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We study a semilinear differential-algebraic equation (DAE) with the focus on the Lagrange stability (instability). The conditions for the existence and uniqueness of global solutions (a solution exists on an infinite interval) of the Cauchy problem, as well as conditions of the boundedness of the global solutions, are obtained. Furthermore, the ob...
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Mathematical modelling of transient current in an Electrical Oscillatory System, a case of Resistive, Inductive and a Capacitive (RLC) circuit is presented in this research work. The governing equation was formulated from the RLC circuit with the use of Kirchoffs Voltage law (KVL) which is a second order differential equation. The model was then so...
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... Below we give definitions [10,24] that will be needed to formulate further results. ...
Two combined numerical methods for solving time-varying semilinear differential-algebraic equations (DAEs) are obtained. These equations are also called degenerate DEs, descriptor systems, operator-differential equations and DEs on manifolds. The convergence and correctness of the methods are proved. When constructing methods we use, in particular, time-varying spectral projectors which can be numerically found. This enables to numerically solve and analyze the considered DAE in the original form without additional analytical transformations. To improve the accuracy of the second method, recalculation (a ``predictor-corrector'' scheme) is used. Note that the developed methods are applicable to the DAEs with the continuous nonlinear part which may not be continuously differentiable in $t$, and that the restrictions of the type of the global Lipschitz condition, including the global condition of contractivity, are not used in the theorems on the global solvability of the DAEs and on the convergence of the numerical methods. This enables to use the developed methods for the numerical solution of more general classes of mathematical models. For example, the functions of currents and voltages in electric circuits may not be differentiable or may be approximated by nondifferentiable functions. Presented conditions for the global solvability of the DAEs ensure the existence of an unique exact global solution for the corresponding initial value problem, which enables to compute approximate solutions on any given time interval (provided that the conditions of theorems or remarks on the convergence of the methods are fulfilled). In the paper, the numerical analysis of the mathematical model for a certain electrical circuit, which demonstrates the application of the presented theorems and numerical methods, is carried out.
... Various conditions for the global solvability of time-invariant semilinear differential-algebraic equations with a regular characteristic pencil are obtained in [5,8,9]. The papers [10] and [11] established Lagrange stability conditions, which also include global solvability conditions, for time-invariant semilinear differential-algebraic equations with a regular and singular (nonregular) characteristic pencils, respectively. ...
... Unlike the theorems on the global solvability and Lyapunov stability (asymptotic stability) in [3], the theorems obtained in the present paper do not require that the differential-algebraic equations in question be regular differential-algebraic equations of tractability index 1, i.e., that the pencil λA(t) + B(t) − ∂f (t, x)/∂x be a regular pencil of index 1 (for a more refined comparison, see the definition of tractability index for regular differential-algebraic equations in [3, pp. 91, 319-320] and the comments regarding its relation to other concepts of index given in [3] and [10,Sec. 2]), but require that the pencil λA(t) + B(t) corresponding to the linear part (left-hand side) of the differential-algebraic equations in question be a regular pencil of index not higher than 1 (see Sec. 1 for the definition). ...
... The conditions of global solvability (including the uniqueness of global solutions) for autonomous degenerate DEs of the form (1.1) ( , are time-invariant) were obtained by the author in [6] and, together with A.G. Rutkas, in [21] in the case of the regular characteristic pencil + , and in [9] in the case of the nonregular (singular) characteristic pencil. The conditions of Lagrange stability for autonomous degenerate DEs of the form (1.1) were obtained by the author in [8,9]. In [7], two combined numerical methods were presented and it was verified that the results of applying the theorems from [8] were consistent with the results of numerical experiments. ...
... The conditions of Lagrange stability for autonomous degenerate DEs of the form (1.1) were obtained by the author in [8,9]. In [7], two combined numerical methods were presented and it was verified that the results of applying the theorems from [8] were consistent with the results of numerical experiments. In the present paper, the nonautonomous degenerate DEs (1.1) and (1.3) with characteristic pencils regular for every are studied. ...
... A solution ( ) of (1.1), (1.2) is called Lagrange unstable if it has a finite escape time [8]. ...
... In Ascher and Petzold (1998), Brenan et al. (1996), Kunkel and Mehrmann (2006), Hairer et al. (1989), Hairer and Wanner (2010), and Knorrenschild (1992), restrictions similar to the above requirement of index 1 for a regular DAE are used locally to prove the local solvability of DAEs. Various notions of an index for a regular DAE and the relationship between them are discussed in [Filipkovska (2018), Remark 2.1]. In what follows, for the sake of generality, the equation (1) with an arbitrary (not necessarily degenerate) linear operator A : R n → R n will be called a semilinear DAE. ...
... In the general case, according to [Vlasenko (2006), Section 6.2], the maximum length of the chain of an eigenvector and adjoint vectors of the matrix pencil A + µB at the point µ = 0 is called the index of the matrix pencil λA + B. Various notions of an index of the pencil, an index of a DAE and their relationship with the mentioned notion of the pencil of index 1 are considered in [Filipkovska (2018), Remark 2.1]. ...
... Definition 3.1: (Rutkas and Filipkovska (2013a), Filipkovska (2018)) An operator function (a mapping) Φ : D → L(W, Z), where W , Z are s-dimensional linear spaces and D ⊂ W , is called basis invertible on an interval [ŵ,w], whereŵ,w ∈ D, if for some additive resolution of the identity {Θ k } s k=1 in the space Z (see [Filipkovska (2018), Definition 2.2] or [Rutkas and Filipkovska (2013a), Definition 2]) and for any set ...
... In [6,10,[15][16][17][18], restrictions similar to the above requirement of index 1 for a regular DAE are used locally to prove the local solvability of DAEs. Various notions of an index for a regular DAE and the relationship between them are discussed in [19,Remark 2.1,section 2]. ...
... In Section 2, we consider the restriction on the operator coefficients of the equation (1) (on the operator pencil) and give the corresponding definition of a regular pencil of index not higher than 1; also, we consider the method of spectral projectors for the reduction of the semilinear DAE to an equivalent semi-explicit form. In Section 3, the theorems proved in earlier works [19,29], which give conditions for the existence and uniqueness of exact global solutions, are presented. In Section 4, the two combined numerical methods for solving the semilinear DAE are obtained and their convergence is proved (note that the method from Subsection 4.1 was proposed by the author in [30] without the theorem on its convergence), as well as the important remarks on the convergence of the methods, when weakening the smoothness requirements for the nonlinear function, are given. ...
... Various notions of an index of the pencil, an index of a DAE and their relationship with the mentioned notion of the pencil of index 1 are considered in [19,Remark 2.1.]. ...
Two combined numerical methods for solving semilinear differential-algebraic equations (DAEs) are obtained and their convergence is proved. The comparative analysis of these methods is carried out and conclusions about the effectiveness of their application in various situations are made. In comparison with other known methods, the obtained methods require weaker restrictions for the nonlinear part of the DAE. Also, the obtained methods enable to compute approximate solutions of the DAEs on any given time interval and, therefore, enable to carry out the numerical analysis of global dynamics of mathematical models described by the DAEs. The examples demonstrating the capabilities of the developed methods are provided. To construct the methods we use the spectral projectors, Taylor expansions and finite differences. Since the used spectral projectors can be easily computed, to apply the methods it is not necessary to carry out additional analytical transformations.
