Abstract
In this study, we introduce a new modification of fractional reduced differential transform method (m-FRDTM) to find exact and approximate solutions for nonhomogeneous linear multiterm time-fractional diffusion equations (MT-TFDEs) of constant coefficients in a bounded domain with suitable initial conditions. Different applications in two and three fractional order terms are given to illustrate our new modification. The approximate solutions are given in the form of series solutions. The results show that the m-FRDTM for MT-TFDEs is a powerful method and can be generalized to other types of multiterm time-fractional equations.
1. Introduction
Researchers are trying to find different techniques to solve linear and nonlinear fractional differential equations. Some of them modify the classical methods to be an effective approach, others related between two or more methods to find numerical or analytical solutions of fractional equations. For instance, in [1, 2] we modified definition of beta fractional derivative to find exact and approximate solutions of time-fractional diffusion equations in different dimensions. Many numerical and analytical methods have been developed for single-term-fractional derivatives (only one fractional power). On the other hand, there are still little works in multi-term-fractional derivatives (more than one fractional power). The time-fractional diffusion-wave equation describes important physical phenomena that arise in amorphous, colloid, glassy, and porous materials, in fractals and percolation clusters, comb structures, dielectrics and semiconductors, biological systems, polymers, random and disordered media, and geophysical and geological processes [3]; Abuasad et al. [4] applied a fractional multistep differential transformed method to find approximate solutions to one of the most important epidemiologies and mathematical ecologies, fractional stochastic SIS epidemic model with imperfect vaccination. Recently, the extension to multiterm time-fractional diffusion equations (MT-TFDEs) has been considered [5]. Edwards et al. [6] showed that the numerical approximation of the solution of a linear multiterm fractional differential equation can be calculated by reduction of the problem to a system of ordinary and fractional differential equations each of order at most unity. Gejji and Bhalekar [7] solved multiterm linear and nonlinear diffusion-wave equations of fractional order using the Adomian decomposition method (ADM). In [8] they used the method of separation of variables to solve multiterm fractional diffusion-wave equation subject to homogeneous/nonhomogeneous boundary conditions. Srivastava and Rai [9] used a new iterative method (NIM) and a modified Adomian decomposition method (MDM) to solve the MT-TFDE for different conditions. El-Sayed et al. [10] compared the Adomian decomposition method (ADM) against the proposed numerical method (PNM) to solve multiterm nonlinear fractional differential equations. Jiang et al. [3] used the method of separating variables to solve the multiterm time-fractional diffusion-wave equation and the MT-TFDE. Jin et al. [11] analyzed a space semidiscrete scheme based on the standard Galerkin finite element method using continuous piecewise linear functions to solve an initial/boundary value problem for a diffusion equation, involving multiple time-fractional derivatives. Dehghan et al. [12] applied a high order difference scheme and Galerkin spectral technique for the numerical solution of multiterm time-fractional partial differential equations. Zheng et al. [13] developed a high order numerical method based on the space-time spectral method for solving MT-TFDEs. Zhao et al. [5] used the finite element method to establish a fully discrete scheme for a class of two-dimensional MT-TFDEs. Chen et al. [14] proposed a unified numerical scheme based on the finite difference method in time and Legendre spectral method in space to solve MT-TFDE and diffusion-wave equations with variable coefficients; in this scheme they employed the Legendre spectral method in space and the weighted and shifted Grünwald difference operators for the discretization of the time-fractional operators.
