# Volume 3 Issue #2 (June 2011)

• Asadullin E.M., Nasyrov F.S. About filtering problem of diffusion processes. Pp. 3 - 9
Abstract: The filtering problem of nonlinear one-dimensional diffusion processes is considered. The structures of observable and nonobservable processes are found. It is shown, that solution of the optimal filtering problem can be reduced to solution of the filtering problem for the case when a nonobservable process has a simpler structure and an observable process is the Wiener process with a random smooth trend. The equation connecting a conditional expectation for the initial filtering problem with a nonnormalized filtering density for the reduced filtering problem is obtained.
• Baltaeva I.I., Urazboev G.U. About the Camassa–Holm equation with a self-consistent source. Pp. 10 - 18
Abstract: This work is devoted to solving the Camassa–Holm equation with a self-consistent source of a special type by the inverse scattering method. The main result consists in determining the evolution of the scattering data for the spectral problem associated with the Camassa–Holm equation with a self-consistent source of a special type. In contrast to the classical Camassa–Holm equation, the eigenvalues of the spectral problem are moving in problem under consideration. The resulting equalities determine the evolution of the scattering data completely; this fact allows us to apply the inverse scattering method for solving the considered problem.
• Imanbaev N.S., Sadybekov M.A. Stability of basis property of a type of problems on eigenvalues with nonlocal perturbation of boundary conditions. Pp. 27 - 32
Abstract: The article is devoted to a spectral problem for a multiple differentiation operator with an integral perturbation of boundary conditions of one type which are regular, but not strongly regular. The unperturbed problem has an asymptotically simple spectrum, and its system of normalized eigenfunctions creates the Riesz basis. We construct the characteristic determinant of the spectral problem with an integral perturbation of the boundary conditions. The perturbed problem can have any finite number of multiple eigenvalues. Therefore, its root subspaces consist of its eigen and (maybe) adjoint functions. It is shown that the Riesz basis property of a system of eigen and adjoint functions is stable with respect to integral perturbations of the boundary condition.
• Krivosheyeva O.A. The convergence domain for series of exponential monomials. Pp. 42 - 55
Abstract: Questions of convergence for exponential series of monomials are studied in this paper. Exponential series, Dirichlet's series and power series are particular cases of these series. The space of coefficients of exponential series of monomials converging in the given convex domain in a complex plane is described. The full analogue of Abel's theorem for these series is formulated with a natural restriction. In particular, results on continuation of convergence of exponential series follow from this analogue. A full analogue of Cauchy–Hadamard's theorem is obtained as well. It provides a formula for finding the convergence domain of these series by their coefficients. The obtained results include all earlier known results connected with Abel and Cauchy–Hadamard's theorems for exponential series, Dirichlet's series and power series as particular cases.
• Merzlyakov S. G. Integrals of exponential functions with respect to Radon measure. Pp. 56 - 78
Abstract: Properties of sets of convergence for integrals of exponential functions in a finite-dimensional Euclidean space are studied in the paper. It is shown that these sets are always convex. In particular, these sets include the sets of absolute convergence of series of exponential functions. A special class of convex sets is introduced and a complete description of sets of convergence is obtained for the case of open and relatively close convex sets in terms of this class. Necessary and sufficient conditions for any set of convergence to be open and independently unbounded are formulated.
• Umarov Kh.G. Explicit solution of the Cauchy problem to the equation for groundwater motion with a free surface. Pp. 79 - 84
Abstract: A linear partial differential equation modelling evolution of a free surface of the filtered fluid $\lambda u_t-\Delta_2u_t=\alpha\Delta_2u-\beta\Delta^2_2u+f$ is considered. Here $u(x,y,t)$ is the searched function characterizing the fluid pressure, $f=f(x,y,t)$ is the given function calculating an external influence on the filtration flow, $\Delta_2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$ is the Laplace differential operator, $\lambda,\alpha,\beta$ are positive constants depending on characteristics of the watery soil. The explicit solution to the Cauchy problem for the above linear partial differential equation is obtained in the space $L_p(R^2)$, \(1
• Khabirov S.V. Explicit solution of the Cauchy problem to the equation for groundwater motion with a free surface. Pp. 85 - 88
Abstract: Group classification of gasdynamic equations by the state equation consists of 13 types of finite-dimensional Lie algebras of different dimensions, from 11 to 14. Some types depend on a parameter. Five pairs of Lie algebras appear to be equivalent. The equivalent transformations for Lie algebras contain the equivalent transformations for gasdynamic equations. The equivalence test resulted in nine nonisomorphic Lie algebras with different structures. One type has 3 different structures for different parameters. Each of these Lie algebras is represented as a semidirect sum of a six-dimensional Abeilian ideal with a subalgebra, which is decomposed into a semidirect or direct sum in its turn. The optimal systems for subalgebras are constructed. The Abeilian ideal is added in 6 cases while constructing the optimal system. There remain 3 Lie algebras of the dimensions 12, 13, 14 for which the optimal systems are not constructed.
• Shabat A.B., Elkanova Z.S. Commuting differential operators in two-dimension. Pp. 89 - 95
Abstract: A generalization to the multi-dimensional case of commutative rings of differential operators is considered. An algorithm for construction of commuting two-dimensional differential operators is formulated for a special kind of operators related to the simple one-dimensional model proposed by Burchnall and Chaundy in 1932. The problem of classifying such commutative pairs is discussed. The suggested algorithm is based on necessary conditions for general commutativity and the reducibility lemma proved in the present paper.