Application of generalized differential transform method to multi-order fractional differential equations

In a recent paper [Odibat Z, Momani S, Erturk VS. Generalized differential transform method: application to differential equations of fractional order, Appl Math Comput. submitted for publication] the authors presented a new generalization of the differential transform method that would extended the application of the method to differential equations of fractional order. In this paper, an application of the new technique is applied to solve fractional differential equations of the form y ( μ ) ( t ) = f ( t , y ( t ) , y ( β 1 ) ( t ) , y ( β 2 ) ( t ) , … , y ( β n ) ( t ) ) with μ > β n > β n - 1 > … > β 1 > 0 , combined with suitable initial conditions. The fractional derivatives are understood in the Caputo sense. The method provides the solution in the form of a rapidly convergent series. Numerical examples are used to illustrate the preciseness and effectiveness of the new generalization.