New exact solutions of the conformable time-fractional Cahn–Allen and Cahn–Hilliard equations using the modified Kudryashov method

Hosseini, K.; Bekir, Ahmet; Ansari, R.

doi:10.1016/j.ijleo.2016.12.032

Cited by 108 publications

(40 citation statements)

References 40 publications

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“…[27][28][29][30][31][32][33] Since most of the complex phenomena are modeled mathematically by nonlinear fractional differential equations, there are many methods in the literature to solve these equations. The Gardner and Cahn-Hilliard equations are studied through distinct techniques such as reduced differential transform method, 34 the modified Kudryashov technique, 35 Adomian decomposition method (ADM), 36 improved (G ′ /G) − expansion method, 37 homotopy perturbation method (HPM), 26 residual power series method (RPSM), 22 and many others. 38,39 In this framework, we employ two distinct and efficient techniques, ie, fractional natural decomposition method (FNDM) and q-homotopy analysis transform methodq-HATM, to find the solution for both cited equations.…”

Section: Introductionmentioning

confidence: 99%

Two novel computational techniques for fractional Gardner and Cahn‐Hilliard equations

Prakasha

Veeresha

Baskonus

2019

Comp and Math Methods

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The numerical solutions for nonlinear fractional Gardner and Cahn-Hilliard equations arising in fluids flow are obtained with the aid of two novel techniques, namely, fractional natural decomposition method (FNDM) and q-homotopy analysis transform method (q-HATM). Both featured techniques are different from each other since FNDM is algorithmic by the aid of Adomian polynomial and q-HATM is defined by the help of homotopy polynomial. The numerical simulations have been conducted to verify that the proposed schemes are reliable and accurate. The outcomes are revealed through the plots and tables. The comparison of solution obtained by proposed schemes with the available solutions exhibits that both the featured schemes are methodical, efficient, and very exact in solving the nonlinear complex phenomena. KEYWORDS fractional Cahn-Hilliard equation, fractional Gardner equation, fractional natural decomposition method, q-homotopy analysis transform methodwhere is real constant. Here, v (x, t) is the wave function with scaling variables space (x) and time (t), the terms vv x and v 2 v x are symbolize by the nonlinear wave steepening, and v xxx denotes the dispersive wave effects.

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Section: Introductionmentioning

confidence: 99%

Two novel computational techniques for fractional Gardner and Cahn‐Hilliard equations

Prakasha

Veeresha

Baskonus

2019

Comp and Math Methods

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“…The amalgamation of Taylor's series expansion and DE algorithm, name as TOM, considered in this attempt, exhibits a remarkable tool to acquire the effective solutions of functions together with the globally optimized values of the error function. In addition to this, the illustrative examples, considered with conformable fractional derivative [5,6,10], elevated the efficiency, stability and appropriateness of TOM.…”

Section: Introductionmentioning

confidence: 99%

“…For the reason that the non-local properties of fractional operator enable these differential models to put the information, about the recent and the historical situation, in a nutshell [1]. For the last decades, many enlargements have been made in this regard to explore enhanced definitions and properties in order to overcome the inadequacies of previous definitions of fractional calculus, such as, He's fractional derivative [3], Atangana-Baleanu fractional derivative [4], Caputo and Fabrizio [5], conformable derivative [6], etc. Consequently, these novel aspects enrich the capabilities of fractional differential models in bringing diverse physical significances to light [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

A heuristic optimization method of fractional convection reaction: An application to diffusion process

et al. 2018

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The convection differential models play an essential role in studying different chemical process and effects of the diffusion process. This paper intends to provide optimized numerical results of such equations based on the conformable fractional derivative. Subsequently, a well-known heuristic optimization technique, differential evolution algorithm, is worked out together with the Taylor's series expansion, to attain the optimized results. In the scheme of the Taylor optimization method (TOM), after expanding the functions with the Taylor's series, the unknown terms of the series are then globally optimized using differential evolution. Moreover, to illustrate the applicability of TOM, some examples of linear and non-linear fractional convection diffusion equations are exemplified graphically. The obtained assessments and comparative demonstrations divulged the rapid convergence of the estimated solutions towards the exact solutions. Comprising with an effective expander and efficient optimizer, TOM reveals to be an appropriate approach to solve different fractional differential equations modeling various problems of engineering.

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“…Nowadays, by developing a specific transformation, a partial differential equation can be converted into an ordinary differential equation. This conversion causes the produced ordinary differential equation to be solved readily by means of a group of powerful approaches, such as functional variable method [1][2][3], first integral method [4][5][6][7], exp-function method [8][9][10], modified trial equation method [11][12][13], the Kudryashov methods [14][15][16][17][18][19] and so on. The modified Kudryashov method is considered as one of the most robust techniques which has been developed for solving nonlinear differential equations.…”

Section: Introductionmentioning

confidence: 99%

New Optical Solitons of the Longitudinal Wave Equation in a Magnetoelectro-Elastic Circular Rod

et al. 2018

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This paper studies a nonlinear partial differential equation known as the nonlinear longitudinal wave equation which describes the propagation of optical solitons in a magnetoelectro-elastic circular rod. Mentioned task is performed by solving this nonlinear equation using the well-designed modified Kudryashov method, producing a series of new optical solitons. The capability of the modified Kudryashov method in handling the nonlinear longitudinal wave equation in a magnetoelectro-elastic circular rod is confirmed.

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New exact solutions of the conformable time-fractional Cahn–Allen and Cahn–Hilliard equations using the modified Kudryashov method

Cited by 108 publications

References 40 publications

Two novel computational techniques for fractional Gardner and Cahn‐Hilliard equations

Two novel computational techniques for fractional Gardner and Cahn‐Hilliard equations

A heuristic optimization method of fractional convection reaction: An application to diffusion process

New Optical Solitons of the Longitudinal Wave Equation in a Magnetoelectro-Elastic Circular Rod

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