1. Optimization of Adams-type difference formulas in Hilbert space W2(2,1)(0, 1).
- Author
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Shadimetov, Kh. M. and Karimov, R. S.
- Subjects
- *
HILBERT space , *ALGEBRAIC equations , *DIFFERENTIAL operators , *DIFFERENCE equations , *EULER method , *CAUCHY problem , *ORDINARY differential equations , *LINEAR systems - 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 d²/dx² - 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. [ABSTRACT FROM AUTHOR]
- Published
- 2024