119
Views
11
CrossRef citations to date
0
Altmetric
Section B

Modified method of fundamental solutions for the Cauchy problem connected with the Laplace equation

Pages 2185-2198 | Received 07 Apr 2013, Accepted 18 Nov 2013, Published online: 26 Mar 2014

References

  • G. Alessandrini, L. Rondi, E. Rosset, and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Probl. 25 (2009), pp. 1–47. doi: 10.1088/0266-5611/25/12/123004
  • D.D. Ang, N.H. Nghia, and N.C. Tam, Regularized solutions of cauchy problem for the Laplace equation in an irregular layer: A three dimensional case, Acta. Math. Vietnam. 23 (1998), pp. 65–74.
  • F. Berntsson and L. Eldén, Numerical solution of a Cauchy problem for the Laplace equation, Inverse Probl. 17(2001), pp. 839–53. doi: 10.1088/0266-5611/17/4/316
  • A.P. Calderon, Uniqueness in the Cauchy problem for partial differential equations, Am. J. Math. 80 (1958), pp. 16–36. doi: 10.2307/2372819
  • W. Chen and Z.J. Fu, Boundary particle method for inverse Cauchy problem of inhomogeneous Helmholtz equations, J. Marine Sci. Technol. 17 (2009), pp. 157–163.
  • G. Fairweather and A. Karageorghis, The method of fundamental solutions for elliptic boundary value problem, Adv. Comput. Math. 9 (1998), pp. 69–95. doi: 10.1023/A:1018981221740
  • R.T. Fenner, A force superposition approach to plane elastic stress and strain analysis, J. Strain Anal. 36 (2001), pp. 517–529. doi: 10.1243/0309324011514674
  • H.S. Gupta, A numerical study of variable coefficient elliptic Cauchy problem via projection method, Int. J. Comput. Math. 89 (2012), pp. 795–809. doi: 10.1080/00207160.2012.659426
  • P.C. Hansen, Analysis of discrete ill-posed problems by means of the L-curve, SIAM Rev. 34 (1992), pp. 561–580. doi: 10.1137/1034115
  • D.N. Háo and D. Lesnic, The Cauchy problem for Laplace's equation via the conjugate gradient method, IMA J. Appl. Math. 65 (2000), pp. 199–217. doi: 10.1093/imamat/65.2.199
  • Y.C. Hon and T. Wei, Backus–Gilbert algorithm for the Cauchy problem of the Laplace equation, Inverse Probl. 17 (2001), pp. 261–71. doi: 10.1088/0266-5611/17/2/306
  • Y.C. Hon and T. Wei, A fundamental solution method for inverse heat conduction problem, Eng. Anal. Bound. Elem. 28 (2004), pp. 489–95. doi: 10.1016/S0955-7997(03)00102-4
  • T. Hrycak and V. Isakov, Increased stability in the continuation of solutions to the Helmholtz equation, Inverse Probl. 20 (2004), pp. 697–712. doi: 10.1088/0266-5611/20/3/004
  • V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verlag, New York, 1998.
  • H.M. Ismail and A.R. Abu-Gammaz, Electric field and right-of-way analysis of Kuwait high-voltage transmission systems, Electr. Power Syst. Res 50 (1999), pp. 213–218. doi: 10.1016/S0378-7796(98)00148-5
  • B.T. Jin and Y. Zheng, Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation, Eng. Anal. Bound. Elem. 29 (2005), pp. 925–935. doi: 10.1016/j.enganabound.2005.05.005
  • B.T. Jin and Y. Zheng, A meshless method for some inverse problems associated with the inhomogeneous Helmholtz equation, Comput. Methods Appl. Mech. Eng. 195 (2006), pp. 2270–2288. doi: 10.1016/j.cma.2005.05.013
  • T. Johansson and D. Lesnic, Reconstruction of a stationary flow from incomplete boundary data using iterative methods, Eur. J. Appl. Math. 17 (2006), pp. 651–663. doi: 10.1017/S0956792507006791
  • T. Johansson and D. Lesnic, An iterative method for the reconstruction of a stationary flow, Numer. Methods Partial Differ. Equ. 23 (2007), pp. 998–1017. doi: 10.1002/num.20205
  • C. Kanali, H. Murase, and N. Honami, Three-dimensional shape recognition using a charge simulation method to process primary image features, J. Agri. Eng. Res. 70 (1998), pp. 195–208. doi: 10.1006/jaer.1998.0265
  • C. Kanali, H. Murase, and N. Honami, Shape identification using a charge simulation retina model, Math. Comput. Simul. 48 (1998), pp. 103–118. doi: 10.1016/S0378-4754(98)00144-X
  • A. Karageorghis, The method of fundamental solutions for the calculation of the eigenvalues of the Helmholtz equation, Appl. Math. Lett. 14 (2001), pp. 837–842. doi: 10.1016/S0893-9659(01)00053-2
  • A. Karageorghis and G. Fairweather, The method of fundamental solutions for axisymmetric elasticity problems, Comput. Mech. 25 (2000), pp. 524–532. doi: 10.1007/s004660050500
  • A. Karageorghis and D. Lesnic. Detectionof cavities using the method of fundamental solutions, Inverse Probl. Sci. Eng. 17 (2009), pp. 803–20. doi: 10.1080/17415970802580263
  • A. Karageorghis and D. Lesnic. The method of fundamental solutions for the inverse conductivity problem, Inverse Probl. Sci. Eng. 18 (2010), pp. 567–83. doi: 10.1080/17415971003675019
  • A. Karageorghis,D. Lesnic. The pressure-stream function MFS formulation for the detection of an obstacle immersed in a two-dimensional Stokes flow, Adv. Appl. Math. Mech. 2 (2010), pp. 183–99.
