# Implementation of the Adams method for solving ordinary differential equations in the Sage computer algebra system

## Abstract

This work is devoted to the implementation and testing of the Adams method for solving ordinary differential equations in the Sage computer algebra system. The Sage computer algebra system has, to some extent, trivial means for numerical integration of ordinary differential equations, but at the same time, it is worth noting that this environment is convenient and practical for conducting computer experiments related to symbolic numerical calculations in it. The article presents the FDM package developed on the basis of the RUDN, which contains the developments of recent years, performed by M. D. Malykh and his students, for numerical integration of differential equations. In this package, attention is paid to the visualization of the calculation results, including the construction of various kinds of auxiliary diagrams, such as Richardson diagrams, as well as graphs of dependence, for example, the value of a function or step from a moment in time. The implementation of the Adams method will be considered from this package. In this article, this implementation of the Adams method will be tested on various examples of input data, and the method will also be compared with the Jacobi system. Exact and approximate values will be found and compared, and an estimate for the error will be obtained.

## Full Text

1. Introduction To describe models in a variety of subject areas from mechanics to economics, ordinary differential equations are used [1]. These equations admit solutions in elementary functions only in some very special cases, therefore they are usually investigated numerically. The finite difference method was proposed by Euler, the Runge-Kutta method of the 4th order is the most popular numerical method for solving initial problems for ordinary differential equations [2]. Old authors, including J. Scarborough [3, ch. XIII], mention numerical methods alternative to the Runge-Kutta method. The method that J. Scarborough has associated with the name of the English theoretical astronomer J. K. Adams, was forgotten for a long time, because it was very inconvenient to implement on a computer: before its use, a number of preparatory calculations had to be carried out on paper. However, with the development of computer computing, it became possible to perform these actions on a computer, which pushes us to study the possibility of implementing the Adams method in modern computer algebra systems. Currently, RUDN University is developing an addition to Sage - the FDM package, which contains the achievements of recent years, made by M. D. Malykh and his students. The goal of the project is to create a convenient environment for numerical experiments with ODES in the Sage computer algebra system. This project is available to everyone on https://github. com/malykhmd/fdm. The general principles of the package are described in [4]. The purpose of this work is to test the implementation of the Adams method in FDM. 2. The Adams method and its implementation in FDM Consider the initial problem
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### Mikhail D. Malykh

RUDN University; Joint Institute for Nuclear Research

Author for correspondence.
Email: malykh-md@rudn.ru
ORCID iD: 0000-0001-6541-6603
Scopus Author ID: 6602318510
ResearcherId: P-8123-2016

Doctor of Physical and Mathematical Sciences, Assistant Professor of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation; 6, Joliot-Curie St., Dubna, Moscow Region, 141980, Russian Federation

### Polina S. Chusovitina

RUDN University

Email: 1032192941@rudn.ru
ORCID iD: 0009-0006-4191-2454

Student of Department of Applied Probability and Informatics

6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation

## References

1. H. Gould and J. Tobochnik, An introduction to computer simulation methods. Applications to physical systems. Addison-Wesley Publishing Company, 1988.
2. A. Baddour and M. D. Malykh, “Richardson-Kalitkin method in abstract description,” Discrete and Continuous Models and Applied Computational Science, vol. 29, no. 3, pp. 271-284, 2021. doi: 10.22363/2658-4670-2021-29-3-271-284.
3. J. B. Scarborough, Numerical methods of mathematical analysis. Oxford book company, 1930.
4. L. Gonzalez and M. D. Malykh, “On a new package for numerical solution of ordinary differential equations in Sage [O novom pakete dlya chislennogo resheniya obyknovennykh differentsial’nykh uravneniy v Sage],” in Proceedings of ITTMM’22, Moscow, Russia, in Russian, 2022, pp. 360-364.

Copyright (c) 2023 Malykh M.D., Chusovitina P.S.