On a spectral problem for the bessel equation of zero order

In the present paper, for part of the eigenfunction system corresponding to a spectral problem with spectral parameter in the boundary condition, we write out a spectral problem for the same differential equation with a nonlocal boundary condition that does not contain the spectral parameter. In addition, we construct the biorthogonal system.     

On a method for constructing algebro-geometric solutions to the zero curvature equation

Under special conditions that hold for a number of applications, we suggest a construction that reduces the calculation of algebro-geometric solutions of the zero curvature equation for 2 × 2 matrices to solving the Jacobi inversion problem on a hyperelliptic Riemann surface and the Riccati equation. An application to the system of equations of the principal chiral field is considered. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 1, pp. 61–72, January, 1997.     

On a two-velocity model of the Boltzmann-Enskog equation

Summary A simple two-velocity model (-model)of the Boltzmann-Enskog equation for a gas of hard spheres is proposed. Physical properties of such a model are analyzed and its differences with respect to the Carleman model are investigated. Global existence theorems and some qualitative properties of solutions of the -model are also proved.On leave from Dept. of Mathematics and Mechanics, University of Warsaw, Poland.     

On a difference equation arising in a learning-theory model

An analysis is presented of the equationf(x+a)−f(x)=e ^−x {f(x)−f(x−b)}. Herea andb denote arbitrary positive constants, and a solution is sought which satisfies the following conditions:f(−∞)=0,f(+∞)=1, 0≦f(x)≦1. Existence and uniqueness of solution are established, and then an analytical form of the solution is obtained by use of bilateral Laplace transform. Research supported by the National Science Foundation, Grant GP-2558.     

On the convergence to zero of solutions of a second-order differential equation with complex-valued potential

We consider the second-order linear differential equation y" + A(t)y = 0 on the semiaxis with complex-valued potential function. Sufficient conditions for the potential function assuring that all solutions of the equation converge to zero at infinity are obtained. It is shown that the conditions imposed on the potential function are close to the necessary ones. One of the results seems to be new even in the case of real-valued function A(·).     

Langevin equation of a fluid particle in wall-induced turbulence

We derive the Langevin equation describing the stochastic process of fluid particle motion in wall-induced turbulence (turbulent flow in pipes, channels, and boundary layers including the atmospheric surface layer). The analysis is based on the asymptotic behavior at a large Reynolds number. We use the Lagrangian Kolmogorov theory, recently derived asymptotic expressions for the spatial distribution of turbulent energy dissipation, and also newly derived reciprocity relations analogous to the Onsager relations supplemented with recent measurement results. The long-time limit of the derived Langevin equation yields the diffusion equation for admixture dispersion in wall-induced turbulence.     

On stability of zero solution of an essentially nonlinear second-order differential equation

Small periodic (with respect to time) perturbations of an essentially nonlinear differential equation of the second order are studied. It is supposed that the restoring force of the unperturbed equation contains both a conservative and a dissipative part. It is also supposed that all solutions of the unperturbed equation are periodic. Thus, the unperturbed equation is an oscillator. The peculiarity of the considered problem is that the frequency of oscillations is an infinitely small function of the amplitude. The stability problem for the zero solution is considered. Lyapunov investigated the case of autonomous perturbations. He showed that the asymptotic stability or the instability depends on the sign of a certain constant and presented a method to compute it. Liapunov’s approach cannot be applied to nonautonomous perturbations (in particular, to periodic ones), because it is based on the possibility to exclude the time variable from the system. Modifying Lyapunov’s method, the following results were obtained. “Action–angle” variables are introduced. A polynomial transformation of the action variable, providing a possibility to compute Lyapunov’s constant, is presented. In the general case, the structure of the polynomial transformation is studied. It turns out that the “length” of the polynomial is a periodic function of the exponent of the conservative part of the restoring force in the unperturbed equation. The least period is equal to four.     

