Boundary-domain integral equations for the diffusion equation in inhomogeneous media based on a new family of parametrices

ABSTRACT

A mixed boundary value problem (BVP) for the diffusion equation in non-homogeneous media partial differential equation is reduced to a system of direct segregated parametrix-based boundary-domain integral equations (BDIEs). We use a parametrix different from the one employed by Mikhailov [Localized boundary-domain integral formulations for problems with variable coefficients. Eng Anal Bound Elem. 2002;26:681–690], Mikhailov and Portillo [A new family of boundary-domain integral equations for a mixed elliptic BVP with variable coefficient. In: Paul Harris, editor. Proceedings of the 10th UK conference on boundary integral methods. Brighton: Brighton University Press; 2015. p. 76–84] and Chkadua, Mikhailov, Natroshvili [Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient. I: equivalence and invertibility. J Integral Eqs Appl. 2009;21:499–543]. We prove the equivalence between the original BVP and the corresponding BDIE system. The invertibility and Fredholm properties of the boundary-domain integral operators are also analysed.

COMMUNICATED BY:

• X. Xie

KEYWORDS:

• Variable coefficient
• parametrix
• remainder
• mixed boundary value problem
• boundary-domain integral equations

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35J25
• 31B10
• 45K05
• 45A05

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by the grants 1636273 from the EPSRC, UK, and also by a PhD studentship from the School of Mathematics of Brunel University London.