Fractional Calculus Fundamentals and Applications in Economic Modeling
                          by
                       Austin McTier
             Athesis submitted to the Georgia College & State University
                     in partial fulfillment of the
                    requirements for the degree of
                   Bachelor of Science in Mathematics
                      Milledgeville, Georgia
                      December 12, 2016
           Keywords: Fractional Calculus, Caputo form, Riemann-Liouville form
                   Copyright 2016 by Austin McTier
                        Approved by
              Dr. Jebessa Mijena, Assistant Professor of Mathematics
                                                        Abstract
                   A relatively untapped branch of calculus, Fractional Calculus deals with integral and
               differential operators of non-integer order, as well as resolving differential equations consisting
               of said operators.  This paper examines certain properties, definitions, and examples of
               fractional integrals, Riemann-Liouville fractional derivatives, Caputo fractional derivatives
               and differential equations, along with various methods in order to solve them. In addition,
               this paper applies a fractional order approach to modeling the growth of the economies of
               the United States and Italy, particularly their gross domestic products (GDPs). Based on
               previous research, we expect to find that the implemented fractional models will have a
               stronger performance than alternative methods of measuring economic growth.
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                       Acknowledgments
         I would like to thank Dr. Mijena for being an incredible capstone adviser, as well as a
       great mentor and integral component in my understanding of the material at hand. I would
       also like to thank the Mathematics Department of Georgia College and State University,
       particularly the professors that have taught me throughout my undergraduate career. I
       would also like to give my thanks to my mother and father for supporting me in all that I
       do.
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                                                        Table of Contents
                Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     ii
                Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .       iii
                1     Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     1
                2     Fractional Calculus: Definitions and Examples . . . . . . . . . . . . . . . . . . .           2
                         2.0.1   Fractional Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . .      2
                         2.0.2   Properties of Riemann-Liouville Integrals and Fractional Derivatives .            3
                         2.0.3   Properties of Riemann-Liouville Integrals and Fractional Derivatives .            7
                3     Fractional Differential Equations      . . . . . . . . . . . . . . . . . . . . . . . . . .    9
                    3.1  Fractional Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . .        11
                4     Application: Economic Growth Modeling . . . . . . . . . . . . . . . . . . . . . .           18
                         4.0.1   Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .     20
                Bibliography    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   22
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