Three-Dimensional Fourth-Order Time-Fractional Parabolic Partial Differential Equations and Their Analytical Solution

In this study, the fractional reduced differential transform method (FRDTM) is employed to solve three-dimensional fourth-order time-fractional parabolic partial differential equations with variable coefficients. /e fractional derivative used in this study is in the Caputo sense. A few important lemmas which are essential to solve the problems using the proposedmethod are proved./e novelty of this method is that it uses appropriate initial conditions and finds the solution to the problems without any discretization, linearization, perturbation, or any restrictive assumptions. Two numerical examples are considered in order to validate the efficiency and reliability of the method. Furthermore, the FRDTM solution when α� 1 is compared with other analytical methods available in the existing literature. Computational results are shown in tables and graphs./e obtained results revealed that themethod is capable and simple to solve fractional partial differential equations. /e software used for the calculations in this study is Mathematica 7.


Introduction
In the last decade, fractional calculus has become an important mathematical tool in several branches of sciences and engineering due to its enormous numbers of applications ( [1] and the references therein). What specifics fractional calculus is the existence of various approaches to define differentiation with the nonintegral order [2]. As a result, several definitions for fractional derivatives appear in the literature to present more accurate models for real life phenomena. Some of known fractional derivatives are Riemann-Liouville, modified Riemann-Liouville, Caputo, Hadmard, Erdélyi-Kober, Riesz, Grunwald-Letnikov, Marchaud, and others ( [3] and the references therein). All known fractional derivatives satisfy one of the well-known properties of classical derivative, namely, the linear property. However, the other properties of classical derivative, such as the derivative of a constant is zero; the product rule, quotient rule, and the chain rule either do not hold or are too complicated for many fractional derivatives.
In recent years, fractional differential equations have received considerable attention owing to their applicability in different fields of sciences such as chemistry, biology, diffusion, control theory, rheology, viscoelasticity, and so on [4]. Consequently, the solution of FPDEs represents nowadays a vigorous research area for scientists, and finding approximate and exact solutions to FPDEs is an important task [5]. However, PDEs are commonly hard to tackle, and their fractional order types are more complicated [2,[6][7][8][9][10].
erefore, several methods such as the homotopy perturbation method [11], sub-ODE method [12,13], generalized tanh method [14], residual power series method [14,15], and so on [16][17][18][19][20][21] are developed to obtain solutions of some nonlinear fractional differential equations. Most of these methods sometimes require complex and huge calculation in order to obtain approximate solutions. To overcome such difficulties and drawbacks, an alternative method, the so called the fractional reduced differential transform method (FRDTM), has been developed by Keskin and Oturanc [22]. FRDTM plays a vital role among all the listed methods because it takes small size computation, easy to implement as compared to other techniques [23]. Using this method, it is possible to find both exact and approximate solutions in a rapidly convergent power series form. FRDTM is a very reliable, efficient, and effective powerful computational technique for solving physical problems [24][25][26].
In the field of modern science and engineering, the fourth-order initial/boundary value problems are of great importance. For example, airplane wings, bridge slabs, floor systems, and window glasses are being modeled as plates subjected to different types of end supports which are successfully described in terms of fourth-order PDEs [27,28]. Time-fractional NLPDEs come from classical NLPDEs by replacing its time derivative with a fractional derivative. Time-fractional partial differential equations have many applications in areas such as electrochemistry, diffusion processes, electromagnetic, material science, chaotic dynamics, and so forth [27,29].
In this work, we consider the class of three-dimensional fourth-order time-fractional parabolic PDEs of the form (1) Subjected to the initial conditions, And boundary conditions u(a, y, z, t) � g 0 (y, z, t), where 0 < α ≤ 1, μ(x, y, z), λ(x, y, z), and η(x, y, z) are positive, and the functions f i , g i , k i , h i , f i , g i , k i , h i for i � 0, 1. e remaining sections of this study are organized as follows: in Section 2, we give some fundamental definitions and lemmas associated with fractional calculus. In Section 3, some basic definitions and properties related to the threedimensional fractional reduced differential transform method taking the fractional derivatives in the Caputo sense are presented, and some lemmas are proved. In Section 4, we present the formulation of the method. Section 5 is devoted to apply the method to solve three-dimensional time-fractional parabolic partial differential equation with variable coefficients. In Section 6, we present the numerical results and discussion. Tables and graphs are used to show the effectiveness, validity, and performance of the FRDTM for some values of α. Finally, the conclusion is presented in Section 7.

Fractional Calculus
In this section, some basic definitions and lemmas associated with fractional calculus are presented. Some of these definitions are due to Riemann-Liouville and Caputo sense; for details, see [2,24,[30][31][32][33].

