Abstract
In this study, we deal with the problem of constructing semianalytical solution of mathematical problems including space-time-fractional linear and nonlinear differential equations. The method, called Shehu Variational Iteration Method (SVIM), applied in this study is a combination of Shehu transform (ST) and variational iteration method (VIM). First, ST is utilized to reduce the time-fractional differential equation with fractional derivative in Liouville-Caputo sense into an integer-order differential equation. Later, VIM is implemented to construct the solution of reduced differential equation. The convergence analysis of this method and illustrated examples confirm that the proposed method is one of best procedures to tackle space-time-fractional differential equations.
1. Introduction
Last couple of decades, employing fractional differential equations in modelling of processes such as dynamical systems, biology, fluid flow, signal processing, electrical networks, reaction and diffusion procedure, and advection–diffusion–reaction process [1–4] has gained great importance since these models reflect the behaviour of the processes better than integer-order differential equations.
Consequently, a great deal of methods such as [3, 4] are established to construct analytical and numerical solutions of fractional differential equations. Moreover, their existence, uniqueness, and stability have been studied by many scientists.
One of the significant integral transformations is Shehu transformation proposed by Maitama and Zhao [5]. This linear transformation is a generalization of Laplace transformation. However, the Laplace transformation is obtained by substituting in Shehu transformation. By this transformation, differential equations are reduced into simpler equations.
Various methods such as the homotopy perturbation method (HPM) and VIM are utilized to establish approximate solutions of differential equations of any kind [6, 7]. As a result, it is employed widely to deal with differential equations in various branches of science [8–11]. VIM has been modified by many researchers to improve this method. By modified VIM, the approximate solutions of initial value problems can be established by making use of an initial condition.
2. Preliminaries
In this section, preliminaries, notations, and features of the fractional calculus are given [12, 13]. Riemann-Liouville time-fractional integral of a real valued function is defined as where denotes the order of the integral.
The -order Liouville-Caputo time-fractional derivative operator of is defined as
The function is called Mittag-Leffler function depending on two parameters and .
The following set of functions has Shehu transformation: and it is defined as which has the following property: The inverse Shehu inverse transform of is defined as where [5].
For the -order of Liouville-Caputo time-fractional derivative of , the Shehu transformation has the following form [14]:
3. Main Results
3.1. Fractional Shehu Variational Iteration Method
To reveal the fundamental notions of this method, let us take the following space-time-fractional initial value problem in the Liouville-Caputo fractional derivative: where , , and denote the nonlinear, linear part of the differential equation, and the source function, respectively.
Utilizing Shehu transformation for Equation (9), we have
Employing the inverse Shehu transformation for Equation (11) leads to where , and so
The following recurrence relation is established by VIM:
Alternately, is called the -order of truncated solution.
If exists, then the analytical solution .
3.2. Convergence Theorem
Now, the convergence of VIM is investigated and required conditions and error estimate [15] are established for Equation (9).
The operator is introduced as where denote the components of the solution satisfying
Theorem 1 [16]. Let , defined in (5), be an operator from a Banach space BS to BS. The series solution as defined in (6) converges if exists such that , (i.e., ), .
Theorem 1, obtained from the Banach fixed-point theorem, is utilized to establish a sufficient condition for the convergence of fractional VIM.
Theorem 2 [16]. The exact solution of nonlinear problem (9) exists under the condition that the series solution defined in (18) converges.
Theorem 3 [16]. Suppose that the series solution defined in (18) converges to the solution . The maximum error for the approximate solution satisfies the following inequality:
The series solution of problem (9) is convergent to an exact solution , if the conditions
, hold where the parameters for are introduced as
Furthermore, the maximum absolute truncation error satisfies the inequality where .
4. Illustrative Examples
Example 1. Let us consider following space-time-fractional initial value problem
Step 1. Implementing Shehu transform for (23), we have
Step 2. Taking the inverse Shehu transform of (25), we get and so
Step 3. Employing the variational iteration method, we obtain
Based on the iteration formula (28), we have
By using the recurrence relation, we obtained the approximate solution of (23) as follows:
As a result, the analytical solution of (23) is reached by taking the limit of (30): where is the two-parameter Mittag-Leffler function.
Notice from Table 1 and Figure 1 that the values of the solution for and exact solution are the same which implies that the method implemented in this study is one of the best one for the solution of space-time-fractional differential equations of any order. Moreover, it is clear from Figure 1 that as and tend to 1, the corresponding solutions tend to exact solution. Three-dimensional graphs of exact solution and a truncated solution are given in Figure 2.
Example 2. Let us consider the space-time-fractional equation with the condition at .
Step 1. Carrying out Shehu transform of (32), we have
Step 2. Enforcing inverse Shehu transform of (34), we obtain and so
Step 3. Utilizing the variational iteration method, we have
Based on the iteration formula (37), we have
By using the recurrence relation, the approximate solution of (32) is obtained as follows:
Hence, the analytical solution of (32) and (33) is reached by taking the limit of (39): which is the same as obtained in [17].
As in Example 1, it is obvious from Table 2 and Figure 3 that the values of the solution for and exact solution are the same and as and tend to 1, the corresponding solutions tend to exact solution which indicates that the method employed in this research is a good choice for the solution of space-time-fractional differential equations of any order in Figure 4, 3-dimensional graphs of exact solution and a truncated solution are presented.
5. Conclusions
In this research, the targeted goal is to construct truncated solutions of linear/nonlinear space-time-fractional initial value problem by employing SVIM, the combination of the Shehu transform and variational iteration method. The main advantage of this method is that its implementation is straightforward and fruitful. Moreover, the illustrated examples reveal that the obtained approximate solutions with high precision converge swiftly to exact analytical solutions.
In the future study, this method and its improved modifications are applied to initial value problems including space-time-fractional linear and nonlinear differential equations.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Authors’ Contributions
All authors contributed equally to this work.
Acknowledgments
The first author would like to thank Scientific and Technological Research Council of Turkey (TUBITAK) for the financial support of the 2228-B Fellowship Program.