On the iterative method for solution of direct and inverse problems for parabolic equations

The paper is devoted to approximate methods for solution of direct and inverse problems for parabolic equations. An approximate method for the solution of the initial problem for multidimensional nonlinear parabolic equation is proposed. The method is based on the reduction of the  initial problem to a nonlinear multidimensional intergral Fredholm equation of the second kind which is approximated by a system of nonlinear algebraic equations with the help of the method of mechanical quadratures. For constructing the computational scheme we use the nodes of the local splines which realize order-optimal approximation of the functional class that contains solutions of parabolic equations. For implementation of the computational scheme we use the generalization of the continuous method for solution of nonlinear operator equations that is described in the paper. We also analyse the inverse problem for parabolic equation with fractional order derivative with respect to the time variable. The approximate methods for defining the fractional order of the time derivative and the coeffcient at spatial derivative are proposed

Stability of solutions for systems of delayed parabolic equations

Background. The study is devoted to the analysis of stability in the sense Lyapunov steady state solutions for systems of linear parabolic equations with coefficients depending on time, and with delays depending on time. The cases of continuous and impulsive perturbations are considered. Materials and methods. A method for studying the stability of solutions to systems of linear parabolic equations is as follows. Applying the Fourier transform to the original system of parabolic equations, we arrive at a system of non-stationary ordinary differential equations defined in the spectral region. First, the stability of the resulting system is studied by the method of frozen coefficients in the metric of the space Rn of n-dimensional vectors. Then the resulting statements are extended to the space L2 . The application of the Parseval equality allows us to return to the domain of the originals and obtain sufficient conditions for the stability of solutions to systems of linear parabolic equations. Results. An algorithm is proposed that allows one to obtain sufficient stability conditions for solutions of finite systems of linear parabolic equations with time-dependent coefficients and with time-dependent delays. Sufficient stability conditions are expressed in terms of the logarithmic norms of matrices composed of the coefficients of the system of parabolic equations. They are obtained in the metric of the space L2 . Algorithms for constructing sufficient stability conditions are efficient, as in the case continuous, and in the case of impulsive perturbations. Conclusions. A method for constructing sufficient stability conditions for solutions of finite systems of linear parabolic equations with time-dependent coefficients and delays. The method can be used in the study non-stationary dynamical systems described by systems of linear parabolic equations with delays depending from time

Approximate methods for solving degenerate singular integral equations

Background. Singular integral equations in degenerate cases describe many processes in natural science and technology. The theory of these equations has been studied quite well, but as far as the authors know, there are currently no analytical methods for solving them. In this regard, there is a need to construct approximate methods for solving singular integral equations in degenerate cases. The article is devoted to the construction of such methods, which determines its relevance. Materials and methods. When constructing approximate methods, iteration-projection methods are used. Results and conclusions. A spline-collocation method for solving a degenerate singular characteristic equation is constructed. A two-stage approximate method is proposed for solving complete singular integral equations in degenerate cases and their characteristic equations