Xuechuan Wang¹, Wei He^1,*, Haoyang Feng¹, Satya N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.139, No.2, pp. 1263-1294, 2024, DOI:10.32604/cmes.2023.043068 - 29 January 2024

Abstract Although predictor-corrector methods have been extensively applied, they might not meet the requirements of practical applications and engineering tasks, particularly when high accuracy and efficiency are necessary. A novel class of correctors based on feedback-accelerated Picard iteration (FAPI) is proposed to further enhance computational performance. With optimal feedback terms that do not require inversion of matrices, significantly faster convergence speed and higher numerical accuracy are achieved by these correctors compared with their counterparts; however, the computational complexities are comparably low. These advantages enable nonlinear engineering problems to be solved quickly and accurately, even with rough… More > Graphic Abstract

Abdulsattar Abdullah Hamad¹, Ahmed S. Al-Obeidi², Enas H. Al-Taiy², Osamah Ibrahim Khalaf³, Dac-Nhuong Le^4,5,*

CMC-Computers, Materials & Continua, Vol.66, No.3, pp. 3311-3327, 2021, DOI:10.32604/cmc.2021.013395 - 28 December 2020

Abstract Synchronization is one of the most important characteristics of dynamic systems. For this paper, the authors obtained results for the nonlinear systems controller for the custom Synchronization of two 4D systems. The findings have allowed authors to develop two analytical approaches using the second Lyapunov (Lyp) method and the Gardano method. Since the Gardano method does not involve the development of special positive Lyp functions, it is very efficient and convenient to achieve excessive system SYCR phenomena. Error is overcome by using Gardano and overcoming some problems in Lyp. Thus we get a great investigation… More >

Xuechuan Wang¹, Satya N. Atluri²

CMES-Computer Modeling in Engineering & Sciences, Vol.111, No.6, pp. 567-585, 2016, DOI:10.3970/cmes.2016.111.567

Abstract This paper compares the variational iteration method (VIM), the Adomian decomposition method (ADM) and the Picard iteration method (PIM) for solving a system of first order nonlinear ordinary differential equations (ODEs). A unification of the concepts underlying these three methods is attempted by considering a very general iterative algorithm for VIM. It is found that all the three methods can be regarded as special cases of using a very general matrix of Lagrange multipliers in the iterative algorithm of VIM. The global variational iteration method is briefly reviewed, and further recast into a Local VIM, More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.98, No.4, pp. 375-407, 2014, DOI:10.3970/cmes.2014.098.375

Abstract In this paper we study the nonlinear dynamical system x^·=f(x,t) from a newly developed theory, viewing the time-varying function of sign(||f||²||x||²− 2(f·x)²) = −sign(cos 2θ) as a key factor, where θ is the intersection angle between x and f. It together with sign(cos θ) can reveal the complexity of nonlinear Duffing oscillator and a quadratic ship rolling oscillator. The barcode is formed by plotting sign(||f||²||x||²− 2(f·x)²) with respect to time. We analyze the barcode to point out the bifurcation of subharmonic motions and the range of chaos in the parameter space. The bifurcation diagram obtained by plotting the More >

T.A. Elgohary^1,2, L. Dong³, J.L. Junkins^2,4, S.N. Atluri^1,4

CMES-Computer Modeling in Engineering & Sciences, Vol.100, No.3, pp. 249-275, 2014, DOI:10.3970/cmes.2014.100.249

Abstract In this study, we consider Initial Value Problems (IVPs) for strongly nonlinear dynamical systems, and study numerical methods to analyze short as well as long-term responses. Dynamical systems characterized by a system of second-order nonlinear ordinary differential equations (ODEs) are recast into a system of nonlinear first order ODEs in mixed variables of positions as well as velocities. For each discrete-time interval Radial Basis Functions (RBFs) are assumed as trial functions for the mixed variables in the time domain. A simple collocation method is developed in the time-domain, with Legendre-Gauss-Lobatto nodes as RBF source points… More >

