Exponentially convergent methods for the nonlocal abstract Cauchy problem and nonlinear boundary value problems
Prof. Volodymyr Makarov
Defended, approved
Institute of mathematics, NAS Ukraine

The thesis is devoted to the development of numerical and analytical methods which determine the existence of the solution to a nonlocal Cauchy problem for linear abstract differential equation of the first order with an unbounded operator coefficient in Banach space. For such kind of Cauchy problems a new exponentially–convergent numerical method has been developed. Aside of that, we further present a general scheme of FD–method for numerical solution to abstract nonlinear functional equations. Using the general framework of FD–method, we develop superexponentially–convergent numerical method for the nonlinear boundary value problem on an interval.
In particular, for the case when non-local condition represented as a linear combination of the unknown solution at different times we have presented an approach allowing us to reduce the non-local problem to the equivalent classical Cauchy problem. As a result, an explicit representation of solution to the original non-local problem have been established. The conditions under which such reduction process can be justified along with the corresponding theorem on the existence of solution to the classical Cauchy problem form the primary criterion for the existence of solution to the given non-local problem. Such conditions essentially depend on the relative position of zeros of an entire function , correspondent to the coefficients of non-local condition, and the spectrum of strongly–positive operator coefficient. For the case when it is impossible to obtain a closed form set of zeros for we developed a new approach, which allows us to study the set of zeros of related polynomials instead. The application of this approach results in several new sufficient conditions for the existence of solution to the original non-local problem which generalizes the sufficient conditions available in the literature.
In order to approximate the solution to the given non-local Cauchy problem we have developed an exponentially convergent method based on the explicit representation of the solution via Dunford–Cauchy integral with subsequent application of Sinc–quadrature formulas.
The condition on the existence of solution to the given non-local problem along with some natural assumptions about the smoothness of initial data combined with the proper choice of integration contour and the quadrature step enabled us to prove the a-priori error estimates for the developed method and show its exponential rate of convergence. We also derived an efficient parallel algorithm for the proposed numerical method, and studied its computational complexity. Some practical aspects of the implementation of fully–discretized version of the algorithm have been studied as well.
Aside of that we proposed a general FD–method scheme for solving abstract nonlinear functional equation and developed a superexponentially convergent numerical method for the nonlinear BVP on an interval using the framework of this scheme. Theorems on the existence of solution of the base problem as well as the convergence of FD-method to a solution of the given problem have been proved. Furthermore, using the method of generating functions we established an a-priory error estimate of the developed method.