# American Institute of Mathematical Sciences

November  2020, 19(11): 5097-5114. doi: 10.3934/cpaa.2020228

## A new family of boundary-domain integral equations for the diffusion equation with variable coefficient in unbounded domains

 School of Engineering, Computing and Mathematics, Wheatley Campus, Oxford Brookes University, OX33 1HX, Wheatley, UK

Received  March 2020 Revised  May 2020 Published  November 2020 Early access  July 2020

Fund Project: This research was supported by the grants 1636273 from the EPSRC

A system of Boundary-Domain Integral Equations is derived from the mixed (Dirichlet-Neumann) boundary value problem for the diffusion equation in inhomogeneous media defined on an unbounded domain. This paper extends the work introduced in [25] to unbounded domains. Mapping properties of parametrix-based potentials on weighted Sobolev spaces are analysed. Equivalence between the original boundary value problem and the system of BDIEs is shown. Uniqueness of solution of the BDIEs is proved using Fredholm Alternative and compactness arguments adapted to weigthed Sobolev spaces.

Citation: Carlos Fresneda-Portillo. A new family of boundary-domain integral equations for the diffusion equation with variable coefficient in unbounded domains. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5097-5114. doi: 10.3934/cpaa.2020228
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