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Engineering, Energy, Transport AIC

space SCIENTIFIC JOURNALS OF VINNITSA NATIONAL AGRARIAN UNIVERSITY

Issue №: 2(101)

Published: 1970.01.01
DOI: XX.XXXXX/XXXXXXXXXXXXX


Description:
The journal addresses the problems of creating and improving machinery and technologies for agriculture, food and processing industries, mechanical engineering, energy and transport.

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ON A MODIFICATION OF THE GAUSS METHOD FOR SOLVING SYSTEMS OF ALGEBRAIC EQUATIONS IN ENERGY PROBLEMS

DOI: XX.XXXXX/XXXXXXXXXXXXX
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Dubchak Victor – PhD, Associate Professor of the Department "Mathematics, physics and computer technologies" of Vinnytsia National Agrarian University (3, Solnechnaya St., Vinnytsia, 21008, Ukraine,
e-mail: viktor_dubchak@rambler.ru).

Novitskaya Lyudmila – PhD, Associate Professor of the Department "Mathematics, physics and computer technologies" of the Vinnytsia National Agrarian University (3, Solnechnaya St., Vinnytsia, 21008, Ukraine, e-mail: li@vsau.vin.ua)

Annotation

In this article a modified approach for finding solutions of a system of equations similar to the classical Gaussian method are investigated. In the proposed modification as well as in the classical method, the essence of the transformations reduces to the possibility of first finding one, usually the last, unknown; here the initial system of the n-th order first reduces to solving some auxiliary system of equations (n-1) -th order, and through its solutions, for example, the last unknown of the initial algebraic system can be found. This method can be applied to any algebraic systems according to the number of their solutions-both joint and incompatible, both definite and indeterminate systems. The determination of the number of solutions is achieved automatically by the algorithm of the calculations given in the work. Similarly, the search for other unknowns that have not yet been found is realized. It can be carried out by the same method according to the recurrence principle, and the order of the next auxiliary algebraic system decreases by one every time. Also, an auxiliary system as to additional unknowns can be solved either by other classical methods, or by applying elementary algebraic transformations. The main advantage of this method is the reduction of the order of the auxiliary system by one.
As an illustration of the proposed method, examples of finding solutions to algebraic systems (definite and indefinite) are offered. The conclusions of the method are drawn in the article. The results of this work can be used to solve specific technical problems, in particular, in the energy sector, in transport tasks, etc.

Keywords: Systems of algebraic equations, Gaussian method, scalar unknowns, matrix rank, Kronecker-Capelli theorem, determinant of matrix, recurrent principles of computations

List of references

[1]          Khomenko, O.V., Hoody, V.S. (2017). Matematychni zadachi enerhetyky. Chastyna 1. Metodychni vkazivky dlia studentiv osvitnoho stupeniu «bakalavr» spetsialnosti 141 «Elektro-enerhetyka, elektrotekhnika ta elektromekhanika» spetsializatsiia «Systemy upravlinnia vyrobnytstvom i rozpodilom elektroenerhii» [Methodical instructions for implementation of practical classes on discipline "Mathematical problems of power engineering. Part 1 »for the students of educational degree «Bachelor» specialty 141«Electro-energetics, electrical engineering and electromechanics» specialization «Systems of production and distribution of electric power management ». Kyiv: NTUU "KPI" [in Ukrainian].

[2]          Kurosh, A.G. (1974). Obshchaia alhebra [General algebra]. Moscow: Nauka [in Russian].

[3]          Artamonov, V.A. (2007). Vvedenye v vysshuiu alhebru y analytycheskuiu heometryiu [Introduction to higher algebra and analytic geometry]. Moscow: Factorial Press[in Russian].

[4]          Latyshev, V.N., Artamonov, V.A. (2004). Lyneinaia alhebra y vypuklaia heometryia [Linear algebra and convex geometry]. Moscow: Factorial Press [in Russian].

[5]          Maltsev, A.I. (1970). Alhebraycheskye systemy [Algebraic systems]. Moscow: Nauka [in Russian].

[6]          Vinberg, E.B. (2002). Kurs alhebry [Algebra course]. Moscow: Factorial Press [in Russian].

