# Volume 57, № 8, 2005

### On Exact Solutions of Nonlinear Diffusion Equations

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1011 – 1019

New classes of the exact solutions of nonlinear diffusion equations are constructed.

### On the Transfer of Generalized Functions by an Evolution Flow

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1020 – 1029

We investigate properties of a solution of a stochastic differential equation with interaction and their dependence on a space variable. It is shown that $x(u, t) − u$ belongs to $S$ under certain conditions imposed on the coefficients, and, furthermore, it depends continuously on the initial measure as an element of S. We also study the problem of the existence of a solution of the equation governed by a generalized function.

### Extended Stochastic Integral and Wick Calculus on Spaces of Regular Generalized Functions Connected with Gamma Measure

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1030–1057

We introduce and study an extended stochastic integral, a Wick product, and Wick versions of holomorphic functions on Kondrat'ev-type spaces of regular generalized functions. These spaces are connected with the Gamma measure on a certain generalization of the Schwartz distribution space \(S'\). As examples, we consider stochastic equations with Wick-type nonlinearity.

### Classification of Quadratic Parastrophically Uncancelable Functional Equations for Five Object Variables on Quasigroups

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1058 – 1068

We continue the investigation of quadratic functional equations over quasigroup operations and prove that every parastrophically uncancelable quadratic equation for five object variables is parastrophically equivalent to one of four given functional equations.

### Properties of the Flows Generated by Stochastic Equations with Reflection

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1069 – 1078

We consider properties of a random set $\varphi_t(\mathbb{R}_+^d)$, where $\varphi_t(x)$ is a solution of a stochastic differential equation in $\mathbb{R}_+^d$ with normal reflection on the boundary starting at the point $x$. We perform the characterization of inner and boundary points of the set $\varphi_t(\mathbb{R}_+^d)$. We prove that the Hausdorff dimension of the boundary $\partial \varphi_t(\mathbb{R}_+^d)$ is not greater than $d - 1$.

### Approximation of Classes of Analytic Functions by Fourier Sums in Uniform Metric

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1079 – 1096

We find asymptotic equalities for upper bounds of approximations by Fourier partial sums in a uniform metric on classes of Poisson integrals of periodic functions belonging to unit balls of spaces $L_p,\quad 1 \leq p \leq \infty$. We generalize the results obtained to classes of $(\psi, \overline{\beta})$-differentiable functions (in the Stepanets sense) that admit analytical extension to a fixed strip of the complex plane.

### Approximation of $(\psi, \beta)$-Differentiable Functions Defined on the Real Axis by Abel-Poisson Operators

Kharkevych Yu. I., Zhyhallo T. V.

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1097 – 1111

We obtain asymptotic equalities for upper bounds of approximations of functions on the classes \(\hat C_{\beta ,\infty }^\psi\) and \(\hat L_{\beta ,1}^\psi\) by Abel-Poisson operators.

### Exact Solutions of a Mathematical Model for Fluid Transport in Peritoneal Dialysis

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1112–1119

A mathematical model for fluid transport in peritoneal dialysis is constructed. The model is based on a nonlinear system of two-dimensional partial differential equations with corresponding boundary and initial conditions. Using the classical Lie scheme, we establish that the base system of partial differential equations (under some restrictions on coefficients) is invariant under the infinite-dimensional Lie algebra, which enables us to construct families of exact solutions. Moreover, exact solutions with a more general structure are found using another (non-Lie) technique. Finally, it is shown that some of the solutions obtained describe the hydrostatic pressure and the glucose concentration in peritoneal dialysis.

### Leonid Pavlovych Nyzhnyk (on his 70-th birthday)

Berezansky Yu. M., Gorbachuk M. L., Gorbachuk V. I., Khruslov E. Ya., Kostyuchenko A. G., Kuzhel' S. A., Marchenko V. O., Samoilenko A. M., Samoilenko Yu. S.

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1120-1122

### On Partially Irregular Almost Periodic Solutions of Weakly Nonlinear Ordinary Differential Systems

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1123 – 1130

For weakly nonlinear almost periodic ordinary differential systems, we obtain conditions for the existence of partially irregular almost periodic solutions and propose algorithms for their construction.

### On the Well-Posedness of a Two-Point Boundary-Value Problem for a System with Pseudodifferential Operators

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1131 – 1136

We investigate the problem of the well-posedness of a boundary-value problem for a system of pseudodifferential equations of arbitrary order with nonlocal conditions. The equation and boundary conditions contain pseudodifferential operators whose symbols are defined and continuous in a certain domain $H ⊂ ℝ_{σ}^m$. A criterion for the existence and uniqueness of solutions and for the continuous dependence of the solution on the boundary function is established.

### On the Lagrange Stability of Motion in the Three-Body Problem

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1137 – 1143

For the three-body problem, we study the relationship between the Hill stability of a fixed pair of mass points and the Lagrange stability of a system of three mass points. We prove the corresponding theorem establishing sufficient conditions for the Lagrange stability and consider a corollary of the theorem obtained concerning a restricted three-body problem. Relations that connect separately the squared mutual distances between mass points and the squared distances between mass points and the barycenter of the system are established. These relations can be applied to both unrestricted and restricted three-body problems.

### Rapidly Decreasing Solution of the Initial Boundary-Value Problem for the Toda Lattice

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1144 – 1152

Using the inverse scattering transform, we investigate an initial boundary-value problem with zero boundary condition for the Toda lattice. We prove the existence and uniqueness of a rapidly decreasing solution and determine a class of initial data that guarantees the existence of a rapidly decreasing solution.