Study of exp(-Φ(ξ))-expansion Method for Solving Nonlinear Partial Differential Equations

In this work we study the Gardner equation or the combined KdV-mKdV equation. We use the exp(-Φ(ξ))-expansion method for a reliable treatment to establish exact traveling wave solutions then the solitary wave solutions for the aforementioned nonlinear partial differential equation (NPDEs). As a result, the traveling wave solutions are obtained in four arbitrary functions including hyperbolic function solutions, trigonometric function solutions, exponential function solutions, and rational function solutions.