A new approach for a class of nonlinear boundary value problems with multiple solutions

Semary, Mourad S.; Hassan, Hany N.

doi:10.1016/j.jaubas.2013.11.002

Cited by 8 publications

(17 citation statements)

References 24 publications

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“…The valid values of h can be determined by means of the so-called h-curves [24] [25] [26]. According to this h-curve, it can be easily determine the valid region of h, which corresponds to the line segment nearly parallel to the horizontal axis.…”

Section: Variation Of Parameters Methods With An Auxiliary Parametermentioning

confidence: 99%

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Approximate Solution of Lane-Emden Type Equations Using Variation of Parameters Method with an Auxiliary Parameter

Khalifa¹,

Hassan²

2019

JAMP

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Lane-Emden type equation is a nonlinear differential equation appears in many fields such as stellar structure, radioactive cooling and modeling of clusters of galaxies. In this work, this equation is investigated using a semi-analytical method called the Variation of parameters method with an auxiliary parameter. In the applied technique, an unknown auxiliary parameter is inserted in Variation of Parameters Method to solve some special cases of these equations. The used algorithm is easy to implement and very effective. The obtained solutions are also fairly accurate.

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Section: Variation Of Parameters Methods With An Auxiliary Parametermentioning

confidence: 99%

“…Now we apply Variation of Parameters Method (VPM) coupled with an unknown auxiliary parameter developed in Section 2 for solving Equation (1). The Journal of Applied Mathematics and Physics recursive formulas (22) and (24) for this problem is obtained respectively as…”

Section: Approximate Solution Of the Lane-emden Equationmentioning

confidence: 99%

Approximate Solution of Lane-Emden Type Equations Using Variation of Parameters Method with an Auxiliary Parameter

Khalifa¹,

Hassan²

2019

JAMP

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“…So, it should be emphasized that the iteration formula (9) is suitable to solve the problem (1) for integer and fractional orders. The variational iteration method (VIM) is one of the famous techniques used to solve linear and nonlinear differential equations (Ghaneai and Hosseini, 2015;He, 1999;Wazwaz, 2009a;Semary and Hassan, 2015). To solve the integer order differential Eq.…”

Section: The Methodologymentioning

confidence: 99%

“…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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“…In recent years, there has appeared an ever increasing interest of scientist and engineers in analytical techniques for studying Klein-Gordon and sine-Gordon equations. The Klein-Gordon equation was solved by many analytical methods such as variational iteration method (Elcin, 2008;Semary and Hassan, 2015) differential transform method (Ravi Kanth and Aruna, 2009), the decomposition method (El-Sayed, 2003), homotopy perturbation method (Chowdhury and Hashim, 2009), new homotopy perturbation method (Biazar and Mostafa, 2014), Local fractional series expansion method (Yang et al, 2014), and the tanh and the sine-cosine methods (Wazwaz, 2005). Also, the sine-Gordon equation has been solved analytically by modified decomposition method (Kaya, 2003), variational iteration method (Batiha, 2007;Abbasbandy, 2007), and differential transform method (Biazar and Mohammadi, 2010).…”

Section: Introductionmentioning

confidence: 99%

Approximate solutions for solving the Klein–Gordon and sine-Gordon equations

Yousif

Mahmood

2017

Journal of the Association of Arab Universities for Basic and A

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In this paper, we practiced relatively new, analytical method known as the variational homotopy perturbation method for solving Klein-Gordon and sine-Gordon equations. To present the present method's effectiveness many examples are given. In this study, we compare numerical results with the exact solutions, the Adomian decomposition method (ADM), the variational iteration method (VIM), homotopy perturbation method (HPM), modified Adomian decomposition method (MADM), and differential transform method (DTM). The results reveal that the VHPM is very effective.

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A new approach for a class of nonlinear boundary value problems with multiple solutions

Cited by 8 publications

References 24 publications

Approximate Solution of Lane-Emden Type Equations Using Variation of Parameters Method with an Auxiliary Parameter

Approximate Solution of Lane-Emden Type Equations Using Variation of Parameters Method with an Auxiliary Parameter

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

Approximate solutions for solving the Klein–Gordon and sine-Gordon equations

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