Abstract
This paper studies the analytical solutions of the fractional fluid models described by the Caputo derivative. We combine the Fourier sine and the Laplace transforms. We analyze the influence of the order of the Caputo derivative the Prandtl number, the Grashof numbers, and the Casson parameter on the dynamics of the fractional diffusion equation with reaction term and the fractional heat equation. In this paper, we notice that the order of the Caputo fractional derivative plays the retardation effect or the acceleration. The physical interpretations of the influence of the parameters of the model have been proposed. The graphical representations illustrate the main findings of the present paper. This paper contributes to answering the open problem of finding analytical solutions to the fluid models described by the fractional operators.
Highlights
This paper aims to propose the analytical solution of the fractional differential equation defined by the above Equations (14) and (15)
We propose the exact solutions of our current model using the Laplace transform and the Fourier sine transformation
The Laplace transform permits obtaining of the exact analytical solution by applying the inverse of the Laplace transform to the linear fractional differential equation obtained after the application of the Fourier sine transform
Summary
We model fluids using the Caputo derivative. In [19], Khan et al investigate using the Laplace transform of the Atangana–Baleanu derivative, the exact solutions of the constructive equations in the study of the second-grade fluids flow with combined gradients of mass concentration, and temperature distribution over a vertical flat plate. In [24], Abro and Atangana give a comparative presentation of convective fluid motion in a rotating cavity described by the non-singular fractal-fractional operators for differentiation. The method permits us to get the exact solutions of the fluid models; we do not need to utilize the integral balance methods and the numerical discretizations.
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