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We consider a class of autonomous functional-differential equations for which the properties of oscillation of all solutions and positiveness of the fundamental solution are mutually complementary.
We obtain new nonimprovable conditions of oscillation of all solutions to a linear differential equation with several variable delays and positive coefficients. These conditions have the form of the upper and lower limits of the sum of integrals of the coefficients over the sets that are determined only by the delay corresponding to this coefficient. These results differ from the well-known results in which a coarsening of the integration set is assumed.
We consider autonomous and periodic differential equations containing two scalar parameters and analyze the construction of boundaries of stability domains of equilibrium points for such equations in the plane of parameters. We indicate conditions under which a unique smooth boundary curve passes through a given point. We propose a scheme of approximate construction of this curve.
This paper is devoted to the theory of approximation of coefficient inverse problems for differential equations of parabolic, elliptic, and hyperbolic types in functional spaces. We present general statements of problems and their approximations and review results obtained earlier in the literature.
In this work, we study approximations of solutions of fractional differential equations of order 1/2. We present a new method of approximation and obtain the order of convergence. The presentation is given within the abstract framework of a semidiscrete approximation scheme, which includes finite-element methods, finite-difference schemes, and projection methods.
We consider the boundary value problem for a differential operator with an integrable potential and study the asymptotic behavior of solutions for large values of the spectral parameter. We derive an equation for eigenvalues and obtain asymptotics of eigenfunctions of the operator.
We prove an existence and uniqueness theorem for the linear differential equation u′(z) = Au(z)+e{sup γz}f(z), where A is a closed operator and f is an entgire vector-valued function of zero exponential type. Bibliography: 10 titles.
We establish conditions for the existence, in the scale of Sobolev spaces, of a unique solution to the problem for partial differential equations of high order with integral conditions in the form of moments. The solution of this problem is constructed with the help of Fourier transforms.
We study linear systems of ordinary differential equations of higher order with an identically singular matrix at the higher derivative of the required vector-valued function in the domain of definition. We give definitions of the index and singular points for these systems, formulate the conditions of solvability, and deduce a formula for the general solution. The algorithms for finding the index and singular points are presented.
We construct and justify asymptotics of linearly independent system of solutions to an algebraic-differential system with large rapidly oscillating coefficients.
In the disk, we consider the first boundary-value problem for a functional differential equation containing transformations of orthotropic contractions of independent variables of the unknown function. We study the smoothness of generalized solutions inside special-type subdomains and near their boundaries and pose strong ellipticity conditions.
For a second-order differential equation whose the right-hand side contains the sum of terms with regularly and rapidly varying nonlinearities, we establish the conditions of existence and asymptotic representations of rapidly varying solutions, as the argument tends to a singular point.
We consider an approximation method based on Steklov functions of the first and second order. We obtain estimates for the norms in the space C of continuous periodic functions and clarify how they connect with numerical differentiation formulas.
We study the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the negative Laplace operator in a fractional power in two dimensions. The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for non Fredholm elliptic operators in unbounded domains are used.
We study the problem for a homogeneous partial differential equation of the second order with respect to time with given homogeneous two-point conditions in this variable and, in general, of the infinite order in the other (space) variable. It is proved that the analyzed problem possesses solely the trivial solution if the characteristic determinant is not identically equal to zero. In the case where the set of zeros of the characteristic determinant of this problem is nonempty, we propose a method for the construction of nontrivial solutions of the problem.
We obtain differential operators for the bivariate Chebyshev polynomials of the first kind associated with the root systems of the simple Lie algebras C{sub 2} and G{sub 2}.
By using the theory of polynomial matrix pencils, we construct the asymptotics of linearly independent solutions of homogeneous singularly perturbed system of linear differential equations of any order m with matrices at higher derivatives degenerating as a small parameter approaches zero. The general case is analyzed. In this case, the limit matrix pencil has several finite and infinite elementary divisors of both the same and different multiplicities. The corresponding asymptotic estimates are presented.
Filled skutterudites are among the most promising thermoelectric materials, and the obtainment of monophasic samples is crucial for the study of their transport properties. In this work, a differential thermal analysis study was performed on samples belonging to the Sm{sub y}(Fe{sub x}Ni{sub 1-x}){sub 4}Sb{sub 12} system with the aim to determine the most proper annealing temperature to possibly obtain skutterudite monophasic samples, and to study the effect of Sm addition on the thermodynamics of the Fe-Ni-Sb system. Samples were prepared through the traditional procedure consisting in melting–quenching–annealing the starting elements. DTA analyses were carried out on annealed samples, and results were interpreted using the Fe{sub 25}Sb{sub 75}-Ni{sub 25}Sb{sub 75} pseudobinary phase diagram taken from the literature as a starting point. They reveal that 870 K is the best annealing temperature, being the formation of a liquid phase just above this point; moreover, the Sm introduction results in widening the one-phase skutterudite stability region toward both higher temperatures and higher Fe concentrations compared with the Sm-free system.