Optimization of difference formulas for solving differential equations in the Hilbert space.

In the numerical solution of Initial Value Problems (IVP) for ordinary differential equations, computational methods serve to approximate the determination of functions representing the solution of these problems. The problem of finding the most convenient numerical expressions for a function and its connection with methods for improving such approximations plays an important role in practical calculations. It is of great interest to consider the so-called discrete methods, i.e. methods that determine the solution for discrete values of the independent variable. Discrete methods are currently the most widely used. One of the discrete methods is difference formulas for the numerical solution of the IVP. In this paper, in the Sobolev space where all derivatives up to the mth order participate in the norms of functions, we consider the problem of constructing optimal difference formulas. In the optimization of difference formulas, i.e. when constructing optimal difference formulas in function spaces, an important role is played by the extremal function of a given difference formula. In this paper, the extremal function of the difference formula is explicitly found in the Sobolev space. Next, the square of the norm of the error functional of difference formulas is calculated. Minimizing this norm with respect to the coefficients of the difference formulas, a system of equations is obtained. In addition, the existence and uniqueness of a solution to the resulting system are proved. [ABSTRACT FROM AUTHOR]