CENTRAL EXPONENTS OF LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS

This paper introduces the concept of first-order systems of linear fuzzy differential equations for \(\mathcal {S}\)-linearly correlated fuzzy processes. These fuzzy processes have range embedded in Banach spaces of fuzzy numbers, and the fuzzy initial value problems studied are given in terms of the Fréchet derivative of these fuzzy functions. An equivalence between the first-order systems of linear fuzzy differential equations and a family of classical first-order systems of linear differential equations is established. Also, conditions on the existence and uniqueness of the solutions are presented. Lastly, an application on the multiple mass-spring system is provided.KeywordsBanach spacesStrongly linearly independence\(\mathcal {S}-\)linearly correlated fuzzy processesSystems of fuzzy differential equationsMass-spring multiple system

The Rothamsted soil-carbon turnover model — discrete to continuous form

3 - LINEAR SYSTEMS

X Sheffer, Systems of linear differential equations of infinite order, with constant coefficients, to appear in the Annals of Mathematics.We shall refer to this as Annals Paper I.% We can equally well consider the system Y.Ati

The purpose of this paper is to derive a means of scaling the dependent variables of a system of linear differential equations so that the normalized system accurately reflects the dependence of the system on initial data. The theory is developed for the special case of a system of first order hyperbolic partial differential equations in two independent variables. Several other systems of differential equations, to which the same normalization could be applied, are also indicated. The scaling is equivalent to row and column scaling commonly used in linear algebra. For the case of positive matrices (that is, for the case where the diagonalizing matrix and its inverse have no zero elements), the theory for the “1” and “$\infty $ ” norms has been thoroughly developed by Bauer [1]. The assumption of positive matrices is too stringent for some applications. Thus, the theory is generalized to show that for irreducible matrices a best scaling exists (for both the “1” and “ $\infty $” norms), while for simply reducible matrices a best scaling exists for at least one of these two norms. Several examples are given.

The questions of instant stability of systems of linear differential equations Ito with delays based on the theory of positively reversible matrices are investigated. The ideas and methods developed by N. V. Azbelev and his students to investigate the sustainability of deterministic linear functional--differential equations are used for this. Are brought sufficient conditions $2p$--stability $(1\le p <\infty)$ systems of linear differential Ito equations with delays in terms of positive reversibility of the matrices, built according to the parameters of the source system. The fulfillment of these conditions for specific equations is checked.

We consider boundary value problems with periodic boundary conditions for first-order linear systems of impulsive ordinary differential equations with unknown right-hand sides and jumps of solutions at the impulse points entering into the statement of these problems which are assumed to be subjected to some quadratic restrictions. From indirect noisy observations of their solutions on a finite system of intervals, we obtain the optimal, in certain sense, estimates of images of their right-hand sides under linear continuous operators. Under the condition that the unknown correlation functions of noises in observations belong to some special sets, it is established that such estimates and estimation errors are expressed explicitly via solutions of special periodic boundary value problems for linear impulsive systems of ordinary differential equations.

The paper introduces the concepts of quasi-periodic and multiperiodic functions. Floquet's theorem is formulated for a system of linear differential equations with periodic coefficients. The problem of finding sufficient conditions for stability and instability of a system of linear differential equations is described. The points of the program of work on the stability analysis of linear systems of differential equations with quasiperiodic and multiperiodic coefficients are given.

Systems of linear differential equations of the type occurring in the theory of radioactive transformations are normally solved by direct integration. For certain complex systems solutions may contain a formidable number of terms and therefore would be extremely laborious to derive. This paper presents a more practical method for solving such systems of equations. By means of a recurrence formula the method provides simple, compact, readily derivable expressions, and considerably reduces the length of numerical computation. The method is well suited for use with digital computers.

We establish sufficient algebraic coefficient conditions for the asymptotic stability of solutions of systems of linear difference equations with continuous time and delay in the case of a rational correlation between delays. We use (n2 + m)-parameter Lyapunov functions (n is the dimension of the system of equations and m is the number of delays).

An approach to solutions of systems of linear partial differential equations with applications

We give a generalization, called a generalized Okubo system, of a system of linear differential equations of the Okubo normal form and define a mapping π from a set of generalized Okubo systems to a set of linear differential systems. We consider the operation, the middle convolution introduced by Katz, using π, and show that any system of linear differential equations, not necessarily of the Fuchsian type, with a regular singularity at infinity, can be transformed into the generalized Okubo system.

The large moment method can be used to compute a large number of moments of physical quantities that are described by coupled systems of linear differential equations. Besides these systems the algorithm requires a certain number of initial values as input, that are often hard to derive in a preprocessing step. Thus a major challenge is to keep the number of initial values as small as possible. We present the basic ideas of the underlying large moment method and present refined versions that reduce significantly the number of required initial values.

J. Bourgain and H. Brezis have obtained in 2002 some new and surprising estimates for systems of linear differential equations, dealing with the endpoint case L¹ of singular integral estimates and the critical Sobolev space W1,n(Rn). This paper presents an overview of the results, further developments over the last ten years and challenging open problems.

We prove that there exists an open set of irreducible systems in the space of systems of linear differential equations with quasiperiodic skew-adjoint matrices and fixed frequency module.