Verification of analytical antiderivatives forms using correlation analysis for mechanical problems

A. Alpatov, V. Kravets, V. Kravets, E. Lapkhanov
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Abstract

An analytical search for antiderivative functions (indefinite integrals) is widely used in the mathematical simulation of various engineering, economic, ecological, biological, social, and other processes. In their turn, mechanical problems have many subproblems whose solution involves analytical integration methods. Among these problems is the problem of development of analytical models for navigation and ballistics support and control theory models in space rocket engineering. The advantage of this approach to mathematical simulation is a fast analysis of the state of dynamic systems on different time intervals without calculating all previous states. In their turn, for some classes of functions, antiderivatives may be found in several different ways, as a result of which there exist several different forms of antiderivatives that are hard to verify by the classical method in standard form. This is mainly due to the choice of various combinations of integration methods used in the development of analytical models, in particular in problems of applied mechanics. Taking into consideration these difficulties in the verification of the set of antiderivative functions, this paper proposes a method to check their analytical forms for correspondence with the use of correlation analysis. In doing so, the arrays of the values of each antiderivative form at certain nodal points are represented as a set of random variables. With this in mind, it is suggested that the verification process be conducted with the use of the standard approach based on correlation analysis (using Pearson’s correlation coefficient). The efficiency of the method is shown by the example of verifying the antiderivatives of the reciprocal of a squared quadratic trinomial. This approach will make it possible to check the adequacy of the i-th candidate antiderivative and to adapt the problem to the standard form.
用相关分析验证力学问题的解析不定积分形式
不定积分的解析搜索在各种工程、经济、生态、生物、社会和其他过程的数学模拟中被广泛应用。力学问题又有许多子问题,这些子问题的解涉及解析积分方法。其中包括航天火箭工程中导航分析模型和弹道支撑控制理论模型的开发问题。这种数学模拟方法的优点是可以快速分析动态系统在不同时间间隔上的状态,而不需要计算所有以前的状态。反过来,对于某些函数,可以用几种不同的方法求不定积分,因此存在几种不同形式的不定积分,这些不定积分很难用标准形式的经典方法来验证。这主要是由于在分析模型的发展中,特别是在应用力学问题中,选择了各种组合的积分方法。考虑到不定函数集验证中的这些困难,本文提出了一种利用相关分析来检验其解析形式是否对应的方法。在这样做时,每个不定积分形式的值在某些节点的数组被表示为一组随机变量。考虑到这一点,建议使用基于相关分析(使用Pearson相关系数)的标准方法进行验证过程。通过对平方二次三项式倒数不定积分的验证,说明了该方法的有效性。这种方法可以检验第i个候选不定积分的充分性,并使问题适应标准形式。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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