2010
DOI: 10.1016/j.apm.2009.09.011
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Solution of nonlinear fractional differential equations using homotopy analysis method

Abstract: a b s t r a c tIn this article, the homotopy analysis method has been applied to solve nonlinear differential equations of fractional order. The validity of this method has successfully been accomplished by applying it to find the solution of two nonlinear fractional equations. The results obtained by homotopy analysis method have been compared with those exact solutions. The results show that the solution of homotopy analysis method is good agreement with the exact solution.Crown

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Cited by 71 publications
(61 citation statements)
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“…Similarly, HPMs were also modified for solving the fractional Riccati differential equation [13][14][15]. Moreover, some modified Homotopy analysis method [11,[16][17][18][19] and variational iteration method [20,21] were also proposed to solve the nonlinear fractional differential equations.…”
Section: Introductionmentioning
confidence: 99%
“…Similarly, HPMs were also modified for solving the fractional Riccati differential equation [13][14][15]. Moreover, some modified Homotopy analysis method [11,[16][17][18][19] and variational iteration method [20,21] were also proposed to solve the nonlinear fractional differential equations.…”
Section: Introductionmentioning
confidence: 99%
“…To find explicit solutions of linear and nonlinear fractional differential equations, many powerful methods have been used such as the homotopy perturbation method (Momani and Odibat, 2007;Wang, 2008;Gupta and Singh, 2011), the Adomain decomposition method (Ray, 2009;Herzallah and Gepreel, 2012;Rida et al, 2008), the variational iteration method (He, 2000(He, , 2004(He, , 2007He and Wang, 2007), the homotopy analysis method (Hemida et al, 2012;Gepreel and Mohamed, 2013;Ganjiani, 2010;Behzadi, 2011), the fractional complex transform (Ghazanfari, 2012;Su et al, 2013), the homotopy perturbation Sumudu transform method (Karbalaie et al, 2014;Mahdy et al, 2015), the local fractional variation iteration method He and Liu, 2013;Yang et al, 2014), the local fractional Adomain decomposition method (Yang et al, 2013b), the Cantor-type Cylindrical-Coordinate method (Yang et al, 2013c), the variational iteration method with Yang-Laplace , the Yang-Fourier transform (Yang et al, 2013a), the Yang-Laplace transform (Zhao et al, 2014;Zhang et al, 2014) and variational homotopy perturbation method by (Noor and Mohyud-Din,2008). The variational homotopy perturbation method (VHPM) is a combination of the variational iteration method and homotopy perturbation method.…”
Section: Introductionmentioning
confidence: 99%
“…For better understanding of the complicated nonlinear physical phenomena, the solution of the fractional differential equation is much involved. In the past, various methods have been proposed to obtain solutions of FPDEs, such as homotopy perturbation method [12][13][14], homotopy perturbation Sumudu transform method [15,16], Adomian decomposition method [17,18], homotopy analysis method [19], fractional variational iteration method [20,21], finite difference method [22], fractional sub-ODE method [23][24][25][26], and so on. Based on these methods, many fractional differential equations have been investigated.…”
Section: Introductionmentioning
confidence: 99%