Filtering Function Method for the Cauchy Problem of a Semi-Linear Elliptic Equation


A Cauchy problem for the semi-linear elliptic equation is investigated. We use a filtering function method to define a regularization solution for this ill-posed problem. The existence, uniqueness and stability of the regularization solution are proven; a convergence estimate of Hölder type for the regularization method is obtained under the a-priori bound assumption for the exact solution. An iterative scheme is proposed to calculate the regularization solution; some numerical results show that this method works well.

Share and Cite:

Zhang, H. and Zhang, X. (2015) Filtering Function Method for the Cauchy Problem of a Semi-Linear Elliptic Equation. Journal of Applied Mathematics and Physics, 3, 1599-1609. doi: 10.4236/jamp.2015.312184.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Kirsch, A. (1996) An Introduction to the Mathematical Theory of Inverse Problems. Applied Mathematical Sciences, Vol. 120, Springer-Verlag, New York.
[2] Engl, H.W., Hanke, M. and Neubauer, A. (1996) Regularization of Inverse Problems. Mathematics and Its Applications, Vol. 375, Kluwer Academic Publishers Group, Dordrecht.
[3] Belgacem, F.B. (2007) Why Is the Cauchy Problem Severely Ill-Posed? Inverse Problems, 23, 823.
[4] Feng, X.L., Ning, W.T. and Qian, Z. (2014) A Quasi-Boundary-Value Method for a Cauchy Problem of an Elliptic Equation in Multiple Dimensions. Inverse Problems in Science and Engineering, 22, 1045-1061.
[5] Hào, D.N., Duc, N.V. and Lesnic, D. (2009) A Non-Local Boundary Value Problem Method for the Cauchy Problem for Elliptic Equations. Inverse Problems, 25, Article ID: 055002.
[6] Hào, D.N., Van, T.D. and Gorenflo, R. (1992) Towards the Cauchy Problem for the Laplace Equation. Partial Differential Equations, 111.
[7] Isakov, V. (2006) Inverse Problems for Partial Differential Equations. Springer Verlag, Berlin.
[8] Lavrentiev, M.M., Romanov, V.G. and Shishatski, S.P. (1986) Ill-Posed Problems of Mathematical Physics and Analysis. Translations of Mathematical Monographs, Vol. 64, American Mathematical Society, Providence.
[9] Zhang, H.W. and Wei, T. (2014) A Fourier Truncated Regularization Method for a Cauchy Problem of a Semi-Linear Elliptic Equation. Journal of Inverse and Ill-Posed Problems, 22, 143-168.
[10] Tuan, N.H., Thang, L.D. and Khoa, V.A. (2015) A Modified Integral Equation Method of the Nonlinear Elliptic Equation with Globally and Locally Lipschitz Source. Applied Mathematics and Computation, 265, 245-265.
[11] Tuan, N.H. and Tran, B.T. (2014) A Regularization Method for the Elliptic Equation with Inhomogeneous Source. ISRN Mathematical Analysis, 2014, Article ID: 525636.
[12] Clark, G.W. and Oppenheimer, S.F. (1994) Quasireversibility Methods for Non-Well-Posed Problems. Electronic Journal of Differential Equations, 1994, 9 p.
[13] Xiong, X.T. (2010) A Regularization Method for a Cauchy Problem of the Helmholtz Equation. Journal of Computational and Applied Mathematics, 233, 1723-1732.
[14] Tuan, N.H. and Trong, D.D. (2010) A Nonlinear Parabolic Equation Backward in Time: Regularization with New Error Estimates. Nonlinear Analysis: Theory, Methods and Applications, 73, 1842-1852.
[15] Evans, L.C. (1998) Partial Differential Equations. American Mathematical Society, Vol. 19.

Copyright © 2023 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.