Advertisement

Boundary-Type RBF Collocation Methods

  • Wen ChenEmail author
  • Zhuo-Jia Fu
  • C. S. Chen
Chapter
Part of the SpringerBriefs in Applied Sciences and Technology book series (BRIEFSAPPLSCIENCES)

Abstract

The mesh generation in the standard BEM is still not trivial as one may imagine, especially for high-dimensional moving boundary problems. To overcome this difficulty, the boundary-type RBF collocation methods have been proposed and endured a fast development in the recent decade thanks to being integration-free, spectral convergence, easy-to-use, and inherently truly meshless. First, this chapter introduces the basic concepts of the method of fundamental solutions (MFS). Then a few recent boundary-type RBF collocation schemes are presented to tackle the issue of the fictitious boundary in the MFS, such as boundary knot method (BKM), regularized meshless method, and singular boundary method. Following this, an improved multiple reciprocity method (MRM), the recursive composite MRM (RC-MRM), is introduced to establish a boundary-only discretization of nonhomogeneous problems. Finally, numerical demonstrations show the convergence rate and stability of these boundary-type RBF collocation methods for several benchmark examples.

Keywords

Meshless Integration-free Collocation Fundamental solutions Singularity Method of fundamental solutions Boundary knot method Regularized meshless method Singular boundary method Boundary particle method 

References

  1. 1.
    V.D. Kupradze, M.A. Aleksidze, The method of functional equations for the approximate solution of certain boundary value problems. USSR Comput. Math. Math. Phys. 4(4), 82–126 (1964)MathSciNetCrossRefGoogle Scholar
  2. 2.
    W. Chen, M. Tanaka, A meshless, integration-free, and boundary-only RBF technique. Comput. Math. Appl. 43(3–5), 379–391 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    D.L. Young, K.H. Chen, C.W. Lee, Novel meshless method for solving the potential problems with arbitrary domain. J. Comput. Phys. 209(1), 290–321 (2005)CrossRefzbMATHGoogle Scholar
  4. 4.
    D.L. Young, K.H. Chen, J.T. Chen, J.H. Kao, A modified method of fundamental solutions with source on the boundary for solving laplace equations with circular and arbitrary domains. CMES Comput. Model. Eng. Sci. 19(3), 197–221 (2007)MathSciNetzbMATHGoogle Scholar
  5. 5.
    D.L. Young, K.H. Chen, T.Y. Liu, L.H. Shen, C.S. Wu, Hypersingular meshless method for solving 3D potential problems with arbitrary domain. CMES Comput. Model. Eng. Sci. 40(3), 225–269 (2009)MathSciNetGoogle Scholar
  6. 6.
    C.S. Chen, A. Karageorghis, Y.S. Smyrlis, The Method of Fundamental Solutions—A Meshless Method (Dynamic Publishers, Atlanta, 2008)Google Scholar
  7. 7.
    W. Chen, Z.J. Fu, X. Wei, Potential problems by singular boundary method satisfying moment condition. CMES Comput. Model. Eng. Sci. 54(1), 65–85 (2009)MathSciNetzbMATHGoogle Scholar
  8. 8.
    W. Chen, F.Z. Wang, A method of fundamental solutions without fictitious boundary. Eng. Anal. Boundary Elem. 34(5), 530–532 (2010)CrossRefzbMATHGoogle Scholar
  9. 9.
    P.W. Partridge, C.A. Brebbia, L.C. Wrobel, The Dual Reciprocity Boundary Element Method (Computational Mechanics Publications, Southampton, 1992)Google Scholar
  10. 10.
    A.J. Nowak, A.C. Neves, The Multiple Reciprocity Boundary Element Method (Computational Mechanics Publication, Southampton, 1994)zbMATHGoogle Scholar
  11. 11.
    K.K. Prem, in Fundamental Solutions for Differential Operators and Applications (Birkhauser Boston Inc., Cambridge, 1996)Google Scholar
  12. 12.
    W. Chen, Z.J. Shen, L.J. Shen, G.W. Yuan, General solutions and fundamental solutions of varied orders to the vibrational thin, the Berger, and the Winkler plates. Eng. Anal. Boundary Elem. 29(7), 699–702 (2005)CrossRefzbMATHGoogle Scholar
  13. 13.
