Analytic approximate solution for the Bratu's problem by optimal homotopy analysis method

Hassan, Hany N.; Semary, Mourad S.

doi:10.5899/2013/cna-00139

Cited by 13 publications

(9 citation statements)

References 19 publications

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“…(3.12) is linear and thus can be easily solved, especially by means of symbolic computation software such as Mathematica, Maple, Matlab. Yabushita et al [29] and Mohamed S. Mohamed et al [30,31,32] applied the homotopy analysis method to nonlinear ODE's and suggested the so called optimization method to find out the optimal convergence control parameters by minimum of the square residual error integrated in the whole region having physical meaning. Their approach is based on the square residual error.…”

Section: The Optimal Homotopy Analysis Methods (Oham)mentioning

confidence: 99%

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Analytical Approximate Solution For Nonlinear Time-Space Fractional Fornberg-Whitham Equation By Fractional Complex Transform

Hamed¹,

Mohamed²,

El‐Zahar³

2015

Communications in Numerical Analysis

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In this article, fractional complex transform with optimal homotopy analysis method (OHAM) is used to obtain numerical and analytical solutions for the nonlinear time-space fractional Fornberg-Whitham. Fractional complex transform is proposed to convert time-space fractional Fornberg-Whitham equation to the nonlinear ordinary differential equations and then applied OHAM to the new obtained equations. This optimal approach has general meaning and can be used to get fast convergent series solutions of the different type of nonlinear fractional differential equation. The results reveal that the method is very effective, simple and the OHAM contains a certain auxiliary parameter h which provides us with a simple way to adjust and control the convergence region of convergence of the series solution.

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Section: The Optimal Homotopy Analysis Methods (Oham)mentioning

confidence: 99%

“…The approximate solutions of Eq. (5.24) take the following form at β = 1, 32) and so on. According to the HAM, we can conclude that…”

Section: Applicationsmentioning

confidence: 99%

Analytical Approximate Solution For Nonlinear Time-Space Fractional Fornberg-Whitham Equation By Fractional Complex Transform

Hamed¹,

Mohamed²,

El‐Zahar³

2015

Communications in Numerical Analysis

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“…3 when a ¼ 2. The integer order case was solved using the semi-analytic and numerical techniques (Hassan and El-Tawil, 2011;Hassan and Semary, 2013;Jalilian, 2010;Kafri and Khuri, 2016;Semary and Hassan, 2015) using spline method (Jalilian, 2010), variational iteration method (Semary and Hassan, 2015), homotopy analysis method (Hassan and El-Tawil, 2011;Hassan and Semary, 2013) and other methods.…”

Section: Numerical Examplesmentioning

confidence: 99%

Single and dual solutions of fractional order differential equations based on controlled Picard’s method with Simpson rule

Semary

Hassan

Radwan

2017

Journal of the Association of Arab Universities for Basic and A

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This paper presents a semi-analytical method for solving fractional differential equations with strong terms like (exp, sin, cos,.. .). An auxiliary parameter is introduced into the well-known Picard's method and so called controlled Picard's method. The proposed approach is based on a combination of controlled Picard's method with Simpson rule. This approach can cover a wider range of integer and fractional orders differential equations due to the extra auxiliary parameter which enhances the convergence and is suitable for higher order differential equations. The proposed approach can be effectively applied to Bratu's problem in fractional order domain to predict and calculate all branches of problem solutions simultaneously. Also, it is tested on other fractional differential equations like nonlinear fractional order Sine-Gordon equation. The results demonstrate reliability, simplicity and efficiency of the approach developed.

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“…al [28] and Mohamed S. Mohamed et. al [29][30][31] applied the homotopy analysis method to nonlinear ODE's and suggested the so called optimization method to find out the optimal convergence control parameters by minimum of the square residual error integrated in the whole region having physical meaning. Their approach is based on the square residual error.…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

Analytic approximate solution for some integral equations by optimal homotopy analysis transform method

Mohamed¹,

Alharthi²,

Alotabi³

2015

Communications in Numerical Analysis

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The main aim of this paper is to propose a new and simple algorithm namely homotopy analysis transform method (HATM), to obtain approximate analytical solutions of integral equations. Integral equation occurs in the mathematical modeling of several models in physics, astrophysics, solid mechanics and applied sciences. The numerical solutions obtained by proposed method indicate that the approach is easy to implement and computationally very attractive. Finally, several numerical examples are given to illustrate the accuracy and stability of this method. Comparison of the approximate solution with the exact solutions also we show that the proposed method is very efficient and computationally attractive. A new efficient approach is proposed to obtain the optimal value of convergence controller parameter ℏ to guarantee the convergence of the obtained series solution.

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Analytic approximate solution for the Bratu's problem by optimal homotopy analysis method

Cited by 13 publications

References 19 publications

Analytical Approximate Solution For Nonlinear Time-Space Fractional Fornberg-Whitham Equation By Fractional Complex Transform

Analytical Approximate Solution For Nonlinear Time-Space Fractional Fornberg-Whitham Equation By Fractional Complex Transform

Single and dual solutions of fractional order differential equations based on controlled Picard’s method with Simpson rule

Analytic approximate solution for some integral equations by optimal homotopy analysis transform method

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