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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Nonlinear Modifications of Perturbation Theory with Applications to Complex System Analysis

Discontinuity, Nonlinearity, and Complexity 7(3) (2018) 327--354 | DOI:10.5890/DNC.2018.09.009

V. V. Uchaikin; V. A. Litvinov; E.V. Kozhemjakina

Ulyanovsk State University, Ulyanovsk, Russia

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The article reviews some nonlinear generalizations of the perturbation theory performed by various authors (including some co-authors of this article) in various years, and can be useful in analyzing complex systems and inverse problems. There is considered the roblem of widening the range of perturbation theory applicability with simultaneous reducing the complexity of computational algorithm. The consideration is based on the duality principle, including adjoint (in the Lagrange sense) operators and importance unctions. The article contains an exposition of perturbation theory of higher orders, variational methods of higher orders, symmetry transformation and some other question. Special attention is paid to such a promising approach as multi-reference methods. It is shown that this approach extends the range of applicability of perturbation theory and saves calculations time. To demonstrate details of the method, some examples of transport problems from reactor engineering and astrophysics are considered.


This work is partially supported by the Russian Foundation of Basic Research (Projects 16-01-00556).


  1. [1]  Marchuk, G.I. (1995), Adjoint Equations and Analysis of Complex Systems, Dordrecht; Boston; Kluwer Academic Publishers.
  2. [2]  Weinberg, A.M. andWigner, E. P. (1958), The Physical Theory of Neutron Chain Reactors. Chicago, In: The University of Chicago Press.
  3. [3]  Ussachoff, L. N. (1955), Equation for the importance of neutrons, reactor kinetics and the theory of perturbations, in International Conference on the Peaceful Uses of Atomic Energy, (Geneva).
  4. [4]  Lewins, J. (1965), Importance, the Adjoint Function. The Physical Basis of Variational and Perturbation Theory in Transport and Diffusion Problems, Elsevier Science & Technology.
  5. [5]  Abdel-Khalik, H. (2012), Adjoint-based sensitivity analysis for multi-component models, Nuclear Engineering and Design, 245, 49-54.
  6. [6]  Pazsit, I. and Dykin, V. (2015), The dynamic adjoint as a Green’s function, Annals of Nuclear Energy, 86, 29-34.
  7. [7]  Marchuk, G.I. (1975), Methods of Numerical Mathematics, New York etc.: Springer.
  8. [8]  Kostin, M.D. and Brooks, H. (1964), Generalization of the variational method of Kahan, Rideau, and Roussopoulos. II. A variational principle for linear operators and its application to neutron-transport theory, J. Math. Phys., 52(1), 1691-1700.
  9. [9]  Marchuk, G.I. and Orlov, V.V. (1961), To the adjoint function theory, In: Neutron Physics, Moscow, Gosatomizdat, 30-45 (in Russian).
  10. [10]  Shichov, S.B. and Shmelev, A.N. (1968), Account of influence of an arbitrary change in reactor sizes on the critical mass of a fast reactor with using the perturbation theory, In: Nuclear Reactor Physics, Moscow, Gosatomizdat, 67-85 (in Rusian).
  11. [11]  Kolchuzhkin, A.M., Ryzhov, V.V., and Uchaikin, V.V. (1970), Application of the perturbation method to study of the transition effect, Acta Physica Academiae Scientiarum Hungaricae, 29(3), 333-335.
  12. [12]  Lewis, E.E. (2008), Fundamentals of Nuclear Reactor Physics, Academic Press.
  13. [13]  Fano, U., Spencer, L. V. and M. J. Berger (1959), Penetration and Diffusion of X-rays, Encyclopedia of Physics, 38, Springer-Verlag: Berlin.
  14. [14]  Fano, U., Zerby, C.D., and Berger, M.J. (1962), Gamma-Ray Attenuation, Reactor Handbook, IIIB.
  15. [15]  Leypunsky, O.I. and Novozhilov, E.E. (2008), Fundamentals of Nuclear Reactor Physics, Academic Press.
  16. [16]  Uchaikin, V.V. (1989), Variational method of interpolation of nuclear-engineer calculations, Atomic Energy, 67, 54-55 (in Russian).
  17. [17]  Marshak, R.E. (1947), The variational method for asymptotic neutron densities, Phys. Rev. 71, 688.
  18. [18]  Litvinov, V.A. (1993), Variational interpolation in the problem of sensitivity of cascade processes characteristics, Nuclear Energy, 56(2), 244-254 (in Russian).
  19. [19]  Uchaikin, V.V. and Litvinov, V.A. (1989), Variational method of interpolation in the radiation transfer theory, Optics of Atmoshere and Ocean, 2, 36-40 (in Russian).
  20. [20]  Broder, D.L., Kayurin, Y.P., and Kutuzov,A.A.(1962), Penetration of gamma-radiation in heterogeneous media, Atomic Energy, 12, 30-35.
  21. [21]  Goldstein, H. (1957), The Attenuation of Gamma Rays and Neutrons in Reactor Shields, U.S. Government Printing Office, Washington 25, D.C., 192-193.
  22. [22]  Kolchuzhkin, A.M., Ryzhov, V.V., and Uchaikin, V.V. (1970), Application of the perturbation method to study the transition effect, Acta Physica Academiae Scientiarum Hungaricae, 29(3), 333-335.
  23. [23]  Gaisser, T.K., Engel, R., and Resconi, E. (2016), Cosmic Rays and Particle Physics, Cambridge University Press.
  24. [24]  Belenky, S.Z. (1948), Cascade Processes in Cosmic Rays, Moscow.
  25. [25]  Crannell, C.J. and et al. (1969), Experimental determination of the transition effect in electromagnetic cascade showers, Phys. Rev., 182, 1435.
  26. [26]  Kolchuzhkin, A.M., Ryzhov, V.V., and Uchaikin V.V. (1970), The perturbation method in the transition effect theory, Bull. Acad. Sci. USSR, Ser. Phys., 34(9), 2019-2023; (1971) Transition effect in electron-photon showers, ibid. 35(10), 2171-2175.
  27. [27]  Uchaikin, V.V. (2012), Statistical mechanics of fragmentation-advection processes and nonlinear measurements problem, Discontinuity, Nonlinearity, and Complexity, 1(1), 79-112; 1(2), 171-196.
  28. [28]  Lagutin, A.A. and Uchaikin, V.V. (1995), The Sensitivity Theory in Cosmic Ray Physics, Barnaul, Altai State University (in Russian).
  29. [29]  Lagutin, A.A. and Uchaikin, V.V. (2013), The Adjont Equations Method in High-Energy Cosmic Ray Transport, ASUpress (in Russian).