Finite difference schemes and classical transcendental functions

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.