About Book

Calculus is a branch of mathematics that explores variables and how they change by looking at them in infinitely small pieces called infinitesimals. Calculus refers to the study of how things change. It provides a framework for modeling systems in which there is change, and a way to deduce the predictions of such models. The most common practical use of calculus is when plotting graphs of certain formulae or functions. Using methods such as the first derivative and the second derivative, a graph and its dimensions can be accurately estimated. These two derivatives are used to predict how a graph may look like, the direction that it is taking on a specific point, the shape of the graph at a specific point (if concave or convex), just to name a few. Among the disciplines that utilize calculus include physics, engineering, economics, statistics, and medicine. This book elucidates new techniques and applications of calculus in a multidisciplinary approach. It picks up individual branches and explains their need and contribution in the context of the growth of this field. It deals with calculus elements, quantum physics, fractional calculus, pulse‐induced calculus, predicate calculus, quantum calculus and differential equations. The various sub-fields of calculus along with technological progress that have future implications are glanced at in this book. It is a complete source of knowledge on the present status of this field. For all those who are interested in calculus, this book can prove to be an essential guide. It will prove to be of great help to those in the fields of mathematics, physics and computational mathematics.

Preface ....................................................................................................................... v

1. Wavelets for Differential Equations and Numerical Operator Calculus .... 1

Abstract ........................................................................................................... 1

1. Introduction ................................................................................................... 1

2. Generalities on PDE ....................................................................................... 3

3. Wavelets .......................................................................................................... 9

4. Some examples of application of wavelets to PDE ................................. 19

5. Conclusions .................................................................................................. 22

2. dT-Calculus: A Formal Method to Specify Distributed Mobile

Real-Time IoT Systems ...................................................................................... 27

Abstract ......................................................................................................... 27

1. Introduction ................................................................................................. 27

2. Related research ........................................................................................... 29

3. Preliminary research .................................................................................. 33

4. dT-Calculus ................................................................................................... 35

5. Example ......................................................................................................... 41

6. Comparative analysis ................................................................................. 52

7. SAVE ............................................................................................................... 55

8. Conclusions and future research .............................................................. 58

Acknowledgements ..................................................................................... 58

3. Quantum Calculus with the Notion ±-Periodicity and Its

Applications ......................................................................................................... 61

Abstract ......................................................................................................... 61

1. Introduction ................................................................................................. 61

2. Preliminaries about time scale calculus .................................................. 63

3. Application of d± -periodicity of Q-calculus ........................................... 70

4. Conclusion .................................................................................................... 77

5. Discussion ..................................................................................................... 78

Acknowledgements ..................................................................................... 78

4. Predicate Calculus as a Tool for AI Problems Solution: Algorithms and

Their Complexity ................................................................................................. 81

Abstract ......................................................................................................... 81

1. Introduction ................................................................................................. 81

2. Logic-predicate approach to some AI problems and number of

steps of these problems solution ............................................................... 83

3. Level description of classes ........................................................................ 87

4. Logic-predicate network ............................................................................ 92

5. Multi-agent description of an object ........................................................ 97

6. Conclusion .................................................................................................. 103

5. Investigation of Laser Pulse-induced Calculus Damage Mechanism

by a High-speed Camera .................................................................................. 107

Abstract ....................................................................................................... 107

1. Introduction ............................................................................................... 108

2. Experimental method and setup ............................................................. 114

3. Results ......................................................................................................... 122

4. Discussions ................................................................................................. 133

Acknowledgements ................................................................................... 138

6. Application of Fractional Calculus to Oil Industry ..................................... 143

Abstract ....................................................................................................... 143

1. Introduction ............................................................................................... 143

2. Background of the approach of models of diffusion on fractal media144

3. Brief history of fractional calculus .......................................................... 145

4. Fractional calculus .................................................................................... 151

5. Fractal geometry and fractional calculus .............................................. 155

6. Fractional calculus for modeling oil pressure ....................................... 157

7. Charathéodory’s “Royal Road” to the Calculus of Variations:

A Possible Bridge Between Classical and Quantum Physics ................... 168

1. Introduction ............................................................................................... 168

2. Carathéodory’s royal road ....................................................................... 171

3. Electromagnetism: The London equations and the Bohr-van

Leeuwen theorem ...................................................................................... 180

4. Hamilton-Jacobi equations without Hamiltonian ............................... 187

5. Gauge invariance in electromagnetism and gravitation .................... 189

6. Summary and conclusions ....................................................................... 193

Acknowledgements ................................................................................... 194

8. Calculus Elements for Mechanical Presses in Oil Industry ....................... 197

1. Introduction ............................................................................................... 197

2. Theoretical elements ................................................................................. 206

3. Application ................................................................................................. 214

4. Conclusions ................................................................................................ 215

Citation ................................................................................................................ 219

Index .................................................................................................................... 221

About Author

Dakota Rogerson obtained her Ph.D. in Mathematics and Statistics from Public State University, Hobart. She is a Guest Lecturer at Public Research University, Aichi. Rogerson has more than a decade of experience in the study of differential calculus, integral calculus and calculus of variations. Her current researches are focused on process calculus, mathematical analysis and lambda calculus. She has also won a “Distinguished Contributions to Lifelong Learning” award for her outstanding contribution especially in the higher education programs.