Editorial briefcase
- Timoshin M.I. Two-dimensional algebras of ODEs dynamic symmetries.
Status: reviewing
Abstract. In this work the effectivity of the dynamical symmetries usage to the investigation of differential equations integrability is shown. It is constructed the generalization of {\bf S. Lie} classification to two-dimensional algebrae of point symmetries for ordinary differential equations of second order. Integrable cases for found second order ODE are indicated. It is noted that a part of found integrability cases is related to {\bf Abel's} type equations and possibly has self-dependent interest.
Date of submission: 02 Avgust 2011 г.
- Belykh V.N. The problem of numerical realization of integral operators of axisymmetric boundary value problems (algorithms without saturation)
Status: accepted, planed in в т.4 №3
Abstract. In the paper a new principle - {\it unsaturated \/} - method of numerical implementation integral operators $C^\infty$-smooth axisymmetric boundary value problems to automatically take into account the specifics of the axisymmetric problems, which is an obstacle to any numerical methods with the principal term of error. The method was extensively tested on the problem of precise evaluation of the integral Gauss theory of harmonic potential in high aspect ratio ellipsoid.
Date of submission: 22 September 2011 г.
- Mishin S.N. On growth characteristics of operator-valued functions.
Status:
Abstract. In the work Liouville's theorem and concept of order and type of entire function find their generalisation to the case of operator-valued function with values from space ${\rm Lec}({\bf H}_1,{\bf H})$ of all linear continuous operators from locally convex space ${\bf H}_1$ to ${\bf H}$ with equicontinuous bornology. Formulae which give order and type of operator-valued function from characterisctics of sequence of coefficients are found. Some properties of order and type of operator-valued function are ascertained.
Date of submission: 27 September 2011 г.
- Bibikov P.V. On automorphic systems of differential equations and $\mathrm{GL}_2(\mathbb{C})$-orbits of binary forms
Status: reviewing
Abstract. In the work we introduce new method for studying classical algebraic problem of classifying $\mathrm{GL}_2(\mathbb{C})$-orbits of binary forms with the help of differential equations. We construct and study automorphic system of differential equations $\mathcal{S}$ of the fourth order, whose solution space coincides with $\mathrm{GL}_2(\mathbb{C})$-orbit of fixed binary form $f$. In cases of order 2 and 3 system $\mathcal{S}$ is integrable. In the most difficult case of order 4 we prove that system $\mathcal{S}$ may be reduced to the system of the differential Abel equation and the linear partial differential equation of order 1.
Date of submission: 30 September 2011 г.
- Bagderina Yu.Yu. Separation of an equation in the system of two second-order ordinary differential equations
Status: accepted, planed in в т.4 №4
Abstract. We consider projectable type systems of two second-order ordinary differential equations with cubic nonlinearity of right-hand side in first derivatives. For such systems we obtain criteria of reducibility by local transformation to a system with separating equation in one of unknown functions. Applications of the criteria and construction of the corresponding transformation is illustrated with a number of examples.
Date of submission: 24 October 2011 г.
- Sharafutdinova G.G. The problem of the forms of the deflection of a freely supported plate under longitudinal force.
Status: accepted, planed in в т.4 №4
Abstract. This paper considers the problem of bifurcation behavior of the elastic plate when the longitudinal compressive strength. A new scheme which allows to determine the critical values of the force at which the plate takes stable curvilinear equilibrium. The scheme also leads to an asymptotic formula which describes the nonlinear deflections of the plate when passing through the critical force.
Date of submission: 24 October 2011 г.
- Bazzaev A.K. A finite difference schemes for diffusion equation of fractional order with third boundary value conditions in multidimensional field.
Status: reviewing
Abstract. In this work we consider the finite difference schemes for diffusion equation of fractional order in the multidimensional field with third boundary value conditions. We prove the stability and convergence of difference schemes for this problem.
Date of submission: 11 November 2011 г.
- Nonlinear hyperbolic differential equations, related with the Klein-Gordon equation by differential substitutions.
Status: reviewing
Abstract. In the work we solved the classification problem of nonlinear hyperbolic differential equations in two independent variables $u_{xy}=f(u,u_x,u_y)$, reduced by differential substitutions of the special form $v = \varphi(u, u_x)$ to the Klein-Gordon equation.
