On an explicit optimal difference formula
Keywords:
Sobolev space, extremal function, discrete analogue, Lagrange function, Sobolev method, optimal coefficientsAbstract
The finite difference method is used to numerically solve many problems in physics and engineering described by mathematical physics equations. The basic concepts of difference methods are approximation, stability, and convergence, which are illustrated by examples of difference schemes for ordinary differential equations. Optimization of the process of finding approximate solutions to ordinary differential equations is essential in a large number of calculations. Therefore, in this paper, we consider the problem of constructing an explicit optimal difference formula in the Sobolev space for an approximate solution of the Cauchy problem posed for an ordinary differential equation of the second order. Here we present an algorithm for the construction of an explicit optimal difference formula. We use the Sobolev method to construct the explicit optimal difference formula of the Sturmer type. For this purpose, using the discrete analogue of the second-order differential operator, we find the appearance of the optimal difference formula and also calculate the error estimate of this formula.
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