No polynomial algorithms are known for finding the coefficients of the characteristic polynomial and characteristic equation of a matrix in max-algebra. The following are proved: (1) The task of finding the max-algebraic characteristic polynomial for permutation matrices encoded using the lengths of their constituent cycles is IVP-complete. (2) The task of finding the lowest order finite term o...

In a previous article, the authors developed two conversion methods to improve the Σ -method for structural analysis (SA) of differential-algebraic equations (DAEs). These methods reformulate a DAE on which the Σ -method fails into an equivalent problem on which this SA is more likely to succeed with a generically nonsingular Jacobian. The basic version of these methods processes the DAE as a w...

in the present paper, a numerical method is considered for solving one-dimensionalheat equation subject to both neumann and dirichlet initial boundaryconditions. this method is a combination of collocation method and radial basis functions (rbfs). the operational matrix of derivative for laguerre-gaussians (lg) radial basis functions is used to reduce the problem to a set of algebraic equations...

The Σ-method for structural analysis of a differential-algebraic equation (DAE) system produces offset vectors from which the sparsity pattern of DAE’s system Jacobian is derived; this pattern implies a fine block-triangular form (BTF). This article derives a simple method for quasilinearity analysis of a DAE and combines it with its fine BTF to construct a method for finding the minimal set of...

We prove a comparison theorem for the solutions of a rational matrix difference equation, generalizing the Riccati difference equation, and existence and convergence results for the solutions of this equation. Moreover we present conditions ensuring that the corresponding algebraic matrix equation has a stabilizing or almost stabilizing solution. AMS Classification: 39A10, 93E03.

The objective of the paper is to present a method, called the sequential regularization method (SRM), for the nonstationary incompressible Navier–Stokes equations from the viewpoint of regularization of differential-algebraic equations (DAEs), and to provide a way to apply a DAE method to partial differential-algebraic equations (PDAEs). The SRM is a functional iterative procedure. It is proved...

Differential algebraic equations (DAEs) are implicit systems of ordinary differential equations, F (x′, x, t) = 0, for which the Jacobian Fx′ is always singular. DAEs arise in many applications. Significant progress has been made in developing numerical methods for solving DAEs. Determination of consistent initial conditions remains a difficult problem especially for large higher index DAEs. Th...

In this paper, the optimal power flow (OPF) problem is augmented to account for the costs associated with the loadfollowing stability of a power network. Load-following stability costs are expressed through the linear quadratic regulator (LQR). The power network is described by a set of nonlinear differential algebraic equations (DAEs). By linearizing the DAEs around a known equilibrium, a line...

We develop a stability theory for time-varying linear differential algebraic equations (DAEs). Well-known stability concepts of ordinary differential equations are generalised to DAEs and characterised. Lyapunov’s direct method is derived as well as the converse of the stability theorems. Stronger results are achieved for DAEs, which are transferable into standard canonical form; in this case t...

This paper describes the quantization–based integration methods and extends their use to the simulation of hybrid systems. Using the fact that these methods approximate ordinary differential equations (ODEs) and differential algebraic equations (DAEs) by discrete event systems, it is shown how hybrid systems can be approximated by pure discrete event simulation models (within the DEVS formalism...