Date of submission: 21 November 2011 г.
- Zhiber A.V., Murtazina R.D., Habibullin I.T., Shabat A.B. Characteristic Lie rings and integrable models in mathematical physics.
Status: accepted, planed in в т.4 №3
Abstract. Review is devoted to a systematic exposition of the algebraic approach to the study of nonlinear integrable partial differential equations and their discrete analogues, based on the concept of the characteristic vector field. A special attention is paid to the Darboux integrable equations. The problem of constructing higher symmetries of the equations, as well as their particular and general solutions is discussed. In particular, it is shown that the partial differential equation of hyperbolic type is integrated in quadratures if and only if its characteristic Lie rings in both directions are of finite dimension. For the hyperbolic type equations integrable by the inverse scattering method, the characteristic rings are of minimal growth. The possible applications of the concept of characteristic Lie rings to the systems of differential equations of hyperbolic type with more than two characteristic directions, to the equations of evolution type, and to ordinary differential equations are discussed.
Date of submission: 25 November 2011 г.
- Losev A.G., Mazepa E.A. On the asymptotic behavior of positive solutions of some quasilinear inequalities on model Riemannian manifolds.
Status: accepted, planed in в т.4 №4
Abstract. In this paper asymptotic behavior of positive solutions of some quasilinear elliptic inequalities on on spherically symmetric noncom\-pact (model) Riemannian manifolds. In particular, we find conditions under which Liouville theorems on no nontrivial solutions, as well as the conditions of existence and cardinality of the set of positive solutions of the studied inequalities on the Riemannian manifolds. The results generalize similar results obtained previously by Naito.~Y. and Usami~H. for the Euclidean space ${\rm\bf R}^n $.
Date of submission: 29 November 2011 г.
- Golichev I.I. Iterative linearization of the evolution Navier-Stokes equations.
Status: accepted, planed in в т.4 №4
Abstract. Constructed and validated an iterative process, which reduces the solution of nonlinear time-dependent Navier-Stokes equations to the solution of a sequence of linear problems. Using a priori estimates of solutions allows us to prove the convergence of the method with any initial approximation. It is shown that the proposed method can be used to prove the existence and uniqueness of the solution.
Date of submission: 06 December 2011 г.
- Karimov SH.T. Solution of a Cauchy problem for the three-dimensional hyperbolic equation with singular coefficients by the method of fractional integrals.
Status: reviewing
Abstract. Is found the explicit formula of a solution of a Cauchy problem for the three-dimensional hyperbolic equation \, with \, three singular coefficients and with spectral parameter by the method of fractional integrals.
Date of submission: 17 December 2011 г.
- Mazalov M. Ya. On uniform approximability by elliptic equations solutions of order higher than two
Status: accepted, planed in в т.4 №3
Abstract. We consider uniform approximation problems on compact subsets of $\real^d$, $d>2$, by solutions of homogeneous constant coefficients elliptic equations of order $n>2$. We construct an example showing that in the general case for compact sets with nonempty interior there is no uniform approximability criteria analogous to well-known Vitushkin's criterion for analytic functions in $\com$. On the contrary, for nowhere dense compact sets the situation is the same as for analytic and harmonic functions, including instability of corresponding capacities.
Date of submission: 18 December 2011 г.
- Imomnazarov Kh.Kh., Mikhailov A.A. Application of a spectral method of Laguerre for numerical solution of seismic fields in porous media for dissipative case.
Status: reviewing
Abstract. The paper presents the algorithm, based on the application of the spectral Laguerre method for approximation of temporal derivatives as applied to the problem of seismic wave propagation in the porous media in the presence of dissipation of energy. The initial system of equations as first order hyperbolic system in terms of velocities, stresses and pore pressure. For the numerical solution of the task in question, the method of combination of analytical Laguerre transformation and a finite difference method is used. The proposed method of the solution can be considered as analogue to the known spectral method on the basis of Fourier transform. However, unlike Fourier transform, application of integral Laguerre transform with respect time allows us to reduce the initial problem to solving a system of equations in which the parameter of division is present only on the right-hand side of equations and has a recurrent dependence. As compared to time-domain method, with the help of an analytical transformation in the spectral method it is possible to reduce an original problem to solving the system of differential equations, in which there are only derivatives with respect to spatial coordinates. This allows us to apply a known stable difference scheme for recurrent solutions to similar systems. Such an approach is effective when solving dynamic problems for porous media. Thus, because of the presence of the second longitudinal wave with a low velocity, the use of difference schemes in all coordinates for stable solutions requires a consistent small step both in time and in space, which inevitably results in an increase of computer costs.
