New theories and applications of tempered fractional differential equations

In this paper, we develop theories, properties and applications of a new technique in tempered fractional calculus called the Tempered Fractional Natural Transform Method. This method can be used to solve a myriad of problems in tempered fractional linear and nonlinear ordinary and partial differential equations in both the Caputo and Riemann–Liouville senses. We prove some theorems and establish related properties of the Tempered Fractional Natural Transform Method. We give exact solutions, with graphical illustrations, to three well-known problems in tempered fractional differential equations including a special case of Langevin equation. Our results are the first rigorous proofs of Tempered Fractional Natural Transform Method. Further, the present work can be considered as an alternative to existing techniques, and will have wide applications in science and engineering fields.