Analysis of differential equations involving Caputo–Fabrizio fractional operator and its applications to reaction–diffusion equations

This manuscript deals with fractional differential equations including Caputo–Fabrizio differential operator. The conditions for existence and uniqueness of solutions of fractional initial value problems is established using fixed point theorem and contraction principle, respectively. As an application, the iterative Laplace transform method (ILTM) is used to get an approximate solutions for nonlinear fractional reaction–diffusion equations, namely the Fitzhugh–Nagumo equation and the Fisher equation in the Caputo–Fabrizio sense. The obtained approximate solutions are compared with other available solutions from existing methods by using graphical representations and numerical computations. The results reveal that the proposed method is most suitable in terms of computational cost efficiency, and accuracy which can be applied to find solutions of nonlinear fractional reaction–diffusion equations.


Introduction
Nowadays, the mathematical models involving fractional order derivative were given noticeable importance because they are more accurate and realistic as compared to the classical order models [22,26,28]. Motivated by the advancement of fractional calculus, many researchers have focused to investigate the solutions of nonlinear differential equations with the fractional operator by developing quite a few analytical or numerical techniques to find approximate solutions [6,10,19,29,30]. These differential equations involves several fractional differential operators like Riemann-Liouville, Caputo, Hilfer etc. [4,15,32].
However, these operators possess a power law kernel and have limitations in modeling physical problems. To overcome this difficulty, recently an alternate fractional differential operator having a kernel with exponential decay has been introduced by Caputo and Fabrizio [9]. This novel approach of fractional derivative is known as the Caputo-Fabrizio (C-F) operator which has attracted many research scholars due to the fact that it has a non-singular kernel. Also the C-F operator is most appropriate for modeling some class of real-world problem which follows the exponential decay law. With the passage of time, developing a mathematical model using the C-F fractional order derivative became a remarkable field of research. In recent times, several mathematicians were busy in development and simulation of CFFDE. One can read the articles of the aforementioned derivative to see further characteristics and applications [3, 8, 11, 16-18, 20, 25, 34].
In the present study, we analyze the following Caputo-Fabrizio fractional differential equations (C-FFDE) to obtain uniqueness and existence criteria of solutions [5,7,35]: (1) Fractional reaction-diffusion equations have been broadly examined as of lately. These equations emerge normally as description models of numerous evolution processes in various branches of science [12,21,33]. Furthermore by continuation to the above literature, we demonstrate the utility of the C-F operator on one-and two-dimensional reactiondiffusion equations, namely the Fitzhugh-Nagumo (FN) equation and the Fisher equation, respectively given by where the nonlinear function g(u) represents the reaction kinetics, h(x, t) is a source term and β denotes the diffusion coefficient. By considering g(u) = λu α (1u ν )(uθ ) and h(x, t) = 0, Eqs. (2) and (3) If we put λ = 1, β = 1, α = 1 and ν = 1, then Eq. (4) reduces to a time fractional FN equation which is one of the most significant reaction-diffusion equation, used to display the transmission of nerve driving forces [14]. The mathematical model of population genetics is also described by using the FN equation [1]. Next, if we take λ = 1 2 , β = 1, α = 1, ν = 1 and θ = 0, then Eq. (5) becomes a time fractional Fisher equation in an infinite domain as suggested by Fisher [13] as a model for the spatial transient propagation of a virile gene. In [2], Atangana studied a nonlinear Fisher reaction-diffusion equation.
In [24] Khan et al. used the homotopy analysis method (HAM) to find approximate analytical solutions of fractional reaction-diffusion equations. The residual power series method (RPSM) was applied by Tchier et al. [31] to find a numerical solution of fractional reaction-diffusion equations. The series solutions of reaction-diffusion equations were obtained by Merdan [27] by using a fractional variational iteration method (FVIM). Motivated by this, in the present manuscript, we compute approximate solutions of nonlinear fractional reaction-diffusion equations by using a reliable and efficient approach known as the iterative Laplace transform method (ILTM). The proposed technique is an amalgamation of the Laplace transform and the new iterative method (NIM) proposed by Jafari et al. [23].
In the light of the above examined writing, the present study concentrates on establishing the uniqueness and existence criteria of solutions of the nonlinear C-FFDE. Also, we demonstrate the effectiveness of the iterative Laplace transform method by obtaining the approximate solutions and 3D plots of the time fractional reaction-diffusion equations.

Definition 2.2 ([8])
The C-FFDO of order 0 < γ < 1 is given by like the usual Caputo derivative, this new operator gives CF D γ t u(t) = 0, if u is a constant function.
The main advantage of the Caputo-Fabrizio operator over the old operator of Caputo is that there is no singularity for t = s in the new kernel.

