Two-Operator Boundary–Domain Integral Equations for a Variable-Coefficient BVP

Abstract

Partial differential equations (PDEs) with variable coefficients often arise in mathematical modeling of inhomogeneous media (e.g., functionally graded materials or materials with damage-induced inhomogeneity) in solid mechanics, electromagnetics, heat conduction, fluid flows through porous media, and other areas of physics and engineering.

Generally, explicit fundamental solutions are not available if the PDE coefficients are not constant, preventing formulation of explicit boundary integral equations, which can then be effectively solved numerically. Nevertheless, for a rather wide class of variable–coefficient PDEs, it is possible to use instead an explicit parametrix (Levi function) taken as a fundamental solution of corresponding frozen–coefficient PDEs, and reduce boundary value problems (BVPs) for such PDEs to explicit systems of boundary–domain integral equations (BDIEs); see, e.g., [Mi02, CMN09, Mi06] and references therein. However this (one–operator) approach does not work when the fundamental solution of the frozen–coefficient PDE is not available explicitly (as, e.g., in the Lam’e system of anisotropic elasticity).