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E-raamat: Mathematical Modeling

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Mathematical ModelingSuurem pilt
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Mathematical models are the decisive tool to explain and predict phenomena in the natural and engineering sciences. With this book readers will learn to derive mathematical models which help to understand real world phenomena. At the same time a wealth of important examples for the abstract concepts treated in the curriculum of mathematics degrees are given. An essential feature of this book is that mathematical structures are used as an ordering principle and not the fields of application.





Methods from linear algebra, analysis and the theory of ordinary and partial differential equations are thoroughly introduced and applied in the modeling process. Examples of applications in the fields electrical networks, chemical reaction dynamics, population dynamics, fluid dynamics, elasticity theory and crystal growth are treated comprehensively.

Arvustused

Mathematical Modeling is very well written and provides a rigorous introduction to the mathematical modeling of problems from physics and engineering. It is well suited as a text for mathematical modeling courses at the undergraduate and graduate levels that focus on classical deterministic models at continuous scales. (Laura A. Miller, SIAM Review, Vol. 61 (2), 2019) The goal of this book (an English translation of a German text first published about ten years ago) is to teach undergraduate and graduate students the basic examples and techniques of mathematical modeling of real-world phenomena. for a very well-prepared reader with a willingness to work hard, there is a wealth of interesting material to be found here. (Mark Hunacek, MAA Reviews, maa.org, July, 2017)

1 Introduction
1(36)
1.1 What Do We Mean by (Mathematical) Modeling?
1(2)
1.2 Aspects of Mathematical Modeling: Example of Population Dynamics
3(5)
1.3 Population Models with Restricted Resources
8(4)
1.4 Dimensional Analysis and Scaling
12(5)
1.5 Asymptotic Expansions
17(10)
1.6 Applications from Fluid Mechanics
27(5)
1.7 Literature
32(1)
1.8 Exercises
32(5)
2 Systems of Linear Equations
37(38)
2.1 Electrical Networks
37(12)
2.2 Space Frames
49(11)
2.3 Constrained Optimization
60(8)
2.4 Literature
68(1)
2.5 Exercises
68(7)
3 Basic Principles of Thermodynamics
75(56)
3.1 The Model of an Ideal Gas and the Maxwell-Boltzmann Distribution
76(4)
3.2 Thermodynamic Systems and the Thermodynamic Equilibrium
80(1)
3.3 The First Law of Thermodynamics
81(4)
3.4 The Second Law of Thermodynamics and the Notion of Entropy
85(10)
3.5 Thermodynamic Potentials
95(3)
3.6 The Legendre Transform
98(1)
3.7 The Calculus of Differential Forms
99(3)
3.8 Thermodynamics of Mixtures and the Chemical Potential
102(7)
3.9 Chemical Reactions in Multi Species Systems
109(5)
3.10 Equilibria of Chemical Reactions and the Mass Action Law
114(5)
3.11 Kinetic Reactions
119(4)
3.12 Literature
123(1)
3.13 Exercises
123(8)
4 Ordinary Differential Equations
131(66)
4.1 One-Dimensional Oscillations
131(10)
4.1.1 Forced Oscillations
137(4)
4.2 The Lagrangian and Hamiltonian Form of Mechanics
141(11)
4.3 Examples from Population Dynamics
152(3)
4.4 Qualitative Analysis, Phase Portraits
155(5)
4.5 The Principle of Linearized Stability
160(2)
4.6 Stability of Linear Systems
162(5)
4.7 Variational Problems for Functions of One Variable
167(16)
4.8 Optimal Control with Ordinary Differential Equations
183(6)
4.9 Literature
189(1)
4.10 Exercises
190(7)
5 Continuum Mechanics
197(106)
5.1 Introduction
197(2)
5.2 Classical Point Mechanics
199(4)
5.3 From Particle Mechanics to a Continuous Medium
203(3)
5.4 Kinematics
206(6)
5.5 Conservation Laws
212(10)
5.6 Constitutive Relations
222(12)
5.7 The Second Law of Thermodynamics in Continuum Mechanics
234(8)
5.8 Principle of Frame Indifference
242(5)
5.9 Constitutive Theory for Viscous Fluids
247(5)
5.10 Modeling of Elastic Solids
252(17)
5.11 Electromagnetism
269(18)
5.12 Dispersion
287(1)
5.13 Literature
288(1)
5.14 Exercises
288(15)
6 Partial Differential Equations
303(124)
6.1 Elliptic Equations
303(41)
6.1.1 Calculus of Variations
304(10)
6.1.2 The Fundamental Solution
314(3)
6.1.3 Mean Value Theorem and Maximum Principles
317(2)
6.1.4 Plane Potential Flows, the Method of Complex Variables
319(6)
6.1.5 The Stokes Equations
325(2)
6.1.6 Homogenization
327(5)
6.1.7 Optimal Control of Elliptic Differential Equations
332(4)
6.1.8 Parameter Identification and Inverse Problems
336(4)
6.1.9 Linear Elasticity Theory
340(4)
6.2 Parabolic Equations
344(41)
6.2.1 Uniqueness of Solutions, the Energy Method
345(2)
6.2.2 Large Time Behavior
347(4)
6.2.3 Separation of Variables and Eigenfunctions
351(3)
6.2.4 The Maximum Principle
354(1)
6.2.5 The Fundamental Solution
355(3)
6.2.6 Diffusion Times
358(2)
6.2.7 Invariant Transformations
360(1)
6.2.8 General Initial Data
361(1)
6.2.9 Brownian Motion
362(3)
6.2.10 Traveling Waves
365(2)
6.2.11 Reaction Diffusion Equations and Traveling Waves
367(7)
6.2.12 Turing Instability and Pattern Formation
374(7)
6.2.13 Cahn--Hilliard Equation and Pattern Formation Processes
381(4)
6.3 Hyperbolic Conservation Laws
385(8)
6.4 The Wave Equation
393(7)
6.5 The Navier--Stokes Equations
400(5)
6.6 Boundary Layers
405(15)
6.7 Literature
420(1)
6.8 Exercises
420(7)
7 Free Boundary Problems
427(62)
7.1 Obstacle Problems and Contact Problems
428(7)
7.2 Free Boundaries in Porous Media
435(11)
7.3 The Stefan Problem
446(7)
7.4 Entropy Inequality for the Stefan Problem
453(1)
7.5 Undercooled Liquids
454(2)
7.6 Gibbs--Thomson Effect
456(2)
7.7 Mullins--Sekerka Instability
458(3)
7.8 A Priori Estimates for the Stefan Problem with Gibbs--Thomson Condition
461(3)
7.9 Phase Field Equations
464(8)
7.10 Free Surfaces in Fluid Mechanics
472(4)
7.11 Thin Films and Lubrication Approximation
476(3)
7.12 Literature
479(1)
7.13 Exercises
480(9)
Appendix A Function Spaces 489(4)
Appendix B Curvature of Hypersurfaces 493(6)
References 499(6)
Index 505
Prof. Dr. Christof Eck, Prof. Dr. Harald Garcke, Universität Regensburg Prof. Dr. Peter Knabner, Universität Erlangen
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