Volume Integral Equation Method in Problems of

Preface In our course we will consider the volume integral equations in the following form) () () () () () (x f dy y u y b y x y x K x u x a Q m = − − + ∫ , 3 ≤ m. Many important classes of the wave scattering problems can be described by equations of this form; for example, this is the case for problems of electromagnetic and acoustic scattering on 3D transparent bodies. The corresponding integral operator is compact (m=1) in acoustic problems and singular (m=3) in electromagnetic problems. Why do we want to consider integral equations though the initial problems usually are formulated as boundary value problems for partial differential equations? For that there are two main reasons. I. One of the ways for the mathematical investigation (proof of the existence and uniqueness theorems, etc.) of the initial problem of mathematical physics is the following. We reduce the initial boundary value problem to an integral equation. Then we establish the equivalence of the differential formulation of the problem and the corresponding integral equation. It means that any solution of the integral equation (maybe with some restriction on the parameters of the problem) satisfies the partial differential equations and boundary condition and back. Based on the integral inequalities which are usually obtained from the differential formulation, we prove the uniqueness theorem of our problem. Then using the theory of solvability of integral equations (Fredholm integral equation or singular integral equation theories) in appropriate (from the physical point of view) functional space we prove the existence and uniqueness theorem and others facts for the initial problem of mathematical physics. In our course we will follow these steps. II. We will construct the methods and algorithms for the numerical solution of the initial problems by using integral equations. At the first glance the partial differential equations are more appropriate for the numerical solution because after discretization we receive the system of linear algebraic equations (SLAE) with sparse matrix in comparison with the full matrix which we obtain in the integral equation case. But for the wave scattering problems the solution must satisfy the radiation condition at infinity. Therefore for the good accuracy we need to find numerically the unknown wave field in the domain which is sufficiently greater than scattering object Q and due to 3D of the initial problem the dimension of the SLAE …