Difference approximation of elementary differential operators

Grids and grid functions. The composition of a difference scheme approximating a differential equation of interest amouts to performing the following operations  

Following these procedures, we are led to a system of algebraic equations, thereby reducing numerical solution of an initial (linear) differential equation to solving an algebraic system. 

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We would like to discuss the questions raised above in more detail. Obviously, in numerical solution of mathematical problems it is unrealistic to reproduce a difference solution for all the values of the argument varying in a certain domain of a prescribed Euclidean space. The traditional way of covering this is to select some finite set of points in this domain and look for an approximate solution only at those points. Any such set of points is called a grid and the isolated points are termed the grid nodes. 

Example 1. An equidistant grid on a segment. The segment [0,1] of unit length is splitted into N equal intervals. The spacing between the adjacent nodes = h = is termed a grid step or simply step. 

On the segment [0,1] we are working with a new function y x, ) of the discrete argument instead of a given function y(x) of the continuous argument. The values of such a function are calculated at the grid nodes 2, j and the function itself depends on the step h as on the parameter. 

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Difference approximation of elementary differential operators where... 

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