Graphical Representation and Possibility of Refinement of the Solution Which are calculated by means of the known algorithms.Ĥ. The linear change of variables transforms integrals with variable limits into integrals with constant limits On the third and the subsequent steps we receive On the second step the Equation (3.1) receives the form After averaging on x of the received equation on the segment δ 1 it is possible to calculate w 1įor further calculations we introduce designations On the first step under integral in (3.1). Thanks to it, it is possible to carry out their comparison and processing. Then the system of the bisecting points of the whole segments of the first grid completely contains in a system of the bisecting points of the second grid. Let’s consider two grids (3.2) with steps of h and h/2. (2.7) is system of the equations for definition of integral average values of unknown function on system of segments (2.3).įor the given h we accept the grid of segments , j is number of the equation, i is number of item in the equation. Let’s carry out to (2.6) integral averaging on segments δ j Where δ i is characteristic function of the corresponding segment. Let’s provide the function y(t) under integral (2.1) in the form

They have the form or functions y(x) and f(x) We will enter concept of integral average values on these segments. We build segments δ i in the vicinity of each point We form on the segment a uniform grid of N segments with a step Which has the unique solution in a class of the integrable functions. On the main segment of the integral equation the system of segments is formed and on each such segment the integral average value of function is defined.

The essence of the offered algorithm is as follows. įor convenience of readers in this work, fundamentals of the article are repeated.

In such situation the alternative version of the quadrature solution of the integral equation is possible. This requirement cannot be fulfilled for the class of integrable functions. In other words, the integrable function has to have a certain value at each point. Quadrature formulas are applied only to continuous functions. Quadrature formulas are quite often used in the solution of the integral equations. Numerical methods of the solution of the equations are universal.

Linear integral equations