Fractional reduced differential transform method (FRDTM) has been used successfully for solving linear and nonlinear fractional differential equations to obtain exact and approximate solutions. FRDTM was firstly introduced by Keskin and Oturance [15]. Gupta [16] considered eight different cases to find the approximate analytical solutions of Benney–Lin equation with fractional time derivative by FRDTM and the homotopy perturbation method (HPM). Srivastava et al. [17] studied the generalized time fractional-order biological population model (GTFBPM) by FRDTM and Rawashdeh [18] employed FRDTM to solve the nonlinear fractional Harry–Dym equation. Singh and Srivastava [19] presented an approximate series solution of the multidimensional (heat-like) diffusion equation with time-fractional derivative using FRDTM. Saravanan and Magesh [20] compared two analytical methods: FRDTM vs fractional variational iteration method (FVIM), to find numerical solutions of the linear and nonlinear Fokker–Planck partial differential equations with space and time fractional derivatives, Singh [21] presented FRDTM to compute an alternative approximate solution of initial valued autonomous system of linear and nonlinear fractional partial differential equations. Rawashdeh [22] proposed FRDTM to solve one dimensional space and time fractional Burgers’ equations (BE) and the time-fractional Cahn–Allen equation and Arshad et al. [23] presented a general form of FRDTM to solve wave-like problems, Zakharove–Kuznetsov equation, and couple BE, and Abuasad et al. [24] proposed FRDTM for finding exact and approximate solutions of the fractional Helmholtz equation.
In our work, we modify the FRDTM for the first time to handle MT-TFDEs. The rest of the paper is organized as follows: Section 2 gives some basic definitions and Section 3 presents the FRDTM for single and multiterm time-fractional equations. Three illustrative examples are given in Section 4. Section 5 is the conclusion.
2. Preliminaries
In this section, we give some basic definitions that we will use in the next sections.
2.1. The Gamma Function
The Gamma function is simply a generalization of the factorial real arguments. The Gamma function can be defined as [25]
2.2. Fractional Derivative
There are various definitions for fractional derivatives. The three most frequently used definitions for the general fractional derivative are the Grünwald-Letnikov (GL) definition, the Riemann-Liouville (RL), and the Caputo definition ([25–27]). In this paper we will use the Caputo fractional derivative, since the initial conditions for fractional order differential equation are in a form involving only the limit values of integer-order derivative at the lower terminal initial time , such as [25], and also the fractional derivative of a constant function is zero. The Caputo fractional derivative is defined as ([28]) where , , , and .
3. Fractional Reduced Differential Transform Method (FRDTM)
To illustrate the FRDTM, we give its definitions and properties. Consider a function of variables , such that Then, from the properties of the one-dimensional differential transform method (DTM), we havewhere is referred to as the spectrum of . Also, the lowercase is used for the original function, while its fractional reduced transformed function is represented by the uppercase , which is called the -function.
3.1. FRDTM for Single-Term Time-Fractional Equations [17, 19, 23, 24]
Let be analytical and continuously differentiable with respect to variables , , in the domain of interest; then FRDTM in –dimensions of is given bywhere .
The inverse FRDTM of is defined byFrom (5) and (6), we have In particular, for , the above equation becomesFrom the above definition, it can be seen that the concept of FRDTM is derived from the power series expansion of a function. Then the inverse transformation of the set of values gives approximate solution aswhere is the order of approximate solution. Therefore, the exact solution is given by Table 1 gives some properties of FRDTM, where is defined bywhere , , , and
3.2. The New Modification
In this section, we modify the definition of FRDTM for single-term time-fractional differential equation to be effective for MT-TFDEs as follows:where , , and , , , , and , , , with initial condition and arbitrary constants not both of them equal to zero.
Let be analytical and continuously differentiable with respect to variables and in the domain of interest ; then the m–FRDTM in –dimensions of is given by where .
The inverse FRDTM of is defined by From (13) and (14), we obtain In particular, for , the above equation becomes Then the inverse transformation of the set of values , , gives the approximate solution of the function as a finite power series, where is the order of approximate solution:where in (17) for MT-TFDEs (12) can be defined as where is the FRDTM of and can be found using the operations from Table 1. This shows that the FRDTM is a special case of the power series expansion.
Equation (12) represents time-fractional diffusion-wave equation if and a fractional diffusion equation if . When , Equation (12) represents a traditional wave equation, while if , it represents a traditional diffusion equation.
4. Application and Numerical Results
In this section, we consider three different applications to demonstrate the applicability of the m-FRDTM to MT-TFDEs. Two applications in two-term (1+1)-dimensional time-fractional equations and the third one in three-term (2+1)-dimensional time-fractional equation.
