Numerical solution of a Cauchy problem for Laplace equation in 3-dimensional domains by integral equations

Ihor Borachok, Roman Chapko, B. Tomas Johansson

    Research output: Contribution to journalArticlepeer-review


    A numerical method based on integral equations is proposed and investigated for the Cauchy problem for the Laplace equation in 3-dimensional smooth bounded doubly connected domains. To numerically reconstruct a harmonic function from knowledge of the function and its normal derivative on the outer of two closed boundary surfaces, the harmonic function is represented as a single-layer potential. Matching this representation against the given data, a system of boundary integral equations is obtained to be solved for two unknown densities. This system is rewritten over the unit sphere under the assumption that each of the two boundary surfaces can be mapped smoothly and one-to-one to the unit sphere. For the discretization of this system, Weinert’s method (PhD, Göttingen, 1990) is employed, which generates a Galerkin type procedure for the numerical solution, and the densities in the system of integral equations are expressed in terms of spherical harmonics. Tikhonov regularization is incorporated, and numerical results are included showing the efficiency of the proposed procedure.

    Original languageEnglish
    Pages (from-to)1550-1568
    Number of pages19
    JournalInverse Problems in Science and Engineering
    Issue number9
    Early online date13 Jan 2016
    Publication statusPublished - 2016

    Bibliographical note



    • Cauchy problem
    • discrete projection method
    • double connected 3D domain
    • integral equation of the first kind
    • Laplace equation
    • single- and double-layer potentials
    • Tikhonov regularization


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