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Nonlinear partial differential equations (NLPDEs) are an inevitable mathematical tool to explore a large variety of engineering and physical phenomena. Due to this importance, many mathematical approaches have been established to seek their traveling wave solutions. In this study, the researchers examine the Gardner equation via two well-known analytical approaches, namely, the improved - expansion method and the wave ansatz method. We derive the exact bright, dark, singular, and W-shaped soliton solutions of the Gardner equation. One can see that the methods are relatively easy and efficient to use. To better understand the characteristics of the theoretical results, several numerical simulations are carried out.

mathematical problems in engineering

Exact Traveling Wave Solutions of the Gardner Equation by the Improved <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi mathvariant="normal">tan</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi mathvariant="normal">Θ</mml:mi><mml:mfenced open="(" close=")" separators="|"><mml:mrow><mml:mi>ϑ</mml:mi></mml:mrow></mml:mfenced></mml:mrow></mml:mfenced></mml:math>-Expansion Method and the Wave Ansatz Method

Exact Traveling Wave Solutions of the Gardner Equation by the Improved tanΘϑ-Expansion Method and the Wave Ansatz Method