The homotopy perturbation method applied to the nonlinear fractional Kolmogorov–Petrovskii–Piskunov equations

Abstract

a b s t r a c tThe fractional derivatives in the sense of Caputo, and the homotopy perturbation method are used to construct approximate solutions for nonlinear Kolmogorov-PetrovskiiPiskunov (KPP) equations with respect to time and space fractional derivatives. Also, we apply complex transformation to convert a time and space fractional nonlinear KPP equation to an ordinary differential equation and use the homotopy perturbation method to calculate the approximate solution. This method is efficient and powerful in solving wide classes of nonlinear evolution fractional order equations.

On the Closed Form Solutions of Linear and Nonlinear Cauchy Reaction-Diffusion Equations Using the Hybrid of Fourier Transform and Variational Iteration Method

“…However, the solutions of these problems are valid only in onedirectional problem domain either in time or space problem domain. In other words, the unsatisfied boundary conditions in the solutions of the VIM and other semi-analytical methods play no role in the final results (Kasozi et al, 2011;Gupta and Singh, 2011;Othman et al, 2010;Gepreel, 2011;Aslanov, 2011;Madani et al, 2011;He et al, 2010;Chen and Wang, 2010;Zhou and Yao, 2010;Odibat, 2010;Zhao and Xiao, 2010;Shang and Han, 2010;Turkyilmazoglu, 2011;Afshari et al, 2009;Aruchunan and Sulaiman, 2010). Therefore, there has been a deficiency as a built in short comes in the solutions using the VIM and other semi-analytical methods.…”

“…Moreover, due to the non-linear nature and variable coefficients of these differential equations, attentions are devoted to the approximate solutions obtained by semi analytical methods such as the Homotopy Perturbation Method (HPM) (He, 2009;Kasozi et al, 2011;Gupta and Singh, 2011;Othman et al, 2010;Gepreel, 2011;Aslanov, 2011) and Variational Iteration Method (VIM) (He et al, 2010;Shakeri, et al, 2009;Chen and Wang, 2010;Zhou and Yao, 2010;Odibat, 2010;Zhao and Xiao, 2010;Shang and Han, 2010;Turkyilmazoglu, 2011). The VIM is developed by employing a correction functional and a general Lagrange multiplier for the differential equation.…”

A novel computational framework to approximate analytical solution of nonlinear fractional elastic beam equation

“…Considerable research work has recently been conducted in application of this method to fractional advection-dispersion equations, multi-order fractional di erential equations, Navier-Stokes equations, nonlinear Schr odinger equations, Volterra integro-di erential equations, nonlinear oscillators, boundary value problems, fractional KdV equations, quadratic Riccati differential equations of fractional order and many others. For more details about the homotopy perturbation method and its applications, the reader is advised to consult the results of research work presented in [14][15][16][17][18][19]. All these successful applications veri ed the e ectiveness, exibility, and validity of the homotopy perturbation method.…”

Modified Trial Equation Method to the Nonlinear Fractional Sharma–Tasso–Olever Equation

“…Also, a remarkable progress has been become in the construction of the approximate solutions for fractional nonlinear partial differential equations [1]- [3]. Several powerful methods have been proposed to obtain approximate and exact solutions of fractional differential equations, such as the Adomian decomposition method [4], [5], the homotopy analysis method [6], [7], the homotopy perturbation method [8], and so on. The exact solutions of these problems, when they exist, are very important in the understanding of the nonlinear fractional physical phenomena.…”