The scalar boundary value problem (BVP) for a nonlinear second order
differential equation on the semiaxis is considered. Under some natural assumptions
it is shown that on an arbitrary finite grid there exists a unique three-point exact
difference scheme (EDS), i.e., a difference scheme whose solution coincides with the
projection of the exact solution of the given differential equation onto the underlying
grid. A constructive method is proposed to derive from the EDS a so-called truncated
difference scheme (n-TDS) of rank n, where n is a freely selectable natural number.
The n-TDS is the basis for a new adaptive algorithm which has
all the advantages known from the modern IVP-solvers. Numerical examples are given
which illustrate the theorems presented in the paper and demonstrate the reliability of
the new algorithm.
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.
We develop a new algorithmic implementation of the exact three-point difference schemes on a nonuniform grid for the Sturm-Liouville problem. It is shown that, in order to find the coefficients of exact scheme for an arbitrary node of the grid, it is necessary to solve two auxiliary Cauchy problems for second-order linear ordinary differential equations: one problem on the interval [x j−1 , x j ] (forward) and one problem on the interval [x j , x j+1 ] (backward). We prove the theorem on the coefficient stability of the exact three-point difference scheme.
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