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Abstract:
Our manuscript is related to use Caputo fractional order derivative (CFOD) to investigate results of non-linear mode in plasma. We establish results for both temporal and spatial approximate solution. For the require results, we use reduction perturbation method (RPM) to find the analytical solution of the dust acoustic shock waves. Further, using the same technique we find the solitary wave potential and compared the solutions obtained with another very useful technique known as Homotopy perturbation method (HPM). The comparison of results for both approaches are more precise and agreed with the exact solution of the problem. Finally, we present graphical representation for different fractional order for both temporal and spatial approximate solution.

Keywords:
Approximate solution, Caputo derivative, Reductive perturbation, Homotopy perturbation method

Abstract:
In our observation, we have used an easy and reliable approach of the reduction perturbation method to obtain the solution of the ion temperature gradient mode driven linear and nonlinear structures of relatively small amplitude. One can use that methodology in the more complex environment of the plasma and can obtain a straightforward approach toward his studies. We have studied different parameter impacts on the linear and nonlinear modes of the ITG by using data from tokamak plasma. Hence, our study is related to the tokamak plasma and one that can apply to the nonlinear electrostatic study of stiller and interstellar regimes where such types of plasma environment occur.

Keywords:
Ion temperature gradient, soliton, shock, electron-ion plasma, reduction perturbation method, linear and non-linear structures