Exact solutions for nonlinear partial fractional differential equations

Abstract

In this article, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. We use the improved (G ′ /G)-expansion function method to calculate the exact solutions to the time-and space-fractional derivative foam drainage equation and the time-and space-fractional derivative nonlinear KdV equation. This method is efficient and powerful for solving wide classes of nonlinear evolution fractional order equations.

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

“…(1) and (2). By using the properties of the homotopy perturbation method [11,13,14,16,19], we have the following 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.…”

Untitled

“…A particular category of nonlinear PDEs are nonlinear fractional PDEs that have continually appeared in physics, chemistry, biology, polymeric materials, electromagnetic, acoustics, neutron point kinetic model, vibration and control, signal and image processing, fluid dynamics and so on [3][4][5][6]. Due to its practicability and complexity, it is important to seek the solutions of nonlinear fractional PDEs and researchers [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] have put considerable effort into it. For the purpose of solving problems in the practical application fields, more exact traveling wave solutions the exp(-Φ(ξ) …”

Travelling wave solutions: A new approach to the analysis of nonlinear physical phenomena

“…Therefore, there have been attempts to develop new approaches for obtaining analytical or numerical solutions which reasonably approximate the exact solutions. For more details see [1][2][3][4][5][6]. Recently, a promising analytical approach called homotopy analysis method (HAM), has successfully been applied to solve many types of linear and nonlinear functional equations [7][8][9][10][11][12].…”