[7]          Zavaylo, S.T. (1985). Kurs alhebry [Algebra course]. Kyiv: Higher school [in Russian].

[8]          Proskuryakov, I.V. (2010). Sbornyk zadach po lyneinoi alhebre: Uchebnoe posobye [A collection of problems in linear algebra]. Izd. "Lan" [in Russian].

[9]          Thomas, W. (2003). Hungerford Algebra. Springer [in USA].

[10]      Herstein, I.N. (1975). Topics in Algebra. John Wiley & Sons [in USA].

[11]      Joseph, J. (2015). Rotman Advanced Modern Algebra. AMS [in USA].

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About journal

Topics of the journal:

The journal "Engineering, Energy, Transport AIC" on its pages presents materials on the following issues:

• Machine use in crop production and in animal husbandry

• Technical service of mobile equipment

• Processes and equipment for processing and food productions

• Mechanical engineering and material processing

• Energy technology and alternative energy sources

• Transport and transport-technological processes and equipment

• Automation of technological processes

• Opinion of a young scientist

Key information:
ISSN (print): 2520-6168
DOI:

ISSN (print): 2520-6168

Certificate of state registration of mass media: No. 21906-11806 P dated 03/12/2016.

Founder of the journal: Vinnytsia National Agrarian University

Editor-in-chief: Tokarchuk Olexiy A.

Ph.D., Assoc., VNAU (Vinnytsia)

Deputy Editor-in-Chief: Veselovska Nataliia R.

Doctor of Technical Sciences, Full Prof., VNAU (Vinnytsia)

Executive Secretary: Polevoda Yuri A.

Ph.D., Assoc., VNAU (Vinnytsia)

Members of the Editorial Board:

Tokarchuk O.A. - Ph.D., Associate Professor, Vinnytsia National Agrarian University

Veselovska N.R. - Doctor of Technical Sciences, Full Professor, Vinnytsia National Agrarian University

Bulgakov V.M. - Doctor of Technical Sciences, Full Professor, Acad. NAAS, National University of Life and Environmental Sciences of Ukraine

Iskovich-Lototsky R.D. - Doctor of Technical Sciences, Full Professor, Vinnytsia National Technical University

Bandura V. N. - Ph.D., Full professor, Vinnytsia National Agrarian University

Polevoda Yu.A. - Ph.D., Associate Professor, Vinnytsia National Agrarian University

Tsurkan A.V. - Ph.D., Associate Professor, Vinnytsia National Agrarian University

Ivanchuk Y.V. - Ph.D., Associate Professor, Vinnytsia National Technical University

Jordan Todorov Maximov - Doctor of Technical Sciences, Full Professor, Technical University of Gabrovo (Bulgaria)

Type of publication: journal

Kind of publication: scientific

Publication status: domestic

Year of foundation: 1997

Periodicity: 4 times per year

Volume: 18.75 conv. print. sheet

ISSN: 2520-6168 (print version), (online version)

Language of publication: (mixed languages) Ukrainian, Russian, English

The sphere of distribution and category of readers: national, foreign, faculty, scientists, entrepreneurs.

The periodical edition is included to the List of scientific professional publications of Ukraine on technical sciences, approved by order of the Ministry of Education and Science of Ukraine in 05.16.2016, No. 515.

The journal "Engineering, Energy, Transport AIC" is included in the "Catalog of Ukrainian Publications".

Subscription to the journal can be issued in each post office. The subscription index is 99720.

History of journal:

It was founded in 1997 under the title "Bulletin of the Vinnitsa State Agricultural Institute." According to the Decree of the Presidium of the Higher Attestation Commission of Ukraine of September 11, 1997, this publication received the status of a professional in which it is allowed to publish the main results of dissertation researches.

In 2001-2014 the journal published under the title “Collection of scientific works of Vinnytsia National Agrarian University. Series: Technical Sciences "(Certificate of State Registration of Mass Media KV No. 16644-5116 PR of 04/30/2010).

Since 2015 journal has title "Engineering, Energy, Transport AIC" (Certificate of State Registration of Mass Media No. 21906-11806 P dated 03/12/2016).