On the solution of time fractional initial value problem by a new method with ARA transform

The purpose of this research is to establish the solution to the time-fractional initial value problem (TFIVP) in Caputo- Fabrizio sense by implementing a new integral transform called ARA transform together with the iterative method. The existence of the ARA transform is investigated. Moreover, it is shown that the ARA integral transform of order n of a continuous function well defined. First, TFIVP is reduced into a simpler problem by utilizing the ARA transform. Secondly, the truncated solution of the reduced problem is obtained through the iterative method. Finally, the application of inverse ARA transform allows us to construct a truncated solution of TFIVP. The novelty of this study is that the first time the ARA transform is applied to obtain the solution of TFIVP in the Caputo-Fabrizio sense. Illustrative examples with the Fokker-Planck equation present that this method works better than other methods which is one of the strong points of this research.