    W. Chen, Z.J. Fu, B.T. Jin, A truly boundary-only meshfree method for inhomogeneous problems based on recursive composite multiple reciprocity technique. Eng. Anal. Boundary Elem. 34(3), 196–205 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    G.H. Koopmann, L. Song, J.B. Fahnline, A method for computing acoustic fields based on the principle of wave superposition. J. Acoust. Soc. Am. 86(6), 2433–2438 (1989)CrossRefGoogle Scholar
  15. 15.
    C. Yusong, W.S. William, F.B. Robert, Three-dimensional desingularized boundary integral methods for potential problems. Int. J. Numer. Meth. Fluids 12(8), 785–803 (1991)CrossRefzbMATHGoogle Scholar
  16. 16.
    K. Amano, A charge simulation method for the numerical conformal mapping of interior, exterior and doubly-connected domains. J. Comput. Appl. Math. 53(3), 353–370 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    C.S. Chen, The method of fundamental-solutions for nonlinear thermal explosions. Commun. Numer. Methods Eng. 11(8), 675–681 (1995)CrossRefzbMATHGoogle Scholar
  18. 18.
    S. Chantasiriwan, Methods of fundamental solutions for time-dependent heat conduction problems. Int. J. Numer. Meth. Eng. 66(1), 147–165 (2006)CrossRefzbMATHGoogle Scholar
  19. 19.
    L.L. Cao, Q.H. Qin, N. Zhao, An RBF-MFS model for analysing thermal behaviour of skin tissues. Int. J. Heat Mass Transf. 53(7–8), 1298–1307 (2010)CrossRefzbMATHGoogle Scholar
  20. 20.
    P.S. Kondapalli, D.J. Shippy, G. Fairweather, Analysis of acoustic scattering in fluids and solids by the method of fundamental-solutions. J. Acoust. Soc. Am. 91(4), 1844–1854 (1992)CrossRefGoogle Scholar
  21. 21.
    J. Antonio, A. Tadeu, L. Godinho, A three-dimensional acoustics model using the method of fundamental solutions. Eng. Anal. Boundary Elem. 32(6), 525–531 (2008)CrossRefzbMATHGoogle Scholar
  22. 22.
    K. Balakrishnan, P.A. Ramachandran, The method of fundamental solutions for linear diffusion-reaction equations. Math. Comput. Model. 31(2–3), 221–237 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    S.P. Hu, D.L. Young, C.M. Fan, FDMFS for diffusion equation with unsteady forcing function. CMES Comput. Model. Eng. Sci. 24(1), 1–20 (2008)MathSciNetzbMATHGoogle Scholar
  24. 24.
    A. Karageorghis, G. Fairweather, The method of fundamental solutions for axisymmetric elasticity problems. Comput. Mech. 25(6), 524–532 (2000)CrossRefzbMATHGoogle Scholar
  25. 25.
    D.L. Young, C.L. Chiu, C.M. Fan, C.C. Tsai, Y.C. Lin, Method of fundamental solutions for multidimensional Stokes equations by the dual-potential formulation. Eur. J. Mech. B Fluids 25(6), 877–893 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    D.L. Young, S.J. Jane, C.M. Fan, K. Murugesan, C.C. Tsai, The method of fundamental solutions for 2D and 3D stokes problems. J. Comput. Phys. 211(1), 1–8 (2006)CrossRefzbMATHGoogle Scholar
  27. 27.
    P.P. Chinchapatnam, K. Djidjeli, P.B. Nair, Radial basis function meshless method for the steady incompressible Navier-Stokes equations. Int. J. Comput. Math. 84, 1509–1526 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    D.L. Young, Y.C. Lin, C.M. Fan, C.L. Chiu, The method of fundamental solutions for solving incompressible Navier-Stokes problems. Eng. Anal. Boundary Elem. 33(8–9), 1031–1044 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    J.T. Chen, I.L. Chen, K.H. Chen, Y.T. Lee, Y.T. Yeh, A meshless method for free vibration analysis of circular and rectangular clamped plates using radial basis function. Eng. Anal. Boundary Elem. 28(5), 535–545 (2004)CrossRefzbMATHGoogle Scholar
  30. 30.
