| Summary: | The present study is concerned with studying the dynamical behavior of two space-dimensional nonlinear time-fractional models governing the unsteady-flow of polytropic-gas (in brief, pGas) that occurred in cosmology and astronomy. For this purpose, two efficient hybrid methods so-called optimal homotopy analysis <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">J</mi></semantics></math></inline-formula>-transform method (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mi>O</mi></msub></semantics></math></inline-formula>HA<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">J</mi></semantics></math></inline-formula>TM) and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">J</mi></semantics></math></inline-formula>-variational iteration transform method (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">J</mi></semantics></math></inline-formula>-VITM) have been adopted. The <sub><i>O</i></sub>HA<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">J</mi></semantics></math></inline-formula>TM is the hybrid method, where optimal-homotopy analysis method (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mi>O</mi></msub></semantics></math></inline-formula>HAM) is utilized after implementing the properties of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">J</mi></semantics></math></inline-formula>-transform (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">J</mi></semantics></math></inline-formula>T), and in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">J</mi></semantics></math></inline-formula>-VITM is the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">J</mi></semantics></math></inline-formula>-transform-based variational iteration method. Banach’s fixed point approach is adopted to analyze the convergence of these methods. It is demonstrated that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">J</mi></semantics></math></inline-formula>-VITM is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">T</mi></semantics></math></inline-formula>-stable, and the evaluated dynamics of pGas are described in terms of Mittag–Leffler functions. The proposed evaluation confirms that the implemented methods perform better for the referred model equation of pGas. In addition, for a given iteration, the proposed behavior via <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow></mrow><mi>O</mi></msub></semantics></math></inline-formula>HA<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">J</mi></semantics></math></inline-formula>TM performs better in producing more accurate behavior in comparison to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">J</mi></semantics></math></inline-formula>-VITM and the methods introduced recently.
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