New Solutions of Gardner's Equation Using Two Analytical Methods

This article introduces and applies new methods to determine the exact solutions of partial differential equations that will increase our understanding of the capabilities of applied models in real-world problems. With these new solutions, we can achieve remarkable advances in science and technology. This is the basic idea in this article. To accurately describe this, some exact solutions to the Gardner's equation are obtained with the help of two new 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 results obtained in this paper are new solutions to this equation that have not been introduced in previous literature. Another advantage of these methods is the determination of the varied solutions involving various classes of functions, such as exponential, trigonometric, and elliptic Jacobian. The three-dimensional diagrams of some of these solutions are plotted with specific values for their existing parameters. By examining these graphs, the behavior of the solution to this equation will be revealed. Mathematica software was used to perform the computations and simulations. The suggested techniques can be used in other real-world models in science and engineering.