Explicit Jacobi elliptic exact solutions for nonlinear partial fractional differential equations

Khaled A Gepreel

doi:10.1186/1687-1847-2014-286

Abstract

In this article, we use the fractional complex transformation to convert nonlinear partial fractional differential equations to nonlinear ordinary differential equations. An algebraic method is improved to construct uniformly a series of exact solutions for some nonlinear time-space fractional partial differential equations. We construct successfully a series of some exact solutions including the elliptic doubly periodic solutions with the aid of computerized symbolic computation software package such as Maple or Mathematica. This method is efficient and powerful in solving a wide classes of nonlinear partial fractional differential equations. The Jacobi elliptic doubly periodic solutions are generated by the trigonometric exact solutions and the hyperbolic exact solutions when the modulus m→0 and m→1, respectively.

Highlights

Nonlinear partial fractional equations are very effective for the description of many physical phenomena such as rheology, the damping law, diffusion processes, and the nonlinear oscillation of an earthquake can be modeled with fractional derivatives [, ]
Many applications of nonlinear partial fractional differential equations can be found in turbulence and fluid dynamics and nonlinear biological systems [ – ]
Zhang and Zhang [ ] have introduced a direct method called the sub-equation method to look for the exact solutions for nonlinear partial fractional differential equations

Summary

Introduction

Nonlinear partial fractional equations are very effective for the description of many physical phenomena such as rheology, the damping law, diffusion processes, and the nonlinear oscillation of an earthquake can be modeled with fractional derivatives [ , ]. Zhang and Zhang [ ] have introduced a direct method called the sub-equation method to look for the exact solutions for nonlinear partial fractional differential equations. We will improve the extended proposed algebraic method to solve the nonlinear partial fractional differential equations. We use the improved extended proposed algebraic method to construct the Jacobi elliptic exact solutions for the following nonlinear time-space partial fractional differential equations:. We give the main steps of the modified extended proposed algebraic method for nonlinear partial fractional differential equations. ±m) are arbitrary constants to be determined later, while φ(ξ ) satisfies the following nonlinear first order Jacobi elliptic differential equation:. ±m), e , e , e , K , L, M, and N and the general solutions of ( ) into ( ), we obtain more new Jacobi elliptic exact solutions for the nonlinear partial fractional derivative equation ( )

Example 1

Example 2

Conclusion