Application of homotopy analysis method for fractional Swift Hohenberg equation – Revisited

Vishal, K.; Kumar, Sunil; Das, S.

doi:10.1016/j.apm.2011.10.001

Cited by 73 publications

(35 citation statements)

References 22 publications

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“…The HAM has been successfully applied by many researchers for solving linear and nonlinear partial differential equations. [14][15][16][17][18][19][20][21][22] In recent years, many researchers have paid attention to obtaining solutions to linear and nonlinear differential and integral equations using various methods by combining the Laplace transform method. Among these, we may mention the following: Laplace decomposition methods 23,24 and homotopy perturbation transform method.…”

Section: Introductionmentioning

confidence: 99%

A modified homotopy analysis method for solution of fractional wave equations

Yin

Kumar

2015

Advances in Mechanical Engineering

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The aim of this article is to introduce a modified analytical approach to obtain quick and accurate solution of wave-like fractional physical models. This modified analytical approach is an innovative adjustment in Laplace transform algorithm and homotopy analysis method for fractional partial differential equations. The proposed technique solves the problems using Adomian's polynomials. The homotopy analysis transform method utilizes a simple and powerful method to adjust and control the convergence region of the infinite series solution using an auxiliary parameter. The numerical solutions obtained by this modified proposed method indicate that the approach is easy to implement, highly accurate, and computationally very attractive. A good agreement between the obtained solutions and some well-known results has been obtained.

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Section: Introductionmentioning

confidence: 99%

A modified homotopy analysis method for solution of fractional wave equations

Yin

Kumar

2015

Advances in Mechanical Engineering

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“…To address this subject, Nassar et al [21] developed the HAM to derive an approximate solution of the Poisson-Boltzmann equation for semiconductor devices. The solutions of the fractional Swift Hohenberg equation can be found in the work of Vishal et al [22]. They could report influence of the real bifurcation parameter on probability density function and then concluded that the proper values of auxiliary and homotopy parameters are needed.…”

Section: Homotopy Analysis Methods (Ham)mentioning

confidence: 99%

“…It is mainly due to the fact that this parameter plays an important role in higher-order governing equations (e.g., Equation (19c)) (see Refs. [22] [33] [53] [54]). It is to be noted that a similar conclusion for increasing t can also be drawn through a similar procedure.…”

Section: Universal Graphs For Convergence Region Of the Presented Itementioning

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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“…Various definitions and basic concept of fractional calculus are present in many books [1][2][3][4]. Therefore, several analytical and numerical methods were developed for solutions of fractional differential equations (both linear and nonlinear), among which Adomian's decomposition method [5][6][7], variation iteration method [8,9], homotopy perturbation method [10][11][12], homotopy analysis method [13][14][15], homotopy asymptotic method [16][17][18],differential transform method [19], and Galerkin method [20].…”

Section: Introductionmentioning

confidence: 99%

A Modified Analytical Approach for Fractional Discrete KdV Equations Arising in Particle Vibrations

Kumar

2017

Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci.

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In this paper, we present an effectiveness of the proposed analytical algorithm for nonlinear time fractional discrete KdV equations. The discrete KdV equation play an important role in modeling of complicated physical phenomena such as particle vibrations in lattices, current flow in electrical flow and pulse in biological chains. The proposed algorithm is amagalation of the homotopy analysis method, Laplace transform method and homotopy polynomials. First, we present an alternative framework of the proposed method which can be used simply and effectively to handle nonlinear problems arising in science and engineering. Then, mainly, we address the convergence study of nonlinear fractional differential equations with Laplace transform. Camparisons are made between the results of the proposed method and exact solutions. Illustrative examples are given to demonstrate the simplicity and efficiency of the proposed method. The new results reveal that the proposed method is explicit, efficient and easy to use.

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Application of homotopy analysis method for fractional Swift Hohenberg equation – Revisited

Cited by 73 publications

References 22 publications

A modified homotopy analysis method for solution of fractional wave equations

A modified homotopy analysis method for solution of fractional wave equations

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

A Modified Analytical Approach for Fractional Discrete KdV Equations Arising in Particle Vibrations

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