High-precision numerical integration of equations in dynamics

Estimates in the Taylor series method for polynomial total systems of PDEs

“…This work directly continues and generalizes what was proposed in the articles [1][2][3][4][5][6] (for ODE and PDE systems) and [7] (for total linear PDE systems) to the case of total polynomial systems of PDEs. First, we consider some preliminaries (largely from [1][2][3][4][5][6][7]): the Cauchy problem for total systems and polynomial total systems; additional variables method [2][3][4]; Taylor coefficients and estimates to total linear systems of PDEs; Cauchy formula for product of multivariate power series; the idea of schemes and the concept of the Taylor Series Method (TSM). In final section the examples of how one arrives at total polynomial systems of PDEs are discussed.…”

“…consisting from u − n pairs (p(r), q(r)) such that r > p(r), q(r) for any r ∈ [(n + 1) : u] (see [2,6]): using the scheme one can propose algorithms which, given the monomials x i(1) = x 1 , . .…”

Estimates for Taylor series method to linear total systems of PDEs

“…On the Taylor series method. The Taylor series method [4][5][6][7][8] for solving the Cauchy problem (3) consists in constructing a table of approximate values x tw = x(t w ) using the formula…”

“…In the general case of integration along a curve in C s all h w,ν are complex numbers, and points τ w lie on this curve. To calculatex τw for some given τ w with high accuracy by formula (5), even for τ w from its domain of convergence (see (5)), the number of steps may turn out to be large, which can cause a fast accumulation of rounding errors and an increased processor time. That is why it is advisable to use the steps as large as possible (in actual fact, one has to find all ρ ν as large as possible see (6) and Proposition).…”