Exact Traveling Wave Solutions of Certain Nonlinear Partial Differential Equations Using the G′/G2

Abstract

We apply the ( / 2 )-expansion method to construct exact solutions of three interesting problems in physics and nanobiosciences which are modeled by nonlinear partial differential equations (NPDEs). The problems to which we want to obtain exact solutions consist of the Benny-Luke equation, the equation of nanoionic currents along microtubules, and the generalized Hirota-Satsuma coupled KdV system. The obtained exact solutions of the problems via using the method are categorized into three types including trigonometric solutions, exponential solutions, and rational solutions. The applications of the method are simple, efficient, and reliable by means of using a symbolically computational package. Applying the proposed method to the problems, we have some innovative exact solutions which are different from the ones obtained using other methods employed previously.

Some Applications of the (G′/G,1/G)-Expansion Method for Finding Exact Traveling Wave Solutions of Nonlinear Fractional Evolution Equations

“…where a = 0, b, c, λ(< 0), µ, k are arbitrary constants. From Equations (10), (20), and 22, we obtain the traveling wave solution of Equation 1as follows:…”

“…where a 0 , a = 0, b, c, λ(> 0), µ are arbitrary constants and σ 2 = A 2 1 + A 2 2 , where A 1 , A 2 are arbitrary constants. From Equations (12), (20), and (29), we obtain the exact solution of Equation (1) as follows:…”

“…If λ < 0, we substitute Equation (20) into Equation (19) along with the use of Equation (9) and Equation (11). Then, the left-hand side of (19) turns out to be a polynomial in φ(ξ) and ψ(ξ).…”

“…Useful methods for solving NLEEs numerically are those such as the finite element method [13], the finite volume method [14], and the finite-difference predictor-corrector method [15]. Several efficient and reliable methods which have recently been developed to obtain exact explicit solutions for NLEEs are, for instance, the Jacobi elliptic function method [16], the (G /G)-expansion method and its various modifications [17][18][19][20], the Exp-function method [21], the sub-equation method [22], the first integral method [23], the modified trial equation method [24,25], and the simplest equation method [26].…”

An improved (G'/G)- expansion method for solving nonlinear evolution equations

“…With the development of solution theory, many powerful methods for obtaining the exact solutions of NLEEs have been presented by many authors. For instances, Tanh method [2,3], F-expansion method [4], Backlund transforms [5], a numerical simulation and explicit solutions [6], Adomian decomposition method (ADM) [3,9], variational iteration method [3,12], the Jacobi elliptic function expansion method [13], ( ′ / , 1/ ) -expansion method [15], inverse scattering transform [20], Modified Pseudospectral method [21], spectral collocation method [22], an efficient algorithm method [24], finite element analysis and numerical method [25], ( ′ / 2 )expansion method [26], Exp-function method [28].…”