Numerical Algorithms

, Volume 59, Issue 2, pp 249–269 | Cite as

An improved non-local boundary value problem method for a cauchy problem of the Laplace equation

  • Hongwu Zhang
  • Ting Wei
Original Paper


In this paper, we propose an improved non-local boundary value problem method to solve a Cauchy problem for the Laplace equation. It is known that the Cauchy problem for the Laplace equation is severely ill-posed, i.e., the solution does not depend continuously on the given Cauchy data. Convergence estimates for the regularized solutions are obtained under a-priori bound assumptions for the exact solution. Some numerical results are given to show the effectiveness of the proposed method.


Ill-posed problem Cauchy problem Laplace equation Regularization method Convergence estimate 


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  1. 1.
    Clark, G.W., Oppenheimer, S.F.: Quasireversibility methods for non-well-posed problems. Electron. J. Differential Equations 1994(8), 1–9 (1994)MathSciNetGoogle Scholar
  2. 2.
    Colli Franzone, P., Guerri, L., Tentoni, S., Viganotti, C., Baruffi, S., Spaggiari, S., Taccardi, B.: A mathematical procedure for solving the inverse potential problem of electrocardiography. Analysis of the time-space accuracy from in vitro experimental data. Math. Biosci. 77(1–2), 353–396 (1985)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Denche, M., Bessila, K.: A modified quasi-boundary value method for ill-posed problems. J. Math. Anal. Appl. 301(2), 419–426 (2005)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Falk, R.S., Monk, P.B.: Logarithmic convexity for discrete harmonic functions and the approximation of the Cauchy problem for Poisson’s equation. Math. Comput. 47(175), 135–149 (1986)MATHMathSciNetGoogle Scholar
  5. 5.
    Hanke, M., Engle, H.W., Neubauer, A.: Regularization of Inverse Problems, Mathematics and its Applications, volume 375. Kluwer, Dordrecht (1996)Google Scholar
  6. 6.
    Hadamard, J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Dover, New York (1953)Google Scholar
  7. 7.
    Hào, D.N., Duc, N.V., Sahli, D.: A non-local boundary value problem method for the Cauchy problem for elliptic equations. Inverse Probl. 25, 055002 (2009)CrossRefGoogle Scholar
  8. 8.
    Hào, D.N., Duc, N.V., Sahli, H.: A non-local boundary value problem method for parabolic equations backward in time. J. Math. Anal. Appl. 345(2), 805–815 (2008)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Kaup, P.G., Santosa F.: Nondestructive evaluation of corrosion damage using electrostatic measurements. J. Nondestr. Eval. 14(3), 127–136 (1995)CrossRefGoogle Scholar
  10. 10.
    Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems. Applied Mathematical Sciences, vol. 120. Springer, New York (1996)CrossRefMATHGoogle Scholar
  11. 11.
    Klibanov, M.V., Santosa, F.: A computational quasi-reversibility method for Cauchy problems for Laplace’s equation. SIAM J. Appl. Math. 51(6), 1653–1675 (1991)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Lattès, R., Lions, J.-L.: The Method of Quasi-reversibility. Applications to Partial Differential Equations. Translated from the French edition and edited by Richard Bellman. Modern Analytic and Computational Methods in Science and Mathematics, No. 18. American Elsevier Publishing Co. Inc., New York (1969)MATHGoogle Scholar
  13. 13.
    Lavrentiev, M.M.: Some Improperly Posed Problems in Mathematical Physics. Springer, Berlin (1967)Google Scholar
  14. 14.
    McIver, M.: An inverse problem in electromagnetic crack detection. IMA J. Appl. Math. 47(2), 127 (1991)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Mel′nikova, I.V.: Regularization of ill-posed differential problems. Sibirsk. Mat. Zh. 33(2), 125–134, 221 (1992)MathSciNetGoogle Scholar
  16. 16.
    Mel′nikova, I.V., Filinkov, A.: Abstract Cauchy Problems: Three Approaches. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 120. Chapman & Hall/CRC, Boca Raton, FL (2001)CrossRefGoogle Scholar
  17. 17.
    Michael, D.H., Waechter, R.T., Collins, R.: The measurement of surface cracks in metals by using ac electric fields. Proc. R. Soc. Lond., A Math. Phys. Sci. 381(1780), 139–157 (1982)CrossRefGoogle Scholar
  18. 18.
    Qian, Z., Fu, C.L., Li, Z.P.: Two regularization methods for a Cauchy problem for the Laplace equation. J. Math. Anal. Appl. 338(1), 479–489 (2008)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Qian, Z., Fu, C.L., Xiong, X.T.: Fourth-order modified method for the Cauchy problem for the Laplace equation. J. Comput. Appl. Math. 192(2), 205–218 (2006)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Reinhardt, H.J., Han, H., Hào, D.N.: Stability and regularization of a discrete approximation to the Cauchy problem for Laplace’s equation. SIAM J. Numer. Anal. 36(3), 890–905 (1999) (electronic)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Showalter, R.E.: Cauchy problem for hyper-parabolic partial differential equations. In: Trends in the Theory and Practice of Non-linear Analysis (1983)Google Scholar
  22. 22.
    Stefanesco, S., Schlumberger, C., Schlumberger, M.: Sur la distribution électrique potentielle autour d’une prise de terre ponctuelle dans un terrain à couches horizontales, homogènes et isotropes (1930)Google Scholar
  23. 23.
    Tikhonov, A.N., Arsenin, V.Y.: Solutions of Ill-Posed Problems. Translated from the Russian, preface by translation editor Fritz John, Scripta Series in Mathematics. V. H. Winston & Sons, Washington, DC (1977)Google Scholar
  24. 24.
    Trong, D.D., Tuan, N.H.: A nonhomogeneous backward heat problem: regularization and error estimates. Electr. J. Differ. Equ. 2008(33), 1–14 (2008)Google Scholar
  25. 25.
    Vabishchevich, P.N.: Numerical solution of nonlocal elliptic problems. Izv. Vyssh. Uchebn. Zaved. Mat., 5, 13–19 (1983) (in Russian)MathSciNetGoogle Scholar
  26. 26.
    Xiong, X.T.: A regularization method for a Cauchy problem of the Helmholtz equation. J. Comput. Appl. Math. 233(8), 1723–1732 (2010)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhou CityPeople’s Republic of China
  2. 2.Department of MathematicsHexi UniversityZhangye CityPeople’s Republic of China

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