Approximate series solution of multi-dimensional, time fractional-order (heat-like) diffusion equations using FRDTM

The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multi-dimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations.


Summary
The main goal of this paper is to present a new approximate series solution of the multi-dimensional (heat-like) diffusion equation with time-fractional derivative in Caputo form using a semi-analytical approach: fractional-order reduced differential transform method (FRDTM). The efficiency of FRDTM is confirmed by considering four test problems of the multidimensional time fractional-order diffusion equation. FRDTM is a very efficient, effective and powerful mathematical tool which provides exact or very close approximate solutions for a wide range of real-world problems arising in engineering and natural sciences, modelled in terms of differential equations.

Introduction
The history of fractional calculus is very long and the first idea appeared in Leibniz's letter in 1695. In the beginning, for up to three centuries, fractional calculus theory was restricted to only pure mathematics. Later on, fractional partial differential equations began receiving great attention among researchers due to their tremendous applications in the fields of physics, chemistry, ecology, biology and engineering [1][2][3][4][5][6][7][8][9][10][11]. It has been found that derivatives of non-integer order are very effective for the description of many physical phenomena such as rheology, damping laws and diffusion process. These findings have invoked the growing interest in studies of fractional calculus in many branches of science and engineering. preliminaries based on fractional derivatives and fractional integrals, which we will use to complete our study. Following are the most reasonable and meaningful definitions due to Liouville [4,11]: Definition 3.1 [4,11]. Let μ ∈ R and m ∈ N. A real valued function f : Definition 3.2 [4,11]. The Riemann-Liouville fractional integral of f ∈ C μ of the order α ≥ 0 is defined as where Γ denotes gamma function: In their work, Caputo & Mainardi [1] proposed a modified fractional differentiation operator D α t to describe the theory of viscoelasticity in order to overcome the discrepancy of the Riemann-Liouville derivative [4,11]. It is mentioned that the proposed Caputo fractional derivative allows the utilization of initial and boundary conditions involving integer order derivatives.

Definition 3.3 [1,11].
The fractional derivative of f ∈ C μ of the order α ≥ 0, in Caputo sense, is defined as The basic properties of Caputo fractional derivative are given as follows: In this work, the Caputo fractional derivative is considered because it includes traditional initial and boundary conditions in the formulation of the physical problems. For more details on fractional derivatives, one can refer to [2][3][4][5][6][7][8][9][10][11].

Fractional reduced differential transform method
In this section, basic properties of FRDTM are described [57][58][59]. Let ψ(x, t) be a function of two variables such that ψ(x, t) = f (x)g(t), then from the properties of the one-dimensional differential transform (DT) method, we have where ψ(i, j) = f (i)g(j) is referred to as the spectrum of ψ(x, t). Throughout the paper, R D and R −1 D denote the operators for fractional reduced differential transform (FRDT) and inverse FRDT, respectively. Furthermore, the lowercase ψ(x, t) is used for the original function, whereas its fractional reduced transformed function is represented by the uppercase Ψ k (x). The basic definitions and properties of FRDTM are described below.
Lemma 3.5 [57,59]. Let ψ(x, t) be an analytic and continuously differentiable with respect to space variable x and time variable t in the domain of interest, then (a) FRDT of ψ is given by In particular, for t 0 = 0, the above equation becomes This shows that FRDTM is a special case of the power series expansion.

Lemma 3.6 [57-60]. Let u(x, t) and v(x, t) be any two analytic and continuously differentiable functions with respect to space variable x and time t such that u(x, t)
where the convolution ⊗ denotes the fractional reduced differential transform version of multiplication and the function δ is defined by

Numerical results and discussion
subject to initial concentration The following recurrence relation is obtained by applying FRDTM to equation (4.1): Now, using FRDTM to the initial condition (4.2), we obtain Using the above equation in equation (4.3), the following recursive values of U k are obtained successively: ; . . . (4.5) Next, using the inverse FRDT of U k (x) and equation (4.5), we have is the Mittag-Leffler function. Equation (4.6) represents the exact solution of (4.1). The same solution was obtained by Momani [44] using ADM. In particular, when α → 1, equation (4.6) reduces to which is the exact solution of the one-dimensional classical heat-like diffusion equation (i.e. equation subject to initial concentration which grows exponentially in x and y as follows: u(x, y, 0) = e x+y . (4.9) The following recurrence relation is obtained by applying FRDTM to equation (4.8): (4.10) Now, using FRDTM to the initial condition (4.9), we have U 0 (x, y) = e x+y . (4.11) Using the above equation in equation (4.10), the following recursive values of U k are obtained successively: Next, using the inverse FRDT of U k (x, y) and equation (4.12), we have subject to initial concentration which grows exponentially in both x and z as follows: u(x, y, z, 0) = (1 − y) e x+z . (4.16) The following recurrence relation is obtained by applying FRDTM to equation (4.15): Next, using the inverse FRDT of U k (x, y, z) and equation (4.19), we have u(x, y, z, t) = U 0 (x, y, z) + U 1 (x, y, z)t α + U 2 (x, y, z)t 2α + · · · + U k (x, y, z)t αk + · · · = (1 − y) e x+z 1 + 2t α Γ (1 + α) (4.20) which is the exact solution of (4.15). For α → 1, equation (4.20) reduces to which is the exact solution for the three-dimensional classical diffusion equation. The same solution was obtained by Kumar et al. [55] using a modified homotopy perturbation method. For α = 1 and t = 1, comparison of exact concentration with approximate concentration as well as the behaviour of concentration of the two-dimensional diffusion equation (4.15) with respect to different axes is depicted in figure 5.

Conclusion
In this study, FRDTM is implemented successfully to find out the analytical solution of the time fractional-order multi-dimensional diffusion equation in terms of an infinite power series for the appropriate initial condition. The proposed approximate solutions are obtained without any transformation, perturbation, discretization or any other restrictive conditions. Four examples are carried out to study the accurateness and effectiveness of the technique. The proposed solutions by FRDTM are in excellent agreement with those obtained Kumar et al. [55] using M-HPM, and Momani [44] using ADM. The small size of calculation in FRDTM in comparison with M-HPM is its main advantage.
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