## Masters Theses & Specialist Projects

#### Publication Date

5-2011

Dr. Ferhan Atici (Director), Dr. Ngoc Nguyen, Dr. Mark Robinson

Our translation of real world problems to mathematical expressions relies on calculus, which in turn relies on the differentiation and integration operations. We differentiate or integrate a function once, twice, or any whole number of times. But one may ask what would be the 1/2-th or square root derivative of x. Fractional calculus generalize the operation of differentiation and integration to non-integer orders. Although it seems not to have significant applications, research on this subject could be valuable in understanding the nature.
The main purpose of this thesis is to develop fractional trigonometry and generalize Wronskian determinant that can be used to determine the linear independence of a set of solutions to a system of fractional differential equations. We also introduce a new set of fractional differential equations whose solutions are fractional trigonometric functions.
The first chapter gives a brief introduction to fractional calculus and the mathematical functions which are widely used to develop the theory of this subject. The second chapter introduces the reader to fractional derivative and fractional integral, and the most important properties of these two operators that are used to develop the following chapters. The third chapter focuses on the Laplace transform which is the means that will be used to solve the fractional differential equations. In the fourth chapter, we develop fractional trigonometric functions, generalized Wronskian, also classes of fractional homogeneous differential equations and the characteristic equations, as well as the solution to these types of equations. Also, we present a new method to solve a system of fractional differential equations, and by using this method we solve a system that generates a solution of a linear combination of two fractional trigonometric functions. The fifth chapter is the conclusion of this thesis and is devoted to demonstrating an application of fractional calculus to pharmacokinetic model, which is used to study the drug concentration in the body.

#### Degree Program

Department of Mathematics and Computer Science

#### Degree Type

Master of Science

Mathematics

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