Abstract
In this article the new extension of the generalized and improved (Gâ/G)-expansion method has been used to generate many new and abundant solitons and periodic solutions, where the nonlinear ordinary differential equation has been used as an auxiliary equation, involving many new and real parameters. We choose the Fisher Equation in order to explain the advantages and effectives of this method. The illustrated results belongs to hyperbolic functions, trigonometric functions and rational functional forms which show that the implemented method is highly effective for investigating nonlinear evolution equations in mathematical physics and engineering science.
Highlights
In physical sciences all essential equations are nonlinear and these are often complicated to interpret
nonlinear evolution equations (NLEEs) is one of the most powerful and important modelled equations among all equations in nonlinear sciences and it plays a vital role in the field of scientific work of engineering sciences such as chemical kinematics, fluid mechanics, chemistry, biology, nonlinear optics, optical fibers, plasma physics, solid state physics, biophysics, geochemistry, quantum mechanics, chemical physics, condensed matter physics, high-energy physics and so on
As they reveal a lot of physical information which help to understand the operation of the physical model better, that is why the explicit solutions of NLEEs play important role in the study of physical phenomena and remains a crucial field for researchers in the ongoing investigation
Summary
In physical sciences all essential equations are nonlinear and these are often complicated to interpret. For example- Zhang et al [22] expanded the original (G′ G) - expansion method and named as the improved (G′ G) - expansion method Using this method many researches have been carried out in order to find travelling wave solutions for NLPDEs [23,24,25,26,27,28,29,30]. By solving Eq (2.6) we obtain a general solution, which is substituted with the values of constants into Eq (2.4) we can achieve more general type and more new travelling wave solutions of NLPDE of Eq (2.1). Step 5: Using the general solution of Eq (2.6), the following solutions for Eq (2.5) are obtained: Family 1: When μ ≠ 0, Ψ= λ − δ and
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