2019
DOI: 10.3390/math7010040
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Abstract: The analysis of Homotopy Perturbation Method (HPM) for the solution of fractional partial differential equations (FPDEs) is presented. A unified convergence theorem is given. In order to validate the theory, the solution of fractional-order Burger-Poisson (FBP) equation is obtained. Furthermore, this work presents the method to find the solution of FPDEs, while the same partial differential equation (PDE) with ordinary derivative i.e., for α = 1 , is not defined in the given domain. Moreover, HPM is app… Show more

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Cited by 64 publications
(29 citation statements)
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“…This creates rooms for memory settings of the systems. The applications of FC are widely seen in [9][10][11]. In financial mathematics, Jumaris [12,13] introduced FC to option pricing with its base in the Black-Scholes pricing model (Financial derivatives).…”
Section: Introductionmentioning
confidence: 99%
“…This creates rooms for memory settings of the systems. The applications of FC are widely seen in [9][10][11]. In financial mathematics, Jumaris [12,13] introduced FC to option pricing with its base in the Black-Scholes pricing model (Financial derivatives).…”
Section: Introductionmentioning
confidence: 99%
“…For example of these methods, the Hirotaʼs bilinear transformation method (Zhou and Ma 2017), the Backlund transform method (Arnous et al 2015), the inverse scattering method (Ablowitz and Musslimani 2016), the first integral method (Tascan and Bekir 2010), the exp-function method (Ma et al 2010), the tanh-function method , Abdel et al 2011, the Jacobi elliptic function method (Ma et al 2018), the ¢ ( ) G G -expansion method (Bekir and Guner 2013;Bekir and Cevikel 2009), the extended ¢ ( ) G G -expansion method (Roshid et al 2014, Alam andBelacem 2015), the improved ¢ ( ) G G -expansion method (Redi et al 2018), the new generalized ¢ ( ) G G -expansion method (Alam 2015, Alam and Stepanyants 2016, Alam and Li 2019, the generalized and improved ¢ ( ) G G -expansion method(Akbar et al 2012aG -expansion method(Akbar et al , 2012b. The Weierstrass elliptic function method (Ping and Li 2008), the truncated Painleve expansion method (Radha et al 2007), the functional variable method (Khan and Akbar 2014), the j x -( ( )) exp -expansion method (Alam and Belacem 2016; Khater 2016, Alam and Tunc 2016, Alam and Alam 2017, the modified simple equation method (Roshid and Roshid 2018), the auxiliary equation method (Kaplan et al 2015, Akbulut et al 2016, the rational exponential function method (Roshid and Alam 2017), the homotopy perturbation method (Javeed et al 2019), the Riccati equation mapping method (Naher et al 2013), the ¢ ( ) G G G , 1 -expansion method (Kaplan et al 2016), etc.…”
Section: Introductionmentioning
confidence: 99%
“…In the present years, fractional calculus has become widespread because of its applications in mathematical biology, electrochemistry, and physics [1][2][3][4][5][6][7][8]. For example, the earthquake model [9] and traffic model [10] with fractional derivatives have been demonstrated. However, sometimes, it is challenging to find the exact and numerical solutions of these models.…”
Section: Introductionmentioning
confidence: 99%
“…However, sometimes, it is challenging to find the exact and numerical solutions of these models. During the last few decades, several analytical and numerical approaches have been established for the solution of such types of models such as homotopy perturbation method (HPM) [10,11], homotopy perturbation transform method [12,13], homotopy analysis method (HAM) [14,15], Adomian decomposition method (ADM) [16,17], sine-cosine method [18] and transform method [19]. Recently, multi-dimensional diffusion equation of fractional order has been solved by Kumar et al [20] by modified HPM (m-HPM).…”
Section: Introductionmentioning
confidence: 99%