Mathematical Package Mathcad Research Articles

Application of Difference Schemes to Decision the Pursuit Problem

The problem of the pursuit curve construction in the case when the tangent to pursuer’s motion trajectory passes at any time through the point representing the pursued is considered. A new approach to construct the pursuit curves using difference schemes is proposed. The proposed technique eliminates the need to derive the differential equations for the description of the pursuit curves, which is quite difficult task in the general case. In addition, the application of difference methods is justified in a situation where it is complicated to find the analytical solution of an existing differential equation and it is possible to obtain the pursuit curve only numerically. Various modifications of difference schemes respectively equivalent to the Euler, to the Adams – Bashforth and to the Milne methods are constructed. Their software implementation is realized by using the mathematical package Mathcad. We consider the case of a uniform rectilinear motion of the pursued whose differential equation describing the path of the pursuer and its analytical solution are known. We compare the numerical solutions obtained by the different methods with the well-known analytical solution. The error of the obtained numerical solutions is examined. Moreover, an application is considered illustrating the construction of the difference schemes for the case of an arbitrary trajectory of the pursued. Also, we extend the proposed method to the case of cyclic pursuit with several participants in the three-dimensional space. In particular, we construct a difference scheme equivalent to the Euler method for a three-dimensional analogue of the "bugs problem". The results obtained are demonstrated by means of animated examples for either two-dimensional or three-dimensional cases.

ANALYSIS OF TRANSIENTS IN ELECTRICAL CIRCUITS WITH INERTIAL ELEMENTS

The problem from the section of electrical engineering on transient processes in electric circuits with two inertial elements is considered, which is reduced to an approximate solution of the integro-differential equation with numerical implementation in the mathematical package MathCad.

RESEARCH AND APPROXIMATE SOLUTION OF A PHYSICAL PROBLEM

The problem from the section of physics about electrolysis is considered, which is reduced to an approximate solution of a differential equation with numerical implementation in the mathematical package MathCad.

ЗОНИ ВИЯВЛЕННЯ РЛС МЕТРОВОГО ДІАПАЗОНУ З ВРАХУВАННЯМ ОСОБЛИВОСТЕЙ ПОБУДОВИ І ВИКОРИСТАННЯ АНТЕННОЇ СИСТЕМИ

Предметом дослідження є методика розрахунку зон виявлення РЛС метрового діапазону, на формування якої істотно впливає Земля. Метою роботи є розробка та аналіз математичної моделі зони виявлення, яка узагальнює відому модель на випадок, коли антена РЛС не має фазового центру або він не збігається з початком обраної системи координат для розрахунку діаграми спрямованості антени. Модель базується на методі відбиттєвої трактовки і враховує особливості побудови антени. Випромінюючі елементи антени у вигляді хвильових каналів розташовані на двох поверхах, що дозволяє підбором висот поверхів і їх збудження сигналами з визначеними амплітудами і початковими фазами зменшити глибину провалів в діаграмі спрямованості в певних кутових секторах.. У таких антенах зазвичай відсутній фазовий центр, тому в математичній моделі зони виявлення враховані не тільки амплітудні, а й фазові характеристики антени. Це дозволило врахувати відмінності фаз полів у напрямках на ціль і точку відображення й уточнити результат інтерференції прямих і відбитих хвиль. Проведено розрахунки зон виявлення, що ілюструють вплив на їх параметри електричних властивостей землі, висот підйому антени і її нахилу відносно горизонту. Математична модель реалізована за допомогою доступного і простого комп'ютерного математичного пакета Mathcad. Комп'ютерна модель дозволяє при змінах конструкції антени або електричних параметрів Землі оперативно оцінювати зони виявлення та можливості їх управління для поліпшення умов виявлення цілей під певними кутами місця.

Mathematical simulation of rocket engine with a composite sectional charge

Due to the need to improve the appearance of the solid-propellant rocket engine, we stated the task of developing the methods allowing us to calculate internal ballistics performance of a composite sectional charge of solid propellant. These methods were implemented on a computer in the mathematical package MathCAD Prime 3.0, it has accuracy that is sufficient for technical evaluation of the rocket engine using solid propellant with the sectional charge.

The development of mathematical model for competitive processes

Mathematical models of competitive processes in the economy using known universal models that describe the behavior of counterparties in the market are found in this article. The object of research is to build mathematical models of competitive processes in a pair – secondhand dealer, and to process and convert the economic parameters that are necessary for the models. The methods of research use analysing and solving equations for the known and improved universal models that describe the behaviour of contractors in the market. Mathematical package for construction of schedules of dependences is used. The goals are to construct and research the modified model on the basis of mathematical model by Lotka- Volterra and its further development, and to create software product for statistical processing the economic information. Results – the mathematical model manufacturer – secondhand dealer is created, its modified version is received, and with the help of mathematical package Mathcad, researches of models were done. Prospects of the further improvement of models are revealed. The software product is developed, allowing to process the statistical economic information. On the basis of mathematical model by Lotka-Volterra and its further development a mathematical model of several producers and one middleman is created. Its modified version is studied using mathematical package Mathcad. The nonstable behavior of the counterparties is found. For this model some prospects for further improvements is identified.

Модель конкуренции в системах типа “производитель-перекупщик”

Mathematical models of competitive processes in the economy using known universal models describing the behavior of counterparties in the market are built. The mathematical model of "producer-second-hand dealer" on the basis of mathematical model by Lotka-Volterra and its further development is created. Its modified version is obtained and model analyses using mathematical package Mathcad is investigated. The behavior instability of the counterparties and some prospects for further improvements of the model are identified.