Publication Date

5-2010

Advisor(s) - Committee Chair

Dr. Ferhan Atici (Director), Dr. John Spraker, Dr. Mark Robinson, Dr. Nancy Rice

Degree Program

Department of Mathematics and Computer Science

Degree Type

Master of Science

Abstract

Almost every theory of mathematics has its discrete counterpart that makes it conceptually easier to understand and practically easier to use in the modeling process of real world problems. For instance, one can take the "difference" of any function, from 1st order up to the n-th order with discrete calculus. However, it is also possible to extend this theory by means of discrete fractional calculus and make n- any real number such that the ½-th order difference is well defined. This thesis is comprised of five chapters that demonstrate some basic definitions and properties of discrete fractional calculus while developing the simplest discrete fractional variational theory. Some applications of the theory to tumor growth are also studied.
The first chapter is a brief introduction to discrete fractional calculus that presents some important mathematical functions widely used in the theory. The second chapter shows the main fractional difference and sum operators as well as their important properties. In the third chapter, a new proof for Leibniz formula is given and summation by parts for discrete fractional calculus is stated and proved. The simplest variational problem in discrete calculus and the related Euler-Lagrange equation are developed in the fourth chapter. In the fifth chapter, the fractional Gompertz difference equation is introduced.
First, the existence and uniqueness of the solution is shown and then the equation is solved by the method of successive approximation. Finally, applications of the theory to tumor and bacterial growth are presented.

Disciplines

Cell Biology | Discrete Mathematics and Combinatorics | Other Applied Mathematics