# Finite Difference Schemes and Classical Transcendental Functions

• Edik A. Ayryan
• Mikhail D. Malykh
• Leonid A. Sevastianov
• Yu Ying
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11189)

## Abstract

In the first part of the article we give a brief review of various approaches to symbolic integration of ordinary differential equations (Liouvillian approach, power series method) from the point of view of numerical methods. We aim to show that all higher transcendental functions were considered in the past centuries as solutions of such differential equations, for which the application of the computational techniques of that time was particularly efficient. Nowadays the finite differences method is a standard method for integration of differential equations. Our main idea is that now all transcendental functions can be considered as solutions of such differential equations, for which the application of this method is particularly efficient.

In the second part of the article we consider an autonomous system of differential equations with algebraic integrals of motion and try to find a totally conservative difference scheme. There are only two cases when the system can be discretized by explicit totally conservative scheme: integrals specify an elliptic curve or unicursal curve. For autonomous systems describing the Jacobi elliptic functions we construct the finite differences scheme, which conserves all algebraic integrals and defines one-to-one correspondence between the layers. We can see that this scheme truly describes the periodicity of the motion.

## Keywords

Conservative difference scheme Elliptic function Symbolic integration Algebraic curve Algebraic correspondence

## References

1. 1.
Goriely, A.: Integrability and Nonintegrability of Dynamical Systems. World Scientific, Singapore (2001)
2. 2.
Schlesinger, L.: Einführung in die Theorie der gewöhnlichen Differentialgleichungen auf funktionentheoretischer Grundlage. Walter de Gruyter & Co., Berlin und Leipzig (1922)
3. 3.
Ayryan, E.A., Malykh, M.D., Sevastianov, L.A.: Finite differences method and integration of the differential equations in finite terms. In: Preprints of the Joint Institute for Nuclear Research (Dubna), no. P11-2018-17 (2018)Google Scholar
4. 4.
Malykh, M.D., Sevastianov, L.A., Ying, Y.: Elliptic functions and finite difference method. In: International Conference Polynomial Computer Algebra 2018, St. Petersburg, pp. 66–68 (2018)Google Scholar
5. 5.
Samarskii, A.A.: The Theory of Difference Schemes. Marcel Dekker, New York (2001)
6. 6.
Malykh, M.D.: On integration of the first order differential equations in finite terms. In: IOP Conference Series: Journal of Physics: Conference Series, vol. 788, p. 012026 (2017)Google Scholar
7. 7.
Sanz-Serna, J.M.: Symplectic Runge-Kutta schemes for adjoint equations, automatic differentiation, optimal control, and more. SIAM Rev. 58(1), 3–33 (2016)
8. 8.
Zeuthen, H.G.: Lehrbuch der abzählenden Methoden der Geometrie. Teubner, Leipzig (1914)
9. 9.
Badr, E.E., Saleem, M.A.: Cyclic Automorphisms groups of genus 10 non-hyperelliptic curves. arXiv:1307.1254 (2013)
10. 10.
Weierstrass, K.: Math. Werke. Bd. 1. Mayer&Müller, Berlin (1894)Google Scholar