    J.T. Chen, I.L. Chen, Y.T. Lee, Eigensolutions of multiply connected membranes using the method of fundamental solutions. Eng. Anal. Boundary Elem. 29(2), 166–174 (2005)CrossRefzbMATHGoogle Scholar
  31. 31.
    C.J.S. Alves, P.R.S. Antunes, The method of fundamental solutions applied to the calculation of eigensolutions for 2D plates. Int. J. Numer. Meth. Eng. 77(2), 177–194 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Y.S. Smyrlis, A. Karageorghis, A linear least-squares MFS for certain elliptic problems. Numer. Algorithms 35(1), 29–44 (2004)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Y.S. Smyrlis, A. Karageorghis, A matrix decomposition MFS algorithm for axisymmetric potential problems. Eng. Anal. Boundary Elem. 28(5), 463–474 (2004)CrossRefzbMATHGoogle Scholar
  34. 34.
    G. Fairweather, A. Karageorghis, Y.S. Smyrlis, A matrix decomposition MFS algorithm for axisymmetric biharmonic problems. Adv. Comput. Math. 23(1–2), 55–71 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Y.S. Smyrlis, The method of fundamental solutions: a weighted least-squares approach. Bit Numer. Math. 46(1), 163–194 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    A. Karageorghis, C.S. Chen, Y.S. Smyrlis, A matrix decomposition RBF algorithm: approximation of functions and their derivatives. Appl. Numer. Math. 57(3), 304–319 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    T.W. Drombosky, A.L. Meyer, L.V. Ling, Applicability of the method of fundamental solutions. Eng. Anal. Boundary Elem. 33(5), 637–643 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    L. Marin, An alternating iterative MFS algorithm for the cauchy problem in two-dimensional anisotropic heat conduction. CMC Comput. Mater. Con. 12(1), 71–99 (2009)Google Scholar
  39. 39.
    J. Lin, W. Chen, F. Wang, A new investigation into regularization techniques for the method of fundamental solutions. Math. Comput. Simul. 81(6), 1144–1152 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    G. Fairweather, A. Karageorghis, The method of fundamental solutions for elliptic boundary value problems. Adv. Comput. Math. 9(1–2), 69–95 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    G. Fairweather, A. Karageorghis, P.A. Martin, The method of fundamental solutions for scattering and radiation problems. Eng. Anal. Boundary Elem. 27(7), 759–769 (2003)CrossRefzbMATHGoogle Scholar
  42. 42.
    C.S. Liu, Improving the ill-conditioning of the method of fundamental solutions for 2D Laplace equation. CMES Comput. Model. Eng. Sci. 28(2), 77–93 (2008)MathSciNetzbMATHGoogle Scholar
  43. 43.
    W. Chen, Symmetric boundary knot method. Eng. Anal. Boundary Elem. 26(6), 489–494 (2002)CrossRefzbMATHGoogle Scholar
  44. 44.
    F. Wang, W. Chen, X. Jiang, Investigation of regularized techniques for boundary knot method. Int. J. Numer. Methods Biomed. Eng. 26(12), 1868–1877 (2010)MathSciNetzbMATHGoogle Scholar
  45. 45.
    W. Chen, Y.C. Hon, Numerical investigation on convergence of boundary knot method in the analysis of homogeneous Helmholtz, modified Helmholtz, and convection-diffusion problems. Comput. Methods Appl. Mech. Eng. 192(15), 1859–1875 (2003)CrossRefzbMATHGoogle Scholar
  46. 46.
    Y.C. Hon, W. Chen, Boundary knot method for 2D and 3D Helmholtz and convection–diffusion problems under complicated geometry. Int. J. Numer. Meth. Eng. 56(13), 1931–1948 (2003)CrossRefzbMATHGoogle Scholar
  47. 47.
    X.P. Chen, W.X. He, B.T. Jin, Symmetric boundary knot method for membrane vibrations under mixed-type boundary conditions. Int. J. Nonlinear Sci. Numer. Simul. 6(4), 421–424 (2005)CrossRefGoogle Scholar
  48. 48.