  • G. Khatiashvili and G. Silagadze, On a numerical solution of two dimensional problems of elasticity for anisotropic medium by the method of fundamental solutions. Reports of the Enlarged Sessions of Seminars of the I. Vekua Institute of Applied Mathematics, Tbilisi State University XIV(3), 1999.
  • A. Kirsch. An Introduction to the Mathematical Theory of Inverse Problems, Springer-Verlag, New York, 1996.
  • V.D. Kupradze and M.A. Aleksdze, The method of functional equations for the approximate solution of certain boundary value problems, USSR Comput. Math. Math. Phys. 4 (1964), pp. 82–126. doi: 10.1016/0041-5553(64)90006-0
  • J.Y. Lee and J.R. Yoon, A numerical method for Cauchy problem using singular value decomposition, Comm. Korean Math. Soc. 16 (2001), pp. 487–508.
  • L. Marin, L. Elliott, P.J. Heggs, D.B. Ingham, D. Lesnic, and X. Wen, An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation, Comput. Methods Appl. Mech. Eng. 192 (2003), pp. 709–722. doi: 10.1016/S0045-7825(02)00592-3
  • L. Marin, L. Elliott, P.J. Heggs, D.B. Ingham, D. Lesnic, and X. Wen, Conjugate gradient-boundary element solution to the Cauchy problem for the Helmholtz-type equations, Comput. Mech. 31 (2003), pp. 367–377.
  • L. Marin and B.T. Johansson, Relaxation procedures for an iterative MFS algorithm for the stable reconstruction of elastic fields from Cauchy data in two-dimensional isotropic linear elasticity, Int. J. Solids Struct. 47 (2010), pp. 3462–3479. doi: 10.1016/j.ijsolstr.2010.08.021
  • L. Marin, A. Karageorghis, and D. Lesnic. TheMFS for numerical boundary identification in two-dimensional harmonic problems, Eng. Anal. Bound. Elem. 35 (2011), pp. 342–354. doi: 10.1016/j.enganabound.2010.09.014
  • L. Marin and D. Lesnic, The method of fundamental solutions for the Cauchy problem in two-dimensional linear elasticity, Int. J. Solids Struct. 41 (2004), pp. 3425–3438. doi: 10.1016/j.ijsolstr.2004.02.009
  • L. Marin and D. Lesnic, The Method of fundamental solutions for the Cauchy problem associated with two-dimensional the Helmholtz-type equations, Comput. Struct. 83 (2005), pp. 267–278. doi: 10.1016/j.compstruc.2004.10.005
  • R. Mathon and R.L. Johnston, The approximate solution of elliptic boundary value problems by fundamental solutions, SIAM J. Numer. Anal. 14 (1977), pp. 638–650. doi: 10.1137/0714043
  • V. Michael and F.S. Klibanov, A computational quasi-reversibility method for Cauchy problems for Laplace's equation, SIAM J. Appl. Math. 51 (1991), pp. 1653–1675. doi: 10.1137/0151085
  • K. Murota, Comparison of conventional and ‘invariant’ schemes of fundamental solutions method for annular domains, Japan J. Indust. Appl. Math. 12 (1995), pp. 61–85. doi: 10.1007/BF03167382
  • A. Poullikkas, A. Karageorghis, and G. Georgiou, Methods of fundamental solutions for harmonic and biharmonic boundary value problems, Comput. Mech. 21 (1998), pp. 416–23. doi: 10.1007/s004660050320
  • C. Rajamohan and J. Ramachandran, Bending of anisotropic plates in charge simulation method, Adv. Eng. Softw. 30 (1999), pp. 369–373. doi: 10.1016/S0965-9978(98)00075-1
  • Y.S. Smyrlis, Applicability and applications of the method of fundamental solutions, Math. Comp. 78 (2009), pp. 1399–1434. doi: 10.1090/S0025-5718-09-02191-7
  • Y. Sun, D. Zhang, and F. Ma, A potential function method for the Cauchy problem of elliptic operators, J. Math. Anal. Appl. 395 (2012), pp. 164–174. doi: 10.1016/j.jmaa.2012.05.038
  • S. Vlad, M. Mihailescu, D. Rafiroiu, A. Iuga, and L. Dascalescu, Numerical analysis of the electric field in plate-type electrostatic separators, J. Electrostat. 48 (2000), pp. 217–229. doi: 10.1016/S0304-3886(99)00067-4
  • T. Wei, Y.C. Hon, and L. Ling, Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators, Eng. Anal. Bound. Elem. 31 (2007), pp. 373–385. doi: 10.1016/j.enganabound.2006.07.010
  • T. Wei, H.H. Qin, and R. Shi, Numerical solution of an inverse 2D Cauchy problem connected with the Helmholtz equation, Inverse Probl. 24 (2008), pp. 1–18. doi: 10.1088/0266-5611/24/3/035003
  • L. Yan, C.L. Fu, and F.L. Yang, The method of fundamental solutions for the inverse heat source problem, Eng. Anal. Bound. Elem. 32 (2008), pp. 216–222. doi: 10.1016/j.enganabound.2007.08.002
  • D.L. Young, C.C. Tsai, C.W. Chen, and C.M. Fan, The method of fundamental solutions and condition number analysis for inverse problems of Laplace equation, Comput. Math. Appl. 55 (2008), pp. 1189–1200. doi: 10.1016/j.camwa.2007.05.015
  • D. Zhang, G. Zhang, and E. Zheng, The harmonic polynomial method for solving the Cauchy problem connected with the Laplace equation, Inverse Probl. 29 (2013), 065008. doi: 10.1088/0266-5611/29/6/065008

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.