Class Of Differential-algebraic Equations Research Articles

Geometric behavior of a class of algebraic differential equations in a complex domain using a majorization concept

In this paper, a type of complex algebraic differential equations (CADEs) is considered formulating by $ \alpha [\varphi(z) \varphi (z) +(\varphi' (z))^2]+ a_m \varphi^m(z)+a_{m-1} \varphi^{m-1}(z)+...+ a_1 \varphi(z)+ a_0 = 0. $ The conformal analysis (angle-preserving) of the CADEs is investigated. We present sufficient conditions to obtain analytic solutions of the CADEs. We show that these solutions are subordinated to analytic convex functions in terms of $e^z.$ Moreover, we investigate the connection estimates (coefficient bounds) of CADEs by employing the majorization method. We achieve that the coefficients bound are optimized by Bernoulli numbers.

Finite-Order Entire Solutions of a Class of Algebraic Differential Equations

We study the properties of finite-order entire solutions of a class of algebraic differential equations. It is shown that, under certain conditions, all such solutions are solutions of linear differential equations whose coefficients are polynomials in the independent complex variable.

A note on differential-algebraic systems with impulsive and hysteresis phenomena

n this note, we single out some promising classes of differential-algebraic equations (DAEs) with hysteresis phenomena, and propose their meaningful generalizations. We consider D Es of index 2 having two features: i) non-linearity of hysteresis type modeled by a sweeping process, and ii) impulsive control represented by a bounded signed Borel measure. For such a DAE we design an equivalent structural form, based on the Kronecker-Weierstrass transformation, and prove a necessary and sufficient condition for the existence and uniqueness of a solution to an initial value problem. We propose a notion of generalized solution to a DAE as a realization of impulsive trajectory relaxation. This relaxation is described by a dynamical system with states of bounded variation and can be equivalently represented as a system of “ordinary” DAEs.

A study on algebraic differential equations of Gamma function and Dirichlet series

The paper concerns the problem of algebraic differential independence between the gamma function and the function in a certain class F, which contains Dirichlet L-functions, L-functions in the extended Selberg class and some periodic functions. It is proved that the gamma function and the function in F cannot satisfy a class of algebraic differential equations with meromorphic coefficients ϕ having Nevanlinna characteristic satisfying T(r,ϕ)=o(r) as r→∞.

Second‐order sliding mode observers for fault reconstruction in power networks

This study proposes a two-sliding mode (SM) observer to detect and reconstruct a certain class of load altering faults in a power network. The observer design is based on the recently proposed multivariable super-twisting structure. The IEEE benchmark power networks used to test the scheme are modelled as a semi-explicit class of differential algebraic equations (DAEs). For the purpose of developing the detection scheme, only the phase angles of the generators are measured, which represent a subset of the differential states of the DAEs. The objective is to estimate the differential states (the phase angles and frequencies of the generators), the algebraic states (the phase angles of the load bus tensions) and to reconstruct a class of load altering faults affecting the network. The proposed observer is assessed in simulation on two IEEE benchmarks: the 9-bus and 14-bus networks, so as to verify its capability to correctly estimate the differential and algebraic states of the network in spite of its complexity and uncertainty. Moreover, the capability of the proposed scheme to detect the presence of a load altering fault, to exactly identify its position in the network, and to precisely reconstruct the shape of the fault itself is shown and discussed.

ON ALGEBRAIC DIFFERENTIAL EQUATIONS FOR THE GAMMA FUNCTION AND -FUNCTIONS IN THE EXTENDED SELBERG CLASS

This paper concerns the problem of algebraic differential independence of the gamma function and ${\mathcal{L}}$-functions in the extended Selberg class. We prove that the two kinds of functions cannot satisfy a class of algebraic differential equations with functional coefficients that are linked to the zeros of the ${\mathcal{L}}$-function in a domain $D:=\{z:0<\text{Re}\,z<\unicode[STIX]{x1D70E}_{0}\}$ for a positive constant $\unicode[STIX]{x1D70E}_{0}$.

Numerical Methods for a Class of Differential Algebraic Equations

This paper is devoted to the study of some efficient numerical methods for the differential algebraic equations (DAEs). At first, we propose a finite algorithm to compute the Drazin inverse of the time varying DAEs. Numerical experiments are presented by Drazin inverse and Radau IIA method, which illustrate that the precision of the Drazin inverse method is higher than the Radau IIA method. Then, Drazin inverse, Radau IIA, and Padé approximation are applied to the constant coefficient DAEs, respectively. Numerical results demonstrate that the Padé approximation is powerful for solving constant coefficient DAEs.

