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Algorithm for constructing optimal explicit finite-difference formulas in the Hilbert space

Kh. M. Shadimetov, R. S. Karimov

TL;DR

This paper introduces a functional framework to optimize explicit finite-difference formulas in the Hilbert space $W_{2}^{(m,m-1)}(0,1)$. It leverages the extremal function and the Green's function $G_m$ via the Riesz representation to express and minimize the error functional, leading to a constrained linear system for the optimal coefficients. An algorithm based on Wiener-Hopf-type convolutions yields representations for the Adams-type coefficients for any $m\ge3$, including explicit boundary formulas. The results establish existence and uniqueness of the optimal solution and provide practical, closed-form constructions for the coefficients needed to form optimal explicit finite-difference formulas in this Hilbert-space setting.

Abstract

This work presents problems of constructing finite-difference formulas in the Hilbert space, i.e., setting problems of constructing finite-difference formulas using functional methods. The work presents a functional statement of the problem of optimizing finite-difference formulas in the space $W_{2}^{\left(m,m-1\right)} \left(0,1\right)$. Here, representations of optimal coefficients of explicit finite-difference formulas of the Adams type on classes $W_{2}^{\left(m,m-1\right)} \left(0,1\right)$ for any $m\ge 3$ will be found.

Algorithm for constructing optimal explicit finite-difference formulas in the Hilbert space

TL;DR

This paper introduces a functional framework to optimize explicit finite-difference formulas in the Hilbert space . It leverages the extremal function and the Green's function via the Riesz representation to express and minimize the error functional, leading to a constrained linear system for the optimal coefficients. An algorithm based on Wiener-Hopf-type convolutions yields representations for the Adams-type coefficients for any , including explicit boundary formulas. The results establish existence and uniqueness of the optimal solution and provide practical, closed-form constructions for the coefficients needed to form optimal explicit finite-difference formulas in this Hilbert-space setting.

Abstract

This work presents problems of constructing finite-difference formulas in the Hilbert space, i.e., setting problems of constructing finite-difference formulas using functional methods. The work presents a functional statement of the problem of optimizing finite-difference formulas in the space . Here, representations of optimal coefficients of explicit finite-difference formulas of the Adams type on classes for any will be found.

Paper Structure

This paper contains 6 sections, 5 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Statement of the problem of optimization of finite-difference formulas
  3. Error of finite-difference formulas on classes $W_{2}^{(m,m-1)} (0,1)$
  4. A system of linear algebraic equations for finding optimal coefficients of finite-difference formulas
  5. Algorithm for finding optimal coefficients of finite-difference formulas
  6. Coefficients of optimal finite-difference formulas of Adams type in space $W_{2}^{(m,m-1)} (0,1)$

Key Result

Theorem 2.2

The maximizing element, i.e. the extremal function of the finite-difference formulas GrindEQ__6_ in the Hilbert space $W_{2}^{(m,m-1)} \left(0,1\right)$ is given by the equality where is the Green's function or the fundamental solution to the following equation $d=const,$$sgnx=\left\{\right.$$P_{m-2} \left(x\right)$- some unknown polynomial of degree $m-2$, $\delta(x)$ is the Dirac delta functi

Theorems & Definitions (12)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • Definition 5.1
  • Definition 5.2
  • Definition 5.3
  • Theorem 6.1
  • proof
  • ...and 2 more