About Localized Boundary- Domain Integro-Differential Formulations for a Quasilinear Problem with Variable Coefficients

Abstract

Application of the boundary integral equation (BIE) method (boundary element method) to the solution of linear boundary value problems (BVPs) for partial differential equations (PDEs) has been intensively developed over recent decades. Using a fundamental solution of an auxiliary linear PDE, a nonlinear BVP can be reduced to a nonlinear boundary-domain integral or integro-differential equation (BDIE or BDIDE) (see, e.g., [1,2,3]). However, a fundamental solution necessary for the reduction is usually highly nonlocal, which generally leads after discretization to a system of nonlinear algebraic equations with a fully populated matrix. Moreover, the fundamental solution is generally not available if the coefficients of the auxiliary PDE are not constant.