Reduction and noncommutative integration of linear differential equations

A new method is proposed for derivation of exactly integrable linear differential equations based on the theory of noncommutative integration. The equations are obtained by reduction from original equations which are integrable in the noncommutative sense, with a large number of independent variables. It is shown that the reduced equations cannot be solved by traditional methods, since they do not possess the required algebraic symmetry.V. V. Kuibyshev Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 55–60, November, 1993.     

Representations of Lie algebras and the problem of noncommutative integrability of linear differential equations

An algorithm is proposed for integrating linear partial differential equations with the help of a special set of noncommuting linear differential operators — an analogue of the method of noncommutative integration of finite-dimensional Hamiltonian systems. The algorithm allows one to construct a parametric family of solutions of an equation satisfying the requirement of completeness. The case is considered when the noncommutative set of operators form a Lie algebra. An essential element of the algorithm is the representation of this algebra by linear differential operators in the space of parameters. A connection is indicated of the given method with the method of separation of variables, and also with problems of the theory of representations of Lie algebras. Let us emphasize that on the whole the proposed algorithm differs from the method of separation of variables, in which sets of commuting symmetry operators are used.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 95–100, April, 1991.     

Quadratic algebras and noncommutative integration of Klein-Gordon equations in non-steckel Riemann spaces

The method of noncommutative integration of linear partial differential equations is used to solve the Klein-Gordon equations in Riemann space, in the case when the set of noncommutating symmetry operators of this equation for a quadratic algebra consists of one second-order operator and several first-order operators. Solutions that do not permit variable separation are presented.Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 83–87, May, 1995.     

Nonlinear differential equations and lie superalgebras

It is shown how the theory of nonlinear ordinary differential equations with superposition formulas can be generalized to the case of superequations involving anticommuting Grassmann variables. As an example, equations based on the osp(1, 2) superalgebra are analyzed in detail. They turn out to be super-Riccati equations and a super-superposition formula is obtained for their solutions.

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Résumé Nous montrons comment la théorie des équations différentielles ordinaires non-linéaires admettant des lois de superposition peut être généralisée au cas de super-équations contenant des variables de Grassmann anticommutantes. Comme exemple les équations basées sur la super-algèbre osp(1, 2) sont analysées en détail. Nous obtenons des super-équations de Riccati et leurs formules de superposition sont mises en évidence.

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Chargé de recherches du FNRS.     

Numerical integration of stochastic differential equations

The stability of incompressible turbulent fluids with respect to weak mean flow perturbations is discussed. It is shown that for a statistically homogeneous, isotropic, and stationary model such perturbations will decay. This is in marked contrast to the compressible case.     

Exponential stability of linear impulsive differential equations

The notion ofexponential stability for linear impulsive differential equations at fixed moments is made precise.     

Narrow bounds for numerical integration of differential equations

Through a detailed analysis of the properties of a system of differential equations, bounds are given for the error affecting the final result of a numerical integration. These bounds appear to be narrower than those obtained with other methods. The key procedure is to consider carefully the linear part of the system and to bound it taking account of all possible errors. No very significant restriction is made on the system.This work was partially supported by the Ministero della Pubblica Istruzione.     

Classification of degenerate solutions of linear differential equations

An algorithm of obtaining partial solutions of linear differential equations in partial derivatives that admit certain nontrivial symmetry algebra but are nonintegrable by the standard methods is described. The notion of degenerate solution is introduced. Natural classification of solutions is suggested.     

Nonlinear fluctuations-induced rate equations for linear birth-death processes

The Fock-space approach to the solution of master equations for one-step Markov processes is reconsidered. It is shown that in birth-death processes with an absorbing state at the bottom of the occupation-number spectrum and occupation-number independent annihilation probability of occupation-number fluctuations give rise to rate equations drastically different from the polynomial form typical of birth-death processes. The fluctuation-induced rate equations with the characteristic exponential terms are derived for Mikhailov’s ecological model and Lanchester’s model of modern warfare. The text was submitted by the author in English.     

A nonlocal Poisson bracket of the sine-Gordon model

It is well known that the classical string on a two-sphere is more or less equivalent to the sine-Gordon model. We consider the non-abelian dual of the classical string on a two-sphere. We show that there is a projection map from the phase space of this model to the phase space of the sine-Gordon model. The corresponding Poisson structure of the sine-Gordon model is nonlocal with one integration.     

