Analysis of Boundary-Domain Integral Equations for Variable-Coefficient Neumann BVP in 2D

T. G. Ayele (), T. T. Dufera () and S. E. Mikhailov ()
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T. G. Ayele: Addis Ababa University
T. T. Dufera: Adama Science and Technology University
S. E. Mikhailov: Brunel University West London

Chapter Chapter 3 in Integral Methods in Science and Engineering, Volume 1, 2017, pp 21-33 from Springer

Abstract: Abstract In this paper we will consider second order Neumann Boundary Value problem for the “stationary heat transfer” partial differential equation with variable coefficient in two-dimension. This equation is reduced to some boundary-domain integral equations (BDIEs). The construction of Boundary-Domain Integral equation in two-dimension is special from the remaining dimension because of the associated fundamental solution to the partial differential equation. Consequently we need to set conditions on the domain or on the spaces to insure the invertibility of layer potentials and hence the unique solvability of Boundary-Domain integral equation. The equivalence of the BDIEs to the original BVPs, BDIEs solvability, solution uniqueness/nonuniqueness, as well as Fredholm property and invertibility of the BDIEs operator are analyzed. It is shown that the BDIE operators for Neumann BVP are not invertible, and appropriate finite-dimensional perturbations are constructed leading to invertibility of the perturbed operators.

Date: 2017
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DOI: 10.1007/978-3-319-59384-5_3

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