Boundary Integral Formulation and Analysis of Higher-Order Thermoelastic Systems in Couple-Stress Elasticity

We investigate the boundary integral formulation of a steady size-dependent thermoelastic system governed by consistent couple-stress theory. The model consists of a coupled higher-order elliptic system incorporating intrinsic material length-scale effects and non-classical boundary conditions involving both force and moment tractions. Using the fundamental solutions of the associated differential operators, we reformulate the governing equations as a system of boundary integral equations posed on appropriate Sobolev trace spaces. The mapping properties of the resulting boundary operators are analyzed in detail. We establish boundedness and characterize their Fredholm structure within a Hilbert space framework, which yields existence and uniqueness of weak solutions under natural coercivity assumptions. A Galerkin boundary element discretization is introduced, and its stability is proved via a discrete inf–sup condition. Convergence of the approximation is derived through Céa-type estimates, providing a priori error bounds in fractional Sobolev norms. The analysis demonstrates that the higher-order structure induced by couple-stress effects preserves the well-posedness of the boundary integral formulation and leads to a stable and convergent numerical scheme. The results furnish a rigorous operator-theoretic foundation for boundary element methods applied to size-dependent continuum models and clarify the analytical structure of boundary operators associated with higher-order thermoelastic systems.