Fractional-Order Investigation of Diffusion Equations via Analytical Approach

This research article is mainly concerned with the analytical solution of diffusion equations within a Caputo fractional-order derivative. The motivation and novelty behind the present work are the application of a sophisticated and straight forward procedure to solve diffusion equations containing a derivative of a fractional-order. The solutions of some illustrative examples are calculated to confirm the closed contact between the actual and the approximate solutions of the targeted problems. Through analysis it is shown that the proposed solution has a higher rate of convergence and provides a closed-form solution. The small number of calculations is the main advantage of the proposed method. Due to a comfortable and straight forward implementation, the suggested method can be utilized to nonlinear fractional-order problems in various applied science branches. It can be extended to solve other physical problems of fractional-order in multiple areas of applied sciences.


INTRODUCTION
Fractional calculus (FC) can be considered as one of the extensions of traditional ordinary calculus. The roots of FC dates back to about 300 years. The idea of fractional calculus arises from the fact that Leibniz used a notation for the nth derivative in his publication as (d n y/dx n ). L ′ H opital posed a question to Leibniz about the derivative of order (1/2) in 1,695 [1]. In nature derivatives that involve an integer order are local, whereas a fractional order is nonlocal. The mathematical groundwork for derivatives of a fractional order was laid through the combined efforts of several mathematicians such as Caputo, Miller and Ross, Liouville, Riemann, Podlubny, and others. Later on, several mathematicians devoted their work to this area. Fractional calculus has attracted a lot of interest due to its potential implementations in different scientific areas such as biology, viscoelasticity, engineering, fluid mechanics, and other fields of science [2][3][4][5][6][7][8][9][10].
From the above applications, mathematicians implemented different methods to solve some vital differential equations of fractional-orders (FDEs), particularly partial differential equations of fractional-orders (FPDEs). FPDEs are considered to be basic mathematical techniques used to model various physical phenomena more precisely than integer-order. The non-linear FPDEs determine different phenomena in engineering and applied sciences. In particular, non-linear FPDEs are the best sources to be utilize in different fields such as material sciences, physics, thermodynamics, chemistry, chemical kinetics, and several other basic processes [11][12][13][14][15][16][17][18][19][20][21]. Various methods to solve FDEs are used in the literature, such as the variational iteration method [22], the modified Adomian decomposition method (MADM) [23], the differential transformation method (DTM) [24], the optimal homotopy asymptotic method (OHAM) [25], and the homotopy perturbation method (HPM) [26].
In 2006, Daftardar-Gejji et al. introduced the iterative method (IM) to solve functional equations [27,28]. IM is used to solve DEs, of both integers as well fractional-order. In this article, we used the Shehu transform method and the iterative technique and introduced a new technique called the Iterative Shehu Transform method [ISTM] to find the approximate solution of FPDEs. It has been observed that the suggested technique has a simple and effective implementation compared to other existing methods such as the homotopy perturbation method, the variational iteration method, and the Laplace Adomian decomposition method [29].
The diffusion equation is a PDE which expresses the phenomenon which transforms atoms or molecules from a higher construction. The Fick's law of diffusion was first described by the physiologist Adolf Fick. Later on, Fick's law was converted into the Diffusion equation. Researchers generalized the classical Diffusion law to obtain modified Diffusion equations such as the hybrid classical wave equation and slow diffusion [30]. There are several diffusion equation implementations such as filtration, electromagnetism, geochemistry, phase transition, cosmology, biochemistry, acoustics, and dynamics of biological groups [31].
In the present paper, we apply ISTM to solve diffusion equations of the form.
I α τ stands for fractional integral.
The Shehu transformation of a function ](τ) is V(s, μ): then ](τ) is called the inverse of V(s, μ) which is defined as
The ST of the fractional derivative is described as
A further generalization of the equation is given in the form of

IDEA OF ISTM
In this section, we use ISTM to determine the general solution of fractional-order diffusion equations [32][33][34][35].
Now, applying the iterative method, In Eq. 9, the linear and nonlinear parts R and N, respectively, are decomposed as follows Eqs. 11, 12 Substituting Eqs. 9, 11 in Eq. 8 yields We set However, the m-term approximate solution of Eqs. 5, 6 is determined by

APPLICATIONS
In this section, we apply ISTM to obtain the solution of fractional order diffusion equation to show the accuracy and appropriateness of ISTM to solve exactly nonlinear FPDEs.

Example
Consider diffusion equation of fractional-order in onedimension [31].

Example
Consider diffusion equation of fractional-order in twodimension [31].

Example
Consider diffusion equation of fractional-order in twodimension [31].
Thus, in the form of a series, the analytical solution can be written as μ(ϑ, ζ, ω, τ) μ 0 (ϑ, ζ, ω, τ) + μ 1 (ϑ, ζ, ω, τ) + μ 2 (ϑ, ζ, ω, τ)  Figure 1, the Exact and ISTM solutions of Example 1, at α 1 are plotted by using sub-graphs A and B respectively. The graphical representation has shown the closed contact between the exact and ISTM solutions.  Figure 3, the Exact and ISTM solutions of Example 2, at α 1 are plotted by using sub-graphs A and B respectively. The graphical representation has shown the closed contact between the exact and ISTM solutions of Example 2.  When α 1, then the ISTM solution in a closed form: The exact solution is: μ(ϑ, ζ, ω, τ) exp − 3τ sin ϑ sin ζ sin ω.

CONCLUSION
In the present article, the fractional view analysis is analyzed using an efficient analytical technique. Fractional derivatives are expressed in a Caputo sense. The present method is tested to solve diffusion equation of fractional-order. After investigation, we show that the present method is the best tool to use to solve nonlinear fractional order problems. The exact and obtained solutions in closed contact has verified the reliability of the present technique. Moreover, the graphical representation has confirmed the convergence of the approximate solutions toward the exact solutions and provide the closed form solutions of the targeted problems. The convergence phenomenon has confirmed the reliability of the suggested method.if  Figure 6, the Exact and ISTM solutions of Example 4, at α 1 are plotted by using subgraphs A and B respectively. The graphical representation has shown the closed contact between the exact and ISTM solutions of Example 4.  Figure 7, the sub-graphs (C) and (D) represent the ISTM Error and solution graphs of Example 4 respectively. The error graph has confirmed the higher accuracy of ISTM.