Novel hyperbolic and exponential ansatz methods to the fractional fifth-order Korteweg–de Vries equations

Park, Choonkil; Nuruddeen, Rahmatullah Ibrahim; Ali, Khalid K.; Muhammad, Lawal; Osman, M. S.; Băleanu, Dumitru

doi:10.1186/s13662-020-03087-w

Cited by 82 publications

(22 citation statements)

References 57 publications

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“…e mapping is a contraction under the assumption 0 < ε < 1. As a result of the Banach contraction fixed point theorem, there is a unique solution to (9). As a result, the proof is complete.…”

Section: Uniqueness and Existence Solutions For The Modified Decompos...mentioning

confidence: 72%

“…Caputo and Fabrizio modified the existing Caputo derivative to develop the Caputo-Fabrizio fractional derivative [1][2][3][4][5] based on a nonsingular kernel. Because of its advantages, numerous researchers utilized this operator to investigate various types of fractional-order partial differential equations [6][7][8][9]. To address this issue, Atangana and Baleanu proposed a new fractional operator called the Atangana-Baleanu derivative, which combines Caputo and Riemann-Liouville derivatives.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Uniqueness and Existence Solutions For The Modified Decompos...mentioning

confidence: 72%

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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show abstract

“…e KE has been solved analytically and numerically in many researches [12][13][14]. e obtained solutions are N-soliton solutions [15], various solitons solutions [16], soliton and breathers [17], and different types of N-soliton and lump solutions [18]. e numerical solutions are obtained by using modi ed variational iteration algorithm-I and II [19,20], di erential quadrature [21], hybridizable discontinuous Galerkin (HDG) [22], and others.…”

Section: Introductionmentioning

confidence: 99%

On Reduce Differential Transformation Method for Solving Damped Kawahara Equation

Aljahdaly

Alharbi

2022

Mathematical Problems in Engineering

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The damped Kawahara equation (KE) is nonintegrable equation and does not have analytical integration. In this work, the powerful numerical method, which is the reduce differential transformation method (RDTM), is devoted to solve the damped KE. The accuracy of the method is proved. The results are compared with the different numerical methods. The numerical solution is axi-symmetric wave and shows the effect of damping term successfully. We confirmed that the RDTM is useful for solving nonintegrable equations.

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“…Earlier, the study of travelling-wave solutions for non-linear equations played a significant role in analyzing non-linear physical processes. The KdV equation has defined a wide variety of physical phenomena used to model the interaction and evolution of non-linear waves [1][2][3][4][5][6][7][8].…”

Section: Introductionmentioning

confidence: 99%

Fractional System of Korteweg-De Vries Equations via Elzaki Transform

Na-Na²,

Dassios

et al. 2021

Mathematics

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In this article, a hybrid technique, called the Iteration transform method, has been implemented to solve the fractional-order coupled Korteweg-de Vries (KdV) equation. In this method, the Elzaki transform and New Iteration method are combined. The iteration transform method solutions are obtained in series form to analyze the analytical results of fractional-order coupled Korteweg-de Vries equations. To understand the analytical procedure of Iteration transform method, some numerical problems are presented for the analytical result of fractional-order coupled Korteweg-de Vries equations. It is also demonstrated that the current technique’s solutions are in good agreement with the exact results. The numerical solutions show that only a few terms are sufficient for obtaining an approximate result, which is efficient, accurate, and reliable.

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Novel hyperbolic and exponential ansatz methods to the fractional fifth-order Korteweg–de Vries equations

Cited by 82 publications

References 57 publications

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

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

On Reduce Differential Transformation Method for Solving Damped Kawahara Equation

Fractional System of Korteweg-De Vries Equations via Elzaki Transform

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