scholarly journals A PARALLEL MULTIRATE ALGORITHM FOR THE NUMERICAL INTEGRATION OF SYSTEM OF NONLINEAR DIFFERENTIAL EQUATIONS

2014 ◽  
pp. 34-41
Author(s):  
Petro Stakhiv ◽  
Serhiy Rendzinyak
Keyword(s):  
Differential Equations ◽  
Numerical Integration ◽  
Dynamic Behavior ◽  
Large Scale ◽  
Nonlinear Differential Equations ◽  
New Approach ◽  
Large Scale Systems ◽  
Computing Process ◽  
Parallelization Efficiency

The new approach to calculate dynamic behavior of large-scale systems, separated on subsystems is presented. Parallelization efficiency of computing process is described.

scholarly journals A New Approach to Approximate Solutions for Nonlinear Differential Equation

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
Safia Meftah
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Small Parameter ◽  
Perturbation Method ◽  
Periodic Solutions ◽  
Asymptotic Expansions ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Approximation Methods ◽  
New Approach

The question discussed in this study concerns one of the most helpful approximation methods, namely, the expansion of a solution of a differential equation in a series in powers of a small parameter. We used the Lindstedt-Poincaré perturbation method to construct a solution closer to uniformly valid asymptotic expansions for periodic solutions of second-order nonlinear differential equations.

ANALYZING DYNAMIC BEHAVIOR OF LARGE-SCALE SYSTEMS THROUGH MODEL TRANSFORMATION

2005 ◽  
Vol 15 (01) ◽  
pp. 35-60 ◽  
Author(s):  
MICHAEL E. SHIN ◽  
ALEXANDER H. LEVIS ◽  
LEE W. WAGENHALS ◽  
DAE-SIK KIM
Keyword(s):  
Dynamic Behavior ◽  
Model Transformation ◽  
Large Scale ◽  
System Model ◽  
Use Case ◽  
Class Model ◽  
Large Scale Systems ◽  
Collaboration Model ◽  
Model Class ◽  
Use Case Model

This paper describes model transformation for analyzing dynamic behavior of large-scale systems. The Unified Modeling Language (UML) based system model is transformed into the Colored Petri Nets (CPN) model, which is used for analyzing the scenarios of the use cases of a system and checking freedom of system deadlock at an early stage of software development. The CPN model that is executable is hierarchically structured on the basis of the functional decomposition of a large-scale system. The UML-based system model consisting of the use case model, class model and collaboration model is not executable so that the dynamic behavior of the system cannot be analyzed until implementation of the system. However, the UML-based system model has no hierarchical structure to be transformed into the hierarchical CPN model as well. The discrepancies of dynamic and structural views in the two models are resolved by transformation of the UML model into the layered, executable CPN model with three layers — the use case layer, object layer and operation layer. The model transformation is carried out using relationships among the use case model, class model, and collaboration model of the UML. With the executable CPN model transformed, the dynamic properties of the system are analyzed using the simulation technique, occurrence graph, and state space report provided by the Design/CPN tool. The approach in this paper is validated through two case studies — the gas station system and the distributed factory automation system.

An implicit differential equation governing lumped capacitance, radiation dominated, unsteady, heat transfer

2012 ◽  
Vol 22 (7) ◽  
pp. 896-906
Author(s):  
Lawrence J. De Chant
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Simple Model ◽  
Differential Equations ◽  
Large Scale ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Implicit Differential Equation ◽  
Content Type ◽  
Integration Schemes

PurposeAlthough most physical problems in fluid mechanics and heat transfer are governed by nonlinear differential equations, it is less common to be confronted with a “so – called” implicit differential equation, i.e. a differential equation where the highest order derivative cannot be isolated. The purpose of this paper is to derive and analyze an implicit differential equation that arises from a simple model for radiation dominated heat transfer based upon an unsteady lumped capacitance approach.Design/methodology/approachHere we discuss an implicit differential equation that arises from a simple model for radiation dominated heat transfer based upon an unsteady lumped capacitance approach. Due to the implicit nature of this problem, standard integration schemes, e.g. Runge‐Kutta, are not conveniently applied to this problem. Moreover, numerical solutions do not provide the insight afforded by an analytical solution.FindingsA predictor predictor‐corrector scheme with secant iteration is presented which readily integrates differential equations where the derivative cannot be explicitly obtained. These solutions are compared to numerical integration of the equations and show good agreement.Originality/valueThe paper emphasizes that although large‐scale, multi‐dimensional time‐dependent heat transfer simulation tools are routinely available, there are instances where unsteady, engineering models such as the one discussed here are both adequate and appropriate.

