Chaos and Relativistic Energy-Momentum of the Nonlinear Time Fractional Duffing Equation

Attia, Raghda A. M.; Lu, Dianchen; Khater, Mostafa M. A.

doi:10.3390/mca24010010

Cited by 39 publications

(36 citation statements)

References 28 publications

(37 reference statements)

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“…Thus, in light of the RPS algorithm for finding the 2 nd unknown coefficients a 2 and b 2 , substitute the 2 nd -truncated series solutions of (11), p 2,1r (t) = α 1r + β 1r t + a 2 t 2 2 and p 2,2r (t) = α 2r + β 2r t + b 2 t 2 2 , into the 2 th -residual functions of (12), such that…”

Section: The Rps Methods For Fuzzy Duffing Oscillatormentioning

confidence: 99%

“…Duffing oscillator has been used to describe dwindling oscillatory motion with more complex capabilities than simple harmonic motion in the physical sense, to show the chaotic behaviors of nonlinear dynamic systems, and to display vibration jumps in the changing frequency phases of the periodically forced oscillator with nonlinear elasticity, along with many applications, including optimal control problems, robotics, electromagnetic pulses, and fuzzy modeling [6][7][8][9]. However, serious studies have been conducted to solve the Duffing equation, such as the study of a flexible pendulum motion that has a stiff spring that does not follow Hooke's law, and the study of non-harmonic external perturbations [10][11][12]. In most cases where the entries may be precise (crisp) or imprecise (fuzzy), the variables, parameters, or conditions were considered in crisp terms.…”

Section: Introductionmentioning

confidence: 99%

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Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

et al. 2020

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The mathematical structure of some natural phenomena of nonlinear physical and engineering systems can be described by a combination of fuzzy differential equations that often behave in a way that cannot be fully understood. In this work, an accurate numeric-analytic algorithm is proposed, based upon the use of the residual power series, to investigate the fuzzy approximate solution for a nonlinear fuzzy Duffing oscillator, along with suitable uncertain guesses under strongly generalized differentiability. The proposed approach optimizes the approximate solution by minimizing a residual function to generate r-level representation with a rapidly convergent series solution. The influence, capacity, and feasibility of the method are verified by testing some applications. Level effects of the parameter r are given graphically and quantitatively, showing good agreement between the fuzzy approximate solutions of upper and lower bounds, that together form an almost symmetric triangular structure, that can be determined by central symmetry at r = 1 in a convex region. At this point, the fuzzy number is a convex fuzzy subset of the real line, with a normalized membership function. If this membership function is symmetric, the triangular fuzzy number is called the symmetric triangular fuzzy number. Symmetrical fuzzy estimates of solutions curves indicate a sense of harmony and compatibility around the parameter r = 1. The results are compared numerically with the crisp solutions and those obtained by other existing methods, which illustrate that the suggested method is a convenient and remarkably powerful tool in solving numerous issues arising in physics and engineering.

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Section: The Rps Methods For Fuzzy Duffing Oscillatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

et al. 2020

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“…The strategy of this paper is summarized as follows: In Sect. 2, we apply the modified Khater method, Adomian decomposition method, and B-spline schemes [31][32][33][34][35] to the Cahn-Allen model [36][37][38][39]. In Sect.…”

Section: Introductionmentioning

confidence: 99%

Analytical, semi-analytical, and numerical solutions for the Cahn–Allen equation

Khater

Park

et al. 2020

Adv Differ Equ

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This paper studies the analytical, semi-analytical, and numerical solutions of the Cahn-Allen equation, which plays a vital role in describing the structure of the dynamics for phase separation in Fe-Cr-X (X = Mo, Cu) ternary alloys. The modified Khater method, the Adomian decomposition method, and the quintic B-spline scheme are implemented on our suggested model to get distinct kinds of solutions. These solutions describe the dynamics of the phase separation in iron alloys and are also used in solidification and nucleation problems. The applications of this model arise in many various fields such as plasma physics, quantum mechanics, mathematical biology, and fluid dynamics. The comparison between the obtained solutions is represented by using figures and tables to explain the value of the error between exact and numerical solutions. All solutions are verified by using Mathematica software.

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“…The conformable derivative can be regarded as a natural extension of the classical differential operator, which satisfies most important properties, such as the chain rule [29][30][31]. Researchers have recently applied conformable derivatives to many scientific fields [32][33][34][35][36][37][38][39][40][41][42].…”

Section: Introductionmentioning

confidence: 99%

Modeling, discretization, and hyperchaos detection of conformable derivative approach to a financial system with market confidence and ethics risk

et al. 2019

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A new chaotic financial system is proposed by considering ethics involvement in a fourdimensional financial system with market confidence. A five-dimensional conformable derivative financial system is presented by introducing conformable fractional calculus to the integerorder system. A discretization scheme is proposed to calculate numerical solutions of conformable derivative systems. The scheme is illustrated by testing hyperchaos for the system. among interest rates, investments, prices, and savings. Chen [12] presented a fractional form of nonlinear financial system. Wang, Huang and Shen [13] established an uncertain fractional-order form of the financial system. Mircea et al. [14] set up a delayed form of the financial system. Xin, Chen, and Ma [15] proposed a discrete form of financial system. Yu, Cai, and Li [16] extended the

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Chaos and Relativistic Energy-Momentum of the Nonlinear Time Fractional Duffing Equation

Cited by 39 publications

References 28 publications

Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

Residual Series Representation Algorithm for Solving Fuzzy Duffing Oscillator Equations

Analytical, semi-analytical, and numerical solutions for the Cahn–Allen equation

Modeling, discretization, and hyperchaos detection of conformable derivative approach to a financial system with market confidence and ethics risk

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