4.1. Example 1
If in Equation (12), then the two-term (1+1)-dimensional time-fractional diffusion equation is [31]subject to initial conditionwhere , , , and
Applying the appropriate properties given in Table 1 to Equation (19), we obtain the following recurrence relation:where .
From (22) the inverse transform coefficients of are given below: Continuing in the same manner and after a few iterations, the differential inverse transform of will give the following series solution:If , then the FRDTM solution (24) gives the exact solution of nonfractional diffusion Equation (19) subject to (20):which is exactly the same result obtained in [31], while for the fractional order we can obtain the approximate solutions for different values of : In general, the approximate solutions of (19) subject to (20) can be written as where and , , and .
The 3-dimensional plots of the FRDTM solutions of (19) with initial condition (20) are shown in Figure 1 for different values of . Figure 2 depicts solutions in 2-dimensional plots for different values of for and . The solutions in 2-dimensional plots for are depicted for different values of in Figure 3.
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4.2. Example 2
If , , and in Equation (12), then the two-term (1+1)-dimensional time-fractional diffusion equation is [11] subject to initial conditionwhere , , , and Applying the appropriate properties given in Table 1 to Equation (28), we obtain the following recurrence relation:where .
From (31) we obtain the inverse transform coefficients of as follows:Continuing in the same manner and after a few iterations, the differential inverse transform of will give the following series solution:If , then the third-approximate solution of the two-term time-nonfractional diffusion Equation (28) subject to (29) is The exact solution of nonfractional diffusion Equation (28) subject to (29) was given in [11] as The approximate solutions for different values of and are For simplicity if , , then the closest approximate solutions of Equation (28) subject to (29) to the exact solution of nonfractional Equation (35) can be obtained when , then, when , then , then . Similarly, if , , then the closest approximate solutions of Equation (28) subject to (29) to the exact solution of nonfractional Equation (35) can be obtained when , then, when , then , then .
We can note easily that the cases and will give the same approximate solutions when and , for any constants and , as we can see from Figure 5 which depicts solutions in 2-dimensional plots for different values of and for and .
The 3-dimensional plots of the FRDTM solutions of (28) with initial condition (29) are shown in Figure 4 for different values of and .
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4.3. Example 3
If , , and in Equation (12), then the three-term (2+1)-dimensional time-fractional diffusion equation is [5] subject to initial condition where , , , , and
Applying the appropriate properties given in Table 1 to Equation (37), we obtain the following recurrence relation:where .
The differential inverse transform of will give the following series solution: The approximate solutions for different values of and , , are The exact solution of nonfractional time-diffusion Equation (37) subject to (38) was given in [5] asThe 3-dimensional plots of the FRDTM solutions of (37) with initial condition (38) are shown in Figure 6 for different values of and , , , and . Figure 7 depicts solutions in 2-dimensional plots for different values of and for , , and .
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5. Conclusions
In this study, we introduced the m-FRDTM for the first time to handle MT-TFDEs. In Example 1 we obtained the exact solution of the nonfractional equation using the m-FRDTM and compared it with the approximate solutions of fractional equation for different orders, while in Examples 2 and 3 we obtained the approximate solutions of fractional equations using the m-FRDTM and compared them with the exact solutions of the nonfractional equations for different orders. Figures 1, 4, and 6 in three dimensions and Figures 2, 5, and 7 in two dimensions demonstrated that the m-FRDTM is very promising. Our future work will include the different types of multiterm fractional equations.
Data Availability
The datasets used and analysed during the current study are available from the corresponding author upon reasonable request.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All authors read and approved the final manuscript.
Acknowledgments
This work is funded by the Deanship of Scientific Research, King Faisal University, through the Nasher track (Grant no. 186087). Salah Abuasad is thankful to the Deanship of Scientific Research, King Faisal University. The authors would like to thank the respected reviewers for their valuable comments and suggestions.