A mollification regularization method for the Cauchy problem of an elliptic equation in a multi-dimensional case. (English) Zbl 1206.65224

A Cauchy problem for an elliptic equation is considered. This problem is ill-posed since the solution, if it exists, does not depend continuously on the data. The considered ill-posed problem is replaced by a problem with mollified data and some error estimates between the solutions of the two problems are established. Moreover, some numerical experiments are presented to verify the proposed regularization method.


65M30 Numerical methods for ill-posed problems for initial value and initial-boundary value problems involving PDEs
35R25 Ill-posed problems for PDEs
35J25 Boundary value problems for second-order elliptic equations
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
Full Text: DOI


[1] Lavrent’ev MM, Translations of Mathematical Monographs 64 (1986)
[2] Tikhonov AN, Solutions of Ill-posed Problems (1977)
[3] DOI: 10.1007/BF01587651 · Zbl 0127.16903
[4] DOI: 10.1007/BF02576643 · Zbl 0452.92008
[5] Johnson CR, Crit. Rev. Biomed. Eng. 25 pp 1– (1997)
[6] Alessandrini G, Proc. R. Soc. Edinb. A 123 pp 497– (1993)
[7] Hadamard J, Lectures on Cauchy’s Problem in Linear Partial Differential Equations (1953) · Zbl 0049.34805
[8] DOI: 10.1088/0266-5611/17/2/306 · Zbl 0980.35167
[9] DOI: 10.1016/j.jmaa.2007.06.035 · Zbl 1135.35093
[10] DOI: 10.1088/0266-5611/22/2/002 · Zbl 1094.35134
[11] DOI: 10.1080/17415970701228246 · Zbl 1258.65094
[12] Ang DD, Acta Math. Vietnam. 23 pp 65– (1998)
[13] Murio DA, The Mollification Method and the Numerical Solution of Ill-posed Problems (1993)
[14] DOI: 10.1088/0266-5611/23/1/024 · Zbl 1119.35110
[15] DOI: 10.1080/01630569808816834 · Zbl 0907.65049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.