Abstract
me exact solutions to the Gardner’s equation are obtained with the help of two analytical methods including the generalized exponential rational function method and a Jacobi elliptical solution finder method. A set of new exact solutions containing four parameters is reported. The graphical interpretation of the solutions is depicted. Mathematica software is used to perform the computations and simulations. The suggested techniques can be used to another sort of real-world models from science and engineering.
Highlights
It is difficult or impossible to determine the exact solution for many partial differential equations
The Gardner equation belongs to the category of integrable non-linear partial differential equations
We aimed to find new solutions for a given problem in Equation (1), and these new solutions should be described graphically
Summary
It is difficult or impossible to determine the exact solution for many partial differential equations In spite of these problems, in recent years a variety of efficient and practical methods have been proposed by mathematicians and physicists. A classification of Lie symmetries for the Gardner equation has been reported in [23] They have used the similarity transformation method to introduce the invariant solutions. In [28], a certain classification of single traveling wave solutions of the time-fraction Gardner equation is investigated These forms of the Gardner equation can be utilized to model various physical phenomena, such as the non-linear propagation of ion acoustic waves in an unmagnetized plasma.
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