On the numerical solution of parabolic Cauchy problem in a domain with cut (in Ukrainian)

Author V. G. Vavrychuk, R. S. Chapko

Ivan Franko National University of Lviv

Abstract Cauchy problem is numerically solved with help of iterative regularization procedure at every step of which mixed nonstationary Dirichlet-Neumann problems for parabolic equation arise. Using Rothe’s method mixed problems are reduced to the boundary integral equations which have three kinds of singularities: square root, logarithmic and hypersingularity. Special techniques are employed to cope with them so that in the case of analytic input data solution of boundary integral equations have exponential error decay.
Keywords heat equation; Cauchy problem; Ladweber method; mixed boundary value problems; single- and double layer potentials; integral equation of the first kind; trigonometrical quadrature method
Reference 1. Chapko R., Johansson B.T. An alternating boundary integral based method for a Cauchy problem for Laplace equation in semi-infinite domains// Inverse Probl. Imaging – 2008. – V.3. – P. 317–333.

 2. Chapko R., Johansson B.T. An alternating potential based approach for a Cauchy problem for the Laplace equation in a planar domain with a cut// Comp. Methods in Appl. Math. – 2008. – Ή8. – P. 315–335.

 3. Chapko R., Johansson B.T., Vavrychuk V. Recovering boundary data in planar heat conduction using a boundary integral equation method// Electronic Journal of Boundary Elements – 2011. – Ή9. – P. 1–15.

 4. Chapko R., Kress R. Rothe’s method for the heat equation and boundary integral equations// J. Integral Equations Appl. – 1997. – V.1. – P. 47–69.

 5. Chapko R., Vavrychuk V. On the numerical solution of a mixed initial boundary value problem for the heat equation in a double-connected planar domain// Journal of Numerical and Applied Mathematics – 2009. – V.97. – P. 26–38.

 6. Engl H.W., Hanke M., Neuebauer A. Regularization of Inverse Problems, Kluwer Academic Publishers, London, 1996.

 7. Johansson B.T. An iterative method for a Cauchy problem for the heat equation// IMA Journal of Applied Mathematics – 2006. – V.71. – P. 262–286.

 8. Johansson B.T. Determining the temperature from Cauchy data in corner domains// International Journal of Computing Science and Mathematics. – 2010. – V.3. – P. 122–131.

 9. Kress R. Inverse scattering from an open arc// Math. Meth. in the Appl. Sci. – 1995. – V.18. – P. 267–293.

 10. Kress R. On the numerical solution of a hypersingular integral equation in scattering theory// J. Comput. Appl. Math. – 1995. – V.61. – P. 345–360.

 11. Kress R. Linear Integral Equations, second ed. – Springer-Verlag, 1999.

 12. Saut J.-C., Scheurer B. Unique continuation for some evolution equations// J. Differential Equations – 1987. – V.66. – P. 118–139.

 13. Yan Y., Sloan I.H. On integral equations of the first kind with logarithmic kernels// J. Integral Equations Appl. – 1988. – V.1. – P. 549–579.
Pages 209-218
Volume 37
Issue 2
Year 2012
Journal Matematychni Studii
Full text of paper PDF
Table of content of issue HTML