Definition 2.3 ([9])
The Laplace transform for the C-FFDO of order 0 < γ ≤ 1 and m ∈ N is given by In particular, we have has a solution in terms of the integral given by

Main results
Prior to expressing and demonstrating the fundamental outcomes, we present the following lemma and notations. Let μ = [0, 1] and C(μ) denote the space of all continuous functions on μ.
Considering the set B = {u(t)|u(t) ∈ C(μ)}, endowed with the norm u(t) B = max t∈μ |u(t)|, is a Banach space. By Lemma (9), IVP (1) is expressed as an integral equation given by Let the operator T : B → B be defined by then the fixed point of operator T is equivalent to the solution of IVP (1).

In this article, let us denote
Theorem 3.1 Let f : μ × R → R be a continuous function. Assume that at least one of the following conditions is fulfilled: Proof We make use of the Schauder fixed point theorem. For this purpose, assume that condition (H1) is satisfied.
Obviously, in the Banach space B, G is a ball. Next, we show that T: For all u ∈ G, we have Therefore, Here Tu(t) is continuous on μ. Now, assume that condition (H2) is satisfied. Select 0 < K ≤ ( 1 Ac 0 ) 1 δ-1 . Similarly, repeating the above arguments we get Consequently, we get T : G → G.
Obviously, the operator T is continuous because of the continuity of f .
Further, we set up the complete continuity of the operator T. Let R = max t∈μ |f (t, u(t))|, for any u ∈ G, let t 1 , t 2 ∈ μ be such that t 1 < t 2 .
Also, let U 1 = 2(1-γ ) (2-γ )M(γ ) and U 2 = 2γ (2-γ )M(γ ) , then we get In view of the uniform continuity of the function (t 2t 1 ) on the interval μ, we see that TG is an equicontinuous set. Also, this function is uniformly bounded as TG ⊆ G; hence T is completely continuous. As a result of Schauder's fixed point theorem, there exists a solution of IVP (1) in G.
Proof We denote the operator T by In view of σ < 1, T is a contraction. Consequently, T has only one fixed point as a result of Banach contraction principle, hence it gives a solution of IVP (1).

Applications
In this section, we shall propose an algorithm to solve a general non-homogeneous C-FFDE and demonstrate it by solving time fractional FN and Fisher equations.
Here f (x, t) is a source term, φ and ψ are given linear and nonlinear operator, respectively. Applying the Laplace transform (8) on two sides of (14) yields where Next, we apply the inverse Laplace transform on (15) then we have where the term obtained from the source term is denoted by χ(x, t). Further, we apply the new iterative method introduced in [10]. We consider the solution as an infinite series given as since φ is linear, The nonlinear operator ψ is decomposed as In view of (17), (18) and (19), Eq. (16) is equivalent to furthermore, we consider the recurrence relation as given by The p-term approximate solution is given by

Illustrative examples
The efficiency of ILTM is validated by simulating time fractional reaction-diffusion equations in this section.

Example 4.1 The time fractional Fitzhugh-Nagumo equation is
with initial condition Here x ∈ R, θ ∈ (0, 1) and the exact solution of Eq. (26) for γ = 1 is .
Applying the Laplace transform on (8) to (26), we get taking the inverse Laplace transform of (27) u( In view of Eq. (21) Tables 1 and 2 display the comparison among approximate solutions of (23) and (26), respectively, using ILTM with known results obtained by RPSM [31] and FVIM [27] for γ = 1, 0.8. The numerical values demonstrate that the region of convergence of approximate solutions depend continuously on the time fractional derivative γ . These tables clarify convergence of the approximate solutions to the exact solutions, and as   the value of the t decreases the absolute error becomes smaller. Hence, it is observed that the proposed technique is most suitable in terms of computational cost and accuracy for obtaining approximate solutions of nonlinear fractional reaction-diffusion equations.

Conclusions
This manuscript presents the existence and uniqueness criteria for nonlinear fractional differential equations involving the Caputo-Fabrizio differential operator. Further, we have developed ILTM to obtain approximate solutions of the Caputo-Fabrizio fractional differential equations successfully. The approximate solutions are compared with exact solutions and also with other existing solutions by other methods. It is observed that the obtained approximate series solutions for the first three terms are very precise and converge very rapidly to the solutions of real physical problems. The proposed approach is reliable, simple and effective to find approximate solutions of many nonlinear reactiondiffusion equations of fractional order.