An Efficient Algorithm for Some Highly Nonlinear Fractional PDEs in Mathematical Physics

Abstract

In this paper, a fractional complex transform (FCT) is used to convert the given fractional partial differential equations (FPDEs) into corresponding partial differential equations (PDEs) and subsequently Reduced Differential Transform Method (RDTM) is applied on the transformed system of linear and nonlinear time-fractional PDEs. The results so obtained are re-stated by making use of inverse transformation which yields it in terms of original variables. It is observed that the proposed algorithm is highly efficient and appropriate for fractional PDEs and hence can be extended to other complex problems of diversified nonlinear nature.

Application of Residual Power Series Method to Fractional Coupled Physical Equations Arising in Fluids Flow

“…On the other hand it becomes important in material sciences [26,27]. The aim of this paper is to study the time-fractional Gardner equation [28][29][30] and time-fractional Cahn-Hilliard equation [31][32][33][34][35][36][37] of this form, ( , ) + 6 ( − 2 2 ) + = 0, (3)…”

Modified wavelets–based algorithm for nonlinear delay differential equations of fractional order

“…Over the past few decades, several researchers had devoted their study in finding numerical solutions of fractional delay differential equations due to nonavailability of exact solutions in most of the cases. Therefore, different numerical and analytical methods [9][10][11][12] have been developed and applied for finding approximate solutions. Method of steps 13 is commonly used to convert a delay differential equation into an ordinary differential equation.…”

An efficient technique for higher order fractional differential equation

“…The investigation of travelling wave solutions (Shawagfeh 2002; Ray and Bera 2005; Yildirim et al 2011; Kilbas et al 2006; He and Li 2010; Momani and Al-Khaled 2005; Odibat and Momani 2007; Abdou 2007; Nassar et al 2011; Misirli and Gurefe 2011; Noor et al 2008; Ozis and Koroglu 2008; Wu and He 2007; Yusufoglu 2008; Zhang 2007; Zhu 2007; Wang et al 2008; Zayed et al 2004; Sirendaoreji 2004; Ali 2011; Liang et al 2011; He et al 2012; Jawad et al 2010; Zhou et al 2003; Yıldırım and Kocak 2009; Elbeleze et al 2013; Matinfar and Saeidy 2010; Ahmad 2014; Bongsoo 2009; Demiray and Pandir 2014, 2015; Lu 2012; Zayed and Amer 2014) of nonlinear evolution equations plays a significant role to look into the internal mechanism of nonlinear physical phenomena. Nonlinear fractional differential equations (FDEs) are a generalization of classical differential equations of integer order.…”

“…In recent years, considerable interest in fractional differential equations (He and Li 2010; Momani and Al-Khaled 2005; Odibat and Momani 2007) has been stimulated due to their numerous applications in different fields. However, many effective and powerful methods have been established and improved to study soliton solutions of nonlinear equations, such as extended tanh-function method (Abdou 2007), tanh-function method (Nassar et al 2011), Exp-function method (Misirli and Gurefe 2011; Noor et al 2008; Ozis and Koroglu 2008; Wu and He 2007; Yusufoglu 2008; Zhang 2007; Zhu 2007), ( G ’ / G )-expansion method (Wang et al 2008), homogeneous balance method (Zayed et al 2004), auxiliary equation method (Sirendaoreji 2004), Jacobi elliptic function method (Ali 2011), Weierstrass elliptic function method (Liang et al 2011), modified Exp-function method (He et al 2012), modified simple equation method (Jawad et al 2010), F-expansion method (Zhou et al 2003), homotopy perturbation method (Yıldırım and Kocak 2009), Fractional variational iteration method (Elbeleze et al 2013), homotopy analysis method (Matinfar and Saeidy 2010), Reduced differential transform method (Ahmad 2014), Generalized Kudryashov method for time-fractional differential equations (Demiray and Pandir 2014), The first integral method for some time fractional differential equations(Lu 2012; Zayed and Amer 2014), New solitary wave solutions of Maccari system (Demiray and Pandir 2015), and so on.…”