Estimates for Taylor series method to linear total systems of PDEs

Abstract

A large number of differential equations can be reduced to polynomial form. As was shown in a number of works by various authors, one of the best methods for the numerical solution of the initial value problem for such ODE systems is the method of Taylor series. In this article we consider the Cauchy problem for the total linear PDE system, and then-a theorem about the accuracy of its solutions by this method is formulated and proved. In the final part of the article, four examples of total systems of partial differential equations to the well-known two-body problem are proposed: two of them are related to the Kepler equation, one to the motion of a point in the orbit plane, and the last to the motion of the orbit plane.

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.…”

“…Note that in the case s = 1, these formulas reduce to ( 13), (14). The estimates for linear equations one can find in our paper [7].…”

“…• four examples of total polynomial systems of partial differential equations for the two-body problem proposed by us in [7];…”

“…For example, you can use α obtained for the linear approximation of the original non-linear equations in a neighborhood of the initial data. In the paper [7], we explained how, in applications, the matrix A ν of the mentioned approximation should be replaced with a square positive matrix A + ν so that Perron's theorem [14] can be used to select scaling factors. Then as scaling factors α 1 , .…”