Definition 1.
e gamma function. e gamma function Γ(z) is simply a generalization of the factorial real arguments. e Gamma function can be defined as Definition 2. e fractional derivative of f(t) in Caputo sense is defined as e basic properties of the Caputo fractional derivative can be given by the following lemma.
Definition 3. For p to be the smallest integer that exceeds α, and m to be the smallest integer that exceeds, β the Caputo time-fractional derivative of order α > 0 is defined as and the space fractional derivative operator with respect to x, y, and z of order β > 0 is, respectively, defined as

Fractional Reduced Differential Transform Method (FRDTM)
In this section, the basic definition and properties related to the three-dimensional fractional reduced differential transform method taking the fractional derivatives in the Caputo sense are presented.
Definition 4 (see [33,34]). If w(x, y, z, t) is analytic and continuously differentiable with respect to space variables x, y, and time variable t in the domain of interest, then the spectrum function is defined as Definition 5 (see [33,34]). e inverse FRDTM of W k (x, y, z) is defined as gives the n-terms approximate solution as follows: erefore, the exact solution of the problem is given by Mathematical Problems in Engineering Proof. By Definition 3 and properties of FRDTM given in Table 1, we have . α, β, μ, and ω are the constants.

Solution of the Problem by FRDTM
e steps to find the general solution of the governing equation (1) subjected to initial conditions (2) by the proposed method are as follows: Applying the properties of FRDTM given in Table 1 to equations (1) and (2), we obtain Solving for U k+2 (x, y, z) from equation (15), we get When k � 0,1,2,3,... by iterative calculations, we obtain the following successive terms: and so on. us, the inverse transform of the set of values of [U k (x, y, z)] n k�0 giving the n-terms approximate solution u n (x, y, z, t) � n k�0 U k (x, y, z)t αk and the FRDTM solution of equation (1) subjected to the initial condition (2) is given by

Numerical Results and Discussion
In this section, the numerical values of the function u (x, y, z, t) of Examples 1 and 2 are computed for various values of the fractional order α, and the exact solutions for the variables x, y, z, and t are presented in tables and figures. Tables 2 and 3 exhibit the behavior of FRDTM solutions for different values of fractional order α and the absolute errors (E) approximated in the 5 th and 6 th order at different values of the independent variables x, y, z, and time t of equations (20) and (25), respectively. As it can be seen from these tables, when the value of the fractional order α becomes close to 1, the obtained FRDTM solutions of Examples 1 and 2 are close to the exact solutions of its respective classical (nonfractional) equations obtained by different methods. In particular, when α � 1, the FRDTM solutions of the mentioned examples are in excellent agreement with the exact solutions of their corresponding classical (nonfractional) form, for details, see [35][36][37]. Figures 1(a)-1(d) show the evaluation results of the approximate analytical solution of equation (20). ese figures also show the behavior of the approximate solution obtained by the proposed method at fixed values x � 0.6 and y � 0.7 and for different values of the fractional order α � 0.5, 0.7, 1 and the exact solution.
Figures 2(a) and 2(b) demonstrate the solutions u (x, y, z, t) verses t of equation (20) for a fixed value of x, y, and z. In Figure 2(a), the FRDTM solutions at α � 0.7, 0.8, 1 and the exact solution are compared for a fixed value x � y � z � 0.4. In Figure 2(b), the FRDTM solutions at α � 0.7, 0.8, 1 and the exact solution are compared for a fixed value x � y � z � 0.6. Figures 3(a)-3(d) show the evaluation results of the approximate analytical solution of equation (25). ese figures also depict the behavior of the approximate solutions obtained by the proposed method at fixed values x � 0.6 and y � 0.7 and for a different time-fractional order α � 0.5, 0.7, 1 and the exact solution.     (25) for a fixed value of x, y, z. In Figure 4(a), the FRDTM solutions at α � 0.8, 0.9, 1 and exact solution are compared for a fixed value x � y � z � 0.8. In Figure 4(b), the FRDTM solutions at α � 0.8, 0.9, 1 and exact solution are compared for a fixed value x � y � z � 0.9.

Conclusion
In this study, we successfully employed the FRDTM to find the approximate and exact solutions of three-dimensional fourth-order time-fractional parabolic partial differential equations with variable coefficients taking the fractional derivative in Caputo sense. It is found that the results we obtained in Examples 1 and 2 when α � 1 are in excellent agreement with the exact solutions obtained by VIM [35], ADM [36], and RDTM [37]. Also, the illustrated examples demonstrate that the suggested technique can be observed as a fast, simple, and efficient tool for computing three-dimensional fourth-order time-fractional parabolic partial differential equations (PDEs). e main advantage of the method is that it can be applied to fractional PDE's and obtain approximate and exact solutions of fractional PDEs without requiring any discretization, perturbation, or restrictive conditions. Moreover, the tables and graphs constructed in this work demonstrate the physical behavior of the FRDTM solutions of the problems for different values of α. As a result, we conclude that the FRDTM is very powerful, straightforward, and effective to obtain approximate and exact solutions of a wide variety problems related to fractional PDEs applied in sciences such as mathematics, physics, and engineering.

Data Availability
e data used to support the findings of this study are included within the article.