Semianalytical Solutions of Some Nonlinear-Time Fractional Models Using Variational Iteration Laplace Transform Method

In this work, we combined two techniques, the variational iteration technique and the Laplace transform method, in order to solve some nonlinear-time fractional partial differential equations. Although the exact solutions may exist, we introduced the technique VITM that approximates the solutions that are difficult to find. Even a single iteration best approximates the exact solutions. The fractional derivatives being used are in the Caputo-Fabrizio sense. The reliability and efficiency of this newly introduced method is discussed in details from its numerical results and their graphical approximations. Moreover, possible consequences of these results as an application of fixed-point theorem are placed before the experts as an open problem.

Nowadays, the techniques of fractional calculus are being employed successfully for better understanding of complex natural phenomena, which not only agree with the ordinary calculus techniques but also give the best results and understanding of the phenomena. Laplace transform is a powerful tool, which has been used in the past decades to solve the ODEs with constant and variable coefficients as well as to solve PDEs. Similarly, in these days, the variational iteration technique, developed by the Chinese mathematician He [15], is also a reliable technique (which was originally developed to solve differential and integrodifferential equations) to solve PDEs. The main drawback of the variational iteration method is that one may have difficulty in calculating the Lagrange multiplier. Currently, much attention is being paid in combining more than one technique to solve a model especially nonlinear models, to get better and rapid results. In this direction the work has been started, and it is observed that the results obtained by combining more than one technique are much better than that of a single technique as discussed in [24,25].
In the current paper, two techniques, variational iteration technique and the Laplace transform, are being utilized, and the combined technique, the variational iteration transform method (VITM), is employed to handle the nonlinear fractional order partial differential equations, like the Kortewegde Vries equation [26], Schrödinger equation [27], and Burger equation [28]. The rapid convergence of the method proves that it is a more reliable technique now more than ever than the existing one to solve FPDEs, and it introduces a new significant improvement. In the proceeding sections, method description along with the validity of the results obtained by the technique is presented.
In the study of the fractional differential equations, the Caputo-Fabrizio fractional derivative [29] will be considered. The Caputo-Fabrizio fractional derivative is the most recent fractional derivative which is more effective than the other fractional derivatives present in the literature, in dealing with the initial value problems. First, let us recall the some definitions from the area of fractional calculus.
(3) Caputo-Fabrizio Derivative. Let us recall one of the most recent definitions of the fractional derivative Caputo-Fabrizio derivative, as follows. Let FðtÞ∈H 1 ða, bÞ, b > a; then, the Caputo-Fabrizio time fractional derivative of FðtÞ is defined as where αϵ½0, 1 and MðaÞ is a normalization function that is Mð0Þ = Mð1Þ = 1.
(4) Laplace Transform. Let f ðtÞ be a function; then, its Laplace transform is defined as and the Laplace transform of f ′ ðtÞ is given by

Methodology of VITM
This section is devoted to present the methodology of the proposed technique. Then, let us consider the general time fractional partial differential equation.
subject to where LðXðφ, tÞÞ, NðXðφ, tÞ, Yðφ, tÞ, Zðφ, tÞÞ, and Fðφ, tÞ are linear, nonlinear, and known functions, respectively. Also D α t is in the Caputo-Fabrizio sense. Further, we apply the variational iteration method on the above equation. Then, we found the following iterative form: Also, if we apply the Laplace transform on this equation, we transform the variable t, to the new variable s, such that whereX n ðφ, tÞ, etc. are restricted values, which means Using the following relations: where Journal of Function Spaces Then, we have The optimization conditions, give The above equation implies λ = −1/s α . On substituting in equation (10), we obtain The inverse Laplace transform gives Substituting n = 0, 1, 2, ⋯, we find the following successive approximations:

Applications of VITM on Various FODE Types
In this section, we present and apply VITM on some important FODEs from related literature. Then, our first application takes into consideration the most general time fractional form of the Korteweg-de Vries equation (see [24]).
subject to where is the nonlinear parameter and is the dispersion parameter. Applying the variational iteration and Laplace transform, we get Also, substituting the following relation and optimality conditions, etc., we get the next results: By substitution, we have Applying the inverse Laplace transform on the above equation and simplifying, we get For n = 0, 1, 2, ⋯, the following approximations are obtained: X 1 , X 2 , X 3 ⋯ , such as and so on. The solution Xðφ, tÞ can be found as As particular examples, let us consider further some versions of time fractional equations. Example 1. The first particular example is a simple time fractional Korteweg-de Vries equation (see [18]).
Application of the proposed VITM step by step gives The optimality conditions, etc. give the following results: Substitution and inverse Laplace transform implies For n = 0, 1, 2, ⋯, we get the approximations X 1 , X 2 , X 3 , ⋯, such as The solution Xðφ, tÞ can be found as that is, For α = 1, it turns out to be which is the expansion of the exact solution Xðφ, tÞ = 6ðφÞ /ð1 − 36tÞ that confirms the validity of the proposed VITM (see [18]). Next, let us give a graphical representation of the approximated solution Xðφ, tÞ, for different values of α using Mathematica. Moreover, we will also give a graphical 3D representation for the exact solution Xðφ, tÞ = 6ðφÞ/ð1 − 36tÞ (Figure 1(b)). In this way, we show how the proposed technique approaches the exact solution; see Figures 2(a), 2(b), and 1(a), which are the approximations of Figure 1(b). The scale for all the four figures is −50 ≥ φ ≤ 50 and −50 ≥ t ≤ 50. [18]).

