Families of Travelling Waves Solutions for Fractional-Order Extended Shallow Water Wave Equations, Using an Innovative Analytical Method

Khan, Hassan; Shoaib,; Băleanu, Dumitru; Alzaidy, J. F.

doi:10.1109/access.2019.2933188

Cited by 30 publications

(22 citation statements)

References 29 publications

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“…To calculate balance number τ, we consider homogeneous balance between V ¢ and V 2 ( ) ¢ , we have τ + 3 = 2τ + 2, so τ = 1. Putting τ = 1 in (7) suggests the following closed form series solution for (10): 11) in (10) and collecting all terms with the same orders of R(χ) gives an expression in R(χ). We get a system of nonlinear algebraic equations by equating the coefficients of the expression to zero.…”

Section: Resultsmentioning

confidence: 99%

A novel analytical technique for analyzing the (3+1)-dimensional fractional calogero- bogoyavlenskii-schiff equation: investigating solitary/shock waves and many others physical phenomena

Noor,

Alyousef,

Shafee

et al. 2024

Phys. Scr.

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This work presents a thorough analysis of soliton wave phenomena in the (3+1)-dimensional Fractional Calogero-Bogoyavlenskii-Schiff equation (FCBSE) with Caputo’s derivatives through the use of a novel analytical technique known as the modified Extended Direct Algebraic Method (mEDAM). By converting nonlinear Fractional Partial Differential equations (FPDE) into integer-order Nonlinear Ordinary Differential equations (NODE), and then using closed-form series solutions to translate the NODE into an algebraic system of equations, this method allows us to derive families of soliton solutions, which include kink waves, lump waves, breather waves, and periodic waves, exposing new insights into the behavior and distinctive features of soliton waves in the FCBSE. By including contour and 3D graphics, the behaviors of a few selected soliton solutions are well depicted, showcasing their amplitude, shape, and propagation characteristics. The results enhance our understanding of the FCBSE and show that the mEDAM is a valuable tool for studying soliton wave phenomena. This work creates new opportunities for studying wave phenomena in more intricately constructed nonlinear FPDEs (NFPDEs).

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Section: Resultsmentioning

confidence: 99%

A novel analytical technique for analyzing the (3+1)-dimensional fractional calogero- bogoyavlenskii-schiff equation: investigating solitary/shock waves and many others physical phenomena

Noor,

Alyousef,

Shafee

et al. 2024

Phys. Scr.

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show abstract

“…where Ω = −30 λ 4 (ln(κ)) 3 √ ν 3 r t + λx β β . Considering Case 2 and using (10), (16), and the corresponding solution of (9), we obtain the following families of singular stochastic soliton solutions for (1):…”

Section: Stochastic Soliton Solutionsmentioning

confidence: 99%

“…Solitons have piqued the curiosity of mathematicians and academics, prompting them to investigate soliton dynamics in both nonlinear FPDEs. As a result of their efforts, several analytical methods have emerged, including the tan-function method [12], exp-function method [13], sub-equation method [14], Kudryashov method [15], (G'/G)-expansion approach [16], Sardar sub-equation method [17], Khater method [18], sin-Gordon method [19], and mEDAM [20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Unraveling the Dynamics of Singular Stochastic Solitons in Stochastic Fractional Kuramoto–Sivashinsky Equation

Al-Sawalha,

Yasmin,

Shah

et al. 2023

Fractal Fract

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This work investigates the complex dynamics of the stochastic fractional Kuramoto–Sivashinsky equation (SFKSE) with conformable fractional derivatives. The research begins with the creation of singular stochastic soliton solutions utilizing the modified extended direct algebraic method (mEDAM). Comprehensive contour, 3D, and 2D visual representations clearly depict the categorization of these stochastic soliton solutions as kink waves or shock waves, offering a clear description of these soliton behaviors within the context of the SFKSE framework. The paper also illustrates the flexibility of the transformation-based approach mEDAM for investigating soliton occurrence not only in SFKSE but also in a wide range of nonlinear fractional partial differential equations (FPDEs). Furthermore, the analysis considers the effect of noise, specifically Brownian motion, on soliton solutions and wave dynamics, revealing the significant influence of randomness on the propagation, generation, and stability of soliton in complex stochastic systems and advancing our understanding of extreme behaviors in scientific and engineering domains.

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“…These nonlinear FPDEs have a high potential for use in a variety of domains, prompting academics to devote substantial time and effort to discovering analytical and numerical solutions [8][9][10][11][12]. To discover precise answers, Numerous efficient and trustworthy methods, including the unified method [13], exp-function approach [14], residual power series method [15], Kudryashov method [16], replicating kernel method [17], and others, have been developed by researchers.These approaches include the first integral method [18], Laplace Adomian decomposition method [19,20], natural transform decomposition method [21,22], homotopy analysis method [23], (G'/G)-enpension method [24][25][26], modified simple equation method [27], and EDAM [28,29].…”

Section: Introductionmentioning

confidence: 99%

Study of traveling soliton and fronts phenomena in fractional Kolmogorov-Petrovskii-Piskunov equation

Ullah,

Shah,

Abdeljawad

2024

Phys. Scr.

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The present research work presents the modified Extended Direct Algebraic Method (m-EDAM) to construct and analyze propagating soliton solutions for fractional Kolmogorov-Petrovskii-Piskunov Equation (FKPPE) which incorporates Caputo's fractional derivatives. The FKPPE has significance in various disciplines such as population growth, reaction-diffusion mechanisms, and mathematical biology. By leveraging the series form solution, the proposed m-EDAM determines plethora of travelling soliton solutions through the transformation of FKPPE into Nonlinear Ordinary Differential Equation (NODE). These soliton solutions shed light on propagation processes in the framework of the FKPPE model. Our study also offers some graphical representations that facilitate the characterization and investigation of propagation processes of the obtained soliton solutions which include kink, shock soliton solutions. Our work advances our understanding of complicated phenomena across multiple academic disciplines by fusing insights from mathematical biology and reaction-diffusion mechanisms.

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Families of Travelling Waves Solutions for Fractional-Order Extended Shallow Water Wave Equations, Using an Innovative Analytical Method

Cited by 30 publications

References 29 publications

A novel analytical technique for analyzing the (3+1)-dimensional fractional calogero- bogoyavlenskii-schiff equation: investigating solitary/shock waves and many others physical phenomena

A novel analytical technique for analyzing the (3+1)-dimensional fractional calogero- bogoyavlenskii-schiff equation: investigating solitary/shock waves and many others physical phenomena

Unraveling the Dynamics of Singular Stochastic Solitons in Stochastic Fractional Kuramoto–Sivashinsky Equation

Study of traveling soliton and fronts phenomena in fractional Kolmogorov-Petrovskii-Piskunov equation

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