Numerical solutions of coupled nonlinear fractional KdV equations using He’s fractional calculus

Lu, Dianchen; Suleman, Muhammad; Ramzan, Muhammad; Rahman, Jamshaid Ul

doi:10.1142/s0217979221500235

Cited by 14 publications

(5 citation statements)

References 31 publications

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“…The method has been extensively used to solve various differential equations, including those with singularities. Some authors have developed different numerical methods for other fractal differential equations [16][17][18]. To the authors' knowledge, the literature still needs to address a variable mesh approach of order three that uses three grid points to approximate fractal singular TBVPs.…”

Section: Introductionmentioning

confidence: 99%

Third(fourth) order accurate method for one dimensional local fractional Viscous Burgers’ and diffusion-convection equations

Pandey,

Mohanty

2024

Preprint

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A Singular fractal boundary value problem whose order term is a Hausdroff Fractal derivative of order δ ∈ (0, 1] is considered. In this article, using three grid points, we discuss the variable mesh method of order three that requires only three function evaluations on some regularity conditions. we give importance to solve a singular problem in our scheme and prove that the recommended scheme is significant. We also discussed the convergence analysis of our method and proved that the method is reliable, like classical numerical methods. Special numerical illustrations for solving linear and nonlinear fractal differential equations are provided to demonstrate the accurate outcome of the recommended method, which has been discussed in convergence analysis.

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Section: Introductionmentioning

confidence: 99%

Third(fourth) order accurate method for one dimensional local fractional Viscous Burgers’ and diffusion-convection equations

Pandey,

Mohanty

2024

Preprint

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“…Therefore, modeling and simulation of dynamical systems require critical mathematical thinking and sophisticated computational tools to simulate their solutions [1][2][3]. Recently, many semi-analytic iterative methods have been put forward by researchers to address the nonlinear oscillatory behaviors of underlying systems [4][5][6]. The deep neural networks (DNNs) have grown as the most effective architectures to deal with complex patterns and relationships present in data making them well-suited for systems with various interacting components [7].…”

Section: Introductionmentioning

confidence: 99%

Amplifying Sine Unit: An Oscillatory Activation Function for Deep Neural Networks to Recover Nonlinear Oscillations Efficiently

2024

Ann Comp Phy Material Sci

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Many industrial and real-life problems exhibit highly nonlinear periodic behaviors and the conventional methods may fall short of finding their analytical or closed-form solutions. Such problems demand cutting-edge computational tools with increased functionality and reduced cost. Recently, deep neural networks have gained massive research interest due to their ability to handle large data and universality to learn complex functions. In this work, we put forward a methodology based on deep neural networks with responsive layers structure to deal nonlinear oscillations in microelectromechanical systems. We incorporated some oscillatory and non-oscillatory activation functions such as growing cosine unit known as GCU, Sine, Mish and Tanh in our designed network to have a comprehensive analysis on their performance for highly nonlinear problems. Integrating oscillatory activation functions with deep neural networks definitely outperform in predicting the periodic patterns of underlying systems. To support oscillatory actuation for nonlinear systems, we have proposed a novel oscillatory activation function called Amplifying Sine Unit denoted as ASU which is more efficient than GCU for complex vibratory systems such as microelectromechanical systems. Experimental results show that the designed network with our proposed activation function ASU is more reliable and robust to handle the challenges posed by oscillations

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“…Long waves characterise geophysical fluid dynamics in shallow and deep oceans [23,24]. Various studies have suggested numerous systems to overcome the fractional-order Korteweg-de Vries equation employing various methodologies, such as the differential transform method [25], Adomian decomposition method [26], natural decomposition method [27], homotopy analysis method [28], Elzaki projected differential transform method [29], variational iteration method [30], new iterative method [31], modified tanh technique [32], and Lie symmetry analysis [33]. Analogously, same solutions for (2) have been suggested by Inc and Cavlak [34], Fan [35], Lin et al [36], Inc et al [37], and Ghoreishi et al [18].…”

Section: Introductionmentioning

confidence: 99%

A Comparative Analysis of the Fractional-Order Coupled Korteweg–De Vries Equations with the Mittag–Leffler Law

Aljahdaly

Akgül

Shah

et al. 2022

Journal of Mathematics

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This article applies efficient methods, namely, modified decomposition method and new iterative transformation method, to analyze a nonlinear system of Korteweg–de Vries equations with the Atangana–Baleanu fractional derivative. The nonlinear fractional coupled systems investigated in this current analysis are the system of Korteweg–de Vries and the modified system of Korteweg–de Vries equations applied as a model in nonlinear physical phenomena arising in chemistry, biology, physics, and applied sciences. Approximate analytical results are represented in the form of a series with straightforward components, and some aspects showed an appropriate dependence on the values of the fractional-order derivatives. The convergence and uniqueness analysis is carried out. To comprehend the analytical procedure of both methods, three test examples are provided for the analytical results of the time-fractional KdV equation. Additionally, the efficiency of the mentioned procedures and the reduction in calculations provide broader applicability. It is also illustrated that the findings of the current methodology are in close harmony with the exact solutions. The series result achieved applying this technique is proved to be accurate and reliable with minimal calculations. The numerical simulations for obtained solutions are discussed for different values of the fractional order.

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Numerical solutions of coupled nonlinear fractional KdV equations using He’s fractional calculus

Cited by 14 publications

References 31 publications

Third(fourth) order accurate method for one dimensional local fractional Viscous Burgers’ and diffusion-convection equations

Third(fourth) order accurate method for one dimensional local fractional Viscous Burgers’ and diffusion-convection equations

Amplifying Sine Unit: An Oscillatory Activation Function for Deep Neural Networks to Recover Nonlinear Oscillations Efficiently

A Comparative Analysis of the Fractional-Order Coupled Korteweg–De Vries Equations with the Mittag–Leffler Law

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