On the method for reconstructing the boundary condition for parabolic linear equations

Background. A problem of determination of unknown boundary condition often appears in different fields of physics and technical sciences in cases when direct measuring of field characteristics at some part of the boundary is difficult or even impossible. Examples of such problems can be found in applications of geophysics, nuclear physics, inverse heat transfer problems etc. Their complexity is mainly due to their ill-posedness, i. e. instability of solutions to different perturbations of initial data. Taking into account this feature while solving such problems leads to necessity for development of special regularization methods. In spite of a lot of results obtained in this direction, until present moment the problem of development of new methods for solution of inverse boundary problems of mathematical physics appears relevant.
Materials and methods. An initial boundary value problem for one-dimensional heat equation is considered in the paper. We consider the problem of approximate recovery of a boundary condition at one end of the interval range on changing in spatial variable while functions determining initial condition and also another boundary condition are assumed to be known. As an additional information about we use functionals of the solution of basic initial boundary value problem at some fixed value of the spatial variable. In order to construct the numerical algorithm for solving the problem we use the approach based on integral representation of the basic problem, approximation of the obtained integral equation by collocation technique and realization of the computational scheme by means of the iteration process that is constructed using continuous operator method for solving equations in Banach spaces. The advantages of the method include its simplicity together with its universality and stability of perturbations of the initial data.
Results. Numerical methods for solving the boundary value problem for one-dimensional linear parabolic equation have been constructed. The boundary value problems of first and second type have been considered. Efficiency of the proposed methods is illustrated with several model examples.
Conclusions. The approach to solving direct and inverse problems of mathematical physics based on application of continuous operator method for solving equations in Banach spaces has been proved to be effective for solving boundary value problem for linear one-dimensional heat equation. Further development of this approach for its application to the problem of simultaneous recovery of several boundary conditions and also to the inverse boundary value problem for multidimensional equations seems to be promising.