Algebraic differential equations with functional coefficients concerning ζ and Γ

We prove that the Riemann zeta function and the Euler gamma function cannot satisfy a class of algebraic differential equations with functional coefficients that are connected to the zeros of the Riemann zeta function on the critical line.

A Symbolic–Numerical Method for Integration of DAEs Based on Geometric Control Theory

This work describes a symbolic–numerical integration method for a class of differential algebraic equations (DAEs) known as semi-explicit systems. Our method relies on geometric theory of decoupling for nonlinear systems combined with efficient numerical analysis techniques. It uses an algorithm that applies symbolic and numerical calculations to build an explicit vector field $$\tau $$ , whose integral curves with compatible initial conditions are the same solutions of the original DAE. Here, compatible initial conditions are the ones that respect the algebraic restrictions and their derivatives up to their relative degree. This extended set of restrictions defines a submanifold $$\varGamma $$ of the whole space, which is formed by all the variables of the system, and all solutions of the DAE lie on this submanifold. Furthermore, even for nonexactly compatible initial conditions, the solutions of this explicit system defined by $$\tau $$ converge exponentially to $$\varGamma $$ . Under mild assumptions, an approximation result shows that the precision of the method is essentially controlled by the distance of the initial condition from $$\varGamma $$ . A scheme to compute compatible initial conditions with the DAE is also provided. Finally, simulations with benchmarks and comparisons with other available methods show that this is a suitable alternative for these problems, specially for nonexactly compatible initial conditions or high-index problems.

Bohl exponent for time-varying linear differential-algebraic equations

We study stability of linear time-varying differential-algebraic equations (DAEs). The Bohl exponent is introduced and finiteness of the Bohl exponent is characterised, the equivalence of exponential stability and a negative Bohl exponent is shown and shift properties are derived. We also show that the Bohl exponent is invariant under the set of Bohl transformations. For the class of DAEs which possess a transition matrix introduced in this article, the Bohl exponent is exploited to characterise boundedness of solutions of a Cauchy problem and robustness of exponential stability.

Corrigendum to On a class of differential-algebraic equations with infinite delay

This paper serves as a corrigendum to the paper titled On a class of differential-algebraic equations with infinite delay appearing in EJQTDE no. 81, 2011. We present here a corrected version of Lemma 5.5 and Corollary 5.7.

On a class of differential-algebraic equations with infinite delay

We study the set of T-periodic solutions of a class of T-periodically perturbed Differential-Algebraic Equations, allowing the perturbation to contain a distributed and possibly infinite delay. Under suitable assumptions, the perturbed equations are equivalent to Retarded Functional (Ordinary) Differential Equations on a manifold. Our study is based on known results about the latter class of equations and on a “reduction” formula for the degree of a tangent vector field to implicitly defined differentiable manifolds.

Computational complexity of numerical solutions of initial value problems for differential algebraic equations (abstract only)

We investigate the cost of solving initial value problems for differential algebraic equations depending on the number of digits of accuracy requested. A recent result showed that the cost of solving initial value problems (IVP) for ordinary differential equations (ODE) is polynomial in the number of digits of accuracy. This improves on the classical result of information-based complexity, which predicts exponential cost. The new theory is based on more realistic assumptions. The algorithm analyzed in this thesis is based on a previously published Taylor series method for solving a general class of differential algebraic equations. We consider DAE of constant index to which the method applies. The DAE is allowed to be of arbitrary index, fully implicit and have derivatives of order higher than one. Similarly, by considering a realistic model, we show that the cost of computing the solution of IVP for DAE with the algorithm adopted and by using automatic differentiation is polynomial in the number of digits of accuracy. We also show that non-adaptation is more expensive than adaptation, giving thus a theoretical justification of the success of adaptivity in practice. A particular case frequently arising in practical applications, the index-1 DAE, is treated separately, in more depth. On the other hand, an analysis of the higher-index DAE is significantly more complicated and applies to a wider class of problems. In both cases, continuous output is also given. These results apply to many important problems arising in practice. We present an interesting theoretical application to polynomial system solving.

Polynomial cost for solving IVP for high-index DAE

We show that the cost of solving initial value problems for high-index differential algebraic equations is polynomial in the number of digits of accuracy requested. The algorithm analyzed is built on a Taylor series method developed by Pryce for solving a general class of differential algebraic equations. The problem may be fully implicit, of arbitrarily high fixed index and contain derivatives of any order. We give estimates of the residual which are needed to design practical error control algorithms for differential algebraic equations. We show that adaptive meshes are always more efficient than non-adaptive meshes. Finally, we construct sufficiently smooth interpolants of the discrete solution.

