Checking system boundedness using ordinary differential equations

Boundedness is one of the most important properties of discrete Petri nets. Determining the boundedness of a Petri net is usually done through building coverability graph or coverability tree. However, the computation is infeasible for complex applications because the size of the coverability graph may increase faster than any primitive recursive functions. This paper proposes a new technique to check the boundedness without causing this problem. Let a concurrent system be represented by a (discrete) Petri net. By relaxing the (discrete) Petri net to a continuous Petri net, we can model the concurrent system by a family of ordinary differential equations. It has been shown that the boundedness of the discrete Petri net is equivalent to the boundedness of the solutions of the corresponding ordinary differential equations. Hence, we can check the boundedness of a (discrete) Petri net by analyzing the solutions of a family of ordinary differential equations. A case study demonstrates the benefits of our technique.     

基于微分方程的程序性能分析

基于连续Petri网模型,用一组常微分方程来描述程序,通过研究微分方程的解来研究程序的性能。每个微分方程描述程序状态的变化,每个状态可由介于0和1之间的数来度量,显示程序到达状态的程度。该方法的好处在于在做程序分析时,可避开状态爆炸问题。     

Laplace transform formula on fuzzy nth-order derivative and its application in fuzzy ordinary differential equations

In this paper, the Laplace transform formula on the fuzzy nth-order derivative by using the strongly generalized differentiability concept is investigated. Also, the related theorems and properties are proved in detail and, it is used in an analytic method for fuzzy two order differential equation. The method is illustrated by solving some examples.     

Time-series forecasting using a system of ordinary differential equations

This paper presents a hybrid evolutionary method for identifying a system of ordinary differential equations (ODEs) to predict the small-time scale traffic measurements data. We used the tree-structure based evolutionary algorithm to evolve the architecture and a particle swarm optimization (PSO) algorithm to fine tune the parameters of the additive tree models for the system of ordinary differential equations. We also illustrate some experimental comparisons with genetic programming, gene expression programming and a feedforward neural network optimized using PSO algorithm. Experimental results reveal that the proposed method is feasible and efficient for forecasting the small-scale traffic measurements data.     

Uniqueness for ordinary differential equations

Various criteria are known for assuring uniqueness of the solution of a system ofn ordinary differential equations,x = f(t, x), with initial conditionx(t ₀) = x₀. Most of these involve some sort of relaxed Lipschitz condition onf(t, x), with respect tox, valid on an open setD R ¹⁺ⁿ which contains the point (t ₀, x₀). The present paper generalizes (and unifies) a number of known uniqueness criteria to cover cases when (t ₀, x₀) lies on the boundary ofD. Research partially supported by NSF Grant GP-37838.     

Solving ordinary differential equations for simulation

Runge-Kutta formulas are given which are suited to the tasks arising in simulation. They are methods permitting interpolation which use overlap into the succeeding step to reduce the cost of a step and its error estimate.     

Singularity-free time integration of rotational quaternions using non-redundant ordinary differential equations

A novel ODE time stepping scheme for solving rotational kinematics in terms of unit quaternions is presented in the paper. This scheme inherently respects the unit-length condition without including it explicitly as a constraint equation, as it is common practice. In the standard algorithms, the unit-length condition is included as an additional equation leading to kinematical equations in the form of a system of differential-algebraic equations (DAEs). On the contrary, the proposed method is based on numerical integration of the kinematic relations in terms of the instantaneous rotation vector that form a system of ordinary differential equations (ODEs) on the Lie algebra \(\mathit{so}(3)\) of the rotation group \(\mathit{SO}(3)\). This rotation vector defines an incremental rotation (and thus the associated incremental unit quaternion), and the rotation update is determined by the exponential mapping on the quaternion group. Since the kinematic ODE on \(\mathit{so}(3)\) can be solved by using any standard (possibly higher-order) ODE integration scheme, the proposed method yields a non-redundant integration algorithm for the rotational kinematics in terms of unit quaternions, avoiding integration of DAE equations. Besides being ‘more elegant’—in the opinion of the authors—this integration procedure also exhibits numerical advantages in terms of better accuracy when longer integration steps are applied during simulation. As presented in the paper, the numerical integration of three non-linear ODEs in terms of the rotation vector as canonical coordinates achieves a higher accuracy compared to integrating the four (linear in ODE part) standard-quaternion DAE system. In summary, this paper solves the long-standing problem of the necessity of imposing the unit-length constraint equation during integration of quaternions, i.e. the need to deal with DAE’s in the context of such kinematical model, which has been a major drawback of using quaternions, and a numerical scheme is presented that also allows for longer integration steps during kinematic reconstruction of large three-dimensional rotations.     

