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Vol 53, No 6 (2017)
- Year: 2017
- Articles: 14
- URL: https://journals.rcsi.science/0012-2661/issue/view/9315
Ordinary Differential Equations
Some properties of the Lozinskii logarithmic norm
Abstract
We study properties of the logarithmic matrix norm. We obtain new estimates for matrix norms as well as for the spectral radius and the spectral abscissa. We give a new proof of the Fiedler theorem and a block test for the Hurwitz property of a matrix based on the theory of nonnegative and off-diagonal-nonnegative matrices.
709-718
719-724
On the exponential stability of solutions of periodic systems of the neutral type with several delays
Abstract
We obtain conditions for the exponential stability of the zero solution of linear periodic systems of differential equations of the neutral type with several constant delays, which are stated in terms of a Lyapunov–Krasovskii functional of a special form. We derive estimates that specify the decay rate of solutions at infinity.
725-735
Lagrange formula for ordinary continual second-order differential equations
Abstract
For ordinary continual second-order differential equations, we derive a Lagrange formula and construct their fundamental solutions. We use the Lagrange formula to determine well-posed forms of initial data for these equations and obtain explicit representations of solutions of these initial value problems.
736-744
Partial Differential Equations
745-755
756-765
Solution of the boundary value problem for the equations of steady-state flow of a viscous incompressible nonisothermal fluid past a heated rigid spherical particle
Abstract
We obtain an analytical solution of a boundary value problem for a viscous incompressible nonisothermal fluid assuming an exponential–power law dependence of the fluid viscosity on temperature. A uniqueness theorem for the Navier–Stokes equation linearized with respect to the velocity is proved. We obtain expressions for the mass velocity components and pressure. The solution of the boundary value problem is sought in the form of an expansion in Legendre polynomials.
766-772
Problem with analogs of the Frankl’ condition on a characteristic and the degeneration segment for an equation of mixed type with a singular coefficient
Abstract
For the equation (sign y)|y|muxx +uyy −m(2y)−1uy = 0, where m > 0, considered in some mixed domain, we prove existence and uniqueness theorems for the solution of the boundary value problem with an analog of the Frankl’ condition on a characteristic and on the degeneration segment of the equation.
773-783
784-795
Group analysis of the one-dimensional boltzmann equation: II. Equivalence groups and symmetry groups in the special case
Abstract
We obtain relations that define the equivalence algebra of the family of one-dimensional Boltzmann equations ft + cfx + F(t, x, c)fc = 0 and show that all equations of that form are locally equivalent. We carry out the group classification of the equation with respect to the function F in the special case where the function F and the transformations of the variables t and x are assumed to be independent of c. We show that, under such constraints for the transformation and the family of equations, the maximum possible symmetry algebra is eight-dimensional, which corresponds to an equation with a linear function F.
796-808
809-817
818-824
On the asymptotics of motion of a viscous incompressible fluid for small viscosity
Abstract
We study a nonstationary initial–boundary value problem on the motion of a viscous incompressible fluid in the case of small viscosity. We prove the convergence of solutions to the corresponding limit relations as the viscosity tends to zero.
825-835
Short Communications
Version of the elimination method for a system of partial differential equations
Abstract
For a system of three first-order partial differential equations with three independent variables, we obtain sufficient conditions for one component of the solution to satisfy a third-order Bianchi equation. We also obtain conditions for the solvability of this system by quadratures.
836-840