" />

using Gronwall’s inequality [15] , we have


then from the above inequality (2.19), the stability result (2.16) can be obtained. □

3. Convergence Estimate

In this section, under an a-priori bound assumption for the exact solution a convergence estimate of Hölder type for the regularization method is derived. The corresponding result is shown in Theorem 3.1.

Theorem 3.1. Suppose that f satisfies the uniform Lipschitz condition (1.2), and u given by (2.1) is the exact solution of problem (1.1), defined by (2.4) is the regularization solution, the measured data satisfies (2.3). If the exact solution u satisfies


and the regularization parameter is chosen as


then for fixed, we have the following convergence estimate


here, , is given in Theorem 2.2.

Proof. Denote be the solution of problem (2.4) with exact data. We know that


From Theorem 2.2, for, we have


By (2.1), (2.4), (2.7), (2.8), we have

For, we get


use Gronwall’s inequality [15] , it can be obtained that



From (3.2), (3.4), (3.5), (3.7) and (2.3), we can obtain the convergence result (3.3). □

4. Numerical Experiments

In this section, we verify the accuracy and efficiency of our given regularization method by the following numerical example


here we take, , , then and.

It is clear that is an exact solution of problem (4.1), thus

,. We choose the measured

data as, where is an error level, and


Let for, the regularization solution with

can be computed by the following iteration scheme


here, and



For a fixed, in order to make the sensitivity analysis for numerical results, we define the relative root mean square error between the exact and approximate solutions as


We adopt the above given algorithms to compute the regularization solution at with,

for Taking for the numerical results for and at, are shown in Figure 1 and Figure 2, respectively. For, the relative root mean square errors for the various error levels and regularization parameters at are shown in Table 1. In the computational procedure, the regulari- zation parameter is chosen by (3.2), and is computed by (4.2).

From Figure 1 and Figure 2 and Table 1, it can be observed that our regularization method is effective and stable. Meanwhile we note that the smaller is, the better the calculation effect is. Table 1 shows that the numerical results become worse when y approaches to 1, which is a common phenomenon in the computation of ill-posed Cauchy problems for the elliptic equation.

(a) (b)(c) (d)

Figure 1. Exact and regularized solutions at. (a); (b); (c); (d).

(a) (b)(c) (d)

Figure 2. Exact and regularized solutions at. (a); (b); (c); (d).

Table 1. The relative root mean square errors for various and the regularization parameters at.

5. Conclusion

We use a filtering function method to solve a Cauchy problem for semi-linear elliptic equation. The results of the well-posedness for the approximation problem are given. Under the a-priori bound assumption, the conver- gence estimate of Hölder type has been derived. Finally, we compute the regularization solution by constructing an iterative scheme. Some numerical results show that this method is stable and feasible.


The authors would like to thank the reviewers for their constructive comments and valuable suggestions that improve the quality of our paper. The work described in this paper was supported by the SRF (2014XYZ08, 2015JBK423), NFPBP (2014QZP02) of Beifang University of Nationalities, the SRP of Ningxia Higher School (NGY20140149) and SRP of State Ethnic Affairs Commission of China (14BFZ004).

Cite this paper

HongwuZhang,XiaojuZhang, (2015) Filtering Function Method for the Cauchy Problem of a Semi-Linear Elliptic Equation. Journal of Applied Mathematics and Physics,03,1599-1609. doi: 10.4236/jamp.2015.312184


  1. 1. Kirsch, A. (1996) An Introduction to the Mathematical Theory of Inverse Problems. Applied Mathematical Sciences, Vol. 120, Springer-Verlag, New York.

  2. 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. 3. Belgacem, F.B. (2007) Why Is the Cauchy Problem Severely Ill-Posed? Inverse Problems, 23, 823.

  4. 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. 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. 6. Hào, D.N., Van, T.D. and Gorenflo, R. (1992) Towards the Cauchy Problem for the Laplace Equation. Partial Differential Equations, 111.

  7. 7. Isakov, V. (2006) Inverse Problems for Partial Differential Equations. Springer Verlag, Berlin.

  8. 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. 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. 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. 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. 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. 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. 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. 15. Evans, L.C. (1998) Partial Differential Equations. American Mathematical Society, Vol. 19.


*Corresponding author.

Journal Menu >>