Using the Elzaki decomposition method to solve nonlinear fractional differential equations with the Caputo-Fabrizio fractional operator

The techniques of fractional calculus are applied successfully in many branches of science and engineering, one of the techniques is the Elzaki Adomian decomposition method (EADM), which researchers did not study with the fractional derivative of Caputo Fabrizio. This work aims to study the Elzaki Adomian decomposition method (EADM) to solve fractional differential equations with the Caputo-Fabrizio derivative. We presented the algorithm of this method with the CF operator and discussed its convergence by using the method of the Cauchy series then, the method has applied to solve Burger, heat-like, and, couped Burger equations with the Caputo -Fabrizio operator. To conclude the method was convergent and effective for solving this type of fractional differential equations


Introduction
Fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order) has grown in popularity and relevance over the last three decades, owing to its proven applications in a wide range of apparently disparate domains of science and engineering.It does, in fact, give some potentially valuable methods for solving differential and integral equations, as well as a variety of other problems requiring mathematical physics special functions, as well as their extensions and generalizations in one or more variables [1][2][3][4] .
In the past decade, Caputo and Fabrizio introduced a new fractional differential operator many researchers studied this operator and researchers are still interested in this operator because of its importance, as some studies have applied methods of approximate solutions to equations that include this fractional operator 5,6 .The requirement for a model that represents the behavior of classic viscoelastic materials, thermal media, electromagnetic systems, and so on has piqued the attention of researchers in this unique technique.Plasticity, fatigue, and damage, as well as electromagnetic hysteresis, appear to benefit most from the original concept of fractional derivative.When these effects are missing, it appears that the new fractional derivative is more appropriate 7 .where ,  ∈ ℋ 1 ( 1 ,  2 ) Definition 2: 15 Over a set of functions , the Elzaki transform is defined, by the following formula where  is a parameter of Elzaki transform.

Lemma 1:
The Elzaki transform for the Caputo-Fabrizio fractional operator is defined as follows if Proof:

Analysis of EDM
In the Caputo-Fabrizio operator sense, consider the following nonlinear partial differential equation: with initial condition (, 0) =  0 (), 5 where  0 ℱ   (, ) is Caputo-Fabrizio operator of (, ) , a linear operator is ℛ, a nonlinear operator is  and a source term is ℊ.
Applying the Elzaki transform to both sides of the Eq.4, from Lemma 1 and Eq.5, }, 8 by using the Elzaki transform's inverse to Eq.8, Suppose that (, ) is a solution of Eq.9, which it expressed as , the nonlinear term can be decomposed as where, where,  = 0,1,2, … .. Substituting Eq.10 and Eq.11 into Eq.9 gives us the result that When the left and right sides of Eq.13 are compared, In its general form, a recursive relation is The approximate solution is given by . 16

Convergence analysis
This section discusses the convergence of Elzaki decomposition method to the exact solution of fractional differential equations as well as the estimated error resulting from the approximate solutions.

Proof:
The sequence of partial sums is defined as Cauchy series in Banach space, , defined in Eq.16 converges.Therefor the above inequality 19 becomes As a result, the approximate solution may be derived using Eq.15 by the above algorithms and after simple steps, ⋮ and so on.
Therefore, the series solution (, ) of Eq.21 is given by If it was  → 1 in Eq.26 then, the exact solution is Table 1 displays the values of the exact and approximate solutions of Eq.21 at the different  values, also includes the absolute error of the approximate solutions for the exact solutions.Fig. 1 shows graphs of the exact and approximate solutions of Eq.21 at the different  values between 0 and 1.
30 by using the inverse Elzaki transform to both sides of Eq.30,  = sin() sin() as a result, the approximate solution may be derived using Eq.15,  0 = sin() sin(), Hence, from Eq 29 and Eq 32, the components give as follows:  0 = sin() sin(), 33 and so on.Therefore, the approximate solution of (, , ) of Eq.28 is given by  = sin() sin() −2 sin() sin() (   2 displays the values of the exact and approximate solutions of Eq.28 at the different  values, also includes the absolute error of the approximate solutions for the exact solutions.Fig. 2 shows graphs of the exact and approximate solutions of Eq.28 at the different  values between 0 and 1.  , 40 and the nonlinear terms may be broken down into , 41 where = sin() When both sides of the Eq.43 are compared,  0 = sin(),  0 = sin(), ]. Therefore, the approximate solution of Eq.36 is given by 44 If put  → 1 and → 1 in Eq.44, the problem solution will be re-created as follows: 45 This is the closed form equivalent of the precise solution: (, ) = sin() ℯ − , (, ) = sin() ℯ − .46 Table 3 displays the values of the exact and approximate solutions of system 36 at the different  and  values, also includes the absolute error of the approximate solutions for the exact solutions.Fig. 3 shows graphs of the exact and approximate solutions of system 36 at the different  and  values between 0 and 1.  Remark.By comparing the results of this method with Yang decomposition method it appears that the results are similar to both methods 20 .

Conclusion
In this article, the Elzaki decomposition method has been presented in terms of its derivation and convergence and its application to fractional differential equations (FDEs).The method was convergent and efficient to solve fractional differential equations with the Caputo-Fabrizio operator.The approximate solution of fractional differential equations (FDEs) with the derivative of Caputo-Fabrizio was convergent to the exact solution.Finally, this method can be adopted to solve fractional differential equations of this type.

Figure 1 .Example 2 :
Figure 1.The graphs of the approximate and the exact solutions among different values of  and  in case (3D) and fixed  in case (2D) when  = 0.9,1 for nonlinear Burger's equation in the CF fractional operator,(A) is the app.sol. of  at  = ., (B) is the app.sol. of u at  = , (C) is the exact sol. of .

Figure 2 .Example 3 :
Figure 2. The graphs of the approximate and the exact solutions of heat-like equation among different values of  and  when  is fixed in case (3D) and ,  are fixed in case (2D) when  = 0.9,1 for nonlinear Burger's equation in the CF fractional operator ,(A) is the app.sol. of  at  = . ,(B) is the app.sol. of u at  = , (C) is the exact sol. of .

Figure 3 .
Figure 3.The graphs of the approximate and the exact solutions of nonlinear system 36 among different values of  and  in case (3D) and  is fixed in case (2D) when  = 0.9,1 for nonlinear system in the CF fractional operator, (A) is the app.sol. of ,  at  = . ,(B) is the app.sol. of ,  at  = , (C) is the exact sol. of , .

Table 2 . The approximate and exact solution of heat-like equation with fractional operator CF,
is app.sol. of  at  = .,   is app.sol. of  at  = ,   is exact sol. of  .