Elementary 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 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.

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

3 - LINEAR SYSTEMS

In a recent paper, Agarwal and Pituk have considered scalar linear difference equations whose coefficients are asymptotically constant and whose corresponding perturbations are exponentially small. Using the method of generating functions and results from complex analysis, they derived an asymptotic representation for solutions, which was then applied to study the asymptotic behaviour of solutions of certain nonlinear autonomous scalar difference equations near a hyperbolic equilibrium. Here, we first show using standard matrix analysis how their results can be extended to systems of linear difference equations and that the error estimates can be made more precise. Our method also can be extended to weakly nonlinear systems. That result can in turn be used to analyze solutions of some autonomous nonlinear difference systems near an equilibrium.

The article considers second-order system of linear stochastic partial differential equations of hyperbolic type with Goursat boundary conditions. Earlier, in a number of papers, representations of the solution Goursat problem for linear stochastic equations of hyperbolic type in the classical way under the assumption of sufficient smoothness of the coefficients of the terms included in the right-hand side of the equation were obtained. Meanwhile, study of many stochastic applied optimal control problems described by linear or nonlinear second-order stochastic differential equations, in partial derivatives hyperbolic type, the assumptions of sufficient smoothness of these equations are not natural. Proceeding from this, in the considered Goursat problem, in contrast to the known works, the smoothness of the coefficients of the terms in the right-hand side of the equation is not assumed. They are considered only measurable and bounded matrix functions. These assumptions, being natural, allow us to further investigate a wide class of optimal control problems described by systems of second-order stochastic hyperbolic equations. In this work, a stochastic analogue of the Riemann matrix is introduced, an integral representation of the solution of considered boundary value problem in explicit form through the boundary conditions is obtained. An analogue of the Riemann matrix was introduced as a solution of a two-dimensional matrix integral equation of the Volterra type with one-dimensional terms, a number of properties of an analogue of the Riemann matrix were studied.

Controllability and observability of linear time delayed systems have been studied, and various definitions and criteria have been presented since the 1960s. However, the lack of an analytical solution approach has limited the applicability of the existing theories. Recently, the solution to systems of linear delay differential equations has been derived using the matrix Lambert W function, in a form similar to the transition matrix in ordinary differential equations. The criteria for controllability and observability, and their Gramians, for systems of delay differential equations using the solution in terms of the matrix Lambert W function are presented for the first time and illustrated with examples.

Solutions of the system of equations describing non-isothermic flow of real gases in porous media are given in the paper. Approximate solutions were obtained by the method of functional series for stationary flows, taking into account the thermal conductivity of the medium. Accurate solutions were obtained for stationary- flows without considering the thermal conductivity. Solutions are given for plane and plane-radial flows. Moreover, a method of solution is described which involves a consideration of the variation of the coefficient of gas compressibility, z, with pressure, p. Considering non-stationary non-isothermic flow, a method is given for solution with the use of the ‘minor parameter’, reducing the system of non-linear partial differential equations to the system of linear partial differential equations of the parabolic type which determine the coefficients of the solving series. The solutions may be used for the temperature testing of gas wells and production from condensate gas fields. INTRODUCTION THE FLOW OF GAS through a porous medium is determined univocally by the functions p/r, t/, T/r, t/,γ/r, t/ and v/r, t/, where p is pressure, T temperature, γ specific gravity, flow rate and r is the position vector. The distribution in time and space of the above-mentioned parameters may be obtained by solving a system of four equations: (Equation Available In Full Paper) On the whole, the flow of gas is defined in practice assuming that T = const, in a stationary state and with simple geometrical contours of the medium. There also exist several theoretical elaborations of the problems of non-stationary flow with the assumption that T = const.(2,3,4,5). The problem of non-isothermic flow has remained so far unsolved, with the exception of a few investigations. These were undertaken by Czekaluk(1,0) Czarny(4) and Gusein-Zade and Kolosovskaja(7). They referred mainly to the stationary flow of incompressible fluids, assuming that the stationary distribution of pressure in a porous medium is known, omitting the thermal conductivity of the medium. Un-dimensional flows (plane and plane-radial) were considered, with the only equation of energy being solved /3/. Thus, Gusein-Zade and Kolosovskaja(7), start from the equations of energy for linear and plane-radial flow under the form: (Equation Available In Full Paper) Equations /5/ and /6/ are directly obtained from equation /3/, assuming that A = O and ∂p / ∂t = O. Setting the original and boundary conditions in the form: (Equation Available In Full Paper) one seeks for a solution in the form of (Equation Available In Full Paper) As may be seen from the solutions obtained, the area of the temperature field is divided into area I, where x < u °t, and area II, where x > u °t; two different formulae are therefore justified. These considerations are substantial in the case of injecting a fluid into a well. Similarly in the case of plane-radial flow at the initial and boundary conditions in the form of (Equation Available In Full Paper) Solutions given in papers (1) and (4) are based on the same equations.

