A Two Point Difference Scheme of an Arbitrary Order of Accuracy for BVPS for Systems of First Order Nonlinear Odes

Abstract

We consider two-point boundary value problems for systems of first-order nonlinear ordinary differential equations. Under natural conditions we show that on an arbitrary grid there exists a unique two-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection onto the grid of the exact solution of the corresponding system of differential equations. A constructive algorithm is proposed in order to derive from the EDS a so-called truncated difference scheme of an arbitrary rank. The m-TDS represents a system of nonlinear algebraic equations with respect to the approximate values of the exact solution on the grid. Iterative methods for its numerical solution are discussed. Analytical and numerical examples are given which illustrate the theorems proved. Keywords: systems of nonlinear ordinary differential equations, difference scheme, exact difference scheme, truncated difference scheme of an arbitrary order of accuracy, fixed point iteration.

Difference schemes for nonlinear BVPs on the half-axis

“…The next lemma shows that the exact solution of problem (1) can be expressed on each subinterval through the solutions of (24), (25). …”

“…In order to obtain the EDS (27), (28) for all x j ∈ω N it is necessary to solve the problems (24), (25) with v j = u j , j = 0(1)N . It is algorithmically more preferable (because of the well developed theory and software) to deal with IVPs instead of BVPs.…”

“…EDS for miscellaneous mathematical problems which are modelled by ODEs were suggested and studied in [8][9][10]13,14,[19][20][21][22][23][25][26][27][28][29][30]35,36,38,39]. The solutions of these schemes coincide with the projection of the exact solution of the given differential equation onto the underlying grid.…”

Difference schemes for nonlinear BVPs using Runge-Kutta IVP-solvers

“…It is easy to show that condition (PI) guarantees the nonsingularity of the matrix Q ≡ B 0 + B 1 U(1,0) (see, e.g., [14]). Some sufficient conditions which guarantee that the linear homogeneous BVP corresponding to (1.1) has only the trivial solution are given in [14].…”

Exact Difference Schemes for Multidimensional Heat Conduction Equations

“…It is worth to mention here paper [7], in which under natural conditions the authors proved the existence of a two-point exact difference scheme for systems of first-order boundary value problems, or papers of Mickens [12 -16], in which a nonstandard finite difference schemes were introduced. Mickens [11] gives certain rules for constructing nonstandard finite difference schemes and emphasizes that an important feature of nonstandard schemes is that they often can provide numerical integration techniques, for which elementary numerical instabilities do not occur.…”