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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. ii 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. iii 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 iv
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