New applications of the two variable (G′/G, 1/G)-expansion method for closed form traveling wave solutions of integro-differential equations

Miah, M. Mamun; Ali, Hegagi Mohamed; Akbar, M. Ali; Seadawy, Aly R.

doi:10.1016/j.joes.2019.03.001

Cited by 43 publications

(25 citation statements)

References 25 publications

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“…Particularly, the values of the time-fractional order used for the following simulations are α = 1, 0.7 and 0.3. Solutions (41) and (46) of Equation ( 1) and solution (65) of Equation (2) are selected to present in terms of 3D, 2D, and contour plots according to the values of α. All of the 3D solution graphs of ( 1) and ( 2) are portrayed on the domain {(x, y, t) : 0 ≤ x, t ≤ 10, y = 1} and {(x, y, z, t) : 0 ≤ x, t ≤ 10, y = z = 1}, respectively.…”

Section: Graphical Representations Of the Selected Solutionsmentioning

confidence: 99%

“…By classifying the shapes of the 3D and 2D graphs in Figure 1, it can be identified that solution ( 41) is a singular singlesoliton solution that is a solitary wave with discontinuous derivatives occurring at some domain regions, as observed in the contour plots of Figure 1. In addition, Figure 2a-c shows the 3D, 2D, and contour plots for solution (46), respectively, when α = 1. Figure 2d-f and Figure 2g-i are drawn in a similar manner to the above plots except using α = 0.7 and α = 0.3, respectively.…”

Section: Graphical Representations Of the Selected Solutionsmentioning

confidence: 99%

“…A recent literature review for constructing explicit exact solutions of the integrodifferential SK equation using various methods such as the Hirota bilinear method, the (G /G, 1/G)-expansion method, and the generalized Kudryashov method (GKM) can be found in [32][33][34][35][36]46,47]. Furthermore, some scientists have devoted substantial efforts to finding exact solutions of the (3 + 1)-dimensional mKdV-ZK equation in the sense of the classical partial, conformable, and Jumarie's modified Riemann-Liouville derivatives using different and reliable approaches.…”

Section: Introductionmentioning

confidence: 99%

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Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations

et al. 2021

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The core objective of this article is to generate novel exact traveling wave solutions of two nonlinear conformable evolution equations, namely, the (2+1)-dimensional conformable time integro-differential Sawada–Kotera (SK) equation and the (3+1)-dimensional conformable time modified KdV–Zakharov–Kuznetsov (mKdV–ZK) equation using the (G′/G2)-expansion method. These two equations associate with conformable partial derivatives with respect to time which the former equation is firstly proposed in the form of the conformable integro-differential equation. To the best of the authors’ knowledge, the two equations have not been solved by means of the (G′/G2)-expansion method for their exact solutions. As a result, some exact solutions of the equations expressed in terms of trigonometric, exponential, and rational function solutions are reported here for the first time. Furthermore, graphical representations of some selected solutions, plotted using some specific sets of the parameter values and the fractional orders, reveal certain physical features such as a singular single-soliton solution and a doubly periodic wave solution. These kinds of the solutions are usually discovered in natural phenomena. In particular, the soliton solution, which is a solitary wave whose amplitude, velocity, and shape are conserved after a collision with another soliton for a nondissipative system, arises ubiquitously in fluid mechanics, fiber optics, atomic physics, water waves, and plasmas. The method, along with the help of symbolic software packages, can be efficiently and simply used to solve the proposed problems for trustworthy and accurate exact solutions. Consequently, the method could be employed to determine some new exact solutions for other nonlinear conformable evolution equations.

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Section: Graphical Representations Of the Selected Solutionsmentioning

confidence: 99%

Section: Graphical Representations Of the Selected Solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations

et al. 2021

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“…Physicists, engineers and mathematicians have established precise solutions for the NLEEs that transform into a key task in the analysis of nonlinear physical incidents. Generally, there is no single approach to resolve all sorts of NLEEs, and a variety of scientific groups have successfully established many techniques, including the method of extended tanh-function [1], the method of tanh function [2,3], the method of sine-cosine [4], the method of modified extended tanh function [5], the method of exp (−ϕ(ξ ))-expansion [6], the method of exp-function [7], the method of Backlund transformation [8], the method of Miura transformation [9], the method of extended F-expansion [10], the method of improved F-expansion [11], the method of ( ′/ / G G G , 1 )-expansion [18], the method of (G′/G)expansion [12][13][14][15], the method of new auxiliary equation [16], the method of auxiliary equation [17], the method of Kudryashov [18][19][20][21], the method of modified Kudryashov [29], the method of extended trial function [31], the method of variational iteration [32], the method of modified simple equation (MSE) [22][23][24][25][26][27][28][29][30][31][32], extended modified direct algebraic method, extended mapping method and Seadawy techniques to find solutions for some nonlinear partial differential equations [33][34]…”

Section: Introductionmentioning

confidence: 99%

Stable solutions to the nonlinear RLC transmission line equation and the Sinh–Poisson equation arising in mathematical physics

et al. 2020

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The Sinh–Poisson equation and the RLC transmission line equation are important nonlinear model equations in the field of engineering and power transmission. The modified simple equation (MSE) procedure is a realistic, competent and efficient mathematical scheme to ascertain the analytic soliton solutions to nonlinear evolution equations (NLEEs). In the present article, the MSE approach is put forward and exploited to establish wave solutions to the previously referred NLEEs and accomplish analytical broad-ranging solutions associated with parameters. Whenever parameters are assigned definite values, diverse types of solitons originated from the general wave solutions. The solitons are explained by sketching three-dimensional and two-dimensional graphs, and their physical significance is clearly stated. The profiles of the attained solutions assimilate compacton, bell-shaped soliton, peakon, kink, singular periodic, periodic soliton and singular kink-type soliton. The outcomes assert that the MSE scheme is an advance, convincing and rigorous scheme to bring out soliton solutions. The solutions obtained may significantly contribute to the areas of science and engineering.

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“…expansion method [30,31], the 1 G expansion method [32,33], the decomposition Sumudu-like-integral transform method [34], and the inverse scattering method [35].…”

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confidence: 99%

Optical Soliton Solutions of the Cubic-Quartic Nonlinear Schrödinger and Resonant Nonlinear Schrödinger Equation with the Parabolic Law

et al. 2019

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In this paper, the cubic-quartic nonlinear Schrödinger and resonant nonlinear Schrödinger equation in parabolic law media are investigated to obtain the dark, singular, bright-singular combo and periodic soliton solutions. Two powerful methods, the m + G ′ G improved expansion method and the exp − φ ξ expansion method are utilized to construct some novel solutions of the governing equations. The obtained optical soliton solutions are presented graphically to clarify their physical parameters. Moreover, to verify the existence solutions, the constraint conditions are utilized.

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New applications of the two variable (G′/G, 1/G)-expansion method for closed form traveling wave solutions of integro-differential equations

Cited by 43 publications

References 25 publications

Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations

Applications of the (G′/G2)-Expansion Method for Solving Certain Nonlinear Conformable Evolution Equations

Stable solutions to the nonlinear RLC transmission line equation and the Sinh–Poisson equation arising in mathematical physics

Optical Soliton Solutions of the Cubic-Quartic Nonlinear Schrödinger and Resonant Nonlinear Schrödinger Equation with the Parabolic Law

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