundary conditions, Comput. & Structures 10 (1979) 703-707.zbMATHMathSciNetCrossRefGoogle Scholar
- [122]H. Zhou and C. Pozrikidis, Adaptive singularity method for Stokes flow past particles, J. Comput. Phys. 117 (1995) 79-89.zbMATHCrossRefGoogle Scholar
- [123]C.S. Chen, M.A. Golberg and Y.C. Hon, Numerical justification of fundamental solutions and the quasi-Monte Carlo method for Poisson-type equations, Engrg. Anal. Boundary Elem., to appear.Google Scholar
- [124]C.S. Chen, M.A. Golberg and Y.C. Hon, Las Vegas method for diffusion equations, Internat. J. Numer. Methods Engrg., to appear.Google Scholar

## Copyright information

© Kluwer Academic Publishers 1998