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In this paper, we apply the Adomian decomposition method (ADM) for solving Fractional Differential Equations (FDEs) with some modifications to the traditional method. The aim of this paper is to make ADM more efficient, rapid in convergence, and easy to use, so we will discuss two modifications. We use the reliable modification to simplify calculations. For difficulties in symbolic integration, we use a numerical implementation method. All these modifications were applied to the integer-order case, so we would apply it to FDEs. Some numerical results are given from solving these cases and comparing the solution with the ADM method.

<abstract><p>In this paper, the Adomian decomposition method (ADM) and Picard technique are used to solve a class of nonlinear multidimensional fractional differential equations with Caputo-Fabrizio fractional derivative. The main advantage of the Caputo-Fabrizio fractional derivative appears in its non-singular kernel of a convolution type. The sufficient condition that guarantees a unique solution is obtained, the convergence of the series solution is discussed, and the maximum absolute error is estimated. Several numerical problems with an unknown exact solution are solved using the two techniques. A comparative study between the two solutions is presented. A comparative study shows that the time consumed by ADM is much smaller compared with the Picard technique.</p></abstract>

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In this research, we employ a newly developed strategy based on a modified version of the Adomian decomposition method (ADM) to solve nonlinear fractional differential equations (FDE) with both differential and nondifferential variables. FDE have disturbed the interest of many researchers. This is due to the development of both the theory and applications of fractional calculus. This track from various areas of fractional differential equations can be used to model various fields of science and engineering such as fluid flows, viscoelasticity, electrochemistry, control, electromagnetic, and many others. Several fractional derivative definitions have been presented, including Riemann-Liouville, Caputo,and Caputo-Fabrizio fractional derivative. We just need to calculate the first Adomain polynomial in this technique avoiding the hurdles in the nondifferentiable nonlinear terms' remaining polynomials. Furthermore, the proposed technique is easy to programme and produces the desired output with minimal work and time on the same processor. When compared to the exact solution, this method has the advantage of reducing calculation steps, while producing accurate results. The supporting evidence proves that modified Adomian decomposition has an advantage over traditional Adomian decomposition method which can be explained very clear with nonlinear fractional differential equations. Our computational examples with difficult issues are used to prove the new algorithm's efficiency. The results show that the modified ADM is powerful, which has a faster convergence solution than the original one. Convergence analysis is discussed, also the uniqueness is explained.