scholarly journals Benchmark: A Nonlinear Reachability Analysis Test Set from Numerical Analysis

10.29007/6dcf ◽  
2018 ◽  
Author(s):  
Hoang-Dung Tran ◽  
Luan Viet Nguyen ◽  
Taylor T Johnson
Keyword(s):  
Numerical Analysis ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Differential Algebraic Equations ◽  
Hybrid Automata ◽  
State Variables ◽  
Algebraic Equations ◽  
Test Set ◽  
Verification Methods ◽  
Highly Nonlinear

The field of numerical analysis has developed numerous benchmarks for evaluating differential and algebraic equation solvers. In this paper, we describe a set of benchmarks commonly used in numerical analysis that may also be effective for evaluating continuous and hybrid systems reachability and verification methods. Many of these examples are challenging and have highly nonlinear differential equations and upwards of tens of dimensions (state variables). Additionally, many examples in numerical analysis are originally encoded as differential algebraic equations (DAEs) with index greater than one or as implicit differential equations (IDEs), which are challenging to model as hybrid automata. We present executable models for ten benchmarks from a test set for initial value problems (IVPs) in the SpaceEx format (allowing for nonlinear equations instead of restricting to affine) and illustrate their conversion to several other formats (dReach, Flow*, and the MathWorks Simulink/Stateflow [SLSF]) using the HyST tool. For some instances, we present successful analysis results using dReach and SLSF.

scholarly journals Treatment for third-order nonlinear differential equations based on the Adomian decomposition method

2017 ◽  
Vol 20 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Xueqin Lv ◽  
Jianfang Gao
Keyword(s):  
Differential Equations ◽  
Decomposition Method ◽  
Nonlinear Differential Equations ◽  
Adomian Decomposition Method ◽  
Differential Algebraic Equations ◽  
Nonlinear Ordinary Differential Equations ◽  
Algebraic Equations ◽  
Third Order ◽  
Adomian Decomposition ◽  
Linear And Nonlinear

The Adomian decomposition method (ADM) is an efficient method for solving linear and nonlinear ordinary differential equations, differential algebraic equations, partial differential equations, stochastic differential equations, and integral equations. Based on the ADM, a new analytical and numerical treatment is introduced in this research for third-order boundary-value problems. The effectiveness of the proposed approach is verified by numerical examples.

Utilization of Mathematical Software Packages in Chemical Engineering Research

2005 ◽  
Vol 5 (2) ◽  
pp. 125
Author(s):  
Ang Wee Lee ◽  
Nayef Mohamed Ghasem ◽  
Mohamed Azlan Hussain
Keyword(s):  
Differential Equations ◽  
Computer Algebra ◽  
Large Scale ◽  
Chemical Engineering ◽  
Nonlinear Differential Equations ◽  
Differential Algebraic Equations ◽  
Good Choice ◽  
Computer Algebra Systems ◽  
Algebraic Equations ◽  
Starting Point

Using Fortran taken as the starting point, we are now on the sixth decade of high-level programming applications. Among the programming languages available, computer algebra systems (CAS) appear to be a good choice in chemical engineering can be applied easily. Until the emergence of CAS, the assistance from a specialized group for large-scale programming is justified. Nowadays, it is more effective for the modern chemical engineer to rely on his/her own programming ability for problem solving. In the present paper, the abilities of Polymath, Maple, Matlab, Mathcad, and Mathematica in handling differential equations are illustrated for differential-algebraic equations, large system of nonlinear differential equations, and partial differential equations. The programming of solutions with these CAS are presented, contrasted, and discussed in relation to chemical engineering problems. Keywords: Computer algebra systems (CAS),computer simulation,Mathcad, Mathematica,Mathlab and numerical methods.

Numerical methods for differential algebraic equations

Acta Numerica ◽  
1992 ◽  
Vol 1 ◽  
pp. 141-198 ◽  
Author(s):  
Roswitha März
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Image Position

Differential algebraic equations (DAE) are special implicit ordinary differential equations (ODE)where the partial Jacobian f′y(y, x, t) is singular for all values of its arguments.

scholarly journals Assessment of Linearization Approaches for Multibody Dynamics Formulations

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Francisco González ◽  
Pierangelo Masarati ◽  
Javier Cuadrado ◽  
Miguel A. Naya
Keyword(s):  
Mechanical System ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Practical Applications ◽  
Linearization Methods ◽  
Highly Nonlinear ◽  
Wide Range ◽  
The Way

Formulating the dynamics equations of a mechanical system following a multibody dynamics approach often leads to a set of highly nonlinear differential-algebraic equations (DAEs). While this form of the equations of motion is suitable for a wide range of practical applications, in some cases it is necessary to have access to the linearized system dynamics. This is the case when stability and modal analyses are to be carried out; the definition of plant and system models for certain control algorithms and state estimators also requires a linear expression of the dynamics. A number of methods for the linearization of multibody dynamics can be found in the literature. They differ in both the approach that they follow to handle the equations of motion and the way in which they deliver their results, which in turn are determined by the selection of the generalized coordinates used to describe the mechanical system. This selection is closely related to the way in which the kinematic constraints of the system are treated. Three major approaches can be distinguished and used to categorize most of the linearization methods published so far. In this work, we demonstrate the properties of each approach in the linearization of systems in static equilibrium, illustrating them with the study of two representative examples.

