Symmetries and exact solutions of the time fractional Harry-Dym equation with Riemann–Liouville derivative

Huang, Qing; Zhdanov, Renat

doi:10.1016/j.physa.2014.04.043

Cited by 125 publications

(45 citation statements)

References 15 publications

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“…Since the question raised by L'Hospital in 1695 on noninteger order derivative, many researchers tried to establish the definition of a fractional derivative and some of them presented it in the integral form. In this section, we recall some of those definitions, and for more details, we refer to other works …”

Section: Preliminariesmentioning

confidence: 99%

Exact solutions for some time‐fractional evolution equations using Lie group theory

Bira

Sekhar

Zeidan

2018

Math Methods in App Sciences

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In this paper, some classes of nonlinear partial fractional differential equations arising in some important physical phenomena are considered. Lie group method is applied to investigate the symmetry group of transformations under which the governing time-fractional partial differential equation remains invariant. The symmetry generators are used for constructing similarity variables, which leads to a reduced ordinary differential equation of Erdélyi-Kober fractional derivatives. Furthermore, a particular exact solution for each governing equation(s) is constructed. Moreover, the physical significance of the solution is investigated graphically based on numerical simulations in order to highlight the importance of the study.

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Section: Preliminariesmentioning

confidence: 99%

Exact solutions for some time‐fractional evolution equations using Lie group theory

Bira

Sekhar

Zeidan

2018

Math Methods in App Sciences

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“…As reported by Marciniak and Błaszak [7], if a classical Stäckel system is applied to construct the coupled Harry-Dym (cHD) hierarchy, both nonlocal and purely differential parts of hierarchies are obtained. Using the RiemannLiouville derivative, Huang and Zhdanov [8] recently investigated the time fractional nonlinear Harry-Dym equation with the Lie symmetry and obtained some invariant solutions explicitly. Brunelli and da Costa [9] studied a large class of nonlocal equations and charges for the Harry-Dym hierarchy.…”

Section: Introductionmentioning

confidence: 99%

A Mathematical Approach Based on the Homotopy Analysis Method: Application to Solve the Nonlinear Harry-Dym (HD) Equation

Ghiasi¹,

Saleh²

2017

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In this paper, the homotopy analysis method (HAM) has been employed to obtain the approximate analytical solution of the nonlinear Harry-Dym (HD) equation, which is one of the most important soliton equations. Utilizing the HAM, thereby employing the initial approximation, variations of the 7th-order approximation of the Harry-Dym equation is obtained. It is found that effect of the nonzero auxiliary parameter on convergence rate of the series solution is undeniable. It is also shown that, to some extent, order of the fractional derivative plays a fundamental role in the prediction of convergence. The final results reported by the HAM have been compared with the exact solution as well as those obtained through the other methods.

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“…In [22], Wang et al studied the invariance properties of the time fractional fifth-order KdV equations using the Lie group analysis method. Huang and Zhdanov, in [13], determined the maximal symmetry group and the corresponding optimal system of subgroups for the time fractional Harry-Dym equation using the Lie group analysis method. Moreover, in [28], Zedan and Alshamrani considered the (2+1)-dimensional higher-order Broer-Kaup System and used Lie symmetry analysis to determine the optimal systems for subalgebras and their corresponding solutions.…”

Section: Introductionmentioning

confidence: 99%

Lie group method and fractional differential equations

Zedan¹,

Abu-Nawas²

2017

J. Nonlinear Sci. Appl.

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Symmetries and exact solutions of the time fractional Harry-Dym equation with Riemann–Liouville derivative

Cited by 125 publications

References 15 publications

Exact solutions for some time‐fractional evolution equations using Lie group theory

Exact solutions for some time‐fractional evolution equations using Lie group theory

A Mathematical Approach Based on the Homotopy Analysis Method: Application to Solve the Nonlinear Harry-Dym (HD) Equation

Lie group method and fractional differential equations

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