A Toeplitz Jacobian Matrix/Fast Fourier Transformation Method for Steady-State Analysis of Discontinuous Oscillators by T. Ge, A. Y. T. Leung

“…A semianalytical algorithm is proposed for the solutions and their stability of a piecewise nonlinear system. The conventional harmonic balance method is modified by the introduction of Toeplitz Jacobian matrices (TJM) and by the alternative applications of fast Fourier transformation (FFT) and its inverse. …”
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Weak solutions of unconditionally stable second-order difference schemes for nonlinear sine-Gordon systems by Ozgur Yildirim

“…This paper presents the existence and uniqueness of the weak solution for the nonlinear system of sine-Gordon equations which describes DNA dynamics. …”
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Third-Order Approximate Solution of Chemical Reaction-Diffusion Brusselator System Using Optimal Homotopy Asymptotic Method by Salem Alkhalaf

“…It is observed that the method is easy to implement and quite valuable for handling nonlinear system of partial differential equations and yielding excellent results at minimum computational cost. …”
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Stability analysis for first-order nonlinear differential equations with three-point boundary conditions by Kamala E. Ismayilova

“…Several theorems are proved concerning the existence and uniqueness of solutions to the boundary value problems for the first order nonlinear system of ordinary differential equations with three-point boundary conditions. …”
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The formal Laplace-Borel transform of Fliess operators and the composition product by Yaqin Li, W. Steven Gray

“…This gives a compact interpretation of the operational calculus of Fliess for computing the output response of an analytic nonlinear system.…”
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A Galerkin Finite Element Method for Numerical Solutions of the Modified Regularized Long Wave Equation by Liquan Mei, Yali Gao, Zhangxin Chen

“…In addition, an extrapolation technique is used to transform a nonlinear system into a linear system in order to improve the time accuracy of this method. …”
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Series Solution of the Multispecies Lotka-Volterra Equations by Means of the Homotopy Analysis Method by A. Sami Bataineh, M. S. M. Noorani, I. Hashim

“…The continuous solution for the nonlinear system is given, which provides a convenient and straightforward approach to calculate the dynamics of the system. …”
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Alternative Legendre Polynomials Method for Nonlinear Fractional Integro-Differential Equations with Weakly Singular Kernel by Guodong Shi, Yanlei Gong, Mingxu Yi

“…The operational matrices of integration and product together with the derived operational matrix are utilized to transform nonlinear fractional integro-differential equations to the nonlinear system of algebraic equations. Furthermore, the proposed method has also been analyzed for convergence, particularly in context of error analysis. …”
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Neural Network Inverse Model Control Strategy: Discrete-Time Stability Analysis for Relative Order Two Systems by M. A. Hussain, Jarinah Mohd Ali, M. J. H. Khan

“…This paper discusses the discrete-time stability analysis of a neural network inverse model control strategy for a relative order two nonlinear system. The analysis is done by representing the closed loop system in state space format and then analyzing the time derivative of the state trajectory using Lyapunov’s direct method. …”
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Self-organization through decoupling by Romar Correa

“…Formally, a conditionally stable nonlinear system of differential equations is examined. …”
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Compact Implicit Integration Factor Method for the Nonlinear Dirac Equation by Jing-Jing Zhang, Xiang-Gui Li, Jing-Fang Shao

“…For the temporal discretization, the implicit integration factor method is applied to deal with the nonlinear system. We therefore develop two implicit integration factor numerical schemes with full discretization, one of which can achieve fourth-order accuracy in both space and time. …”
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The New Semianalytical Technique for the Solution of Fractional-Order Navier-Stokes Equation by Nehad Ali Shah, Mounirah Areshi, Jae Dong Chung, Kamsing Nonlaopon

“…Then, we apply this new method to analyze the fractional-order nonlinear system of Navier-Stokes models. Finally, we provide 3-D graphical plots to support the impact of the fractional derivative acting on the behavior of the obtained profile solutions to the suggested models.…”
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Newfangled Linearization Formula of Certain Nonsymmetric Jacobi Polynomials: Numerical Treatment of Nonlinear Fisher’s Equation by W. M. Abd-Elhameed, Afnan Ali, Y. H. Youssri

“…The presented method is used to convert the equation governed by its underlying conditions into a nonlinear system of equations. The solution of the resulting system can be obtained through any suitable standard numerical scheme. …”
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The Exact Linearization and LQR Control of Semiactive Connected Hydropneumatic Suspension System by Xuyang Cao, Linlin Cao, Dianlong Wang

“…Based on differential geometry theory, the nonlinear system of connected hydropneumatic suspension was transformed to a linear one. …”
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Controllability of a fluid-structure interaction system coupling the Navier–Stokes system and a damped beam equation by Buffe, Rémi, Takahashi, Takéo

“…Then, the result on the nonlinear system is obtained by a fixed-point argument.…”
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Existence and uniqueness of solutions for nonlinear impulsive differential equations with three-point boundary conditions by Mısır J. Mardanov, Yagub A. Sharifov, Kamala E. Ismayilova

“…Sufficient conditions are found for the existence and uniqueness of solutions for the boundary value problems for the first order nonlinear system of the impulsive ordinary differential equations with three-point boundary conditions. …”
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New Fractional Derivative Expression of the Shifted Third-Kind Chebyshev Polynomials: Application to a Type of Nonlinear Fractional Pantograph Differential Equations by Y. H. Youssri, W. M. Abd-Elhameed, H. M. Ahmed

“…The role of the tau method is to convert the pantograph differential equation with its governing initial/boundary conditions into a nonlinear system of algebraic equations that can be treated with the aid of Newton’s iterative scheme. …”
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Buckling suppression of a cantilever structure with a modified fuzzy sliding mode control by Jianhua Pan, Lin Sun, Xiaopei Liu, Yuyuan Zhang, Yili Duo

“…Subsequently, a second-order Galerkin discretization is applied to establish a two-dimensional nonlinear system for the cantilever structure. Numerical simulations demonstrate that a two-dimensional nonlinear dynamic model of the laminated composite structure is essential for precise buckling vibration estimations. …”
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Solving a system of nonlinear difference equations with bilinear dynamics by Hashem Althagafi, Ahmed Ghezal

“…This paper presented a comprehensive study of a three-dimensional nonlinear system of difference equations, which can be reduced to a two-dimensional bilinear system. …”
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Nonlinear versus Ordinary Adaptive Control of Continuous Stirred-Tank Reactor by Jiri Vojtesek, Petr Dostal

“…Proposed controller is tested by the simulations on the mathematical model of the continuous stirred-tank reactor with cooling in the jacket as a typical nonlinear system.…”
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