On the numerical solution of the coefficient inverse problem for hyperbolic equations

Background. Partial differential equations of hyperbolic type hold a prominent place in mathematical modeling of different processes and phenomena in physical and technical sciences. In particular, hyperbolic equations are widely used in such areas as acoustics, elasticity theory, aerodynamics and electrodynamics. At present time the theory of inverse and incorrect problems for partial differential equations is intensively developing and finds expanding applications in wide variety of application areas. In addition to that, there are important needs in further devising of precise and stable methods which allow us to solve different types of inverse problems. The goal of this paper is constructing of the mentioned methods for solution of one class of inverse coefficient problems for the simplest hyperbolic equations such as wave equation. Materials and methods. Construction of algorithms for solution of inverse initial and boundary coefficient problems for one- and two-dimensional wave equation is based on application of continuous method for solution of nonlinear operator equations in Banach spaces. The important feature of this method is that its implementation do not require construction of inverse operator. At the method's core is substitution of the original nonlinear operator equation for the differential equation of special type and its subsequent approximate solution using methods of stability theory for systems of ordinary differential equations. Results. A problem of numerical solution of inverse coefficient problems for one- and two-dimensional wave equations is studied in the paper. Both initial and boundary value problems for wave equation are considered. As a result algorithms for numerical solution of the mentioned problem are proposed. Solving of several model problems demonstrates the effectiveness of the proposed algorithms. Conclusions. On the base of continuous method for solution of nonlinear operator equation simple and effective algorithms for numerical solution of inverse coefficient problems for wave equation are proposed.