Example 2. Let us consider another version of time fractional version of Korteweg-de Vries equation (see
Applying VITM step by step, we obtain Journal of Function Spaces Using optimality conditions, etc., we get the following results: Substitution and inverse Laplace transform implies For n = 0, 1, 2, ⋯, we get the approximations X 1 , X 2 , X 3 , ⋯, such as The solution Xðφ, tÞ can be found as Then, For α = 1, we have which is the expansion of the exact solution, Xðφ, tÞ = −2 sec h 2 ðφ − 4tÞ (see [18]). Example 3. Consider the simple time fractional Burgers equation (see [18]).
Applying the proposed VITM step by step, we get The optimality conditions, etc. give the following results:

Journal of Function Spaces
Substitution and inverse Laplace transform implies For n = 0, 1, 2, ⋯, we get the approximations X 1 , X 2 , X 3 , ⋯, such as The solution Xðφ, tÞ can be found as It means For α = 1, we have which resembles with the expansion of the exact solution Xðφ, tÞ = φ/ð1 + tÞ, confirming the validity of the proposed VITM (see [18]). Further, let us draw some approximations of Xðφ, tÞ = φ − φt + φt 2 + ⋯, for different values of α. Then, see   [18]) as follows: Applying the variational iteration and Laplace transform, we get Using optimality conditions, etc., we obtain the following results:  Journal of Function Spaces Substitution and inverse Laplace transform implies For n = 0, 1, 2, ⋯, we get the approximations X 1 , X 2 , X 3 , ⋯, such as The solution is in the series form, such as which turns out to be the expansion of the exact solution Xðφ, tÞ = −2 tan φ for α = 1, as discussed earlier (see [18]).
Example 5. Let us consider the time fractional version of the nonlinear simple Schrödinger equation [18] as follows Applying the variational iteration and Laplace transform, we obtain Using optimality conditions, etc., we get the following results: Substitution and inverse Laplace transform implies For n = 0, 1, 2, ⋯, we get the approximations X 1 , X 2 , X 3 , ⋯, such as 7 Journal of Function Spaces The solution is in the series form, such as which turns out to be the exact solution Xðφ, tÞ = e iðφ+tÞ for α = 1 (see [18]). Open question: as an application of the VITM on nonlinear-time fractional differential equations towards fixed-point theorem. One can obtain some approximations of the solution X 1 , X 2 , X 3 ⋯ . Moreover, it can be asked whether these approximations of the solution are equivalent with the iterations of a sequence of successive approximations which are convergent to a fixed point or not? What are the minimum hypotheses imposed which lead us to the existence and the uniqueness of a fixed point in this case?

Discussions and Concluding Remarks
The proposed method VITM being the combination of two basic techniques, VIM and Laplace transform, is understandable by just having the formal knowledge of advanced calculus; indeed, it is understandable even for the reader who has no strong background and base in calculus of variations. It is simple and can be easily applied as compared to the more traditional VIM for fractional differential equations.
Using the proposed method in the present paper, we study the convergence for some nonlinear fractional order partial differential equations as the Korteweg-de Vries equation, Schrödinger equation, and Burger equation. The rapid convergence of the method proves that it is a more reliable technique now more than ever than the existing ones to solve FPDEs, and it introduces a new significant improvement. The reliability and efficiency of this simple and newly introduced method is discussed by giving some numerical results and their graphical approximations.
Moreover, we propose an interesting connection between the approximations of the solution X 1 , X 2 , X 3 ⋯ and the iterations of a sequence (convergent or not), from the area of fixed-point theory.
The main advantage of this proposed variational method together with Laplace transform helps to speed up the computational work and may easily be applied to nonlinear dynamical systems using software like Mathematica™, MATLAB™, and Maple™.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The authors declare that they have no competing interests.