In this paper we present a technique to study the existence of rational solutions for systems of differential equations — for an ordinary differential equation, in particular. The method is relatively straightforward; it is based on a rationality characterisation that involves matrix Pade approximants. It is important to note that, when the solution is rational, we use formal power series “without taking into account” their circle of convergence; at the end of this paper we justify this. We expound the theory for systems of linear first-order ordinary differential equations in the general case. However, the main ideas are applied in numerical resolution of partial differential equations.

We consider a system of two hybrid vector equations containing linear difference (defined on a discrete set) and functional differential (defined on a half-axis) parts. To study it, a model system of two vector equations is chosen, one of which is linear difference with aftereffect (LDEA), and the other is a linear functional differential with aftereffect (LFDEA). Two equivalent representations of this system are shown: the first representation in the form of LFDEA, the second — in the form of LDEA. This allows us to study the stability issues of the system under consideration using the well-known results on the stability of LFDEA and LDEA. Using the results of the article [Gusarenko S. A. On the stability of a system of two linear differential equations with delayed argument // Boundary value problems. Interuniversity collection of scientific papers. Perm: PPI, 1989. P. 3–9], two examples are shown when a joint system of four equations will be stable with respect to the right side. In the first example, we use the LFDEA for which sufficient conditions for the sign-definiteness of the elements of the 2 Ч 2 Cauchy matrix function are known (in terms of the LFDEA coefficients). In the second example, LFDEA is given such that LFDEA is a system of linear ordinary differential equations (LODE) of the second order. In both cases, estimates of the components of the Cauchy matrix function are known. An exponential estimate with a negative exponent is given for the components of the Cauchy matrix function of LDEA.

The article considers a nonlinear boundary value problem for a linear delay differential equation. To solve the problem, the idea of parametrization method, namely, the interval at which the problem is being considered, is divided into subintervals whose lengths do not exceed the values of the constant delay; constant parameters are introduced at the left ends of these intervals; a new unknown function is introduced at each subinterval. Thus, the problem under consideration is reduced to an equivalent multipoint boundary value problem for differential equations with delay containing parameters. Auxiliary Cauchy problems without delay with zero initial conditions at the left ends of the subintervals are consistently considered on each of the subintervals. Using an analog of the Cauchy formula to represent the solution of a system of linear differential equations on each of the subintervals and given nonlinear boundary conditions, an algebraic system with respect to unknown parameters is obtained. The article proposes an algorithm for finding a solution to a multipoint boundary value problem for differential equations with a delay containing parameters. At each step of the algorithm, a system of nonlinear algebraic equations is solved to determine the values of the parameters, and an analog of the Cauchy formula is used to obtain solutions to auxiliary Cauchy problems. The results obtained are demonstrated on a test problem.

The dynamic models of processes in transport logistics systems are considered. In the literature, such complex systems as the military transport logistics system or the combat vehicle support system are often modeled as a set of typical system states. These states are interconnected by a large number of transitions of a given intensity, which are carried out with given probabilities. Graphically, this is represented using the so-called graph of states and transitions, and the probabilities of the system being in a particular state are the subject of research in such a graph. The methods available in the literature for studying the dynamic characteristics of state graphs and transitions are analyzed. A description of the changes in probabilities as a function of time is made using systems of differential equations, usually linear. Based on practical requirements, approximate solutions to such systems are usually sought. One of the approximate methods is the decomposition method, in which, instead of a system of coupled equations, a set of independent equations is considered, the solution of which is not difficult. The results of the solution have an accuracy satisfactory from the point of view of practical use. The assumptions based on which the decomposition method can be used are analyzed. It is shown that the accuracy of the obtained results substantially depends on the given initial conditions and should increase over time, when this dependence weakens. A method is proposed for the exact solution of a system of differential equations, free of any assumptions. The use of operational calculus is substantiated, which reduces the solution of a system of linear differential equations to the solution of a system of linear algebraic equations for unknown images of the sought-for Laplace functions. The method is used to describe the process of technical support for the restoration of the transport flow of military logistics. The boundaries of the possibility of applying the results of a simpler approximate solution are established.

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.

This paper considers a system of uncertain linear differential equations under the Neutrosophic uncertain environment. Suppose the mutual dependency of the dynamics of two variables is set on the mathematical manipulation under the scenario where the available information regarding initial states is imprecise and is given in the Neutrosophic sense of uncertainty. The concept of Neutrosophic differential equations can play a very effective role in this regard. The present chapter is engaged with a brief introduction of the theory of the system of Neutrosophic differential equations and the possible solution approaches. Combining different cases of Neutrosophic differentiability and the signs of the coefficients involved in the differential equations are taken for the synergetic study of the theory. At the end of the theory, a few physical phenomena are explored as possible applications of the proposed theory. For a better understanding and reliability of the proposed theory, numerical simulations and graphical visualizations are added to different pockets of this paper.