Bifurcations of traveling wave solutions for a two-component Fornberg–Whitham equation

Fan, Xinghua; Yang, Shouxiang; Yin, Jiuli; Li, Tian

doi:10.1016/j.cnsns.2011.02.010

Cited by 14 publications

(13 citation statements)

References 16 publications

(25 reference statements)

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“…Let us surmise the infinite sum f( 1 , ϑ; ζ) = ∞ ∑ q=0 f q ( 1 , ϑ; ζ), (q = 0, 1, 2, ...) accompanying it with (45) and affirming the non-linearity. Therefore, (52) takes the form…”

Section: Case I (For the Fuzzy Caputo Fractional Derivative)mentioning

confidence: 99%

“…The FWE has been discovered to need peakon outcomes as a model for controlling wave heights and wave break frequency. Recently, various sorts of FWE models in physics have been investigated by Abidi and Omrani [37], Gupta and Singh [38], Lu [39], Sakar et al [40], Chen et al [41], Yin et al [42], Zhou and Tian [43], He et al [44], Fan et al [45], Jiang and Bi [46]. This research creates a modified fuzzy EADM framework to assess the fuzzy time fractional FWE.…”

Section: Introductionmentioning

confidence: 99%

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Analytic Fuzzy Formulation of a Time-Fractional Fornberg–Whitham Model with Power and Mittag–Leffler Kernels

et al. 2021

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This manuscript assesses a semi-analytical method in connection with a new hybrid fuzzy integral transform and the Adomian decomposition method via the notion of fuzziness known as the Elzaki Adomian decomposition method (briefly, EADM). Moreover, we use the aforesaid strategy to address the time-fractional Fornberg–Whitham equation (FWE) under gH-differentiability by employing different initial conditions (IC). Several algebraic aspects of the fuzzy Caputo fractional derivative (CFD) and fuzzy Atangana–Baleanu (AB) fractional derivative operator in the Caputo sense, with respect to the Elzaki transform, are presented to validate their utilities. Apart from that, a general algorithm for fuzzy Caputo and AB fractional derivatives in the Caputo sense is proposed. Some illustrative cases are demonstrated to understand the algorithmic approach of FWE. Taking into consideration the uncertainty parameter ζ∈[0,1] and various fractional orders, the convergence and error analysis are reported by graphical representations of FWE that have close harmony with the closed form solutions. It is worth mentioning that the projected approach to fuzziness is to verify the supremacy and reliability of configuring numerical solutions to nonlinear fuzzy fractional partial differential equations arising in physical and complex structures.

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Section: Case I (For the Fuzzy Caputo Fractional Derivative)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Analytic Fuzzy Formulation of a Time-Fractional Fornberg–Whitham Model with Power and Mittag–Leffler Kernels

et al. 2021

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“…Consequently, these two systems have the same topological phase portraits except for the straight line = 0. Thus, we can understand the phase portraits of system (9) from those of system (10). In order to state conveniently, for given constants and , let 3 ,…”

Section: Bifurcation Phase Portraitsmentioning

confidence: 99%

“…They investigated the existence of the smooth and nonsmooth traveling wave solutions and gave some analytic expressions of smooth solitary wave, periodic cusp wave, and peakon solutions for (3). Fan et al [10] presented a two-component Fornberg…”

Section: Introductionmentioning

confidence: 99%

Kink-Like Wave and Compacton-Like Wave Solutions for a Two-Component Fornberg-Whitham Equation

Song

2014

Abstract and Applied Analysis

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Using bifurcation method and numerical simulation approach of dynamical systems, we study a two-component Fornberg-Whitham equation. Two types of bounded traveling wave solutions are found, that is, the kink-like wave and compacton-like wave solutions. The planar graphs of these solutions are simulated by using software Mathematica; meanwhile, two new phenomena are revealed; that is, the periodic wave solution can become the kink-like wave or compacton-like wave solution under some conditions, respectively. Exact implicit or parameter expressions of these solutions are given.

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“…Study on two-component equations has drawn a lot of interest among researchers [1][2][3][4][5][6]. The two-component equation we are going to discuss in the present paper is the dual Ito equation [7]…”

Section: Introductionmentioning

confidence: 98%

Bifurcation of Traveling Wave Solutions of the Dual Ito Equation

Fan

2014

Abstract and Applied Analysis

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The dual Ito equation can be seen as a two-component generalization of the well-known Camassa-Holm equation. By using the theory of planar dynamical system, we study the existence of its traveling wave solutions. We find that the dual Ito equation has smooth solitary wave solutions, smooth periodic wave solutions, and periodic cusp solutions. Parameter conditions are given to guarantee the existence.

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Bifurcations of traveling wave solutions for a two-component Fornberg–Whitham equation

Cited by 14 publications

References 16 publications

Analytic Fuzzy Formulation of a Time-Fractional Fornberg–Whitham Model with Power and Mittag–Leffler Kernels

Analytic Fuzzy Formulation of a Time-Fractional Fornberg–Whitham Model with Power and Mittag–Leffler Kernels

Kink-Like Wave and Compacton-Like Wave Solutions for a Two-Component Fornberg-Whitham Equation

Bifurcation of Traveling Wave Solutions of the Dual Ito Equation

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