    J. Shi, W. Chen, C. Wang, Free vibration analysis of arbitrary shaped plates by boundary knot method. Acta Mech. Solida Sin. 22(4), 328–336 (2009)CrossRefGoogle Scholar
  49. 49.
    Y.C. Hon, Z. Wu, A numerical computation for inverse boundary determination problem. Eng. Anal. Boundary Elem. 24(7–8), 599–606 (2000)CrossRefzbMATHGoogle Scholar
  50. 50.
    R.C. Song, W. Chen, An investigation on the regularized meshless method for irregular domain problems. CMES Comput. Model. Eng. Sci. 42(1), 59–70 (2009)MathSciNetGoogle Scholar
  51. 51.
    L.L. Sun, W. Chen, C.Z. Zhang, A new formulation of regularized meshless method applied to interior and exterior anisotropic potential problems. Appl. Math. Model. 37(12–13), 7452–7464 (2013) Google Scholar
  52. 52.
    K.H. Chen, J.T. Chen, J.H. Kao, Regularized meshless method for solving acoustic eigenproblem with multiply-connected domain. CMES Comput. Model. Eng. Sci. 16(1), 27–39 (2006)MathSciNetGoogle Scholar
  53. 53.
    D.L. Young, K.H. Chen, C.W. Lee, Singular meshless method using double layer potentials for exterior acoustics. J. Acoust. Soc. Am. 119(1), 96–107 (2006)CrossRefGoogle Scholar
  54. 54.
    K.H. Chen, J.T. Chen, J.H. Kao, Regularized meshless method for antiplane shear problems with multiple inclusions. Int. J. Numer. Meth. Eng. 73(9), 1251–1273 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    K.H. Chen, J.H. Kao, J.T. Chen, Regularized meshless method for antiplane piezoelectricity problems with multiple inclusions. CMC Comput. Mater. Con. 9(3), 253–279 (2009)MathSciNetGoogle Scholar
  56. 56.
    W. Chen, Z.J. Fu, A novel numerical method for infinite domain potential problems. Chin. Sci. Bull. 55(16), 1598–1603 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Y. Gu, W. Chen, C.-Z. Zhang, Singular boundary method for solving plane strain elastostatic problems. Int. J. Solids Struct. 48(18), 2549–2556 (2011)CrossRefGoogle Scholar
  58. 58.
    X. Wei, W. Chen, Z.J. Fu, Solving inhomogeneous problems by singular boundary method. J. Marine Sci. Technol. Taiwan 21(1), 8–14 (2013)Google Scholar
  59. 59.
    W. Chen, Y. Gu, Recent advances on singular boundary method. Joint international workshop on Trefftz method VI and method of fundamental solution II (Taiwan 2011)Google Scholar
  60. 60.
    W. Chen, Y. Gu, An improved formulation of singular boundary method. Adv. Appl. Math. Mech. 4(5), 543–558 (2012)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Z.J. Fu, W. Chen, C.S. Chen, Singular boundary method for radiation and wave scattering: numerical aspects and applications. Paper presented at the 23rd international congress of theoretical and applied mechanics (ICTAM2012), BeijingGoogle Scholar
  62. 62.
    Z.J. Fu, W. Chen, J. Lin, Improved singular boundary method for various infinite-domain wave applications. Paper presented at the global Chinese workshop in conjunction with 10th national conference on computational methods in engineering, ChangshaGoogle Scholar
  63. 63.
    Y. Gu, W. Chen, X.Q. He, Singular boundary method for steady-state heat conduction in three dimensional general anisotropic media. Int. J. Heat Mass Transf. 55, 4837–4848 (2012)Google Scholar
  64. 64.
    Y. Gu, W. Chen, J. Zhang, Investigation on near-boundary solutions by singular boundary method. Eng. Anal. Boundary Elem. 36(8), 1173–1182 (2012)MathSciNetCrossRefGoogle Scholar
  65. 65.
    Y. Gu, W. Chen, Infinite domain potential problems by a new formulation of singular boundary method. Appl. Math. Model. 37(4), 1638–1651 (2013)MathSciNetCrossRefGoogle Scholar
  66. 66.
