Corrigendum on ‘Modified quasi-boundary value method for a Cauchy problem of semi-linear elliptic equation’ (Int. J. Comput. Math., 89(12):1689–1703, 2012. 10.1080/00207160.2012.693174)

This article refers to:

Modified quasi-boundary value method for a Cauchy problem of semi-linear elliptic equation

1. Correction

The author regrettably finds that there are some minor errors in our paper [2], which include the a-priori bound condition (29) and convergence estimate result (31) in Theorem 3.1. These errors mainly are caused during the procedure of using the Gronwall's inequality on P.1697. It is necessary to make a correction for them, for convenience, here we also give the complete proof process of Theorem 3.1.

Firstly, for Theorem 3.1, we make the following two corrections when other conditions were the same (see [2]).

The a-priori bound assumption (29) should be given by

and the final convergence estimate (31) becomes as

where

Now, we give its complete proof process.

Proof It is clear that

From the Theorem 2.2 in [2], we know

By Equations (7), (14), (17), (18) in [2], we have For , we get

Using Gronwall's inequality [1], it can be obtained that thus

From Equations (3), (4), (6) and (12), (30) in [2], we can obtain the convergence estimate (2).

Acknowledgements

The author would like to thank the reviewers for their constructive comments and valuable suggestions that improve the quality of our paper. This work is supported by the NSF of China (10971089, 11171136) and the YTSRF of Hexi university (QN2011-10).