A new generalized differential transform method for fractional ODEs with statistical applications

This paper presents a new generalized differential transform method (NGDTM) in the solution of fractional-order differential equations. The technique is based on the generalized Taylor formula and the Riemann–Liouville fractional derivative. Theorems of fundamental transformation are developed based on rigorous proofs, and the convergence and uniqueness of the solutions obtained are proven. A number of linear and nonlinear examples such as models of statistical relevance are provided to demonstrate the accuracy and efficiency of the proposed approach. Besides, the classical exponential distribution is derived using the proposed NGDTM, and a new fractional exponential distribution is proposed on the same basis with the use of the same framework. The findings reveal that the technique provides very good approximate solutions and, in few instances, the exact solution by only few iterations, thus validating it as a tool of fractional differential equations as well as use in statistics.