    C.S. Chen, Y.C. Hon, R.S. Schaback, Radial basis functions with scientific computation. Department of Mathematics, University of Southern Mississippi, USA (2007)Google Scholar
  67. 67.
    Z.J. Fu, W. Chen, A novel boundary meshless method for radiation and scattering problems, ed. by C.Z. Zhang, M.H. Aliabadi, M. Schanz. Advances in Boundary Element Techniques XI, Berlin, Germany (EC Ltd, United Kingdom, 12–14 July 2010), pp. 83–90. Google Scholar
  68. 68.
    Y. Gu, W. Chen, X.Q. He, Singular boundary method for steady-state heat conduction in three dimensional general anisotropic media. Int. J. Heat Mass Transf. 55(17–18), 4837–4848 (2012)CrossRefGoogle Scholar
  69. 69.
    W. Chen, Z.J. Fu, Y. Gu, Burton-Miller-type singular boundary method for acoustic radiation and scattering. J. Sound Vib. submitted (2013)Google Scholar
  70. 70.
    K.E. Atkinson, The numerical evaluation of particular solutions for Poisson’s equation. IMA J. Numer. Anal. 5, 319–338 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    M.A. Golberg, The method of fundamental solutions for Poisson’s equation. Eng. Anal. Boundary Elem. 16(3), 205–213 (1995)CrossRefGoogle Scholar
  72. 72.
    M.A. Golberg, C.S. Chen, S.R. Karur, Improved multiquadric approximation for partial differential equations. Eng. Anal. Boundary Elem. 18(1), 9–17 (1996)CrossRefGoogle Scholar
  73. 73.
    C.S. Chen, C.A. Brebbia, H. Power, Dual reciprocity method using compactly supported radial basis functions. Commun. Numer. Methods Eng. 15(2), 137–150 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  74. 74.
    M.A. Golberg, C.S. Chen, M. Ganesh, Particular solutions of 3D Helmholtz-type equations using compactly supported radial basis functions. Eng. Anal. Boundary Elem. 24(7–8), 539–547 (2000)CrossRefzbMATHGoogle Scholar
  75. 75.
    S. Chantasiriwan, Cartesian grid methods using radial basis functions for solving Poisson, Helmholtz, and diffusion-convection equations. Eng. Anal. Boundary Elem. 28(12), 1417–1425 (2004)CrossRefzbMATHGoogle Scholar
  76. 76.
    C.J.S. Alves, C.S. Chen, A new method of fundamental solutions applied to nonhomogeneous elliptic problems. Adv. Comput. Math. 23(1–2), 125–142 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  77. 77.
    C. Erdonmez, H. Saygin, Conduction heat transfer problem solution using the method of fundamental solutions with the dual reciprocity method. HT2005: Proceedings of the ASME Summer Heat Transfer Conference 2005, vol. 3 (2005), pp. 853–858Google Scholar
  78. 78.
    C.C. Tsai, The method of fundamental solutions with dual reciprocity for thin plates on Winkler foundations with arbitrary loadings. J. Mech. 24(2), 163–171 (2008)CrossRefGoogle Scholar
  79. 79.
    W. Chen, L.J. Shen, Z.J. Shen, G.W. Yuan, Boundary knot method for Poisson equations. Eng. Anal. Boundary Elem. 29(8), 756–760 (2005)CrossRefzbMATHGoogle Scholar
  80. 80.
    W. Chen, J. Lin, F. Wang, Regularized meshless method for nonhomogeneous problems. Eng. Anal. Boundary Elem. 35(2), 253–257 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  81. 81.
    W. Chen, Meshfree boundary particle method applied to Helmholtz problems. Eng. Anal. Boundary Elem. 26(7), 577–581 (2002)CrossRefzbMATHGoogle Scholar
  82. 82.
    W. Chen, Z.J. Fu, Boundary particle method for inverse cauchy problems of inhomogeneous Helmholtz equations. J. Marine Sci. Technol. Taiwan 17(3), 157–163 (2009)Google Scholar
  83. 83.
