Сritical density and integrals of liminal dislocation equation.
S. N. Nagornykh1, E. V. Nagornykh2
| Annotation | The paper deals with partial differential equation where unknown function is dislocation scalar density for a thin plate with large bending. Singular points and integrals of this equation are considered. It is shown that usage of characteristics method is necessary to obtain ordinary differential equations and their singular points. Two critical values of the dislocation scalar density for isolated singular point are found. They are sufficient for the conversion of initial equation into identity. Verhulst equation’s bifurcation is important for analysis of different kinds of singular points in determined form as well as under excitation by the white noise. The consequence is given for stationary singular points, for another ordinary differential equation, for critical plate parameters, for critical parameter of Verhulst equation excited by the noise, for dislocation effects, for hardening and fracture of the plate. Zeldovich problem is formulated as a problem of obtaining integrals for partial differential equations with singular points and topological invariant of the plate dislocation structure. |
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| Keywords | Singular points, first order partial differential equation, ordinary differential equation, classification of integrals, bifurcation point of Verhulst equation, white noise. |
1Associate professor of Applied Mathematics Department, Nizhniy Novgorod State Technical University n.a. R.E. Alekseev, Nizhniy Novgorod; algoritm@sandy.ru
2Associate professor of Numerical Simulation of Physical and Mechanical Processes Department, Lobachevsky State University of Nizhniy Novgorod, Nizhniy Novgorod
Citation: S. N. Nagornykh, E. V. Nagornykh, "[Сritical density and integrals of liminal dislocation equation.]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,18:4 (2016) 41–45 (In Russian)
