Calculation Of Definite Integrals Research Articles

Comparison of the Cauchy Integral Formula and Residue Theorem on the Calculation of Definite Integrals

Integration is a useful mathematical tool which is applied in a wide range of studies and assists people to solve many problems. Despite methods of real analysis, complex integrations in complex analysis are more beneficial and more convenient than real integrations under certain circumstances. Cauchy’s Residue Theorem and Cauchy Integral Formula are two methods used to solve complex integrals. Cauchy’s Residue Theorem helps to solve complex integrals via the calculation involving singular points, whereas Cauchy Integral Formula is implemented to solve complex integrals by finding out poles enclosed by the contour. Herein, the comparison and analysis of the process of solving both real and complex integrations between Cauchy’s Residue Theorem and Cauchy Integral Formula are demonstrated through sample questions involving exponential integrations and polynomial integrations. This passage also includes definitions for Cauchy’s Residue Theorem, Cauchy Integral Formula, corollary of Cauchy Integral Formula, steps and solutions of sample contour integral questions.

RESEARCH OF THE TEMPERATURE FIELD IN THE SYSTEM OF MULTILAYER CYLINDRICAL SOLID BODIES UNDER FIRE CONDITIONS

Introduction. The current urgent task is to find the temperature field distribution in cylindrical structures such as "solid cylinder inside a multilayer cylindrical shell". A characteristic feature of such structures is different mechanical and thermophysical characteristics of the layers combination, which makes them more perfect. However, this approach causes significant difficulties in developing analytical methods for their study. Therefore, new research methods development for multilayer, in particular, cylindrical structures is an urgent task today.Purpose. Direct method is used to study the heat transfer processes in the system "one-piece cylinder inside a multilayer cylindrical shell".Methods. To solve the initial parallel, the auxiliary problem of determining the distribution of a nonstationary temperature field in a multilayer hollow cylindrical structure with a "removed" cylinder of a sufficiently small radius is set. The solution of the auxiliary problem is realized by applying the method of reduction using the concept of quasi-derivatives. The Fourier schemeis used by using a modified method of eigenfunctions.Results. To find the solution to the problem, we used the idea of a boundary transition by directing the radius of the removedcylinder to zero. It is established that in this approach, all eigenfunctions of the corresponding problem have no singularities atzero, which means that the solutions of the original problem are limited in the whole structure. To illustrate the proposed method,a model example of finding the temperature field distribution in a four-layer column of circular cross-section (tubular concretecolumn) under the influence of the standard temperature of the fire. The results of the calculations are presented in the form of athree-dimensional graph of temperature changes depending on time and spatial coordinates.Conclusions. A direct method was used to solve the initial problem, using the idea of a boundary transition for the first time.In the general formulation (the function of changing the temperature of the environment is considered arbitrary, no restrictionsare imposed on the thickness of the shell and the number of layers) such a problem is solved for the first time.The structure of the obtained explicit exact formulas allows creating an algorithm for calculating the temperature field inthe form of automated programs, where it is enough to enter the initial data. Note that such algorithms include: a) calculating theroots of the characteristic equation; b) multiplication of a finite number of known matrices; c) calculation of definite integrals; d)summation of the required number of members of the series to obtain a given accuracy of the calculation.

GRAPH ANALYTICAL METHOD FOR DETERMINING THE COMPLEX QUALITY INDICATOR OF QUALIMETRY OBJECTS

The subject of research in the article is a method for determining a complex quality indicator of qualimetry objects that have quality indicators with different units of measurement. The purpose of the work is proposing a method for determining a complex quality indicator of multicriteria objects of qualimetry, using integration methods. The article solves the following task: to investigate the possibility of assessing the quality of qualimetry objects by graph analytical method, ie to apply the principle of determining the area and volume under curved surfaces both on the plane and in space, created by combining estimates of unit quality indicators on dimensionless scale. The following methods are used: mathematical statistics; qualimetry. The following results were obtained: the analysis of scientific literature on quality assessment of qualimetry objects in terms of existing mathematical relationships between the measured quality indicators and their assessment on a dimensionless scale was carried out. It was shown that, as a rule, mathematical dependences are nonlinear and their research is reduced to the development of universal methods that could be applied to objects of qualimetry, regardless of their nature, complexity, importance, etc. Research that involves nonlinear relationships between measured quality scores and their estimates on a dimensionless scale should be based on quality ideology and have a physical explanation. Having single quality indicators in a single (dimensionless) rating scale, it is proposed to determine a single comprehensive quality indicator of the object of qualimetry using integration methods. It is proposed to find the area under the broken curve, which is built as a result of combining estimates of quality indicators on a dimensionless scale over a period of observations. Since the calculation of definite integrals by the Newton-Leibniz formula is not always possible, to solve practical problems, it is enough to know the approximate value of a certain integral with a given degree of accuracy, so it is proposed to use methods of numerical integration, methods: rectangles and trapezoids. A method for determining a generalized indicator of the quality of the object of qualimetry, which can be considered universal, as it can be used for multi-criteria assessment of the quality of objects of qualimetry of different nature. Conclusions: To determine the complex quality indicator of multicriteria qualimetry objects, it is proposed to apply integration by the method of average rectangles. The method of definition of a complex indicator of quality of object of qualimetry with application of integration which considers receipt of estimations of unit indicators of quality is proposed.

Application of Monte Carlo Method Based on Matlab: Calculation of Definite Integrals and Simulation of Heston’s Model

This paper discusses Monte Carlo method in three aspects: pi-approximation, an algorithm to calculate definite integral and simulation to generate financial time series. The first two calculations are based on geometric probability: to calculate the probability that the random points fall within the certain area. The third one is to transfer a stochastic differential equation into a difference equation and realize these equations on matlab to derive a time series which has the properties of the corresponding stochastic differential equation. By analyzing these time series, one can make further analysis on these data, e.g. density function. The paper shows the applicability of Monte Carlo method. The method gives practitioners accessible means of solving complicated models and is easy to operate on computers.

Approximating of functions from Holder classes Hα[0, 1] by Haar wavelets

In the present work, a new direct computational method for solving definite integrals based on Haar wavelets is introduced. The definite integral of the functions from Holder classes is replaced with the approximation of the function by Haar wavelets and the the calculation of definite integrals is reduced to the problem of solving algebraic equation formed by the Fourier coefficients in terms of Haar wavelets. Based on the properties of the Haar wavelets it is shown that the such approximations much better approximate the value of the integrals for the functions from Holder classes. The Error analysis of the approximation method are worked out in the classes of Holder to show the efficiency of the new method and connection of the module of difference with smoothness of the function is established. Finally, some numerical examples of the implementation the method for the functions from Holder classes are presented.