On differential independence of the Riemann zeta function and the Euler gamma function

It is well-known ([2] and [4]) that Γ , as well as ζ, is not a solution of any algebraic differential equation with coefficients in C. In other words, if P (u0, u1, . . . , um) is any polynomial in u0, u1, . . . , um over C, and P (Γ, Γ ′, . . . , Γ )(z) ≡ 0 or P (ζ, ζ ′, . . . , ζ)(z) ≡ 0, for all z ∈ C, then the polynomial P is identically zero. This answers Hilbert’s conjecture [1] in his 18th problem. For a detailed discussion of this subject and other related topics, we refer the reader to [5]. Let h(z) = ζ(sin(2πz)). Recently, Markus [3] proved that if P (h, h′, . . . , h;Γ, Γ ′, . . . , Γ )(z) ≡ 0 for z ∈ C, then the polynomial P (u0, u1, . . . , um; v0, v1, . . . , vn) is identically zero. Thus, in the terminology of differential algebraic theory, Γ and h are differentially independent over C (hence, over C(z)). Furthermore, Markus [3] conjectured that if P (ζ, ζ ′, . . . , ζ;Γ, Γ ′, . . . , Γ )(z) ≡ 0 for z ∈ C, then the polynomial P (u0, u1, . . . , um; v0, v1, . . . , vn) is identically zero, i.e. Γ and ζ are differentially independent. In this short note, we prove that ζ and Γ cannot satisfy a class of algebraic differential equations. Let P (u0, u1, . . . , um; v0, v1, . . . , vn) be any polynomial with coefficients in C. For a non-negative integer μ, we let Λ = Λ(μ) = {(λ0, λ1, . . . , λμ) : λj is a non-negative integer and 0 ≤ j ≤ μ <∞}

Characterizing differential algebraic equationswithout the use of derivative arrays

For a large class of differential algebraic equations with properly stated leading termsand nonsmooth data, an index characterization in terms of the original coefficients and their only first partial derivatives is given. No higher derivatives are used, but several constant rank conditions seem to be essential.

Stable Underlying Equations for Constrained Hamiltonian Systems

AbstractConstrained Hamiltonian systems represent a special class of differential algebraic equations appearing in many mechanical problems. We survey some possibilities for exploiting their rich geometric structures in the numerical integration of the systems. Our main theme is the construction of underlying equations for which the constraint manifold possesses good stability properties. As an application we compare position and momentum projections for systems with externally imposed holonomic constraints. (© 2004 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Trajectories of a DAE near a pseudo-equilibrium

We consider a class of differential-algebraic equations (DAEs) defined by analytic nonlinearities and study its singular solutions. The main assumption used is that the linearization of the DAE represents a Kronecker index-2 matrix pencil and that the constraint manifold has a quadratic fold along its singularity.From these assumptions we obtain a normal form for the DAE where the presence of the singularity and its effects on the dynamics of the problem are made explicit in the form of a quasi-linear differential equation. Subsequently, two distinct types of singular points are identified through which there pass exactly two analytic solutions: pseudo-nodes and pseudo-saddles. We also demonstrate that a singular point called a pseudo-node supports an uncountable infinity of solutions which are not analytic in general.Moreover, akin to known results in the literature for DAEs with singular equilibria, a degenerate singularity is found through which there passes one analytic solution such that the singular point in question is contained within a quasi-invariant manifold of solutions. We call this type of singularity a pseudo-centre and it provides not only a manifold of solutions which intersects the singularity, but also a local flow on that manifold which solves the DAE.

Modified extended backward differentiation formulae for the numerical solution of stiff initial value problems in ODEs and DAEs

For many years the methods of choice for the numerical solution of stiff initial value problems and certain classes of differential algebraic equations have been the well-known backward differentiation formulae (BDF). More recently, however, new classes of formulae which can offer some important advantages over BDF have emerged. In particular, some recent large-scale independent comparisons have indicated that modified extended backward differentiation formulae (MEBDF) are particularly efficient for general stiff initial value problems and for linearly implicit DAEs with index ⩽3. In the present paper we survey some of the more important theory associated with these formulae, discuss some of the practical applications where they are particularly effective, e.g., in the solution of damped highly oscillatory problems, and describe some significant recent extensions to the applicability of MEBDF codes.

PERIODIC MOTIONS AND NONLINEAR DYNAMICS OF A WHEELSET MODEL

Periodic motions and nonlinear dynamics of a wheelset model are investigated numerically. The equations of motion of this multibody system belong to a special class of differential-algebraic equations (DAEs). In contrast to previous investigations of wheelset models the equations are treated directly as DAEs and are not reduced by simplifications to an explicit ODE. Further, it is shown how basic tools for the analysis of Hopf bifurcations and stability of periodic solutions can be transferred to this class of DAEs.