    Z.J. Fu, W. Chen, C.Z. Zhang, Boundary particle method for Cauchy inhomogeneous potential problems. Inverse Probl. Sci. Eng. 20(2), 189–207 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  84. 84.
    Z.J. Fu, W. Chen, A truly boundary-only meshfree method applied to kirchhoff plate bending problems. Adv. Appl. Math. Mech. 1(3), 341–352 (2009)MathSciNetGoogle Scholar
  85. 85.
    Z.J. Fu, W. Chen, W. Yang, Winkler plate bending problems by a truly boundary-only boundary particle method. Comput. Mech. 44(6), 757–763 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  86. 86.
    R. Gospavic, N. Haque, V. Popov, C.S. Chen, Comparison of two solvers for the extended method of fundamental solutions, ed. by L. Skerget. Boundary Elements and Other Mesh Reduction Methods XXX, vol. 47, pp. 191–199 (2008)Google Scholar
  87. 87.
    D.L. Young, M.H. Gu, C.M. Fan, The time-marching method of fundamental solutions for wave equations. Eng. Anal. Boundary Elem. 33(12), 1411–1425 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  88. 88.
    L. Marin, D. Lesnic, The method of fundamental solutions for nonlinear functionally graded materials. Int. J. Solids Struct. 44(21), 6878–6890 (2007)CrossRefzbMATHGoogle Scholar
  89. 89.
    Z.J. Fu, W. Chen, Q.H. Qin, Boundary knot method for heat conduction in nonlinear functionally graded material. Eng. Anal. Boundary Elem. 35(5), 729–734 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  90. 90.
    Z.J. Fu, W. Chen, Q.H. Qin, Three boundary meshless methods for heat conduction analysis in nonlinear FGMs with Kirchhoff and Laplace transformation. Adv. Appl. Math. Mech. 4(5), 519–542 (2012)MathSciNetzbMATHGoogle Scholar
  91. 91.
    J.T. Katsikadelis, The analog equation method: a boundary-only integral equation method for nonlinear static and dynamic problems in general bodies. Theor. Appl. Mech. 27, 13–38 (2002)Google Scholar
  92. 92.
    H. Wang, Q.H. Qin, Y.L. Kang, A new meshless method for steady-state heat conduction problems in anisotropic and inhomogeneous media. Arch. Appl. Mech. 74(8), 563–579 (2005)CrossRefzbMATHGoogle Scholar
  93. 93.
    Z.-J. Fu, W. Chen, H.-T. Yang, Boundary particle method for Laplace transformed time fractional diffusion equations. J. Comput. Phys. 235, 52–66 (2013)MathSciNetCrossRefGoogle Scholar
  94. 94.
    S.P. Timoshenko, S. Woinowsky-Krieger, Theory of Plates and Shells (McGraw-Hill, New York, 1959)Google Scholar
  95. 95.
    D. Lesnic, L. Elliott, D.B. Ingham, An iterative boundary element method for solving numerically the Cauchy problem for the Laplace equation. Eng. Anal. Boundary Elem. 20(2), 123–133 (1997)MathSciNetCrossRefGoogle Scholar
  96. 96.
    L. Marin, Relaxation procedures for an iterative MFS algorithm for two-dimensional steady-state isotropic heat conduction Cauchy problems. Eng. Anal. Boundary Elem. 35(3), 415–429 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  97. 97.
    J. Hadamard, Lectures on cauchy problem in linear partial differential equations (Yale University Press, New Haven, 1923)zbMATHGoogle Scholar
  98. 98.
    P. Hansen, REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems. Numer. Alg. 6(1), 1–35 (1994)CrossRefzbMATHGoogle Scholar
  99. 99.
    A. Farcas, L. Elliott, D.B. Ingham, D. Lesnic, The dual reciprocity boundary element method for solving Cauchy problems associated to the Poisson equation. Eng. Anal. Boundary Elem. 27(10), 955–962 (2003)CrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.College of Mechanics and MaterialsHohai UniversityNanjingPeople’s Republic of China
  2. 2.College of Mechanics and MaterialsHohai UniversityNanjingPeople’s Republic of China
  3. 3.University of Southern MississippiHattiesburgUSA

Personalised recommendations