Date of Award


Document Type


Degree Name

Doctor of Philosophy (PhD)


Mathematical Sciences

First Advisor

Daniel E. Bentil


In Fractional Calculus (FC), the notion of fractional derivatives and integrals arise from convolutions with a power law, which, when multiplied by an exponential factor, one obtains tempered fractional derivatives and integrals.

The purpose of this dissertation is to develop theories and applications of a new technique in FC called the Tempered Fractional Natural Transform Method (TFNTM). This method can be used to solve a myriad of problems in linear and nonlinear ordinary and partial differential equations.

We present convergence analysis, give error estimates, and provide exact solutions, with graphical illustrations, to many well-known problems in tempered fractional differential equation, such as the time-space tempered fractional convection-diffusion equation (FCDE) and tempered fractional Black-Scholes equation (FBSE) arising in the financial market. To further show the accuracy and efficiency of our approach, we also apply our methodology to a nonlinear time-fractional biological population model for the dispersal of animals within an enclave.

Indeed, finding exact solutions to tempered fractional differential equations (TFDEs) is far from trivial. The proposed methodology, which has wide applications in science and engineering fields, is an excellent addition to the myriad of techniques for solving TFDE problems, and an alternate method with considerable promise for further practical applications.



Number of Pages

160 p.

Available for download on Thursday, April 13, 2023