English version
ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Дифференциальные Уравнения
и
Процессы Управления

О журнале История Журнала Издатель Адреса Тематика Редакторский состав Рецензирование Для авторов Издательская этика Коллекция номеров English version

Exterior calculus note on the additive separation of variables 3D solution to a dynamical neutron diffusion BVP

Автор(ы):

Nassar H. S. Haidar

Center for Research in Applied Mathematics and Statistics
AUL, Cola Str., Beirut, Lebanon

nhaidar@suffolk.edu

Аннотация:

The boundary value approach to dynamic optimization is still an active area of research in many domains of process engineering. The particular domain of interest in the present work is a certain nonlinear optimization problem, constrained by a 3D neutron diffusion partial differential equation (PDE) and controlled, in time, by the boundary conditions. An analytical solution to the associated linear boundary value problem (BVP), which is a principal activity of this optimization, has been reached in 2019 by this author, based on application of an additive separation of variables (ASOV) principle, and was published in the International Journal of Dynamical Systems and Differential Equations. The sole objective of this paper is to examine and verify the consistency of the ASOV method in solving the BVP of this optimization process. The justification of this method is based on reversibility of a pertaining generalized Euclidean transformation, and on asserting this reversibility in the context of an exterior differential framework for the BVP.

Ключевые слова

Ссылки:

  1. Bluman, R. W., Kumei, S. Symmetries and Differential Equations. Springer, New York, 1989.
  2. Bluman, G. W. Application of the general similarity solution of the heat equation to boundary value problems. Quarterly of Applied Mathematics, 31: 403-415, 1974.
  3. Bryant, R. L., Chem, S. S., Gardner, R. B., Goldschmidt, H. L., Griffiths, P. A. Exterior Differential Systems, Springer, New York, 1991.
  4. Cherniha, R. Conditional symmetries for BVP : New definition and its applications for nonlinear problems with Neumann conditions. Miskolc Mathematical Notes, 14(2) 637-646, 2013.
  5. Edelen, D. G. B. Applied Exterior Calculus. Courier Corporation, 2005.
  6. Gragert, P. K. H., Kresten, P. H. M., Martini, R. Symbolic computations in applied differential geometry. Acta Applicandae Mathematicae, 1: 43-77, 1983.
  7. Haidar, N. H. S. Eigenfunctions for a class of parametric Sturm-Liouville problems with an eigenvalue continuum. Journal of Mathematical Analysis and Applications, 161(1): 20-27, 1991.
  8. Haidar, N. H. S. On dynamical (B/Gd ) neutron cancer therapy: an accelerator-based single neutron beam. Pacific Journal of Applied Mathematics, 9(1): 9-26, 2018.
  9. Haidar, N. H. S. An additive separation of variables 3D solution to the dynamical BVP of neutron cancer therapy. International Journal of Dynamical Systems and Differential Equations, 9(2):140-163, 2019.
  10. Haidar, N. H. S. Optimization of two opposing neutron beams parameters in dynamical (B/Gd)neutron cancer therapy. Nuclear Energy and Technology, 5(1):1-7, 2019.
  11. Haidar, N. H. S. On the why of dynamical, and not stationary, (B/Gd) neutron beam cancer therapy. Nuclear Physics and Atomic Energy, Accepted for publication, 2019.
  12. Haidar, N. H. S. A resonated and synchrophased three beams neutron cancer therapy installation. ASME Journal of Nuclear Radiation Science, N. 4, 2019 Accepted for publication, 2019.
  13. Harrison, B. K., Estabrook, F. B. Geometric approach to invariance groups and solution of partial differential systems. Journal of Mathematical Physics, 12(4): 653-666, 1971.
  14. Helmbe, U., Moore, J. B. Optimization and Dynamical Systems. Springer-Verlag, Berlin, 1994.
  15. Henry, A. F. Nuclear-Reactor Analysis. The MIT Press, Cambridge, Massachusetts, 1975.
  16. Herzog, R., Kunisch, K. Algorithms for PDE-constrained optimization. GAMM-Mitteilungen, 31: 3-16, 2014.
  17. Holton, P. A. Affine-Invariant Symmetry Sets. PhD thesis, University of Liverpool, Liverpool, UK, 2000.
  18. Hydon, P. E. Symmetry Methods for Differential Equations: A Beginner's Guide, Vol. 22. Cambridge University Press, 2000.
  19. Khesin, B. A., Tabachnikov, S. L. Arnold : Swimming Against the Tide. AMS, Providence, 2014.
  20. Logsdon, J. S. Efficient Determination of Optimal Control Profiles for Differential Algebraic Systems. PhD Thesis, CMU, Pittsburgh, 1990.
  21. Makai, M. Group Theory Applied to Boundary Value Problems With Applications to Reactor Physics. Nova Science Publishers, New York, 2011.
  22. Miller, Jr, W. Mechanism for variable separation in partial differential equations and their relationship to group theory. In Symmetries and Nonlinear Phenomena, World Scientific, Singapore, 1988.
  23. Ovsiannikov, L.V. Group Analysis of Differential Equations. Academic Press, New York, 2014.
  24. Pontryagin, L. S., Boltyanski, V., Gamkrelidze, R., Mischenko, E. The Theory of Optimal Processes. Interscience Publishers, New York, 1962.
  25. Ramsey, S. D., Tellez, J. A., Riewski, E. J., Temple, B. A. Symmetry and separability of the neutron diffusion equation. Journal of Physics Communications 2(10): No.105009, 2018.
  26. Stackel, P. Uber Die Integration Der Hamilton-Jacobischen Differen- ¨ tialgeichung Mittels Separation Der Variabeln. Habilitationschrift, Halle, 1891.
  27. Stoll, M., Watten, A. All-At-Once Solution of Time-Dependent PDE-Constrained Optimization Problems. Tech. Rep. University of Oxford, Oxford, 2010.
  28. Suhubi, E. Exterior Analysis: Using Applications of Differential Forms. Academic Press, New York, 2013.
  29. Tanartbit, P. Boundary Value Approach for Dynamic Optimization. EDRC06-170-94, CMU, Pittsburgh, 1994.
  30. Wheeler, N. W. Electrodynamical Applications of Exterior Calculus. Lecture Notes, Department of Physics, Reed College, Oregon, 1996.

Полный текст (pdf)