Numerical Algorithms

, Volume 81, Issue 3, pp 833–851 | Cite as

Generalized Tikhonov-type regularization method for the Cauchy problem of a semi-linear elliptic equation

  • Hongwu ZhangEmail author
  • Xiaoju Zhang
Original Paper


This paper considers the Cauchy problem of a semi-linear elliptic equation and uses a generalized Tikhonov-type regularization method to overcome its ill-posedness. The existence, uniqueness, and stability for regularized solution are proven. Under an a priori bound assumption for exact solution, we derive the convergence estimate of Hölder type for this method. An application of this method to the Cauchy problem of Helmholtz equation is discussed, and we investigate the stability and convergence estimates for different wave numbers. Finally, an iterative scheme is constructed to calculate the regularization solution, numerical results show that this method is stable and feasible.


Ill-posed problem Cauchy problem Semi-linear elliptic equation Generalized Tikhonov-type regularization method Convergence estimate 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors would like to thank the reviewers for their constructive comments and valuable suggestions that improve the quality of our paper.

Funding information

The work was supported by the YSRP (2017KJ33), NSF of China (11761004, 11371181), and SRP (2017SXKY05) at North Minzu University.


  1. 1.
    Belgacem, F.B.: Why is the Cauchy problem severely ill-posed? Inverse Prob. 23, 823 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beskos, D.E.: Boundary element method in dynamic analysis: Part II (1986-1996). ASME Appl. Mech. Rev. 50, 149–197 (1997)CrossRefGoogle Scholar
  3. 3.
    Chen, J.T., Wong, F.C.: Dual formulation of multiple reciprocity method for the acoustic mode of a cavity with a thin partition. J. Sound Vib. 217, 75–95 (1998)CrossRefGoogle Scholar
  4. 4.
    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems, Volume 375 of Mathematics and Its Applications. Kluwer Academic Publishers Group, Dordrecht (1996)CrossRefzbMATHGoogle Scholar
  5. 5.
    Evans, L.C.: Partial Diferential Equations, American Mathematical Society, vol. 19 (1998)Google Scholar
  6. 6.
    Feng, X.L., Ning, W.T., Qian, Z.: A quasi-boundary-value method for a Cauchy problem of an elliptic equation in multiple dimensions. Inverse Prob. Sci. Eng. 22(7), 1045–1061 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fokas, A.S., Pelloni, B.: The dirichlet-to-neumann map for the elliptic sine-Gordon equation. Nonlinearity 25(4), 1011–1031 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gutshabash, E.S., Lipovskii, V.D.: Boundary value problem for the two-dimensional elliptic sine-Gordon equation and its applications to the theory of the stationary josephson effect. J. Math. Sci. 68, 197–201 (1994)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hào, D.N., Duc, N.V., Lesnic, D.: A non-local boundary value problem method for the Cauchy problem for elliptic equations. Inverse Prob. 25, 055002 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hào, D.N., Van, T.D., Gorenflo, R.: Towards the Cauchy problem for the Laplace equation. Partial differential equations pp. 111 (1992)Google Scholar
  11. 11.
    Isakov, V: Inverse Problems for Partial Differential Equations. Springer, Berlin (2006)zbMATHGoogle Scholar
  12. 12.
    Khoa, V.A., Truong, M.T., Duy, N.H.: A general kernel-based regularization method for coupled elliptic sine-Gordon equations with Gevrey regularity. Comput. Phys. Commun. 183(8), 1813–1821 (2015)Google Scholar
  13. 13.
    Khoa, V.A., Truong, M.T., Duy, N.H., Tuan, N.H.: The Cauchy problem of coupled elliptic sine-Gordon equations with noise: Analysis of a general kernel-based regularization and reliable tools of computing. Computers & Mathematics with Applications 73(1), 141–162 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems, Volume 120 of Applied 2 Sciences. Springer, New York (1996)CrossRefGoogle Scholar
  15. 15.
    M.M. Lavrentiev, Romanov, V.G., Shishatski, S.P.: Ill-posed problems of mathematical physics and analysis, volume 64 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI. Translated from the Russian by J.R. Schulenberger, Translation edited by Lev J. Leifman (1986)Google Scholar
  16. 16.
    Tautenhahn, U.: Optimal stable solution of Cauchy problems of elliptic equations. Journal for Analysis and its Applications 15(4), 961–984 (1996)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Tran, Q.V., Kirane, M., Nguyen, H.T., Nguyen, V.T.: Analysis and numerical simulation of the three-dimensional Cauchy problem for quasi-linear elliptic equations. J. Math. Anal. Appl. 446(1), 470–492 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Trong, D.D., Quan, P.H., Tuan, N.H.: A quasi-boundary value method for regularizing nonlinear ill-posed problems. Electronic Journal of Differential Equations 2009(109), 1–16 (2009)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Tuan, N.H., Thang, L.D., Khoa, V.A.: A modified integral equation method of the nonlinear elliptic equation with globally and locally lipschitz source. Appl. Math. Comput. 265, 245–265 (2015)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Tuan, N.H., Thang, L.D., Khoa, V.A., Tran, T.: On an inverse boundary value problem of a nonlinear elliptic equation in three dimensions. J. Math. Anal. Appl. 426, 1232–1261 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Tuan, N.H., Thang, L.D., Lesnic, D.: A new general filter regularization method for Cauchy problems for elliptic equations with a locally Lipschitz nonlinear source. J. Math. Anal Appl. 434, 1376–1393 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Tuan, N.H., Thang, L.D., Trong, D.D., Khoa, V.A.: Approximation of mild solutions of the linear and nonlinear elliptic equations. Inverse Prob. Sci. Eng. 23(7), 1237–1266 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Tuan, N.H., Trong, D.D.: A nonlinear parabolic equation backward in time Regularization with new error estimates. Nonlinear Anal. Theory Methods Appl. 73 (6), 1842–1852 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Tuan, N.H., Trong, D.D., Quan, P.H.: A note on a Cauchy problem for the Laplace equation Regularization and error estimates. Appl. Math. Comput. 217, 2913–2922 (2010)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Zhang, H.W., Wang, R.H.: Modified boundary Tikhonov-type regularization method for the Cauchy problem of a semi-linear elliptic equation. Inverse Prob. Sci. Eng. 24(7), 1249–1265 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhang, H.W., Wei, T.: A Fourier truncated regularization method for a Cauchy problem of a semi-linear elliptic equation. Journal of Inverse and Ill-posed Problems 22(2), 143–168 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zhang, H.W., Zhang, X.J.: Filtering function method for the Cauchy problem of a semi-linear elliptic equation. Journal of Applied Mathematics and Physics 3(2), 1599–1609 (2015)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and Information ScienceNorth Minzu UniversityYinchuanPeople’s Republic of China

Personalised recommendations