The Wick-type explicit solutions of the nonlinear stochastic Wick-type fractional Gardner equation

Abstract

In this article, we consider the nonlinear stochastic Wick-type fractional Gardner equation and the generalized fractional Gardner equation with coefficient variables. We obtain new Wick-type explicit solutions of the nonlinear stochastic Wick-type fractional Gardner equation and new explicit solutions of the generalized fractional Gardner equation with coefficients variables by employing the improved system method. Additionally, dynamical behaviors of the obtained solutions are provided graphically under suitable functions of white noises and physical parameters. We show that the improved system method is powerful and effective to find explicit solutions of the considered equations. It is also recommended to use the improved system method to solve the fractional nonlinear partial differential equations as well as nonlinear partial differential equations.

Keywords

• Gardner equation
• Wick product
• Hermite transform
• Caputo fractional derivative
• White noise functional