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

TL;DR

This paper develops and solves a system for optimal Adams-type difference formulas in a Hilbert space, demonstrating their superior accuracy over the Euler method through numerical experiments.

Contribution

It introduces a novel method to construct optimal explicit and implicit Adams-type difference formulas in a Hilbert space, minimizing error functional norms.

Findings

Optimal formulas outperform Euler method in accuracy

System of equations reduced to convolution form

Numerical experiments confirm improved approximation

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.