Date of submission: 19 December 2011 г.
- Atnagulova R.A. Variety of classical Yang-Baxter.
Status: reviewing
Abstract. In the theory of integrable differential equations play an important role of the Yang-Baxter equation with the square, that is, the equation $R([R(a),b]-[R(b),a])=R^{2}([a,b])+[R(a),R(b)]$, where $a,b\in g$, $g$ --- Lie algebra, and $R$ --- linear operator space $g$. To construct examples of operators $R$ in use of the Lie subalgebra $h$ in the matrix algebra, additional to subspace of matrices with zero last row. Are two series of examples of such subalgebras $h$.
Date of submission: 19 December 2011 г.
- Khabirov S.V. Extention of the canonic flows
Status: accepted, planed in в т.4 №4
Abstract.
Date of submission: 20 December 2011 г.
- Mamatova N.H., Haetov A.R., Shadimetov Sh.M.
Status: accepted, planed in в т.4 №4
Abstract.
Date of submission: 20 December 2011 г.
- Kirillov K.A., Noskov M.V. A version of the discrete Haar transform with nodes of $\Pi_0$-grids.
Status: reviewing
Abstract. A version of the two-dimensional discrete Haar transform with $2^D$ nodes forming $\Pi_0$-grid is proposed. This version is associated with the triangular partial sums of Fourier\,--\,Haar series of a given function. The computation of coefficients of the discrete transform is based on cubature formula with $ 2 ^ D $ nodes exact for Haar polynomials of degree at most $ D $, so that all the coefficients $A_{m_1,m_2}^{(j_1, j_2)}$ of the constructed transform are equal to the corresponding coefficients of Fourier\,--\,Haar series $c_{m_1, m_2}^{(j_1, j_2)}$ for Haar polynomials of degree at most $D-\max \{m_1, m_2 \}$ \ ($ 0 \leqslant m_1 + m_2 \leqslant d $, where $ d \leqslant D $). The standard two-dimensional discrete Haar transform with $ 2 ^ D $ nodes does not possess this property.
Date of submission: 20 December 2011 г.
- Аbylayeva А.М., Baiarystanov А.О. CRITERION OF COMPACTNESS FOR INFINITESIMAL ORDER FRACTIONAL INTEGRATION OPERATOR
Status: reviewing
Abstract.
Date of submission: 23 December 2011 г.
- Kozhevnikova L.M., Leontiev A.A. Estimates of solutions of anisotropic doubly nonlinear parabolic equation.
Status: accepted, planed in в т.4 №4
Abstract. This work is devoted by some class of parabolic equations with double nonlinearity whose representative by a model equation $$(|u|^{k-2}u)_t=\sum_{\alpha=1}^n(|u_{x_{\alpha}}|^{p_{\alpha}-2}u_{x_{\alpha}})_{x_\alpha},\quad p_n\geq \ldots \geq p_1>k,\quad k\in(1,2).$$ For solution of the first mixed problem in a cylindrical domain $ D=(0,\infty)$ $\times\Omega, \;\Omega\subset \mathbb{R}_n,\;n\geq 2$ with homogeneous Dirichlet boundary condition and finite initial function established precise estimates the rate of decay as $t\rightarrow\infty$. Earlier these results were obtained by the authors for $k\geq 2$. The case $k\in(1,2)$ is different by method of constructing Galerkin's approximations, which for an isotropic model equation was proposed by E.R. Andriyanova and F.Kh.Mukminov.
Date of submission: 23 December 2011 г.
- Galikhanov I.F., Pavlenko V.N. Periodic solutions telegraph equation with
Status: accepted, planed in в т.4 №4
Abstract. We consider telegraph equations with discontinuous by phase variable inner energy and homogeneous Dirichlet boundary condition. Question of existence of general periodic solutions in the resonant case, when operator created by linear part of the equation with homogeneous Dirichlet boundary condition and condition of periodicity has non zero kernel, and nonlinearity appearing in the equation is limited. Topological method obtained an existence theorem for general periodic solution. The proof is based on the principle of Leray-Schauder for convex compact mappings. The main difference from similar results of other authors - an assumption breaks by phase variable inner energy in telegraph equation.