Multiple large-scale coherent mode interactions in a developing round jet

1993 ◽  
Vol 248 ◽  
pp. 383-401 ◽  
Author(s):  
Sang Soo Lee ◽  
J. T. C. Liu
Keyword(s):  
Differential Equations ◽  
Energy Method ◽  
Large Scale ◽  
Initial Conditions ◽  
Nonlinear Differential Equations ◽  
Wave Mode ◽  
Mean Flow ◽  
Round Jet ◽  
The Mean ◽  
Integral Energy

The integral energy method has been used to study the nonlinear interactions of the large-scale coherent structure in a spatially developing round jet. The streamwise development of a jet is obtained in terms of the mean flow shear-layer momentum thickness, the wave-mode kinetic energy and the wave-mode phase angle. With the energy method, a system of partial differential equations is reduced to a system of ordinary differential equations. The nonlinear differential equations are solved with initial conditions which are given at the nozzle exit. It is shown that the initial wave-mode energy densities as well as the initial phase angles play a significant role in the streamwise evolution of the large-scale coherent wave modes and the mean flow.

Dynamic Analysis of Mechanical Systems With Intermittent Motion

1982 ◽  
Vol 104 (4) ◽  
pp. 778-784 ◽  
Author(s):  
R. A. Wehage ◽  
E. J. Haug
Keyword(s):  
Dynamic Analysis ◽  
Numerical Integration ◽  
Large Scale ◽  
Mechanical Systems ◽  
Integration Algorithm ◽  
Large Scale Systems ◽  
Jump Discontinuities ◽  
Impulsive Forces ◽  
Computer Generation ◽  
Intermittent Motion

A method is presented for dynamic analysis of systems with impulsive forces, impact, discontinuous constraints, and discontinuous velocities. A method of computer generation of the equations of planar motion and impulse-momentum relations that define jump discontinuities in system velocity for large scale systems is presented. An event predictor, working in conjunction with a new numerical integration algorithm, efficiently controls the numerical integration and allows for automatic equation reformulation. A weapon mechanism and a trip plow are simulated using the method to illustrate its capabilities.

Optimal Inverse Kinematic Solutions for Redundant Manipulators by Using Analytical Methods to Minimize Position and Velocity Measures

2013 ◽  
Vol 5 (3) ◽  
Author(s):  
M. Kemal Ozgoren
Keyword(s):  
Differential Equations ◽  
Numerical Integration ◽  
Nonlinear Differential Equations ◽  
Inverse Kinematic ◽  
Performance Criterion ◽  
Performance Criteria ◽  
Redundant Manipulators ◽  
Joint Variable ◽  
Velocity Level ◽  
Kinematic Solution

Two methods are presented to obtain optimal inverse kinematic solutions for redundant manipulators, according to two different performance criteria stipulated in the position and velocity levels. Both methods are analytical throughout except their final stages, which involve the numerical solution of a simplified minimization problem in a position-level case and the numerical integration of a set of differential equations derived optimally in a velocity-level case. Owing to the analytical nature of the methods, the multiple and singular configurations of the manipulator of concern can be identified readily and studied in detail. The methods are applicable for both serial and parallel redundant manipulators. However, they are demonstrated here for a humanoid manipulator with seven revolute joints. In the demonstrations, the first performance criterion is stipulated in the position level as the minimization of the potential energy. In that case, the optimal inverse kinematic solution is first obtained in the position level for a specified position of the hand. Then, it is compatibly extended to the velocity level for a specified motion of the hand. In the main analytical part of the solution, six of the joint variables are expressed in terms of the selected seventh one. Then, the optimal value of the selected joint variable is determined numerically by a simple one dimensional scanning. The second performance criterion is stipulated in the velocity level as the minimization of the kinetic energy. In that case, the optimal inverse kinematic solution is first obtained in the velocity level and then extended to the position level by integration. The main analytical part of the solution provides an optimally determined set of nonlinear differential equations. These differential equations are then integrated numerically in order to obtain the corresponding solution in the position level. However, the corrections needed to eliminate the numerical integration errors are still obtained analytically. The distinct optimal behaviors of the manipulator according to the mentioned criteria are also illustrated and compared for a duration, in which the hand moves in the same specified way.

scholarly journals An approximation solvability method for nonlocal differential problems in Hilbert spaces

2017 ◽  
Vol 19 (02) ◽  
pp. 1650002 ◽  
Author(s):  
Irene Benedetti ◽  
Nguyen Van Loi ◽  
Luisa Malaguti ◽  
Valeri Obukhovskii
Keyword(s):  
Differential Equations ◽  
Hilbert Spaces ◽  
Nonlinear Differential Equations ◽  
Degree Theory ◽  
Nonlocal Problems ◽  
Abstract Result ◽  
New Approach

A new approach is developed for the solvability of nonlocal problems in Hilbert spaces associated to nonlinear differential equations. It is based on a joint combination of the degree theory with the approximation solvability method and the bounding functions technique. No compactness or condensivity condition on the nonlinearities is assumed. Some applications of the abstract result to the study of nonlocal problems for integro-differential equations and systems of integro-differential equations are then showed. A generalization of the result by using nonsmooth bounding functions is given.

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