Stability of a conditional Cauchy equation on a set of measure zero

In this paper, we prove the Hyers–Ulam stability theorem when \({f, g, h : \mathbb{R} \to \mathbb{R}}\) satisfy $$|f(x + y) - g(x) - h(y)| \leq \epsilon$$ in a set \({\Gamma \subset \mathbb{R}^{2}}\) of measure \({m(\Gamma) = 0}\) , which refines a previous result in Chung (Aequat Math 83:313–320, 2012) and gives an affirmative answer to the question in the paper. As a direct consequence we obtain that if \({f, g, h : \mathbb{R} \to \mathbb{R}}\) satisfy the Pexider equation $$f(x + y) - g(x) - h(y) = 0$$ in \({\Gamma}\) , then the equation holds for all \({x, y \in \mathbb{R}}\) . Using our method of construction of the set, we can find a set \({\Gamma \subset \mathbb{R}^{2n}}\) of 2n-dimensional measure 0 and obtain the above result for the functions \({f, g, h : \mathbb{R}^{n} \to \mathbb{C}}\) .     

KDV equation on a half-line with the zero boundary condition

We solve the mixed problem for the KdV equation with the boundary condition u|_x=0=0, u_xx|_x=0=0 using the inverse scattering method. The time evolution of the scattering matrix is efficiently defined from the consistency condition for the spectra of two differential operators giving the L-A pair. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 3, pp. 397–404, June, 1999     

On a stochastic differential equation arising in a price impact model

We provide sufficient conditions for the existence and uniqueness of solutions to a stochastic differential equation which arises in the price impact model developed by Bank and Kramkov (2011)  and . These conditions are stated as smoothness and boundedness requirements on utility functions or Malliavin differentiability of payoffs and endowments.     

Attractors for a deconvolution model of turbulence

We consider a deconvolution model for 3D periodic flows. We show the existence of a global attractor for the model.     

Slow convergence to zero for a parabolic equation with a supercritical nonlinearity

We study the behaviour of solutions of the Cauchy problem for a supercritical semilinear parabolic equation which converge to zero from above as t → ∞. We show that any algebraic decay rate slower than the self-similar one occurs for some initial data.     

On polynomials with a prescribed zero

Let us denote byΛ _{n, 1} the supremum of (max_∥z∥=1∥p′ _n (z)∥)/ (max_∥z∥=1∥p _n (z)∥) taken over all polynomialsp _n of degree at mostn having a zero on the unit circle {z ∈ C∶∥z∥=1}. We show that Λn.1=n-(π ²/16)(1/n)+O(1/n ².     

Multiplicity of solutions of a zero mass nonlinear equation on a Riemannian manifold

The relation between the number of solutions of a nonlinear equation on a Riemannian manifold and the topology of the manifold itself is studied. The technique is based on Ljusternik-Schnirelmann category and Morse theory.     

The zero curvature equation for rigid CR-manifolds

In this paper, we present the general analytic solution to the zero curvature equation for rigid three-dimensional CR-manifolds. The solutions are uniquely determined by one function and four real parameters.     

Application of the Kac equation to turbulence simulation

The possibility of applying the Kac equation to the simulation of small-scale turbulence is explored. The hypothesis is substantiated that the formation of a flow regime similar to the actual turbulent one can be qualitatively described as based on the analysis of the properties of the Kac equation.     

Rates of convergence to zero for a semilinear parabolic equation with a critical exponent

We study nonnegative solutions to the Cauchy problem for a semilinear parabolic equation with a nonlinearity which is critical in the sense of Joseph and Lundgren. We establish the rate of convergence to zero of solutions that start from initial data which are near the singular steady state. In the critical case, this rate contains a logarithmic term which does not appear in the supercritical case and makes the calculations more delicate.     

On a Penrose-Fife model with zero interfacial energy leading to a phase-field system of relaxed Stefan type

In this paper we study an initial-boundary value Stefan-type problem with phase relaxation where the heat flux is proportional to the gradient of the inverse absolute temperature. This problem arise naturally as limiting case of the Penrose-Fife model for diffusive phase transitions with nonconserved order parameter if the coefficient of the interfacial energy is taken as zero. It is shown that the relaxed Stefan problem admits a weak solution which is obtained as limit of solutions to the Penrose-Fife phase-field equations. For a special boundary condition involving the heat exchange with the surrounding medium, also uniqueness of the solution is proved.     

Bernoulli jets and the zero mean curvature equation

We consider an elliptic PDE problem related with fluid mechanics. We show that level sets of rescaled solutions satisfy the zero mean curvature equation in a suitable weak viscosity sense. In particular, such level sets cannot be touched from below (above) by a convex (concave) paraboloid in a suitably small neighborhood.