| Summary: | The main objective of the present study is to analyze the nature and capture the corresponding consequences of the solution obtained for the Gardner–Ostrovsky equation with the help of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>q</mi></semantics></math></inline-formula>-homotopy analysis transform technique (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>q</mi></semantics></math></inline-formula>-HATT). In the rotating ocean, the considered equations exemplify strong interacting internal waves. The fractional operator employed in the present study is used in order to illustrate its importance in generalizing the models associated with kernel singular. The fixed-point theorem and the Banach space are considered to present the existence and uniqueness within the frame of the Caputo–Fabrizio (CF) fractional operator. Furthermore, for different fractional orders, the nature has been captured in plots. The realized consequences confirm that the considered procedure is reliable and highly methodical for investigating the consequences related to the nonlinear models of both integer and fractional order.
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