Skip to main content
SpringerLink
Log in

A Numerical Method for Solving Ordinary Differential Equations by Converting Them into the Form of a Shannon

  • Published:
Mathematical Models and Computer Simulations Aims and scope

Abstract

A numerical solution method based on the reduction of systems of ordinary differential equations to the Shannon form is considered. Shannon’s equations differ in that they contain only multiplication and summation operations. The absence of functional transformations makes it possible to simplify and unify the process of numerical integration of differential equations in the form of Shannon. To do this, it is sufficient in the initial equations given in the normal form of Cauchy to make a simple replacement of variables. In contrast to the classical fourth-order Runge-Kutta method, the numerical method under consideration may have a higher order of accuracy.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.

Similar content being viewed by others

REFERENCES

  1. C. E. Shannon, “Mathematical theory of the differential analyzer,” J. Math. Phys. 20 (4), 337–354 (1941).

    Article  MathSciNet  Google Scholar 

  2. C. Shannon, Papers in Information Theory and Cybernetics (Izd. Inostr. Lit., Moscow, 1963) [in Russian].

    Google Scholar 

  3. A. V. Kalyaev, Theory of Digital Integrating Computers and Structures (Sovetskoe Radio, Moscow, 1970) [in Russian].

    MATH  Google Scholar 

  4. E. Hairer, S. P. Nørset S., and G. Wanner, Solving Ordinary Differential Equations. I. Nonstiff Problems (Springer, Berlin, 1987; Mir, Moscow, 1990).

  5. A. M. Gofen, “Fast Taylor-series expansion and the solution of the Cauchy problem,” USSR Comput. Math. Math. Phys. 22 (5), 74–88 (1982).

    Article  MathSciNet  Google Scholar 

  6. A. A. Samarskii and A. V. Gulin, Numerical Methods. Tutorial (Nauka, Moscow, 1989) [in Russian].

    Google Scholar 

  7. E. K. Zholkovskii, A. A. Belov, and N. N. Kalitkin, “Solution of stiff Cauchy problems with explicit schemes with geometrical-adaptive step selection,” KIAM Preprint No. 227 (Keldysh Inst. Appl. Math. RAS, Moscow, 2018) [in Russian]. /https://doi.org/10.20948/prepr-2018-227

  8. Yu. V. Rakitskij, S. M. Ustinov, and I. G. Chernorutskii, Numerical Methods for Solving Stiff Systems (Nauka, Moscow, 1979) [in Russian].

    Google Scholar 

  9. N. G. Chikurov, “Stability and accuracy of implicit methods for stiff systems of linear differential equations,” Differ. Equations 42 (7), 1057–1067 (2006).

    Article  MathSciNet  Google Scholar 

  10. N. N. Kalitkin, Numerical Methods (Nauka, Moscow, 1979) [in Russian].

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ufa State Aviation Technical University, Ufa, Russia

    N. G. Chikurov

Authors
  1. N. G. Chikurov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. G. Chikurov.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chikurov, N.G. A Numerical Method for Solving Ordinary Differential Equations by Converting Them into the Form of a Shannon. Math Models Comput Simul 13, 274–285 (2021). https://doi.org/10.1134/S2070048221020058

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S2070048221020058

Keywords:

Search

Navigation