Numerical solution of nonlinear ordinary differential equations using flatlet oblique multiwavelets

This paper is concerned with the construction of a biorthogonal multiwavelet basis in the unit interval to form a biorthogonal flatlet multiwavelet system. The system is then used to solve nonlinear ordinary differential equations. The biorthogonality and high vanishing moment properties of this system result in efficient and accurate solutions. Finally, numerical results for some test problems with known solutions are presented and the absolute errors are compared with the errors resulting from B-spline bases.     

Parallel methods for ordinary differential equations

Remarkably few methods have been proposed for the parallel integration of ordinary differential equations (ODEs). In part this is because the problems do not have much natural parallelism (unless they are virtually uncoupled systems of equations, in which case the method is obvious). In part it is because the subproblems arising in the solution of ODEs (for example, the solution of linear equations) are the ones that have provided the challenges for parallelism. This paper surveys some of the methods that have been proposed, and suggests some additional methods that are suitable for special cases, such as linear problems. It then looks at the possible application of large-scale parallelism, particularly across the method. If efficiency is of no concern (that is, if there is an arbitrary number of proceessors) there are some ways in which the solution of stiff equations can be done more rapidly; in fact, a speed up from a parallel time of 0(N ²) to 0(logN) forN equations might be possible if communication time is ignored. This is obtained by trying to perform as much as possible of the matrix arithmetic associated with the solution of the linear equations at each step in advance of that step and in parallel with the integration of earlier steps.     

New solutions for ordinary differential equations

This paper introduces a new method for solving ordinary differential equations (ODEs) that enhances existing methods that are primarily based on finding integrating factors and/or point symmetries. The starting point of the new method is to find a non-invertible mapping that maps a given ODE to a related higher-order ODE that has an easily obtained integrating factor. As a consequence, the related higher-order ODE is integrated. Fixing the constant of integration, one then uses existing methods to solve the integrated ODE. By construction, each solution of the integrated ODE yields a solution of the given ODE. Moreover, it is shown when the general solution of an integrated ODE yields either the general solution or a family of particular solutions of the given ODE. As an example, new solutions are obtained for an important class of nonlinear oscillator equations. All solutions presented in this paper cannot be obtained using the current Maple ODE solver.     

Multistep-multistage-multiderivative methods for ordinary differential equations

This paper studies a general method for the numerical integration of ordinary differential equations. The method, defined in part 1, contains many known processes as special case, such as multistep methods, Runge-Kutta methods (“multistage”), Taylor, series (“multiderivative”) and their extensions (section 2). After a short section on trees and pairs of trees we derive formulas for the conditions to be satisfied by the free parameters in order to equalize the numerical approximation with the solution up to a certain order. Next we extend the reuslts of Kastlunger [6]. The proof given here is shorter than the original one. Finally we discuss formulas, with the help of which the conditions for the parameters can be reduced considerably and give numerical examples.     

Characteristic sets for ordinary differential equations

In this paper, the problem of the construction of a characteristic set in the sense of Kolchin for a radical differential ideal is considered. Algorithms for constructing such sets in the ordinary case for arbitrary radical differential ideals, which are based on the estimate of the orders of their elements, are presented. These algorithms are applicable in the case of an orderly ranking on the set of the derivatives. Advantages of the regular and characteristic decompositions of radical differential ideals are discussed.Translated from Programmirovanie, Vol. 31, No. 2, 2005.Original Russian Text Copyright © 2005 by Kondratieva, Ovchinnikov.     

Linear system identification using numerical computation of Laplace transforms

A method for system identification using sampled values of the initial transient step or impulse response is described. A polynomial fit of the sampled values is made using Lagrange interpolation and the Laplace transform of the output observed is determined. Then the coefficients of the numerator and denominator polynomials of the system transfer function are determined by minimizing the square of the difference between the observed and calculated values of the Laplace transform of the output variable at a number of discrete points. This process is considerably simplified by the use of tables of coefficients for the numerical calculation of Laplace transforms.