APPROACH TO MATHEMATICAL SIMULATION
OF TRANSMUTATIONAL PROCESSES IN NUCLEAR POWER PLANTS 

Belozerova Alla Ravil'evna, Candidate of physical and mathematical sciences, senior staff scientist, State Scientific Center of the Research Institute of Nuclear Reactors (10-Dimitrovgrad, Ulyanovsk region, Russia), lmni@niiar.ru
Mel'nikov Boris Feliksovich, Doctor of physical and mathematical sciences, professor, sub-department of applied mathematics and informatics, Togliatti branch of Samara State University (31g Yubileynaya street, Togliatti, Russia), B.Melnikov@tltsu.ru 

519.176, 519.622.2, 519.688 

Background. At the present time the study and simulation of nuclear kinetics or nuclide transformations in the process neutron irradiation in power reactor facilities are especially topical for solution of the problem of nuclear fuel cycle abridgement to provide profitability of nuclear power. The purpose of this article consists in describing
and investigating essentially new mathematical models in simulation of nuclear-physical processes in reactor materials irradiation.
Materials and methods. The initial model of the problem is similar to a linear system of the ordinary differential equations. Relations of nuclide reciprocity and accumulation on a nuclide set of nuclide strong-coherent subsets were entered. With the help of relations a transition from the initial continuous model to a discrete model was made. The discrete model describes nuclear kinetics as an oriented multigraph. The authors averaged the estimation of neutron-physical characteristics of reactor irradiation by means of averaging a stream at certain micro-campaigns in the process of power reactor facility functioning.
Results. The problem of discrete optimization of the nuclide transformations was formulated in terms of the graph theory. The discrete problem is original in the discrete optimization theory. The problem concerns a class of NP-difficult problems. The authors author a method of averaging the estimation of neutron-physical characteristics of reactor irradiations for the period of several micro-campaigns in the problem of nuclear transmutations.
Conclusions. The priority direction of modernization and technological development of the Russian economy consists in development of nuclear technologies, namely in realization of the closed fuel cycle with mixed fuel in fast power reactors. The considered model of calculation of nuclide transformations into materials at reactor irradiation can be successfully used for mathematical modelling of nuclearphysical processes with accumulation and burning out actinides in a fast reactor core. 

nuclear transmutation problem, mathematical model, directed multigraph, ordinary differential equations system. 

1. Ma B. M. Materialy yadernykh energeticheskikh ustanovok [Materials of nuclear power plants]. Moscow: Energoatomizdat, 1987, 408 p.
2. Gromkovich Yu. Teoreticheskaya informatika. Vvedenie v teoriyu avtomatov, teoriyu vychislimosti, teoriyu slozhnosti, teoriyu algoritmov, randomizatsiyu, teoriyu svyazi i kriptografiyu [Theoretical informatics. Introduction into the theory of automata, the theory of computability, the theory of complexity, the theory of algorithms, randomization, the theory of communication cryptography]. Saint Petersburg: BKhV-Peterburg, 2010, 336.
3. Hromkovič J. Algorithmics for Hard Problems: Introduction to Combinatorial Optimization, Randomization, Approximation, and Heuristics. 2nd Ed. Springer, 2001.
4. Gerasimov A. S., Zaritskaya T. S., Rudik A. P. Spravochnik po obrazovaniyu nuklidov v yadernykh reaktorakh [Reference book on nuclide formation in nuclear reactors]. Moscow: Energoatomizdat, 1989, 575 p.
5. Kharari F. Teoriya grafov [Graph theory]. Moscow: Mir, 1973, 301 p.
6. Shimansky G. A. J. Nucl. Mater. 1999, no. 271–272, pp. 30–34.
7. Belozerova A. R., Shimanskiy G. A. Fizicheskoe modelirovanie izmeneniya svoystv reaktornykh materialov v nominal'nykh i avariynykh usloviyakh: sb. referatov seminara [Physical simulation of reactor material properties changes in nominal emergency conditions: collected reports of the seminar]. Dimitrovgrad: FGUP «GNTs RF NIIAR», 2005, pp. 75–77.
8. Mel'nikov B. Kibernetika i sistemnyy analiz (NAN Ukrainy) [Cybernetics and system analysis (National Academy of Sciences of Ukraine)]. 2006, no. 3, pp. 32–42.
9. Mel'nikov B., Panin A. Vektor nauki Tol'yattinskogo gosudarstvennogo universiteta [Science vector of Togliatti State University]. 2012, no. 4, pp. 83–86.