A Smooth Solution to a Linear System of Singular ODEs

3 - LINEAR SYSTEMS

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.

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

It is well known that a model of any electrical circuit can be translated into a system of ordinary differential equations. Usually, the simulation is performed in the time-domain using classical discrete approaches convenient for both linear and non-linear systems. An other method consists in the quantization on the variable magnitude which is based on discrete events. The discrete event system specification formalism is adapted to simulate any electrical circuit described with a system of ordinary differential equations using the system state quantification. In this paper, a solution to make more efficient the magnitude quantification for a system of linear ordinary differential equations is given. The proposed technique is used to simulate a simple three phase coupled power systems and it is compared to other solutions given by the explicit integration using formal language and the time-domain discretization using fixed-step fifth order Runge-Kutta integration method.

An effective modification of the homotopy perturbation method for stiff systems of ordinary differential equations

Consider linear systems of ordinary differential equations \[\frac{{dy}}{{dt}} = A(t)y + F(t),\quad 0 \leqq t \leqq T,\quad,y(0){\text{ given}},\] which contain two different time scales, a slow one and a fast one. In many applications such systems possess special smooth solutions and one would like to compute these smooth solutions using a step size which only resolves the slow time scales in the equation. There are several different strategies to use in order to compute these smooth solutions. One of these strategies is to filter the numerical solution. In this paper we introduce a large class of filtering techniques which combines a digital filter interactively with a linear multistep method. We also introduce various stability concepts and make a thorough investigation of the stability properties of the filtering methods.

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.

In this paper, we define sufficient conditions for the solvability in quadratures of systems of linear ordinary differential equations of higher orders with variable coefficients. The method of variation of constants for equations of higher orders and for systems of equations of first order is generalized to the case of such systems. Results are obtained by reducing the systems under consideration to matrix equations of higher orders studied by methods of the theory of Lie algebras.

Although every iterative scalar linear ordinary differential equation is of maximal symmetry algebra, the situation is different and far more complex for systems of linear ordinary differential equations, and an iterative system of linear equations need not be of maximal symmetry algebra. We illustrate these facts by examples and derive families of vector differential operators whose iterations are all linear systems of equations of maximal symmetry algebra. Some consequences of these results are also discussed.

10 - Applications of Systems of Ordinary Differential Equations

In this study, a comparative analysis of two Runge-Kutta methods; fourth-order Runge-Kutta method and Butcher’s Fifth Order Runge-Kutta method are presented and used to solve systems of first-order linear Ordinary Differential Equations (ODEs). The main interest of this work is to test the accuracy, convergence rate and computational efficiency of these methods by using different numerical problems of ODEs. Empirical conclusions are drawn after close observation of the results presented by the two methods, which further highlights their limitations and enabling researchers to make informed decisions in choosing the appropriate technique for specific systems of ODEs problems.

In this paper, an algorithm for researching the stability of linear systems using a computer is proposed. The stability of systems with delay is reduced to finding the limit conditions for the negatives of the real parts of the zeros of the corresponding quasi-polynomials. Verification in practice of such conditions is possible only in the simplest cases. The algorithm proposed in this paper is based on schemes for approximating linear systems with a delay by used sequence special of systems of linear ordinary differential equations. The equivalence of the exponential stability of systems with delay and of the proposed system of ordinary differential equations is established. This allowed us to build an algorithm for studying the location of non-asymptotic roots of quasi-polynomials, which are implemented on a computer.

Nonperiodical sustained fluctuations of electrical potential are characteristic of the rhythms of cortical electric activities. A nonperiodical negatively or positively damped oscillation would be a general solution of any first order linear system of differential equations. Attention is called to a somewhat specialized type of a system of ordinary linear differential equations, which are used by Vito Volterra (Lecons Sur la Theorie Mathematique de la Lutte Pour la Vie, Paris, Gauthier-Villars, 1931) in his theory of interdevouring species. This system of linear equations leads to nonperiodicalundamped oscillations and admits several plausible interpretations in terms of possible biophysical reactions in the brain. A number of new problems is posed and discussed by this study.

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 Rothamsted soil-carbon turnover model — discrete to continuous form