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arXiv · 2307.05026

Optimization of Adams-type difference formulas in Hilbert space $W_2^{(2,1)}(0,1)$

Kh. M. Shadimetov·R. S. Karimov
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Abstract

In this paper, we consider the problem of constructing new optimal explicit and implicit Adams-type difference formulas for finding an approximate solution to the Cauchy problem for an ordinary differential equation in a Hilbert space. In this work, I minimize the norm of the error functional of the difference formula with respect to the coefficients, we obtain a system of linear algebraic equations for the coefficients of the difference formulas. This system of equations is reduced to a system of equations in convolution and the system of equations is completely solved using a discrete analog of a differential operator $d^2/dx^2-1$. Here we present an algorithm for constructing optimal explicit and implicit difference formulas in a specific Hilbert space. In addition, comparing the Euler method with optimal explicit and implicit difference formulas, numerical experiments are given. Experiments show that the optimal formulas give a good approximation compared to the Euler method.

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Kh. M. Shadimetov, R. S. Karimov. 2023-07-11. Optimization of Adams-type difference formulas in Hilbert space $W_2^{(2,1)}(0,1)$. https://arxiv.org/abs/2307.05026

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