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A Boundary-Domain Integral Equation Method for an Elliptic Cauchy Problem with Variable Coefficients

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Analysis, Probability, Applications, and Computation

Part of the book series: Trends in Mathematics ((RESPERSP))

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Abstract

We consider an integral based method for numerically solving the Cauchy problem for second-order elliptic equations in divergence form with spacewise dependent coefficients. The solution is represented as a boundary-domain integral, with unknown densities to be identified. The given Cauchy data is matched to obtain a system of boundary-domain integral equations from which the densities can be constructed. For the numerical approximation, an efficient Nyström scheme in combination with Tikhonov regularization is presented for the boundary-domain integral equations, together with some numerical investigations.

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References

  1. M.A. Al-Jawary, L.C. Wrobel, Numerical solution of two-dimensional mixed problems with variable coefficients by the boundary-domain integral and integro-differential equation methods. Eng. Anal. Bound. Elem. 35, 1279–1287 (2011)

    Article  MathSciNet  Google Scholar 

  2. W.T. Ang, J. Kusuma, D.L. Clements, A boundary element method for a second order elliptic partial differential equation with variable coefficients. Eng. Anal. Bound. Elem. 18, 311–316 (1996)

    Article  Google Scholar 

  3. A. Beshley, R. Chapko, B.T. Johansson, An integral equation method for the numerical solution of a Dirichlet problem for second-order elliptic equations with variable coefficients. J. Eng. Math. 112, 63–73 (2018)

    Article  MathSciNet  Google Scholar 

  4. F. Cakoni, R. Kress, Integral equations for inverse problems in corrosion detection from partial Cauchy data. Inverse Probl. Imag. 1, 229–245 (2007)

    Article  MathSciNet  Google Scholar 

  5. R. Chapko, B.T. Johansson, A direct integral equation method for a Cauchy problem for the Laplace equation in 3-dimensional semi-infinite domains. Comput. Model. Eng. Sci. 85, 105–128 (2012)

    MathSciNet  MATH  Google Scholar 

  6. R. Chapko, B.T. Johansson, A boundary integral approach for numerical solution of the Cauchy problem for the Laplace equation. Ukr. Math. J. 68, 1665–1682 (2016)

    MathSciNet  Google Scholar 

  7. D.L. Clements, A boundary integral equation method for the numerical solution of a second order elliptic equation with variable coefficients. J. Aust. Math. Soc. 22, 218–228 (1980)

    Article  MathSciNet  Google Scholar 

  8. D.L. Clements, A fundamental solution for linear second-order elliptic systems with variable coefficients. J. Eng. Math. 49, 209–216 (2004)

    Article  MathSciNet  Google Scholar 

  9. R. Kress, Linear Integral Equations, 3rd edn. (Springer, Berlin, 2013)

    MATH  Google Scholar 

  10. R. Kress, On Trefftz’ integral equation for the Bernoulli free boundary value problem. Numer. Math. 136, 503–522 (2017)

    Article  MathSciNet  Google Scholar 

  11. R. Kress, W. Rundell, Nonlinear integral equations and the iterative solution for an inverse boundary value problem. Inverse Probl. 21, 1207–223 (2005)

    Article  MathSciNet  Google Scholar 

  12. S.E. Mikhailov, Localized boundary-domain integral formulations for problems with variable coefficients. Eng. Anal. Bound. Elem. 26, 681–690 (2002)

    Article  Google Scholar 

  13. S.E. Mikhailov, Analysis of united boundary-domain integro-differential and integral equations for a mixed BVP with variable coefficients. Math. Methods Appl. Sci. 29, 715–739 (2006)

    Article  MathSciNet  Google Scholar 

  14. C. Miranda, Partial Differential Equations of Elliptic Type (Springer, New York, 1970)

    Book  Google Scholar 

  15. A. Pomp, The Boundary-Domain Integral Method for Elliptic Systems. With Applications in Shells. Lecture Notes in Mathematics, vol. 1683 (Springer, Berlin, 1998)

    Chapter  Google Scholar 

  16. J. Ravnik, L. Skerget, Integral equation formulation of an unsteady diffusion-convection equation with variable coefficient and velocity. Comput. Math. Appl. 66, 2477–2488 (2014)

    Article  MathSciNet  Google Scholar 

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Correspondence to B. Tomas Johansson .

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Beshley, A., Chapko, R., Johansson, B.T. (2019). A Boundary-Domain Integral Equation Method for an Elliptic Cauchy Problem with Variable Coefficients. In: Lindahl, K., Lindström, T., Rodino, L., Toft, J., Wahlberg, P. (eds) Analysis, Probability, Applications, and Computation. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-04459-6_47

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