Application of fractional sub-equation method to nonlinear evolution equations

Abdelkawy, Mohamed; El-Kalaawy, Omar H.; Al-Denari, Rasha B.; Biswas, Anjan

doi:10.15388/10.15388/na.2018.5.5

Cited by 3 publications

(4 citation statements)

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“…The solutions of the VOF PDEs are investigated by many authors using powerful numerical techniques. Several numerical methods, such as the homotopy perturbation method [36], the Adomian decomposition method [37], the variational iteration method [38], the differential transform method [39], the fractional Riccati expansion method [40], and the fractional subequation method [41][42][43][44], have been suggested for solving FDEs. However, solutions obtained through all these methods are local, and it is important to explore other techniques to find exact analytical solutions of FDEs [45].…”

Section: Literature Review 21 Backgroundmentioning

confidence: 99%

The analytical interface coupling of arbitrary-order fractional nonlinear hyperbolic scalar conservation laws

et al. 2020

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In this paper, the existence and uniqueness of the interface coupling (IC) of time and spatial (TS) arbitrary-order fractional (AOF) nonlinear hyperbolic scalar conservation laws (NHSCL) are investigated. The technique of arbitrary fractional characteristic method (AFCM) is used to accomplish this task. We apply Jumarie’s modification of Riemann–Liouville and Liouville–Caputo’s definition to extend some formulae to the arbitrary-order fractional calculus. Then these formulae are utilized to prove the main theorem. In this process, we develop an analytic method, which gives us the ability to find the solution of IC AOF NHSCL. The feature of this method is that it enables us to verify that the obtained solution satisfies the fractional partial differential equation (FPDE), and the solution is unique. Furthermore, a few examples illustrate the implementation of this technique in the application section.

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Section: Literature Review 21 Backgroundmentioning

confidence: 99%

The analytical interface coupling of arbitrary-order fractional nonlinear hyperbolic scalar conservation laws

et al. 2020

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“…Furthermore, there are many methods to construct solutions for FPDEs. [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] Here, we considered the three-dimensional non-linear time-fractional generalized Z-K equation in the following form:…”

Section: Introductionmentioning

confidence: 99%

“…These definitions enable us to construct the partial differential equations of fractional order (FPDEs), which are used to modulate the non‐linear phenomena in different fields supported by explaining their behaviors and physical properties. Furthermore, there are many methods to construct solutions for FPDEs 18–34 …”

Section: Introductionmentioning

confidence: 99%

The time‐fractional generalized Z‐K equation: Analysis of Lie group, similarity reduction, conservation laws, and explicit solutions

Al-Denari

Ahmed

Tharwat

2022

Math Methods in App Sciences

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In this paper, we consider the three-dimensional non-linear time-fractional generalized Zakharov-Kuznetsov (Z-K) equation that describes the dust-ion acoustic solitary waves in a magnetized dusty plasma. Based on the definition of the Riemann-Liouville (R-L) fractional derivatives, the analysis of the Lie group method can be applied to find the symmetries for the considered equation. After that, these symmetries enable us to transform the equation under study into a two-dimensional non-linear ordinary differential equation of fractional order in the sense of ErdLélyi-Kober (E-K) fractional operator. Also, we obtain a set of new analytical solutions for the time-fractional generalized Z-K equation via the power series and the fractional sub-equation methods. Furthermore, we compute the conservation laws for this equation with a detailed derivation based on the formal Lagrangian.

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“…The solutions of the FPDEs are investigated by many authors using powerful analytical methods. Several numerical methods such as the homotopy perturbation method [22,23], the Adomian decomposition method [24], the variational iteration method [25], the differential transform method [26], the fractional Riccati expansion method [27], the fractional sub-equation method [28][29][30][31][32][33][34] have been suggested for solving FDEs. However, solutions obtained through all these methods are of a local nature and it is important to explore other techniques in order to find exact analytical solutions of FDEs.…”

Section: Introductionmentioning

confidence: 99%

Conservation laws, analytical solutions and stability analysis for the time-fractional Schamel–Zakharov–Kuznetsov–Burgers equation

et al. 2019

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In this paper, we consider the (3 + 1)-dimensional time-fractional Schamel-Zakharov-Kuznetsov-Burgers (SZKB) equation. With the help of the Riemann-Liouville derivatives, the Lie point symmetries of the (3 + 1)-dimensional time-fractional SZKB equation are derived. By applying the Lie point symmetry method as well as Erdélyi-Kober fractional operator, we get the similarity reductions of the time-fractional SZKB equation. Conservation laws of the time-fractional SZKB are constructed. Moreover, we obtain its power series solutions with the convergence analysis. In addition, the analytical solution is obtained by modified trial equation method. Finally, stability is analyzed graphically in different planes.

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Application of fractional sub-equation method to nonlinear evolution equations

Cited by 3 publications

References 0 publications

The analytical interface coupling of arbitrary-order fractional nonlinear hyperbolic scalar conservation laws

The analytical interface coupling of arbitrary-order fractional nonlinear hyperbolic scalar conservation laws

The time‐fractional generalized Z‐K equation: Analysis of Lie group, similarity reduction, conservation laws, and explicit solutions

Conservation laws, analytical solutions and stability analysis for the time-fractional Schamel–Zakharov–Kuznetsov–Burgers equation

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