An initial-boundary value problem for an inhomogeneous system of Maxwell’s equations in the case of a ferromagnetic conducting body with anisotropy and internal defects

Abstract

Background. The initial-boundary value problems for the system of Maxwell’s equations are needed in the context of describing and calculating a non-stationary electromagnetic field (non-harmonic time dependent field). A non-stationary electromagnetic field are induced during transient processes in electrical and radio-engineering devices, nonstationary methods of electrical exploration and eddy-current flaw detection are associated with its use. These circumstances substantiate the relevance and applied significance of the initial boundary-boundary problems of electrodynamics. The purpose of this work is to prove the existence, uniqueness and continuous dependence on the initial data of the solution of the initial boundary value problem for Maxwell’s equations in the case of an anisotropic defective ferromagnet. Materials and methods. In this work methods and techniques of the theory of evolutionary problems in Banach spaces are used. Results. To formulate the studied initial-boundary value problem, a functional class, which takes into account the properties of differential operations, appearing in Maxwell’s equations, and also takes into account the boundary conditions at the boundaries of the ferromagnet and its internal defects, are chosen. With using the general theorem about correctness of the Cauchy problem in Banach space, it is proved, that the proposed functional class guarantees the existence of unique solution for the studied initial-boundary problem, which continuously depends on the initial data. Conclusions. The initial-boundary value problem for the system of Maxwell’s equations in the case of an anisotropic defective ferromagnet, with a certain choice of functional class for its formulation, satisfies the conditions for correctness of the evolutionary problem in Banach space.

Keywords

• initial-boundary value problem
• maxwell's equations
• anisotropy
• banach space
• closed operator
• integro-differential equations