Date of submission: 24 December 2011 г.
- Saks R.S.
Status: reviewing
Abstract.
Date of submission: 26 December 2011 г.
- Zakirova Z.Kh. On one special solution of the Eisenhart equation.
Status: reviewing
Abstract. In this note we find a 6-dimensional $h$-spaces of the \noindent $[(21\ldots1)(21\ldots1)\ldots(1\ldots1)]$ type and then determine quadratic first integrals of the geodesic equations of these $h$-spaces.
Date of submission: 27 December 2011 г.
- Vasilyev A.V., Vasilyev V.B. Approximate solutions of multi-dimensional singular integral equations and their fast algorithms
Status: reviewing
Abstract. In the work the error estimate for continuous singular integral and the discrete ones in multi-dimensional space is obtained. One suggests to use fast Fourier transform for finding approximate solutions for equations with such operators.
Date of submission: 30 December 2011 г.
- Matyoqubov M.M., Yakhshimuratov A.B. Integrating the higher Korteweg-de Vries equation with a self-consistent source in the class of periodic functions
Status: accepted, planed in в т.4 №4
Abstract. In this work we use the inverse spectral problem method for integrating the higher Korteweg-de Vries equation with a self-consistent source in the class of periodic functions
Date of submission: 30 December 2011 г.
- Gapechkina E.V., Nasyrov F.S. On solutions of the first-order PDE with multidimensional symmetric integral and their modelling
Status: reviewing
Abstract. The deterministic analog of multidimensional Stratonovich integral is constructed. Method of solution of equations system with multidimensional symmetric integral is elaborated. The method of characteristics for solving of Cauchy problem for first-order partial differential equations with multidimensional symmetric integral is developed. This method reduces the solving of initial-value problem of above-mentioned equations to solution of equation system with multidimensional symmetric integral.
Date of submission: 10 January 2012 г.
- Khabirov S.V. Reductions of the partial invariant solutions
Status: reviewing
Abstract.
Date of submission: 10 January 2012 г.
- Yulmukhametova Yu.V. The radial expansion of the gas from the vortex.
Status: accepted, planed in в т.4 №3
Abstract. In this paper we consider a submodel of the gas with a linear velocity field. It forms a system of nonlinear differential equations of order 25 with initial data. More than one of the first integrals of the system. As a result the order of the system is reduced to 19 order. For special initial data of the approximate solution of differential equations of the submodel. Such solutions correspond to world lines describing the radial expansion of the gas particles from the vortex. Constructed trajectories of gas particles.
Date of submission: 13 January 2012 г.
- Meshkov A.G., Sokolov V.V. Integrable evolution equations with the constant separant.
Status: accepted, planed in в т.4 №3
Abstract. The survey contains results of classification for integrable one-field evolution equations of orders 2, 3 and 5 with the constant separant. The classification is based on neccesary integrability conditions that follow from the existence of the formal recursion operator for integrable equations. Recursion formulas for the whole infinite sequence of these conditions are presented for the first time. The most of the classification statements can be found in papers by S.I. Svinilupov and V.V. Sokolov but the proofs never been published before. The result concerning the fifth order equations is stronger then obtained before.
Date of submission: 20 January 2012 г.
- Broyan M.F., Khachatryan K.A.
Status: reviewing
Abstract.
Date of submission: 25 January 2012 г.
- Khats' R.V., Vynnyts'kyi B.V. Completeness and minimality of systems of Bessel functions
Status: reviewing
Abstract. We find the necessary and sufficient conditions for the completeness and minimality in the space $L^2(0;1)$ of system $(\sqrt{x\rho_k}J_{\nu}(x\rho_k):k\in\Bbb N)$ generated by Bessel function of the first kind of index $\nu\ge -1/2$. Moreover, we establish a criterion for the completeness and minimality of system $(x^{-2}\sqrt{x\rho_k}J_{3/2}(x\rho_k):k\in\Bbb N)$ in the space $L^2((0;1);x^2 dx)$.
