Combined Adomian Decomposition Method with Integral Transform

At present, three numerical solution methods have mainly been used to solve fractional-order chaotic systems in the literature: frequency domain approximation, predictor–corrector approach and Adomian decomposition method (ADM). Based on the literature, ADM is capable of dealing with linear and nonlinear problems in a time domain. Also, the Adomian decomposition method (ADM) is among the efficient approaches for solving linear and non-linear equations. Numerical solution method is one of the critical problems in theoretical research and in the applications of fractional-order systems. The solution is decomposed into an infinite series and the integral transformation to a differential equation is implemented in this work. Furthermore, the solution can be thought of as an infinite series that converges to an exact solution. The aim of this study is to combine the Adomian decomposition approach with a different integral transformation, including Laplace, Sumudu, Natural, Elzaki, Mohand, and Kashuri-Fundo. The study's key finding is that employing the combined method to solve fractional ordinary differential equations yields good results. The main contribution of our study shows that the combined numerical methods considered produce excellent numerical performance for solving fractional ordinary differential equations. Therefore, the proposed combined method has practical implications in solving fractional order differential equations in science and social sciences, such as finding analytical and numerical solutions for secure communication system, biological system, financial risk models, physics phenomenon, neuron models and engineering application.

Some literature about Laplace transform and Adomian decomposition methods can be seen in [18]- [20]. Yindoula et al. [18] proposed the convection diffusion-dissipation equation using ADM and Laplace transform method. They show that the approximate solution of the equation has an exact solution. Zhen and Song [19] studied synchronization of FitzHugh-Nagumo neurons model using Laplace Transform and the ADM. They show that the Adomian decomposition approach yielded an exact analytical result for the two FitzHugh-Nagumo neurons models. Chen and Liu [20] solved a HIV infection model of CD4+ T Cells via Padé approximation process, ADM and Laplace transform method. They found that the Padé approximation and Laplace transform for obtaining the solutions of ADM is the most preferred approach for solving HIV infection model. Doğan [21] presented combined Laplace Transform and ADM for solution of the ordinary differential equations (ODEs). They found that the Combined Adomian decomposition-Laplace transform can be applied in the linear and non-linear systems of ODEs.
Previous studies about the Adomian-Natural decomposition approach and Elzaki decomposition method can be seen in [29]- [32]. Veeresha et al. [29] presented natural decomposition technique for solving fractional forced KdV equation. Khan et al. [30] proposed natural-transformation decomposition approach for fractional-order hyperbolic telegraph equation. Shah et al [31] analyzed fractional-order diffusion equations' transform decomposition method. Nuruddeen [32] solved linear and nonlinear Schrodinger using Elzaki decomposition method. Varsoliwala and Singh [33] studied model of atmospheric internal waves phenomenon using Elzaki decomposition method. Khan et al. [34] applied the Elzaki transform decomposition method and Caputo operator to solve the Navier-Stokes equations. In addition, the study of Mohand decomposition method and Kashuri-Fundo decomposition method can be seen in [35]- [38].
Based on the background study of previous research, this paper presents review the combination of Adomian decomposition methods with other integral transforms, such as Sumudu, Laplace, Natural, Elzaki, Mohand, and Kashuri-Fundo.

Integral Transform and Fractional
This section describes the basic theories related to Laplace, Sumudu, Natural, Elzaki, Mohand, and Kashuri-Fundo transforms.
Definition 2.1 [39]. Let be a real or complex function of the value > also, let be a real or complex coefficient. Then, the Laplace transformation is defined as Furthermore, ℒ −1 [ ( )] = ( ), ≥ 0 is the inverse of the Laplace transform. If α is a fractional number (FN), then Definition 2.2 [40]. The definition of Laplace transformation of the Caputo fractional derivative (CFD) is given as Definition 2.3 [41]. Given a set of functions Sumudu transform is defined as (2) Definition 2.4 [42]. The Sumudu transform of CFD is expressed as Definition 2. 5 [43]. Given a set of functions Natural transform is defined as Furthermore, the inverse of the Natural transform is denoted as Definition 2.6 [29]. The natural transform of the CFD is expressed as Definition 2.7 [44]. Given a set of functions Elzaki transform is defined as

