Exact Traveling Wave Solutions for Fitzhugh-Nagumo (FN) Equation and Modified Liouville Equation

Abstract

In this paper, we employ the exp(−ϕ(ξ))-expansion method to find the exact traveling wave solutions involving parameters of nonlinear evolution equations Fitzhugh-Nagumo (FN) equation and Modified Liouville equation. When these parameters are taken to be special values, the solitary wave solutions are derived from the exact traveling wave solutions. It is shown that the proposed method provides a more powerful mathematical tool for constructing exact traveling wave solutions for many other nonlinear evolution equations.

Extended Jacobian Elliptic Function Expansion Method and Its Applications in Biology

“…Exact solutions for these equations play an important role in many phenomena in physics such as fluid mechanics, hydrodynamics, Optics, Plasma physics and so on. Recently many new approaches for finding these solutions have been proposed, for example, tanh-sech method [2]- [4], extended tanh-method [5]- [7], ( ) ( ) −ϕ ξ exp [8]- [11], homogeneous balance method [12], F-expansion method [13]- [15], exp-function method [16] [17], trigonometric function series method [18], ′       G G -expansion method [19]- [22], Jacobi elliptic function method [23]- [26] and so on.…”

A Generalization of the Modified Liouville Equation

“…The Bendixson-Dulac criterion consists of a sufficient number of conditions for the nonexistence of periodic orbits in planar dynamical systems (Farkas, 1994). The modified Liouville equation (Abdelrahman et al, 2015;Salam et al, 2012) plays an important role in various areas of mathematical physics, from plasma physics and field theoretical modeling to fluid dynamics, using various transformations the differential equation can be written as a dynamic system that under some conditions does not have periodic orbits 2013a;Osuna and Villaseñor, 2011). The system in (Marin-Ramirez et al, 2015) coincides to our system.…”