Relativistic kinetic momentum operators, half-rapidities and noncommutative differential calculus

It is shown that the generating function for the matrix elements of irreps of Lorentz group is the common eigenfunction of the interior derivatives of the noncommutative differential calculus over the commutative algebra generated by the coordinate functions in the Relativistic Configuration Space (RCS). These derivatives commute and can be interpreted as the quantum mechanical operators of the relativistic momentum corresponding to the half of the non-Euclidean distance in the Lobachevsky momentum space (the mass shell).     

Differential homomorphisms of linear homogeneous systems of differential equations

Linear differential operators with complex-valued infinitely differentiable coefficients, linear homogeneous systems of differential equations, and modules over algebras of scalar linear differential operators are considered. Linear differential changes of variables and homomorphisms of special quotient modules (differential homomorphisms) generated by these changes are studied. In terms of differential homomorphisms, relationships between Maxwell equations and equations of electromagnetic potential and between Dirac equations and the Klein-Gordon system of independent equations are described. It is proved that all ordinary nondegenerate linear homogeneous differential equations of some common order and the homogeneous normal systems of the same common order are differentially isomorphic.     

On noncommutative minisuperspace and the Friedmann equations

In this Letter we present a noncommutative version of scalar field cosmology. We find the noncommutative Friedmann equations as well as the noncommutative Klein–Gordon equation, interestingly the noncommutative contributions are only present up to second order in the noncommutative parameter. Finally we conclude that if we want a noncommutative minisuperspace with a constant noncommutative parameter as viable phenomenological model, the noncommutative parameter has to be very small.     

Stochastic differential calculus,the Moyal *-product,and noncommutative geometry

A reformulation of the Itô calculus of stochastic differentials is presented in terms of a differential calculus in the sense of noncommutative geometry (with an exterior derivative operator d satisfying d² = 0 and the Leibniz rule). In this calculus, differentials do not commute with functions. The relation between both types of differential calculi is mediated by a generalized Moyal *-product. In contrast to the Itô calculus, the new framework naturally incorporates analogues of higher-order differential forms. A first step is made towards an understanding of their stochastic meaning.     

Kinematical similarity and exponential dichotomy of linear abstract impulsive differential equations

The notions of kinematical similarity and exponential dichotomy for linear abstract differential equations are extended to impulsive equations. The fundamental properties of these notions for Banach and Hilbert spaces are investigated.     

Dichotomies for linear impulsive differential equations with variable structure

The notions ofordinary andexponential dichotomy for linear impulsive differential equations are made precise.     

线性常微分方程的保线性变换及其应用

研究了线性常微分方程的保线性变换，得到任意两个二阶线性常微分方程等价的条件，并用于求解一类二阶线性变系数齐次常微分方程．对数学物理方法教学中怎样通过适当的变换把给定的二阶线性变系数齐次常微分方程化为可解的方程给出了合理解释。     

Multi-component hierarchies of double bracket equations

A new integrable hierarchy, with equations defined by double brackets of two matrix pseudo-differential operators (Lax pairs), is constructed. Some algebraic properties are demonstrated. It is also shown that each equation is equivalent to a certain gradient flow. A new version of the Zakharov-Shabat type equations is proved. Formal solutions of this hierarchy are constructed using a matrix “double bracket bilinear identity”.     

Relativistic differential-difference momentum operators and noncommutative differential calculus

The relativistic kinetic momentum operators are introduced in the framework of the Quantum Mechanics (QM) in the Relativistic Configuration Space (RCS). These operators correspond to the half of the non-Euclidean distance in the Lobachevsky momentum space. In terms of kinetic momentum operators the relativistic kinetic energy is separated as the independent term of the total Hamiltonian. This relativistic kinetic energy term is not distinguishing in form from its nonrelativistic counterpart. The role of the plane wave (wave function of the motion with definite value of momentum and energy) plays the generating function for the matrix elements of the unitary irreps of Lorentz group (generalized Jacobi polynomials). The kinetic momentum operators are the interior derivatives in the framework of the noncommutative differential calculus over the commutative algebra generated by the coordinate functions over the RCS.     

Covolume-upwind finite volume approximations for linear elliptic partial differential equations

In this paper, we study covolume-upwind finite volume methods on rectangular meshes for solving linear elliptic partial differential equations with mixed boundary conditions. To avoid non-physical numerical oscillations for convection-dominated problems, nonstandard control volumes (covolumes) are generated based on local Peclet’s numbers and the upwind principle for finite volume approximations. Two types of discretization schemes with mass lumping are developed with use of bilinear or biquadratic basis functions as the trial space respectively. Some stability analyses of the schemes are presented for the model problem with constant coefficients. Various examples are also carried out to numerically demonstrate stability and optimal convergence of the proposed methods.