Deterministic Chaos of N Stochastic Waves in Two Dimensions

DOI: 10.4236/ajcm.2014.44025   PDF   HTML     4,738 Downloads   5,128 Views  

Abstract

Kinematic exponential Fourier (KEF) structures, dynamic exponential (DEF) Fourier structures, and KEF-DEF structures with time-dependent structural coefficients are developed to examine kinematic and dynamic problems for a deterministic chaos of N stochastic waves in the two-dimensional theory of the Newtonian flows with harmonic velocity. The Dirichlet problems are formulated for kinematic and dynamics systems of the vorticity, continuity, Helmholtz, Lamb-Helmholtz, and Bernoulli equations in the upper and lower domains for stochastic waves vanishing at infinity. Development of the novel method of solving partial differential equations through decomposition in invariant structures is resumed by using experimental and theoretical computation in Maple?. This computational method generalizes the analytical methods of separation of variables and undetermined coefficients. Exact solutions for the deterministic chaos of upper and lower cumulative flows are revealed by experimental computing, proved by theoretical computing, and justified by the system of Navier-Stokes PDEs. Various scenarios of a developed wave chaos are modeled by 3N parameters and 2N boundary functions, which exhibit stochastic behavior.

Share and Cite:

Miroshnikov, V. (2014) Deterministic Chaos of N Stochastic Waves in Two Dimensions. American Journal of Computational Mathematics, 4, 289-303. doi: 10.4236/ajcm.2014.44025.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] Lamb, S.H. (1945) Hydrodynamics. 6th Edition, Dover Publications, New York.
[2] Pozrikidis, C. (2011) Introduction to Theoretical and Computational Fluid Dynamics. 2nd Edition, Oxford University Press, Oxford.
[3] Miroshnikov, V.A. (2005) The Boussinesq-Rayleigh Series for Two-Dimensional Flows away from Boundaries. Applied Mathematics Research Express, 2005, 183-227.
http://dx.doi.org/10.1155/AMRX.2005.183
[4] Korn, G.A. and Korn, T.A. (2000) Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review. 2nd Revised Edition, Dover Publications, New York.
[5] Miroshnikov, V.A. (2014) Nonlinear Interaction of N Conservative Waves in Two Dimensions. American Journal of Computational Mathematics, 4, 127-142. http://dx.doi.org/10.4236/ajcm.2014.43012
[6] Kochin, N.E., Kibel, I.A. and Roze, N.V. (1964) Theoretical Hydromechanics. John Wiley & Sons Ltd., Chichester.
[7] Miroshnikov, V.A. (2009) Spatiotemporal Cascades of Exposed and Hidden Perturbations of the Couette Flow. Advances and Applications in Fluid Dynamics, 6, 141-165.
[8] Miroshnikov, V.A. (2012) Dual Perturbations of the Poiseuille-Hagen Flow in Invariant Elliptic Structures. Advances and Applications in Fluid Dynamics, 11, 1-58.

  
comments powered by Disqus

Copyright © 2020 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.