The exact solution of the composite fractional differential equation

The result of the fractional derivative of a function which is the fractional differential equation, has been used to describe many physical phenomena such as composite fractional oscillation equation (CFOE), as it provides memory and hereditary properties of the CFOE. The solution of the CFOE is essential and is at the interest of every researcher. The numerical methods used in obtaining the solution of the CFOE are prone to errors and are time-consuming in terms of the number of iterations before the desired solution is obtained. However, the analytic methods provide the exact solution to the CFOE and additionally, serves as a benchmark for which numerical solution of the CFOE is compared to obtain reliable and good approximated solution. Surprisingly, no researcher has ever applied analytic method to obtain the exact solution of the CFOE. In this paper, both the Homotopy Analysis Method (HAM) and the Variational Iteration Method (VIM) are used to obtain the exact solution of the CFOE in a suitable functional space. In using the HAM, it is flexible to choose the value of the convergence control parameter to adjust the emanated series from the CFOE to converge to the exact solution. On the other hand, the VIM is endowed with the Lagrangian multiplier which facilitates the convergence of emanated series from the CFOE to the exact solution. In addition, it is established in this paper that the use of HAM requires fewer iterations for the series emanating from the CFOE to converge to the exact solution as compared with the use of VIM. With this observation, the HAM is easier and more efficient to use as compared with the VIM.