The use of mathematical structures : modelling real phenomena

“The Use Of Mathematical Structures: Modelling Real Phenomena” is an edited book consisting of 16 contemporaneous open-access articles that are devoted to the mathematical modelling of natural phenomena. To summarize, this book is about the use of applied mathematics and mathematical analysis in the context of its applications to real-world problems. It includes a selection of real-world problems in fluid dynamics, mechanical engineering, biology, and biochemistry. The last chapters include the mathematical modelling of the COVID-19 virus. The intended audience of this book is undergraduate and graduate students, as well as junior researchers. The reader must have a good knowledge of ordinary differential equations, boundary value problems, fractional calculus, stability theory, and wavelets in order to fully understand the real-world problems and their mathematical modelling included in this book. Cover 1 Title Page 5 Copyright 6 DECLARATION 7 ABOUT THE EDITOR 9 TABLE OF CONTENTS 11 List of Contributors 17 List of Abbreviations 21 Preface 23 Chapter 1 Models, Structures, and the Explanatory Role of Mathematics in Empirical Science 25 Abstract 25 Introduction 26 Why Think That Mathematics Does Genuine Explanatory Work? 29 Mathematical Explanations as Structural Explanations 32 From Structural Explanations to Structural Model Explanations 43 Conclusion 52 References 57 Chapter 2 The Real and the Mathematical in Quantum Modeling: From Principles to Models and from Models to Principles 61 Introduction 62 Physical Principles and Mathematical Models in Quantum Mechanics 64 From Models to Principles in Q-Modeling Outside Physics 75 Conflict of Interest Statement 88 Acknowledgments 88 Footnotes 88 References 91 Chapter 3 The Nature and Mathematical Basis for Material Stability in the Chemical and Biological Worlds 95 Abstract 95 Report 96 Discussion 99 Conclusion 108 Endnotes 108 Authors’ Contributions 108 Acknowledgements 109 References 110 Chapter 4 A Riccati-Bernoulli sub-ODE method for Nonlinear Partial Differential Equations and its Application 115 Abstract 115 Introduction 116 Bäcklund transformation of the Riccati-Bernoulli equation 120 Application to the Eckhaus Equation 120 Application to the Nonlinear Fractional Klein-Gordon Equation 124 Application to the Generalized Ostrovsky Equation 126 Application to the generalized ZK-Burgers equation 129 Comparisons and Explanations of the Solutions 131 Conclusions 134 Acknowledgements 135 References 136 Chapter 5 Mathematical Modelling of Mantle Convection at a high Rayleigh number with Variable Viscosity and Viscous Dissipation 139 Abstract 139 Introduction 140 Methods 142 Result and Discussion 150 Conclusion 158 References 160 Chapter 6 Extending the Persistent Primary Variable Algorithm to Simulate Non-Isothermal Two-Phase Two-Component Flow with Phase Change Phenomena 165 Abstract 165 Background 166 Method 168 Numerical Scheme 179 Numerical Solution of the Global Equation System 181 Handling Unphysical Values during the Global Iteration 182 Results and Discussions 182 Nomenclature 194 Authors’ Contributions 196 Acknowledgements 196 References 197 Chapter 7 Modelling and Dynamic Characteristics for a Non-metal Pressurized Reservoir with Variable Volume 201 Abstract 201 Introduction 202 Reservoir Description 204 Modelling and Simulation 210 Experimental Results and Discussion 219 Conclusions 228 Acknowledgements 229 Authors’ Information 229 Author Contributions 230 Funding 230 References 231 Chapter 8 Dynamic Modelling and Natural Characteristic Analysis of Cycloid Ball Transmission Using Lumped Stiffness Method 235 Abstract 235 Introduction 236 Lumped Stiffness Modelling 237 Translational–Torsional Coupling Model 242 Natural Characteristic Analysis 246 Conclusion 250 Authors’ contributions 250 Acknowledgements 250 References 251 Chapter 9 Modelling of Flowslides and Debris Avalanches in Natural and Engineered Slopes: A Review 253 Background 253 Introduction 254 Background 257 Methods 267 Results and Discussion for Natural Slopes 274 Results and Discussion for Engineered Slopes 286 Conclusions 292 Acknowledgements 294 References 295 Chapter 10 On Some Wavelet Solutions of Singular Differential Equations Arising in the Modeling of Chemical and Biochemical Phenomena 307 Abstract 307 Introduction 308 Jacobi Wavelet 314 Bernoulli Wavelet 317 Methods for Solution 320 Error Bounds 325 Numerical Simulation 327 Conclusion 335 Acknowledgements 336 References 337 Chapter 11 A Mathematical Analysis of Hopf-Bifurcation in a Prey-Predator Model with Nonlinear Functional Response 341 Abstract 341 Introduction 342 Mathematical Model Formulation 344 Mathematical Analysis 346 Numerical Experiments and Biological Explanations 363 Conclusion 369 Acknowledgements 371 Funding 371 Authors’ Contributions 371 References 372 Chapter 12 Multiscale Modelling Tool: Mathematical Modelling of Collective Behaviour Without the Maths 375 Abstract 375 Introduction 376 Design and Implementation 378 Results 390 Availability and Future Directions 392 Acknowledgments 393 References 394 Chapter 13 Effects of Greenhouse Gases and Hypoxia on the Population of Aquatic Species: A Fractional Mathematical Model 399 Abstract 399 Introduction 400 Preliminaries 403 Model Dynamics 404 Fractional-Order Analysis on the Proposed Model 411 Experimental Simulations 416 Conclusion 420 Acknowledgements 421 Funding 421 Authors’ Contributions 421 References 422 Index 427 Back Cover 432