Calculation of integrals in MathPartner

We present the possibilities provided by the MathPartner service of calculating definite and indefinite integrals. MathPartner contains software implementation of the Risch algorithm and provides users with the ability to compute antiderivatives for elementary functions. Certain integrals, including improper integrals, can be calculated using numerical algorithms. In this case, every user has the ability to indicate the required accuracy with which he needs to know the numerical value of the integral. We highlight special functions allowing us to calculate complete elliptic integrals. These include functions for calculating the arithmetic-geometric mean and the geometric-harmonic mean, which allow us to calculate the complete elliptic integrals of the first kind. The set also includes the modified arithmetic-geometric mean, proposed by Semjon Adlaj, which allows us to calculate the complete elliptic integrals of the second kind as well as the circumference of an ellipse. The Lagutinski algorithm is of particular interest. For given differentiation in the field of bivariate rational functions, one can decide whether there exists a rational integral. The algorithm is based on calculating the Lagutinski determinant. This year we are celebrating 150th anniversary of Mikhail Lagutinski.

Degree estimates of one class cut-offs for improper integrals

The paper considers a normalized non-integral integral of the first kind with a variable lower bound. In this case the integrand is a generalization of the standard Gaussian distribution density. Such integrals are often called cutoffs or incomplete functions. The purpose of this paper is to obtain power inequalities for this kind of integrals. The necessity of obtaining this type of estimations is due to the fact that incomplete functions have become widespread in applications and theoretical studies. The peculiarity of the results established in the article consists in the fact that arbitrary degrees of a given integral for any value of an argument are evaluated from above not by means, of the value of integrable functions at a certain point, but by the value of the integral in question at some point proportional to this argument. The coefficient of proportionality, a parameter, can take any value from some closed interval. The main difficulty in obtaining these inequalities is that the integrand is a logarithmically concave function, that is, its logarithm is a concave function. The paper also proves that both limits of the closed interval for the parameter cannot be extended. This shows that the obtained estimates are unimprovable.

Mathematical Modelling from the Instrumentation of Improper Integrals

Background: There is little clarity in the application of content related to improper integrals in university students, due to the absence of meaning, which prevents them from making a connection with everyday problem situations. Methods: we designed a mathematical modelling proposal where a specific situation involving the instrumentation, use and application of this type of integrals is experimented and solved with a population of engineering students, who learn to use them. Results: The importance of using mathematical modelling as a didactic-dynamic resource is highlighted because it helps students to reach an understanding of real situations involving improper integrals in different contexts. Conclusions: Despite the numerous errors detected in the students, this strategy made it possible to demonstrate the development of advanced mathematical thinking skills in young people.

On a family of the incomplete H-functions and associated integral transforms

Recently, Srivastava, Saxena and Parmar [H. M. Srivastava, R. K. Saxena and R. K. Parmar, Some families of the incomplete H-functions and the incomplete H ¯ {\overline{H}} -functions and associated integral transforms and operators of fractional calculus with applications, Russ. J. Math. Phys. 25 2018, 1, 116–138] suggested incomplete H-functions (IHF) that paved the way to a natural extension and decomposition of H-function and other connected functions as well as to some important closed-form portrayals of definite and improper integrals of different kinds of special functions of physical sciences. In this article, our key aim is to present some new integral transform (Jacobi transform, Gegenbauer transform, Legendre transform and 𝖯 δ {\mathsf{P}_{\delta}} -transform) of this family of incomplete H-functions. Further, we give several interesting new and known results which are special cases our key results.

Approximating the Sum of Infinite Series of Non Negative Terms with reference to Integral Test

This paper describes a method of obtaining approximate sum of infinite series of positive terms by using integrals under its historical background. It has shown the application of improper integrals to determine whether the given innate series is convergent or divergent. Here, the limits of the integrals and the series usually extend to infinity though they may be slowly convergent. We have also established a relation to approximate the sum of infinite series of positive terms with a suitable example.

FEM-simulation of superconducting linear acceleration system for pellet injection

In order to simulate the high-temperature superconducting (HTS) linear acceleration (SLA) system for the pellet injection, the integration method of the applied magnetic field generated from the acceleration coil has been proposed. To this end, the regularization technique is used in the evaluation of the improper integrals, and simultaneously, a FEM code is developed for analyzing the shielding current density in an HTS film. In addition, the SLA system has been simulated using the code. The results of the computations show that the accuracy of the applied magnetic field is considerably improved. In this sense, the regularization technique is a useful tool. Also by locating the outer coil, the acceleration time during which the pellet speed reaches 5 km/s is about 3.5 times shorter than that of the only use of the inner coil. These results mean that the outer coil is effective in the improvement of the acceleration performance for the SLA system.

Existence and Asymptotic Behaviors of Nonoscillatory Solutions of Third Order Time Scale Systems

Nonoscillation theory with asymptotic behaviors takes a significant role for the theory of three-dimensional (3D) systems dynamic equations on time scales in order to have information about the asymptotic properties of such solutions. Some applications of such systems in discrete and continuous cases arise in control theory, optimization theory, and robotics. We consider a third order dynamical systems on time scales and investigate the existence of nonoscillatory solutions and asymptotic behaviors of such solutions. Our main method is to use some well-known fixed point theorems and double/triple improper integrals by using the sign of solutions. We also provide examples on time scales to validate our theoretical claims.

On the Theory of Determining the Transport Characteristics of Gas Mixtures

The paper considers the identification of properties of real gases and creation of nanomaterials on the basis of molecular and kinetic theory of gases, namely the Boltzmann equation. The collision term of the Boltzmann equation is used in the algorithm for the identification of transport properties of media. The article analyses the uniform convergence of improper integrals in the collision term of the Boltzmann equation depending on the conditions for the connection between the kinetic and potential energy of interacting molecules. This analysis allows to soundly identify the transport coefficient in macro equations of heat and mass transfer.

Mathematical analysis in examples and tasks. Part 1

The purpose of the textbook is to help students to master basic concepts and research methods used in mathematical analysis. In part 1 of the proposed cycle of workshops on the following topics: theory of sets, theory of limits, theory of continuous functions; differential calculus of functions of one variable, its application to the study of the properties of functions and graph; integral calculus of functions of one variable: indefinite, definite, improper integrals; hyperbolic functions; applications of integral calculus to the analysis and solution of practical problems. For the development of each topic the necessary theoretical and background material, reviewed a large number of examples with detailed analysis and solutions, the options for independent work. For self-training and quality control of the obtained knowledge provides exercises and problems with answers and guidance. For teachers, students and postgraduate students studying advanced mathematics.