In this paper, we present a class of numerical schemes and apply it to the diffusion equations. The objective is to obtain numerical solutions of the constructive equations of a type of Casson fluid model. We investigate the solutions of the free convection flow of the Casson fluid along with heat and mass transfer in the context of modeling with the fractional operators. The numerical scheme presented in this paper is called the fractional version of the Adams Basford numerical procedure. The advantage of this numerical technique is that it combines the Laplace transforms and the classical Adams Basford numerical procedure. Note that the usage of the Laplace transforms makes possible the applicability of the numerical approach to diffusion equations in general. The Caputo derivative will be used in the investigations. The influence of the considered Casson fluid model parameters as the Prandtl number

Fractional calculus has received much attraction over this last decade and grew many papers with many applications in sciences and engineering [

Many papers in the literature address the analytical and semianalytical solutions of fluid and nanofluid models in general. In [

Modeling fluid models using fractional-order derivative is also addressed in the literature (see in [

We recall the fractional operators necessary for the rest of our investigations. We mean the Riemann-Liouville integral and the Caputo fractional derivative. These two operators are classical fractional calculus operators and are known to be with singularities. In the literature, there exist many other operators which can be used for the present investigations and are without singular kernels, like the derivative with Mittag-Leffler kernel and the derivative with the exponential kernel.

We describe the Riemann-Liouville fractional integral [

Its associated derivative can also be represented; therefore, we describe the Riemann-Liouville fractional derivative [

The derivative which can be used in the modeling is called Caputo derivative [

The Laplace transform, which helps in general to solve the fractional differential equation, will also be used in this paper, and we try recalling it in this section. The Laplace transform of the Caputo derivative [

In the rest of this paper, the Laplace transform in Equation (

In this section, we consider the fluid model described by fractional diffusion equations. The model investigated in this paper can be found in [

In the above equation,

In this section, the numerical scheme will be applied to obtain the solutions of the model presented in Equations (

We now begin the discretization of the Equation (

The difference between Equation (

We now give the discretization of the fractional integral part of the previous equation; we have the following formula:

Using Equation (

Similarly by the discretization of the integral form

Combining Equation (

We set

Similarly, with the assumption

We can observe when the order

Before closing this section, we focus on the stability of the proposed method. The necessary and sufficient condition will be to prove the function defined by

This subsection will apply the numerical scheme previously described to obtain solutions of the diffusion equations presented in this section. The main idea of the application is for the first time to use the Laplace transforms. The obtained equation after applying the Laplace transform will be of one variable, and in this case, the numerical procedure can be used.

We begin the application of Equation (

For simplification in the style of writing, we suppose

The numerical discretization of the previous Equation (

Applying the inverse of the Laplace transform to both sides of Equation (

We continue the application. We consider Equation (

We consider that the function

The numerical scheme of Equation (

Applying the inverse of the Laplace transform to both sides of Equation (

We finish this section by giving the numerical scheme of the fluid model Equation (

We suppose the following functions which will help us in understanding the numerical scheme,

The numerical procedure of the previous Equation (

Utilizing the inverse of the Laplace transform on both sides of Equation (

This section presents the numerical procedure to obtain the solutions of the fluid model shown in section

In this section, we apply the Adams Bashford numerical scheme presented in the previous quarter. The numerical technique will be used to obtain the graphical representations. Additionally, the profile of the solution will be analyzed according to the variation of the model’s parameters.

In our analysis, we begin with the constructive equation described in Equation (

Dynamics of concentration in the fractional diffusion equation (Equation (

Dynamics of temperature in the fractional diffusion equation (Equation (

In Figures

Dynamics of velocity using the fractional diffusion equation (Equation (

In Figure

Dynamics of the temperature using the fractional diffusion equation (Equation (

We now repeat the same analysis with the Schmidt number of the dynamics of the considered fluid concentration. We fix the order to

Dynamics of the concentration using the fractional diffusion equation (Equation (

We also notice that the Schmidt number has an acceleration effect in the diffusion process. We notice that when the order is fixed and when the Schmidt number increases, then the concentration profile decreases. Referring to Figures

Dynamics of velocity of the fractional diffusion equation (Equation (

We remark that from Figure

Dynamics of velocity using the fractional diffusion equation (Equation (

In Figure

The main findings of this section are summarized as follows. The increase in the mass Grashof number generates an increase in the dynamics of the Casson fluid’s velocity. The increase in the Casson fluid’s material parameters decreases the dynamics of the Casson fluid’s velocity. The increase in the Prandtl number and Schmidt number generates a reduction in the temperature distribution and the concentration distribution.

We have proposed a numerical scheme to obtain the profile of the velocity, temperature, and concentration of the considered fluid. The method proposed in this paper is novel for this fluid model in the context of fractional-order derivative. We have found the Caputo derivative has an acceleration effect in the diffusion processes. Furthermore, the parameters used in our model play an important role in increasing or decreasing the Casson fluid dynamics considered in this paper. In this paper, we find that the Prandtl number

No data were used to support this study.

The authors declare that they have no conflict of interest.