Furthermore, Elzaki transform inverse is represented by
Definition 2.8 [44]. Elzaki transform of CFD is given as Definition 2.9 [45]. Given a set of functions Mohand transform is defined as Definition 2.10 [45]. The Mohand transform of CFD is given as Definition 2.11 [37]. Given a set of functions Kashuri-Fundo transform is expressed as Definition 2.12 [37]. The Kashuri-Fundo transformation of CFD is given as

Adomian Decomposition Method
The following discussion is about the basic concepts of the ADM Given the following equation where L represents the linear operator inverse, N is the nonlinear operator, and P denotes the remaining linear part. Equation (7) can be rewritten with Ly as the subject Since L is inverse, the inverse operator of L is defined as L −1 . So that if both sides of equation (8) are inverted by L −1 , we get The ADM supposes that y should be disintegrated into an infinite series where y n can be determined iteratively. ADM also supports the decomposition of nonlinear operator Ny into an infinite series of polynomial form: where the Adomian polynomial (AP) is defined by A n = A n (y 0 , y 1 , y 2 , … , y n ) and the parameter is denoted by λ. A n is expressed as (10) and (11) are substituted into (9), then, we have Describing both sides of (12) successively will give Thus, in general, the recursive relation obtained from the solution is as follows Thus, equation (7) approximate solution becomes:

Result and Discussion
In this section, we applied the various numerical methods and definitions discussed above to solve fractional ordinary differential equations (FODE) and report the results as follows.

Laplace Decomposition Method
Consider the following FODE with initial condition y(0) = c , and is the CFD operator with 0 < ≤ 1, is a linear operator, is a nonlinear operator, the function displays the non-homogeneity of the differential equation, and the function is said to be determine respect to . Using definition 2.2 and equation (14), we get Then, applying Laplace transform inverse in (15) will produce If we substitute (10) and (11) into (16), we obtain Describing both sides of (17) will successively produce ⋮ Thus, in general, the following recursive relation is obtained from the solution of FODE via the Laplace decomposition approach on (14)

Sumudu Decomposition Method
Consider fractional ordinary differential equation defined in equation (14), Using definition 2.4 and equation (14), we obtain y(u) = y(0) Also, applying the Sumudu transform inverse in (19) will give If we substitute (10) and (11) into (20), we have Describing both sides of (21) will successively produce Thus, in general, the following recursive relation is obtained from the solution of FODE via Sumudu decomposition approach on (14)

Natural Decomposition Method
Consider fractional ordinary differential equation defined in equation (14), Using definition 2.6 on equation (14), we obtain the solution using the Natural Decomposition method as follows: Applying the Natural transform inverse in equation (25) will produce If we substitute (10) and (11) into (24), we will have Describing both sides of (25) will successively produce Thus, in general, the following recursive relation is obtained from the solution of FODE via Natural decomposition technique on (14)

Elzaki Decomposition Method
Consider fractional ordinary differential equation defined in equation (14), Using definition 2.8 on equation (14), we obtain the solution using the Elzaki decomposition approach as follows: Applying the Elzaki transform inverse in (27) will give Next, we substitute (10) and (11) into (28) to obtain By describing both sides of (29) will successively produce

Mohand Decomposition Method
Consider fractional ordinary differential equation defined in equation (14), Using Mohand transform on (14) and based on definition 2.10, we have Applying the Mohand transform inverse in (31) will produce Next is substituting (10) and (11) into (32) to obtain Describing both sides of equation (33) will successively produce Therefore, in general, the following recursive relation is obtained from the solution of FODE via Mohand decomposition method (14)

Conclusions
This paper is structured to combine the Adomian decomposition method with various integral transforms, including Natural, Sumudu, Laplace, Elzaki, Mohand, and Kashuri-Fundo. The main finding of our study shows that the combine numerical methods considered produce excellent numerical performance for solving fractional ordinary differential equations. For future evaluations, the results of this study can be applied to various disciplines. Such as communication security systems and electronic circuits.