RESEARCH OF MULTICONNECTED DYNAMIC SYSTEM BY SOLVING OF SYSTEMS OF VOLTERRA INTEGRAL EQUATIONS OF THE FIRST KIND BY THE METHOD OF COLOCATIONS

Abstract

Multiconnected dynamic systems are characterized by a large number of interconnected input and output values, which determines the complexity of their mathematical description. Modeling of such systems in the form of systems of Volterra integral equations is an effective approach, but requires the implementation of Volterra operators. Their feature is the accumulation of calculations at each next approximation step. As a result, the reduction of the approximation step leads to a significant increase in the number of computational operations. This puts special demands on the simplicity and speed of the corresponding algorithms. The efficiency of applying the collocation method based on piecewise smooth polynomials to the solution of this class of equations is studied. The collocation method is based on obtaining a solution on sections whose length is selected, and on each of them an approximating expression with a small number of coordinate functions is applied. A great advantage of algorithms based on the method of collocations is great flexibility in choosing parameters for replacing functions with piecewise smooth polynomials. The proposed algorithms are implemented in the MATLAB computer mathematics system in the function called slvie1colloc. The constituent parts of the system of integral equations (kernels, right-hand parts) are transmitted by program arguments in the form of anonymous functions or in the form of a table of numerical values of the functions in the approximation nodes. Additional arguments are the numerical values of the nodes of the approximation grid, the degree of the polynomial, and the initial conditions of the integral equation. The program checks the input data, in case of incorrect values, an error code is displayed and the program is interrupted. The computer program was tested using computational experiments. The described results demonstrated the effectiveness of the proposed solutions. The absolute error of calculations for the considered model in the form of a system of Volterra integral equations of the first kind with a kernel of the general form under the given parameters did not exceed 3,5 * 10–3.