Fast Track to Differential Equations: Applications-Oriented―Comprehensible―Compact

Description

The second edition of this successful textbook includes a significantly extended chapter on Climate Change with an analysis of the CO2 budget. It also contains a completely new part on Epidemiology, treating the SEIR-model which describes the behavior and dynamics of epidemics. In particular, COVID-19 with actual data is discussed. This compact introduction to ordinary differential equations and their applications is aimed at anyone who in their studies is confronted voluntarily or involuntarily with this versatile subject. Numerous applications from physics, technology, biomathematics, cosmology, economy and optimization theory are given. Abstract proofs and unnecessary formalism are avoided as far as possible. The focus is on modelling ordinary differential equations of the first and second orders as well as their analytical and numerical solution methods, in which the theory is dealt with briefly before moving on to application examples. In addition, program codes show exemplarily how even more challenging questions can be tackled and represented meaningfully with the help of a computer algebra system. The first chapter deals with the necessary prior knowledge of integral and differential calculus. 103 motivating exercises together with their solutions round off the work.   “I am happy to see such a book. It will serve as a support for many students, professors and faculty.”   Dr. Alessio Figalli, Professor at the ETH Zürich and Fields medalist 2018 Preface to the German Original Preface to the English Translation Preface to the Second Edition Contents Chapter 1 Prerequisites from Calculus 1.1 Exponential and Logarithm Functions 1.1.1 Exponential Function exp(x) as a Power Series 1.1.2 Properties of exp(x) 1.1.3 Exponential Function 1.1.4 Change of Base and Hyperbolic Functions 1.1.5 Logarithm Function 1.2 Integral Calculus 1.2.1 Definite Integral 1.2.2 Fundamental Theorem of Integral Calculus 1.2.3 Computing Integrals 1.3 Applications of Integral and Differential Calculus 1.3.1 Escape Velocity 1.3.2 Elasticity in Economics 1.3.3 Harmonic Sum and Harmonic Series 1.3.4 Optimal Stopping 1.3.5 Fermat’s Principle, Snell’s Law of Refraction 1.4 Parametrized Curves or Vector-Valued Functions 1.4.1 Definition and Examples 1.4.2 Ellipse 1.4.3 Tangents of a Curve, Vectors of Velocity 1.5 Exercises Chapter 1 Chapter 2 First Order Differential Equations 2.1 Concepts and Definitions 2.2 Geometrical Aspects, Direction Fields, Isoclines 2.3 Linear First Order Differential Equations 2.3.1 Homogeneous Case 2.3.2 Nonhomogeneous Case 2.4 Existence and Uniqueness of Solutions 2.5 Separable Differential Equations 2.6 Autonomous Differential Equation and Stability 2.7 Exercises Chapter 2 Chapter 3 First Order Applications 3.1 Population Model 3.2 Newton’s Law of Cooling 3.3 Radioactivity and Radiocarbon Dating 3.3.1 Radioactivity 3.3.2 Radiocarbon Dating 3.3.3 More on Radiocarbon Dating 3.4 Compound Interest 3.5 Specified Elasticity Function in Economics 3.6 Evaporation of a Rain Drop 3.7 Mixing Problem 3.8 Vertical Launch of a Rocket Without Air Drag 3.9 Gravitational Funnel 3.10 Nuclear Waste Disposal by Use of Computer 3.11 Barometric Formulas 3.11.1 Isothermal Model 3.11.2 Model for Linear Temperature Decrease 3.11.3 General Model 3.12 Liquid Container 3.13 Electric Circuit 3.14 Catenary 3.15 GlobalWarming 3.16 Brachistochrone Problem 3.17 Cosmology 3.17.1 Historical Facts 3.17.2 Extent and Age of the Universe 3.17.3 Olbers’ Paradox 3.17.4 Friedmann-Lemaˆıtre Equation 3.17.5 Einstein-De Sitter Model 3.17.6 Model for a Universe with Positive Space Curvature 3.18 Orthogonal Trajectories 3.19 Exercises Chapter 3 Chapter 4 Second Order Differential Equations and Systems with Applications 4.1 Second Order Differential Equation 4.2 System of First Order Differential Equations 4.3 Projectile Motion 4.4 Modeling With Air Drag 4.5 Coriolis Force in Meteorology 4.6 Vector Fields and Field Lines 4.7 Helmholtz Coils 4.8 Oscillations and Resonance 4.9 Curve of Pursuit 4.10 Coupled Pendulums 4.11 Predator-Prey Problem 4.12 Periodic Solutions and Limit Cycles 4.13 Two-Body Problem in Celestial Mechanics 4.13.1 Historical Remarks 4.13.2 Proof of Kepler’s Second Law 4.13.3 Proof of Kepler’s First Law 4.13.4 Proof of Kepler’s Third Law 4.14 Exercises Chapter 4 Chapter 5 Numerical Methods With Applications 5.1 Basics 5.2 Euler’s Method 5.3 Error Considerations 5.4 Heun’s Method 5.5 Runge-Kutta Method 5.6 Numerical Considerations on Systems of Differential Equations 5.7 Trajectories of Tennis Balls 5.7.1 Modeling 5.7.2 Data and Comparison of Forces 5.7.3 Mathematica Program for Computations and Diagrams 5.7.4 Topspin Trajectories 5.7.5 Slice Trajectories 5.7.6 Comparing Topspin Trajectory with Trajectory in a Vacuum 5.8 Mathematical Model for Skydiving 5.8.1 Analytical Model 5.8.2 Numerical Model 5.9 Near-Earth Satellite Orbits 5.10 Brownian Motion and Langevin Equation 5.11 Kalman Filter 5.11.1 Theory 5.11.2 Simulation of an Example 5.12 Exercises Chapter 5 Chapter 6 Climate Change and Epidemics 6.1 Climate Change 6.1.1 The Zero-Dimensional Energy Balance Model 6.1.2 Linearized Model 6.1.3 Increase of Greenhouse Gases 6.1.4 Carbon Budget for the 2 Degree Target 6.2 Epidemiology 6.2.1 The SIR Model 6.2.2 The SEIR Model 6.2.3 Computing S_∞ and R_∞ for the Two Models. 6.2.4 SEIR Model for the Coronavirus Epidemic in Germany 6.2.5 Coronavirus Data and SEIR Model in the First Wave 6.2.6 World Data in the Second or Third Wave 6.3 Exercises Chapter 6 Chapter 7 Solutions 7.1 Solutions Chapter 1 7.2 Solutions Chapter 2 7.3 Solutions Chapter 3 7.4 Solutions Chapter 4 7.5 Solutions Chapter 5 7.6 Solutions Chapter 6 References