On one method of multiplicative models elaboration during experiments

Abstract

Description of functions in the vicinity of a point within the domain of a function is most often used for problems solution of mathematical physics as the Taylor series approximation. The reason for this approximation seems to be the application of function derivatives. The greater the order of derivatives, the more accurately the function in vicinity of the selected point could be presented. However, there is a necessity exists to define functions at the point of the mathematical models during experimental studies in a variety field of science. Mostly, two types of the models are used - additive and multiplicative ones. The multiplicative model is distinguished by the practical sense as well as widespread use. The fact is that the nature of a particular function research for technology industrial problems consists in the sequential change of its parameters. The study of function change upon condition of a single parameter change involves the retention of other parameters of certain pre-selected values, i.e. at a certain point in the functional space of parameters. It is not always clear that the result of the experimental study is finding the values of the function exceptionally in the vicinity of this point, not within the function domain. Neglecting this circumstance along with attempts to find the values of the function far out beyond the vicinity of selected point leads to the values of the function with inappropriate error. The approximate representation of scalar functions in the multiplicative form in the vicinity of the point has a wide range of applications, especially for geomechanics. It turned out that the approximate representation of scalar functions in a multiplicative form at a point within the domain could be extended to the whole domain. Moreover, the maximum error of a representation at the boundary of the domain for geotechnical problems, as a rule, does not exceed 5-7%, which is acceptable for engineering calculations. To test an efficiency of the successive approximations method an applied geomechanical problem has been solved. The conclusion on the efficiency of method for geomechanical problem is made. Keywords: mathematical model, successive approximations method, active experimental study, function.