Fractional Calculus for Skeptics I: The Fractal Paradigm (Fractional Order Thinking in Exploring the Frontiers of STEM)

This book is the first of its kind on fractional calculus (FC), dedicated to advocating for FC in STEM education and research.Fractional calculus is increasingly used today, but there remains a core population of skeptics regarding the utility of this'new'calculus. This book is intended for those who are skeptical about the need for fractional calculus to describe dynamic complex networks and must be convinced of its use on a case-by-case basis. It is a one-stop resource to rapidly read and replace the appropriate skepticism with new knowledge. It offers compelling reasons from the perspectives of the physical, social, and life sciences as to why fractional calculus is needed when addressing the complexity of an underlying STEM phenomenon. The six chapters are accompanied by useful and essential appendices and chapter-end references. Each includes new (fractional-order) ways of thinking about statistics, complexity dynamics, and what constitutes a solution to a complexity science problem.The book will appeal to students and researchers in all STEM-related fields, such as engineering, physics, biology and biomedicine, climate change, big data, and machine learning. It is also suitable for general readers interested in these fields. Cover Half Title Series Page Title Page Copyright Page Dedication Contents Preface Author Bios Nomenclature CHAPTER 1: We Know That How? 1.1. BRIEF HISTORY 1.2. TECH-EVOLUTION PATTERNS 1.3. LANGUAGE OF SET BIBLIOGRAPHY CHAPTER 2: Complex ⇒ Nonsimple 2.1. NEWTON’S WAY OF THINKING 2.2. FRACTIONAL DISSIPATION 2.3. THE INFINITESIMAL BIBLIOGRAPHY CHAPTER 3: Nonsimple ⇔ FOT 3.1. WHAT MANDELBROT SAW 3.2. SCALING THEORY 3.3. THE GAME HAS CHANGED BIBLIOGRAPHY CHAPTER 4: FOO for Skeptics 4.1. OPERATOR INTERPRETATION OF FOD 4.2. RATE EQUATIONS 4.3. FRACTIONAL-ORDER RATE EQUATIONS BIBLIOGRAPHY CHAPTER 5: FOU for Skeptics 5.1. PHYSICS-BASED STATISTICS 5.2. INVERSE POWER LAWS (IPLS) 5.3. NONSIMPLICITY HYPOTHESIS BIBLIOGRAPHY CHAPTER 6: FOT for One, Two, Many 6.1. “ONE VARIABLE” 6.2. “TWO VARIABLES” 6.3. “MANY VARIABLES” BIBLIOGRAPHY APPENDIX A: Fractional-Order Operators A.1. RL− DERIVATIVE A.2. FRACTIONAL-ORDER STOCHASTIC EQUATION & SUBORDINATION A.3. FRACTIONAL BROWNIAN MOTION A.4. DECISION-MAKING MODEL (DMM) A.5. FRACTIONAL-ORDER KINETIC EQUATION BIBLIOGRAPHY Index