Determination of the differential-taylor spectrum of a complex function for a variant of superposition in analysis of the accuracy of dynamic systems
Keywords:differential-Taylor transformations, differential Taylor spectrum, superposition of functions
In the article, dependences are obtained for determining the differential-Taylor spectrum of a complex function, given in the form of a superposition of functions. Namely, for the case when the original function is given by the Taylor series in the degree of some variable - the first argument, and the final function is given by the Taylor series in the degree of the original function. Next, we solve the problem of determining the differential Taylor spectrum - the coefficients of the Taylor series of a finite function in powers of the first argument. In the classical literature on differential-Taylor transformations, the specified differential-Taylor spectrum (individual terms of the Taylor series) is presented as an infinite sum in powers of the first argument. In the article, dependences are obtained that define the differential-Taylor spectrum of a superposition of functions as a finite sum in powers of the first argument. At the same time, dependences are given in two different forms, which makes it possible to choose a form that is more convenient for specific practical use. A feature of the obtained formulas is the use of reduced algebraic convolution when calculating the differential-Taylor spectrum of the power function for the first argument - the convolution does not take into account the zero discrete of the differential-Taylor spectrum of the original function with respect to the first argument. At the same time, dependences are given in two different forms, which makes it possible to choose a form that is more convenient for specific practical use. A feature of the obtained formulas is the use of reduced algebraic convolution when calculating the differential-Taylor spectrum of the power function for the first argument - the convolution does not take into account the zero discrete of the differential-Taylor spectrum of the original function with respect to the first argument. The obtained relations are essential for the problems of analyzing the dependence of the accuracy of the representation of a finite function on a given number of taken into account discrete differential-Taylor spectrum of the original function, as well as solving the problem of estimating the dependence of the accuracy of the solution of the Cauchy problem for ordinary differential equations by the method of differential-Taylor transformations when the number of taken into account discrete Taylor spectrum that take part in the calculations. The obtained dependences are a further development of the theoretical foundations of the domestic mathematical apparatus of Pukhov's differential-Taylor transformations.
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