Factorization of Overdetermined Systems of Linear Partial Differential Equations with Finite-Dimensional Solution Space

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

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

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.

A gradient method for solving the optimal control problem for the system of telegraph equations

The paper establishes conditions of existence and uniqueness of the bounded solution of a special system of linear partial differential equations of the first order. The system arises in the problem of a finite difference scheme of finding an approximate solution is elaborated.

3 - LINEAR SYSTEMS

In this paper, an application of the Riquer-Thomas-Janet theory is described for the problem of transforming a system of partial differential equations into a passive form, i.e., to a special form which contains explicitly both the equations of the initial system and all their nontrivial differential consequences. This special representation of a system markedly facilitates the subsequent integration of the corresponding differential equations. In this paper, the modern approach to the indicated problem is presented. This is the approach adopted in the Knuth-Bendix procedure [13] for critical-pair/completion and then Buchberger’s algorithm for completion of polynomial ideal bases [13] (or, alternatively, for the construction of Groebner bases for ideals in a differential operator ring [14]). The algorithm of reduction to the passive form for linear system of partial differential equations and its implementation in the algorithmic language REFAL, as well as the capabilities of the corresponding program, are outlined. Examples illustrating the power and efficiency of the system are presented.

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.

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

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.

Предложен балансно-характеристический метод решения систем линейных дифференциальных уравнений в частных производных гиперболического типа, обладающий четвертым порядком аппроксимации на равномерных сетках и вторым порядком и улучшенными дисперсионными свойствами на неравномерных сетках. Метод основан на известной схеме КАБАРЕ, балансные фазы которой модифицированы путем добавления антидисперсионных членов особого вида. Ранее метод, обладающий схожими свойствами, предлагался только для простейшего одномерного линейного уравнения переноса. Приведенная модификация схемы позволяет улучшить дисперсионные свойства переноса сразу всех инвариантов Римана рассматриваемой системы уравнений. Схема бездиссипативна при отключенных процедурах монотонизации и устойчива при числах Куранта CFL ≤ 1. Точность метода и его порядок сходимости продемонстрированы на серии расчетов задачи о переносе волны, промодулированной гауссианом, на последовательности сгущающихся сеток. Предложенный метод планируется использовать в качестве основы для построения схемы КАБАРЕ с улучшенными дисперсионными свойствами для систем нелинейных дифференциальных уравнений. A conservative-characteristic method to solve systems of linear hyperbolic-type partial differential equations is proposed. This method has the fourth order of approximation on uniform grids and the second approximation order and improved dispersion properties on non-uniform grids. The proposed method is based on the well-known CABARET scheme whose conservative phases are modified by adding anti-dispersive terms of a special type. Previously, a method with similar properties was proposed only for the simplest one-dimensional linear advection equation. The modification of the scheme allows us to improve the dispersion properties of the advection for all Riemann invariants of the system of equations under consideration at once. The scheme is non-dissipative when the monotonization procedures are not used and is stable at Courant numbers CFL ≤ 1. The accuracy of the method and its order of convergence are shown in a series of solving the problem of advection of a wave modulated by a Gaussian on a sequence of condensing grids. The proposed method is planned to be used as a basis for constructing a CABARET scheme with improved dispersion properties for systems of nonlinear differential equations.

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.

A new technique for solving the large system of linear algebraic equations associated with implicit differencing of multidimensio nal partial differential equations is presented. The coefficient matrix of the equations is factored, and then approximations to certain terms in the matrix are obtained from series expansions. The resulting system of equations is solved easily. The method is developed and demonstrated using a simple representative two-dimensional equation. Very good results are obtained when one direction is dominant. MPLICIT finite-difference schemes for the solution of multidimensional partial differential equations are usually stable and therefore applicable to a large class of problems. However, they are difficult to implement and may require an excessive amount of computer storage and time. The long computing time arises from the need to solve the large system of linear algebraic equations that result from the differencing. The computing time can be reduced significantly by approximating the coefficient matrix of the linear equations with a matrix that produces a system of equations that are relatively easy to solve. Among such methods are the alternating direction method (ADI)l used by Beam and Warming2 and Stone's strongly implicit method,3 which has been tested by Linetal.4 In this paper, a new technique for solving the large system of linear algebraic equations associated with implict differencing of multidimensional partial differential equations is presented. This method, called the pseudo-elimination method (PE), is shown to be faster than Stone's method for certain problems. The method is directly applicable to linear and linearized nonlinear systems of parabolic or elliptic partial differential equations. In order to discuss the method, a simple linear partial differential equation will be used; however, it should be kept in mind that the PE method is applicable to much more complicated problems. The question of whether the method will work when applied to difficult problems is not addressed. The scope of this paper is limited to presenting the method and illustrating, via a simple problem, that the method has some merit and deserves further study.

In this paper, we consider the controllability and the observability of a networked system of linear hyperbolic partial differential equations with coupled boundary conditions. Using the method of characteristics, we characterize them by the controllability/observability of a discrete-time system which is defined on the boundaries of the PDE system, has low-dimension, and hence the analysis is easy for. We also show the effectiveness of our approach using an simple example.

Differential transform method for solving linear and non-linear systems of partial differential equations