Mayko N. The weighted estimates of the functional-discrete methods for solving boundary value problems

The dissertation is devoted to the construction and study of the approximate methods for solving the problems of mathematical physics, namely to obtaining weighted accuracy estimates of these methods with taking into account the influence of boundary and initial conditions. The weighted error estimates of the finite-difference approximation for Poisson's equation and for one- and two-dimensional heat equations in canonical domains under different types of boundary conditions with allowance for the initial and boundary effect are obtained. For the second-order ODE with a fractional derivative in the case of constant and variable coefficients, a number of sufficient conditions for exact solutions to belong to certain functional spaces are found and weighted estimates that take into account the impact of the Dirichlet boundary condition are obtained. To solve the problem numerically, we construct a grid scheme and obtain the accuracy estimate in various discrete norms with taking into account the boundary effect. We also find the weighted estimates for exact and approximate solutions of the first boundary value problem for Poisson's equation in a unit square with a fractional derivative and for the Goursat problem for a differential equation with variable coefficients and fractional derivatives. Next, the weighted estimates in the integral norm for the accuracy of the Cayley transform method for solving the abstract Cauchy problem for the first-order differential equation with a self-adjoint positive definite operator in a Hilbert space are obtained, and their unimprovability in order is studied. The Cayley transform methods without saturation of accuracy for solving the second-order differential equations in Hilbert and Banach spaces are constructed, and the weighted error estimates with taking into account the influence of the boundary conditions and the smoothness of the input data are obtained.