On the approximate method for determination of heat conduction coefficient

The problem of recovering a value of the constant coefficient in heat equation for one- and two-dimensional cases is considered in the paper. This inverse coefficient problem has broad range of applications in physics and engineering, in particular, for modelling heat exchange processes and for studying properties of materials and designing of engineering constructions. In order to solve the problem an approximate method is constructed; it is based on the continuous operator method for solving nonlinear equations. The advantages of the proposed method are its simplicity and universality. The last property allows to apply the method to a wide range of problems. In particular, in constructing and justifying a continuous operator method, in contrast to the Newton–Kantorovich method, the continuous reversibility of Frechet or Gato derivatives is not required. Moreover, derivatives may not exist on sets of measure zero. The application of continuous operator method to the solution of an inverse coefficient problem with a constant coefficient makes it possible to minimize additional conditions – there is enough information about the exact solution at a single point $x^*, t^*$. Solving several model problems illustrates the high efficiency of the proposed method.