Nonstationary fronts in the singularly perturbed power-society model. (English) Zbl 1304.35049

Summary: The theory of contrasting structures in singularly perturbed boundary problems for nonlinear parabolic partial differential equations is applied to the research of formation of steady state distributions of power within the nonlinear “power-society” model. The interpretations of the solutions to the equation are presented in terms of applied model. The possibility theorem for the problem of getting the solution having some preassigned properties by means of parametric control is proved.


35B25 Singular perturbations in context of PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
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[1] Butuzov, V. F.; Vasil’eva, A. B., Asymptotic behavior of a solution of contrasting structure type, Akademiya Nauk SSSR, 42, 6, 831-841 (1987) · Zbl 0725.34061
[2] Butuzov, V. F.; Vasil’eva, A. B.; Nefedov, N. N., Asymptotic theory of contrasting structures. A survey, Automation and Remote Control, 58, 7, 1068-1091 (1997) · Zbl 0921.34054
[3] Vasil’eva, A. B.; Butuzov, V. F.; Nefedov, N. N., Singularly perturbed problems with boundary and internal layers, Proceedings of the Steklov Institute of Mathematics, 268, 1, 258-273 (2010) · Zbl 1207.35039
[4] Vasil’eva, A. B.; Butuzov, V. F.; Nefëdov, N. N., Contrast structures in singularly perturbed problems, Fundamental’naya i Prikladnaya Matematika, 4, 3, 799-851 (1998) · Zbl 0963.34043
[5] Bozhevol’nov, Yu. V.; Nefëdov, N. N., Front motion in a parabolic reaction-diffusion problem, Computational Mathematics and Mathematical Physics, 50, 2, 264-273 (2010) · Zbl 1224.35197 · doi:10.1134/S0965542510020089
[6] Vasil’eva, A.; Nikitin, A.; Petrov, A., Stability of contrasting solutions of nonlinear hydromagnetic dynamo equations and magnetic fields reversals in galaxies, Geophysical and Astrophysical Fluid Dynamics, 78, 1-4, 261-279 (1994) · doi:10.1080/03091929408226582
[7] Moss, D.; Petrov, A.; Sokoloff, D., The motion of magnetic fronts in spiral galaxies, Geophysical and Astrophysical Fluid Dynamics, 92, 1-2, 129-149 (2000) · doi:10.1080/03091920008203714
[8] Vasil’eva, A. B.; Petrov, A. P.; Plotnikov, À. À., Alternating contrast structures, Computational Mathematics and Mathematical Physics, 38, 9, 1471-1480 (1998) · Zbl 0965.35011
[9] Mikhailov, A. P., Mathematical modeling of the authority in the hierarchical structures, Mathematical Modeling, 6 (1994)
[10] Samarskii, A. A.; Mikhailov, A. P., Principles of mathematical modeling, Ideas, methods, examples. Principles of mathematical modeling, Ideas, methods, examples, Numerical Insights, 3 (2002), London, UK: Taylor & Francis, London, UK · Zbl 1031.00009
[11] Mikhailov, A. P., Mathematical modeling of power distribution in state ierarchical structures interacting with civil society, Proceedings of 14th IMACS World Congress
[12] Mikhailov, A. P., The research of the “Power-Society” system
[13] Butuzov, V. F.; Nedelko, I. V., On the global influence domain of stable solutions with internal layers, Differential Equations, 192, 5, 13-52 (2001) · Zbl 1009.35008 · doi:10.1070/SM2001v192n05ABEH000563
[14] Nefëdov, N. N.; Nikitin, A. G., The method of differential inequalities for step-type contrast structures in singularly perturbed integro-differential equations in the spatially two-dimensional case, Differential Equations, 42, 5, 739-749 (2006) · Zbl 1206.45013 · doi:10.1134/S0012266106050132
[15] Nefedov, N. N.; Nikitin, A. G.; Petrova, M. A.; Rekke, L., Moving fronts in integro-parabolic reaction-advection-diffusion equations, Differential Equations, 47, 9, 1318-1332 (2011) · Zbl 1269.45008 · doi:10.1134/S0012266111090096
[16] Nefëdov, N. N.; Nikitin, A. G., Boundary and internal layers in the reaction-diffusion problem with a nonlocal inhibitor, Computational Mathematics and Mathematical Physics, 51, 6, 1011-1019 (2011) · Zbl 1249.35159 · doi:10.1134/S0965542511060157
[17] Dmitriev, M.; Petrov, A.; Zhukova, G., The nonlinear “Authority-Society” model, WSEAS Transactions on Mathematics, 2, 1-2, 31-36 (2003) · Zbl 1156.91459
[18] Vasil’eva, A. B.; Butuzov, V. F., The method of boundary layer functions, Differential Equations, 21, 10, 1107-1112 (1985) · Zbl 0644.35008
[19] Dmitriev, M. G.; Pavlov, A. A.; Petrov, A. P., Optimal power volume in social-economical hierarchy by consumption per capita criterion, Informacionnye Technologii i Vychislitelnye Sistemy, 4, 4-11 (2007)
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