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Application of the Adams-Bashfort-Mowlton Method to the Numerical Study of Linear Fractional Oscillators Models

Roman Parovik

TL;DR

A numerical analysis of the class of mathematical models of linear fractional oscillators is presented, it is shown that the ABM method is more accurate and converges faster to an exact solution than the ENFDS method.

Abstract

The paper presents a numerical analysis of the class of mathematical models of linear fractional oscillators, which is the Cauchy problem for a differential equation with derivatives of fractional orders in the sense of Gerasimov-Caputo. A method based on an explicit nonlocal finite-difference scheme (ENFDS) and the Adams-Bashfort-Moulton (ABM) method is considered a numerical analysis tool. An analysis of the errors of the methods is carried out, and it is shown that the ABM method is more accurate and converges faster to an exact solution than the ENFDS method.

Application of the Adams-Bashfort-Mowlton Method to the Numerical Study of Linear Fractional Oscillators Models

TL;DR

A numerical analysis of the class of mathematical models of linear fractional oscillators is presented, it is shown that the ABM method is more accurate and converges faster to an exact solution than the ENFDS method.

Abstract

The paper presents a numerical analysis of the class of mathematical models of linear fractional oscillators, which is the Cauchy problem for a differential equation with derivatives of fractional orders in the sense of Gerasimov-Caputo. A method based on an explicit nonlocal finite-difference scheme (ENFDS) and the Adams-Bashfort-Moulton (ABM) method is considered a numerical analysis tool. An analysis of the errors of the methods is carried out, and it is shown that the ABM method is more accurate and converges faster to an exact solution than the ENFDS method.

Paper Structure

This paper contains 6 sections, 46 equations, 2 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Some reduction from the theory of fractional calculus
  3. Statement of the problem
  4. Methods of solution
  5. Computational Accuracy Analysis
  6. Conclusion

Figures (2)

  • Figure 1: Numerical results using the Adams-Bashfort-Moulton (ABM) and Explicit Nonlocal Finite Difference Scheme (ENFDS) methods compared to the exact solution (ES) for Example 1 at $N = 20$
  • Figure 2: Numerical results using Adams-Bashfort-Moulton (ABM) and Explicit Nonlocal Finite Difference Scheme (ENFDS) versus Exact Solution (ES) for Example 2 at $N = 20$