Jun-Sheng Duan^1,2, Randolph Rach³, Abdul-Majid Wazwaz⁴

CMES-Computer Modeling in Engineering & Sciences, Vol.94, No.1, pp. 77-118, 2013, DOI:10.3970/cmes.2013.094.077

Abstract In this paper, we propose a new modification of the Adomian decomposition method for solution of higher-order nonlinear initial value problems with variable system coefficients and solutions of systems of coupled nonlinear initial value problems. We consider various algorithms for the Adomian decomposition series and the series of Adomian polynomials to calculate the solutions of canonical first- and second-order nonlinear initial value problems in order to derive a systematic algorithm for the general case of higher-order nonlinear initial value problems and systems of coupled higher-order nonlinear initial value problems. Our new modified recursion scheme is More >

Hung-Chang Lee¹, Chein-Shan Liu²

CMES-Computer Modeling in Engineering & Sciences, Vol.41, No.1, pp. 1-26, 2009, DOI:10.3970/cmes.2009.041.001

Abstract The group-preserving schemes developed by Liu (2001) for integrating ordinary differential equations system were adopted the Cayley transform and Padé approximants to formulate the Lie group from its Lie algebra. However, the accuracy of those schemes is not better than second-order. In order to increase the accuracy by employing the group-preserving schemes on ordinary differential equations, according to an efficient technique developed by Runge and Kutta to raise the order of accuracy from the Euler method, we combine the Runge-Kutta method on the group-preserving schemes to obtain the higher-order numerical methods of group-preserving type. They More >

Nadipuram R. Prasad¹, Salvador Almanza-Garcia¹, Thomas T. Lu²

CMC-Computers, Materials & Continua, Vol.14, No.1, pp. 1-22, 2009, DOI:10.3970/cmc.2009.014.001

Abstract The paper presents a revolutionary framework for the modeling, detection, characterization, identification, and machine-learning of anomalous behavior in observed phenomena arising from a large class of unknown and uncertain dynamical systems. An evolved behavior would in general be very difficult to correct unless the specific anomalous event that caused such behavior can be detected early, and any consequence attributed to the specific anomaly following its detection. Substantial investigative time and effort is required to back-track the cause for abnormal behavior and to recreate the event sequence leading to such abnormal behavior. The need to automatically More >

Chein-Shan Liu¹

CMES-Computer Modeling in Engineering & Sciences, Vol.12, No.2, pp. 83-108, 2006, DOI:10.3970/cmes.2006.012.083

Abstract The system we consider consists of two parts: a purely algebraic system describing the manifold of constraints and a differential part describing the dynamics on this manifold. For the constrained dynamical problem in its engineering application, it is utmost important to developing numerical methods that can preserve the constraints. We embed the nonlinear dynamical system with dimensions n and with k constraints into a mathematically equivalent n + k-dimensional nonlinear system, which including k integrating factors. Each subsystem of the k independent sets constitutes a Lie type system of X^˙_i = A_iX_i with A_i ∈ so(n_i,1) and n₁ +···+n_k = n.… More >

John L. Junkins¹, Kamesh Subbarao², Ajay Verma³

CMES-Computer Modeling in Engineering & Sciences, Vol.1, No.4, pp. 99-118, 2000, DOI:10.3970/cmes.2000.001.551

Abstract Model reference adaptive control formulations are presented that rigorously impose the dynamical structure of the state space descriptions of several distinct large classes of dynamical systems. Of particular interest, the formulations enable the imposition of exact kinematic differential equation constraints upon the adaptation process that compensates for model errors and disturbances at the acceleration level. Other adaptive control formulations are tailored for redundantly actuated and constrained dynamical systems. The utility of the resulting structured adaptive control formulations is studied by considering examples from nonlinear oscillations, aircraft control, spacecraft control, and cooperative robotic system control. The More >