Date of submission: 30 January 2012 г.
- Potapov D.K.
Status: reviewing
Abstract.
Date of submission: 04 February 2012 г.
- Asymptotic analysis of the surfing acceleration model
Status: reviewing
Abstract. A mathematical model of acceleration of the charge particles by the electromagnetic waves is under study. Averaging equations which describe the resonance interaction of the particle with the electromagnetic wave is obtained. We show that each particle leaves the resonance sone under growth of the time. The time length of the stay in resonance depends on the initial data and one is calculated.
Date of submission: 06 February 2012 г.
- Tikhov M.S. Nonparametric estimates of the effective dose at quantal response.
Status: reviewing
Abstract. For the binary response model we propose a new direct method for the nonparametric estimation of the effective dose level $ED_{100\lambda}$ ($0 < \lambda < 1$). This method yields a simple and reliable monotone estimate of the effective dose level curve $\lambda\to ED_{100\lambda}$ and is appealing to users of conventional smoothing methods as kernel estimators. Moreover it is computationally very efficient, because it does not require a numerical inversion of a monotonized estimate of the quantile dose response curve. We prove asymptotic normality of this new estimator and compare it with an DNP-estimate.
Date of submission: 16 February 2012 г.
- Jumabayeva A.A., Tleukhanova N.T. ABOUT THE PROBLEM OF MULTIPLIERS IN THE ANISOTROPIC LORENTZ SPACES
Status: reviewing
Abstract. The problem of multipliers defined on the anisotropic Lorentz space is considered. We obtain sufficient conditions on a multiplier $\varphi$ ensuring that it belongs to space $M_{\bar{p}_0\bar{q}_0}^{\bar{p}_1\bar{q}_1}$. These conditions are expressed in terms of anisotropic total variation.
Date of submission: 17 February 2012 г.
- Yumagulov M.G. Localization of Arnold tongues of discrete dynamical systems
Status: reviewing
Abstract. The work is devoted to presenting the method localization of Arnold tongues finite-dimensional dynamical systems with discrete time -- sets the corresponding rational relations parameters of the system is synchronized. Such sets correspond to regions of parameter values, for which the system has cycles of certain periods. Method allows to obtain an approximate representation of Arnold tongues, study their properties in the major and minor resonances.
Date of submission: 26 February 2012 г.
- Baizaev S., Vositova D.A. About solutions of one system partial differential equations with two independent variables
Status: reviewing
Abstract. In paper for linear elliptic and hyperbolic system of the first order with constant coefficients and two independent variables is considered. For this systems is investigated a problem about of general solutions and solutions of sedate growth.
Date of submission: 27 February 2012 г.
- Karamov I.I., Napalkov V.V. The generalized Dunkl operator.
Status: reviewing
Abstract.
Date of submission: 28 February 2012 г.
- Suleimanov B.I. The ``quantum'' linearization of the Painleve equations as the component of theier $L-A$ pairs.
Status: reviewing
Abstract. The procedure of the ``quantum'' linearization of the hamiltonian ordinary differential equations with one degree of freedom is entered. It is offered to be used for the classification of the integrable equations of Painleve type. For the Hamiltonian $H = (p^2+q^2)/2$ and all natural numbers $n $ the new solutions $ \Psi (\hbar, t, x, n) $ of the non-stationary the Shr\"{o}dinger equation are constructed. The solutions tend to zero at $x\to\pm\infty$. On curves $x=q_n (\hbar, t) $, defined by the old Bohr- Zommerfeld rule, the solutions satisfy the relation \linebreak $i\hbar \Psi ' _x\equiv p_n (\hbar, t) \Psi $. In this relation $p_n (\hbar, t) = (q_n (\hbar, t)) ' _t $ is the classical momentum corresponding to the harmonic $q_n (\hbar, t) $.
Date of submission: 01 Mart 2012 г.
- Borisov D.I., Golovina A.M. On the resolvents of periodic operators with distant perturbations.
Status: reviewing
Abstract. We consider an abstract operator satisfying to certain conditions with distant perturbations in an arbitrary periodic domain. We obtain an explicit formula for the resolvent.
Date of submission: 01 Mart 2012 г.
- Bobkov V.E. On the existence of nodal solution of elliptic equations with convex-concave nonlinearity.
