Improved (G'/G)-Expansion Method for the Time-Fractional Biological Population Model and Cahn–Hilliard Equation

Băleanu, Dumitru; Uğurlu, Yavuz; Kılıç, Bülent

doi:10.1115/1.4029254

Cited by 40 publications

(25 citation statements)

References 14 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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“…In this paper, we use conformable derivative properties and apply them to three nonlinear biological models. These models are a fractional biological population model [30][31][32][33][34], a fractional equal width equation [35][36][37][38], and a fractional modified equal width equation [39][40][41][42], respectively.…”

Section: Introductionmentioning

confidence: 99%

Modified Auxiliary Equation Method versus Three Nonlinear Fractional Biological Models in Present Explicit Wave Solutions

Khater

Attia

2018

MCA

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In this article, we present a modified auxiliary equation method. We harness this modification in three fundamental models in the biological branch of science. These models are the biological population model, equal width model and modified equal width equation. The three models represent the population density occurring as a result of population supply, a lengthy wave propagating in the positive x-direction, and the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes, respectively. We discuss these models in nonlinear fractional partial differential equation formulas. We used the conformable derivative properties to convert them into nonlinear ordinary differential equations with integer order. After adapting, we applied our new modification to these models to obtain solitary solutions of them. We obtained many novel solutions of these models, which serve to understand more about their properties. All obtained solutions were verified by putting them back into the original equations via computer software such as Maple, Mathematica, and Matlab.

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“…Many complex phenomena and dynamic processes in physics, mechanics, chemistry and biology can be represented by nonlinear evolution equations (NEEs) [1][2][3][4][5][6][7]. When we want to understand the physical mechanism of nature phenomena, described by NEEs, exact solutions for the NEEs have to be explored.…”

Section: Introductionmentioning

confidence: 99%

Multi-soliton rational solutions for some nonlinear evolution equations

Osman

2016

Open Physics

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The Korteweg-de Vries equation (KdV) and the (2+ 1)-dimensional Nizhnik-Novikov-Veselov system (NNV) are presented. Multi-soliton rational solutions of these equations are obtained via the generalized unied method. The analysis emphasizes the power of this method and its capability of handling completely (or partially) integrable equations. Compared with Hirota's method and the inverse scattering method, the proposed method gives more general exact multi-wave solutions without much additional e ort. The results show that, by virtue of symbolic computation, the generalized uni ed method may provide us with a straightforward and e ective mathematical tool for seeking multi-soliton rational solutions for solving many nonlinear evolution equations arising in di erent branches of sciences.

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Improved (G'/G)-Expansion Method for the Time-Fractional Biological Population Model and Cahn–Hilliard Equation

Cited by 40 publications

References 14 publications

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

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

Modified Auxiliary Equation Method versus Three Nonlinear Fractional Biological Models in Present Explicit Wave Solutions

Multi-soliton rational solutions for some nonlinear evolution equations

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