2019-4 (33)

Nuclear, radiation and environmental safety

Article Name10.26583/GNS-2019-04-02
Unambiguity of Decisions When Using Linear Functional Equation in the Radiation Protection Model
AuthorsV.P. Cherniavsky
Address

Sarov Рhysical Technical Institute the branch of National Research Nuclear University “MEPhI”,
Dukhov St., 6, Sarov, Nizhny Novgorod region, Russia 607186

ORCID iD: 0000-0002-1628-2271

e-mail: mattutor@mail.ru

AbstractThe paper considers a functional linear equation with a shift in the radiation protection model for the transport of charged particles and ionizing radiation. The aim of the work is to study the questions of the existence and uniqueness of solutions for various cases that arise when the initial parameters of the model change. The analysis of the system accompanying the functional equation is carried out by linear algebra methods. For the case of the inequality to zero of the main determinant of the accompanying system, the correctness of the obtained solutions is shown; the functional equation has a unique solution. If the determinant vanishes, the problem is completely solved for cycles of length 2. The functional equation has no solutions if the determinant is zero and the rank of the extended matrix is 2. For the case of a joint system with a degenerate matrix, analytical formulas for the general solution of a homogeneous and inhomogeneous functional equation are obtained . These solutions depend on the coefficients of the initial equation, the initial function generating the cycle, and contain an arbitrarily chosen function. To eliminate the ambiguity that arises, one can go to a model with a non-degenerate matrix by changing the system of weight coefficients of the model equation, or use additional initial conditions.
Keywordslinear functional equation, shift equation, iteration, cycle, associated system, rank, homogeneous and inhomogeneous functional equation.
LanguageRussian
References
  1. Kuchin N.L. Matematicheskoe modelirovanie radiatzionnogo vozdeistvia atomnikh objectov morskoi tecniki na okrudjaiushuiu sredu [Mathematical Modeling of the Radiation Effect of Atomic Objects of Marine Technology on the Environment and Humans]. Dissertaciya doktora fiziko-matematicheskix nauk [Dissertation of Doctor of Physical and Mathematical Sciences]. Sankt-Peterburg [Saint-Petersburg]. 2002. 297 p. (in Russian).
  2. Chirskaya N.P., Voronina E.N., Mileev V.N., Novikov L.S., Sinolits V.V. Matematicheskoe modelirovanie svoistv neodnorodnikh struktur dlia system radiatzionnoi zashiti [Mathematical Modeling of the Properties of Heterogeneous Structures for Radiation Protection Systems]. Moskva. Trudi XXI Mezdunarodnoi konferentsii «Radiatsionnaia phyzika tverdogo tela» [Moscow. Proceedings of the XXI International Conference «Radiation Solid State Physics»]. 2011. v. 2. P. 436-443 (in Russian).
  3. Kriuk J.E., Kunets I.E. Matematicheskie metodi modelirovania v optimizatsii radiatzionnoi zashiti [Mathematical Modeling Methods in Optimization of Radiation Protection]. Kharkov. Vestnik NTU KhPI. Seria: Informatika i modelirovanie [Bulletin of the National Technical University Kharkov Polytechnic Institute. Series: Computer Science and Modeling]. 2011. № 36. P. 95-100 (in Russian).
  4. Сarleman Т. Sur la theorie des equations integrates et ses applications, Verhandl. Internat. Math. Kongr. Zurich. 1 (1932). P.138-151 (in French).
  5. Litvinchuk G.S. Kraevie zadachi i singuliarnie uravnenia so sdvigom [Boundary Value Problems And Singular Equations with Shift]. Moscow. Pub. Nauka [Publishing House Science]. 1977. 448 p. (in Russian).
  6. Karapetiants N.K., Samko S.G. Uravnenia s involiutivnimi operatorami i ikh prilodjenia [Equations with Involutive Operators and their Applications]. Postov-na-Donu. Izdatelstvo Rostovskogo universiteta [Rostov on Don. Rostov University Publishing House]. 1988. 187 p. (in Russian).
  7. Vasilevsky N.L., Karelin A.A., Kereksha P.V., Litvinchuk G.S. Ob odnom klasse singuliarnikh uravneni’ i ego primeneniakh v teorii kraevikh zadach dlia differentsialnikh uravneni’ v chastnikh proizvodnikh. II [A Class of Singular Integral Equations with Involution and its Applications in the Theory of Boundary Value Problems for Partial Differential Equations. II]. Differencial`ny`e uravneniya [Differential equation]. 1977, 13:11. P. 2051-2062 (in Russian).
  8. Antonevich A.B. Lineinie functional'nie uravnenia: operatorni podhod [Linear Functional Equations: Operator Approach]. Minsk. Izdatelstvo Universitetskoe [Publishing House University]. 1988. 232 p. (in Russian). 
  9. Moskovsky Gosudarstvenny Universitet. Spravochnik 2000 [Moscow State University. Handbook 2000]. Moskva. Izdatelstvo Moskovskogo universiteta [Moscow. Moscow University Press]. 2000. 240 p. (in Russian).
  10. Agakhanov N.H., Bogdanov I.I., Kodjevnicov P.A., Podlipsky O.K., Tereshin D.A. Vserossi’skie olympiadi shkolnikov po matematike 1993-2006 [All-Russian Mathematics Olympiads 1993-2006]. Moscow. MCCME. 2007. 472 p. (in Russian).
  11. Brodsky J.S., Slipenko A.K. Funktsional’nie uravnenia [Functional Equations]. Kiev. Pub. Visha shkola [High school]. 1983. 96 p. (in Russian).
  12. Poliayin A.D., Manzhirov A. V. Spravochnik po integral’nim uravneniam: Tochnie reshenia [Handbook of Integral Equations: Exact Solutions]. Moscow. Pub. Factorial. 1998. 432 p.
     (in Russian).
  13. Prasolov V.V. Zadachi po algebre, arifmetike i analizu [Problems in Algebra, Arithmetic and Analysis]. Moscow. MCCME. 2007. 608 p. (in Russian).
  14. Mal’tsev A.I. Osnovi lineinoi algebra [The Basics of Linear Algebra]. Moscow. Pub. Nauka [Publishing House Science]. 2005. 470 p. (in Russian).
  15. Kurosh A. G. Kurs vishei algebri [Course in Higher Algebra]. Sankt-Peterburg [Saint-Petersburg]. Pub. Lan’ [Publishing House «Doe»]. 2019. 432 p. (in Russian).
  16. Faddeev D.K. Lektsii po algebre [Lectures on Algebra]. Sankt-Peterburg [Saint-Petersburg]. Pub. Lan’ [Publishing House "Doe"]. 2002. 416 p. (in Russian).
  17. Faddeev D.K., Sominsky I.S. Sbornik zadach po vishei algebre [Collection of Problems in Higher Algebra]. Moscow. Pub. Nauka [Publishing House Science]. 1977. 288 p. (in Russian).
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