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

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 initial guesses from elementary predictors.… 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

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 into the convergence of error… 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, which is much more convenient… 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 percentage… 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 as well as collocation 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 designed to decelerate the Adomian… 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… More >