Sidorov M. Two-sided approximations methods for solving certain classes of nonlinear problems in mathematical physics

The object of study is processes whose mathematical models are problems for semilinear elliptic equations, systems of semilinear elliptic equations, and semilinear parabolic equations. The purpose of the work is to develop two-sided iterative methods for solving the first boundary-value problem for the semilinear elliptic equation and the system of semilinear elliptic equations, and the method for solving the first initial-boundary-value problem for the semilinear parabolic equation based on the joint application of the Rothe method and two-sided approximations. Research methods: methods of theory of nonlinear operator equations in semi-ordered spaces, methods of mathematical physics, constructive apparatus of the R-functions theory. Theoretical and practical results – the obtained results are a solution of the scientific problem of constructing methods of two-sided approximations for solving problems for nonlinear equations of mathematical physics. Namely, the work developed two-sided iterative methods for solving the first boundary-value problem for a semilinear elliptic equation and a system of semilinear elliptic equations and a semidiscrete method for solving the first initial-boundary value problem for a semilinear parabolic equation based on the joint use of Rothe method and two-sided approximations. The scientific novelty of the results obtained is that the concept of the Green-Rvachev quasifunction of the first boundary value problem for a nondegenerate second-order elliptic operator was introduced for the first time and with its help it was obtained an integral equation equivalent to the first boundary-value problem for a semilinear elliptic equation and a system of integral equations equivalent to the first boundary problem for a system of semilinear elliptic equations, in domains, whose geometry can be analytically described using structural tools of R-functions theory; it was firstly distinguished the class of semilinear ordinary differential equations, the first boundary-value problem for which allows its representation (using the Green function) as a nonlinear operator equation with a heterotone type operator and the class of semilinear elliptic equations and systems of semilinear elliptic equations, the first boundary-value problem for which allows its representation (using the Green function or the Green-Rvachev quasifunction) in the form of a nonlinear operator equation with a heterotone type operator, which allows build two-sided iterative methods for finding positive solutions of these problems; it was further developed the method of two-sided approximations for solving the first boundary-value problem for semilinear ordinary differential equations and for solving the first boundary-value problem for a semilinear elliptic equation based on the use of the Green function in its application to equations of a more general form; it was developed for the first time the method of two-sided approximations for solving the first boundary-value problem for a semilinear elliptic equation based on the use of the Green-Rvachev quasifunction; it were developed for the first time the methods for two-sided approximations for solving the first boundary-value problem for a system of semilinear elliptic equations based on the use of the Green function or the Green-Rvachev quasifunction; it was developed for the first time a semidiscrete method for solving the first initial-boundary value problem for a semilinear parabolic equation based on the joint use of Rothe methods and two-sided approximations; for the first time from the equations with a non-linear coefficient of thermal conductivity a class of equations has been singled out whose solution to the first boundary-value problem can be found by the method of two-sided approximations, which made it possible to obtain the conditions for the existence of a unique positive solution to the problem and the convergence of successive approximations to it; for the first time, the method of two-sided approximations was applied to the solution of the nonlinear Navier problem, on the basis of which it were obtained the conditions for the existence of a unique positive solution to the problem and the convergence of successive approximations to it; it was improved the method of constructing a strongly invariant cone segment in terms of using the apparatus of the theory of R-functions to select its lower and upper ends, chosen as initial approximations in the implementation of two-sided iterative methods. The results of the thesis are introduced in the educational process of the Kharkov National University of Radio Electronics. The results can be used in mathematical modeling of the processes described by the first boundary-value problem for a semilinear elliptic equation and a system of semilinear elliptic equations and the first initial-boundary value problem for a semilinear parabolic equation.