Status: reviewing
Abstract. In the bounded connected domain $\Omega \subset \mathbb{R}^N$, $N \geq 1$ with piecewise smooth boundary we consider the Dirichlet boundary value problem for elliptic equation with convex-concave nonlinearity \begin{equation*} \begin{cases} -\Delta_p u = \lambda |u|^{q-2} u + |u|^{\gamma-2} u, \quad x \in \Omega \\ u|_{\partial \Omega} = 0, \end{cases} \end{equation*} where $\Delta_p$ is the $p$-Laplacian and $1< q< p< \gamma < p^*$. The main result is a the proof of the existence of nodal solution of this equation on the nonlocal interval $\lambda \in (-\infty, \lambda_0^*)$, where $\lambda_0^*$ is calculated by variational principle of nonlinear spectral analysis of fibering method.
Date of submission: 05 Mart 2012 г.
- Salimov R.B., Shabalin P.L. About resolvability of a homogeneous problem of Riemann-Hilbert with a countable set of coefficient discontinuities and two - side curling at infinity of the order less than 1/2
Status: reviewing
Abstract. In the present paper we study the homogeneous Riemann--Hilbert boundary value problem in the upper half of complex plane with a countable set of coefficient discontinuities and two - side curling at infinity. In the case, when the problem index has a sedate singularity of an order less 1/2, we obtain general solution. Also we investigate the solvability conditions.
Date of submission: 09 Mart 2012 г.
- Yaremko O.E.
Status: reviewing
Abstract. The analytical solution the problem to continue building unit $ N $-dimensional ball of its values on the inner sphere is finds in article. The classical Poisson formula generalization is obtained. The results of the two-dimensional case were obtained by I. Bavrin by the function theory methods.The solution similar to the extension problem in the case known certain operator expression from given potential on the inner sphere is obtained. New result of the moment problems , making a connection between the moment problems on segment and moment problems on half-axis has been found.
Date of submission: 13 Mart 2012 г.
- Abdulaev O.H., Begimkulov F.H.
Status: reviewing
Abstract.
Date of submission: 28 Mart 2012 г.
- Nasibullin R.G., Tuhvatullina A.M.
Status: reviewing
Abstract.
Date of submission: 30 Mart 2012 г.
- , Generalized functions, asymptotically homogeneous with respect to one--parametric group at the origine.
Status: reviewing
Abstract.
Date of submission: 25 Aprel 2012 г.
- Sakieva A.U. Characteristic Lie ring of the Zhiber-Shabat-Tzitzeica equation
Status: reviewing
Abstract. A complete description of the characteristic Lie ring for the Zhiber-Shabat-Tzitzeica equation is given. For the linear space of multiple commutators of arbitrary order a basis is constructed. It is proved that the characteristic Lie ring is a ring of slow growth.
Date of submission: 25 Aprel 2012 г.
- Krivosheyev A.S., Krivosheyeva O.A. The special interpolation in space of exponential type functions.
Status: reviewing
Abstract. In this paper it is studied the special interpolation problem in space of entire exponential type functions. The conjugate diagrams of these functions belong to a given convex domain in complex plane. It is obtained sufficient conditions for interpolation in the case of arbitrary bounded domain. With a help of this result there are obtained sufficient conditions for existence of basis in invariant subspace of analytic functions. The basis consists of linear combinations of eigen and associated functions for differentiation operator. Exponentials of these functions are divided into relatively small groups. Also it is given a method for dividing of exponentials into relatively small groups. The method is based on inner properties of the exponentials sequence.
Date of submission: 30 Aprel 2012 г.
- , Gadyl'shin R.R. Perturbation of an elliptic operator by a narrow potential in $n$-dimensional domain.
Status: reviewing
Abstract. We study a discrete spectrum of an elliptic operator of the second order in $n$-dimensional domain, $n\geq 2$, perturbed by a potential depending on two parameters, one of the parameters describes the length of support of the potential and inverse of the other corresponds to the magnitude of the potential. We give the relation between these para\-me\-ters, under which the generalized convergence of the perturbed operator to the unperturbed one holds. Under this relation we construct the asymptotics w.r.t. small parameters of the eigenvalues of perturbed operators.
Date of submission: 03 May 2012 г.