Haar wavelet operational matrix method for system of fractional nonlinear differential equations

2017 ◽  
Vol 15 (05) ◽  
pp. 1750043 ◽  
Author(s):  
Umer Saeed
Keyword(s):  
Differential Equations ◽  
Reliable Method ◽  
Nonlinear Differential Equations ◽  
Operational Matrix ◽  
Implementation Process ◽  
Haar Wavelet ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Operational Matrix Method ◽  
Fractional Differential

In this paper, we present a reliable method for solving system of fractional nonlinear differential equations. The proposed technique utilizes the Haar wavelets in conjunction with a quasilinearization technique. The operational matrices are derived and used to reduce each equation in a system of fractional differential equations to a system of algebraic equations. Convergence analysis and implementation process for the proposed technique are presented. Numerical examples are provided to illustrate the applicability and accuracy of the technique.

Trajectory Tracking of Underactuated Multibody Systems

2011 ◽  
Author(s):  
Stefan Reichl ◽  
Wolfgang Steiner
Keyword(s):  
Trajectory Tracking ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Feedforward Control ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Differential Algebraic Equations ◽  
State Variables ◽  
Algebraic Equations

This work presents three different approaches in inverse dynamics for the solution of trajectory tracking problems in underactuated multibody systems. Such systems are characterized by less control inputs than degrees of freedom. The first approach uses an extension of the equations of motion by geometric and control constraints. This results in index-five differential-algebraic equations. A projection method is used to reduce the systems index and the resulting equations are solved numerically. The second method is a flatness-based feedforward control design. Input and state variables can be parameterized by the flat outputs and their time derivatives up to a certain order. The third approach uses an optimal control algorithm which is based on the minimization of a cost functional including system outputs and desired trajectory. It has to be distinguished between direct and indirect methods. These specific methods are applied to an underactuated planar crane and a three-dimensional rotary crane.

scholarly journals Investigating Jeffery-Hamel flow with high magnetic field and nanoparticle by HPM and AGM

2014 ◽  
Vol 4 (4) ◽  
Author(s):  
A. Rostami ◽  
M. Akbari ◽  
D. Ganji ◽  
S. Heydari
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Homotopy Perturbation Method ◽  
Volume Fraction ◽  
Nonlinear Differential Equations ◽  
Reynolds Numbers ◽  
Solution Procedure ◽  
Algebraic Equations ◽  
Divergent Channel ◽  
Mathematical Operations

AbstractIn this study, the effects of magnetic field and nanoparticle on the Jeffery-Hamel flow are studied using two powerful analytical methods, Homotopy Perturbation Method (HPM) and a simple and innovative approach which we have named it Akbari-Ganji’s Method(AGM). Comparisons have been made between HPM, AGM and Numerical Method and the acquired results show that these methods have high accuracy for different values of α, Hartmann numbers, and Reynolds numbers. The flow field inside the divergent channel is studied for various values of Hartmann number and angle of channel. The effect of nanoparticle volume fraction in the absence of magnetic field is investigated.It is necessary to represent some of the advantages of choosing the new method, AGM, for solving nonlinear differential equations as follows: AGM is a very suitable computational process and is applicable for solving various nonlinear differential equations. Moreover, in AGM by solving a set of algebraic equations, complicated nonlinear equations can easily be solved and without any mathematical operations such as integration, the solution of the problem can be obtained very simply and easily. It is notable that this solution procedure, AGM, can help students with intermediate mathematical knowledge to solve a broad range of complicated nonlinear differential equations.

Nonlinear Bending of Rectangular Magnetoelectroelastic Thin Plates with Linearly Varying Thickness

2018 ◽  
Vol 19 (3-4) ◽  
pp. 351-356
Author(s):  
Feng Wang ◽  
Yu-fang Zheng ◽  
Chang-ping Chen
Keyword(s):  
Differential Equations ◽  
Variable Thickness ◽  
Plate Theory ◽  
Nonlinear Differential Equations ◽  
Thin Plates ◽  
Thickness Variation ◽  
Algebraic Equations ◽  
Nonlinear Bending ◽  
Nonlinear Load ◽  
Magnetic Potentials

AbstractWith employing the von Karman plate theory, and considering the linearly thickness variation in one direction, the bending problem of a rectangular magnetoelectroelastic plates with linear variable thickness is investigated. According to the Maxwell’s equations, when applying the magnetoelectric load on the plate’s surfaces and neglecting the in-plane electric and magnetic fields in thin plates, the electric and magnetic potentials varying along the thickness direction for the magnetoelectroelastic plates are determined. The nonlinear differential equations for magnetoelectroelastic plates with linear variable thickness are established based on the Hamilton’s principle. The Galerkin procedure is taken to translate a set of differential equations into algebraic equations. The numerical examples are presented to discuss the influences of the aspect ratio and span–thickness ratio on the nonlinear load–deflection curves for magnetoelectroelastic plates with linear variable thickness. In addition, the induced electric and magnetic potentials are also presented with the various values of the taper constants.

Sign in / Sign up

Export Citation Format

Share Document