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E-raamat: Introduction to Mathematical Modeling

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  • Formaat: PDF+DRM
  • Ilmumisaeg: 19-Sep-2017
  • Kirjastus: Chapman & Hall/CRC
  • Keel: eng
  • ISBN-13: 9781498728010
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Introduction to Mathematical ModelingSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 19-Sep-2017
  • Kirjastus: Chapman & Hall/CRC
  • Keel: eng
  • ISBN-13: 9781498728010
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Introduction to Mathematical Modeling helps students master the processes used by scientists and engineers to model real-world problems, including the challenges posed by space exploration, climate change, energy sustainability, chaotic dynamical systems and random processes.

Primarily intended for students with a working knowledge of calculus but minimal training in computer programming in a first course on modeling, the more advanced topics in the book are also useful for advanced undergraduate and graduate students seeking to get to grips with the analytical, numerical, and visual aspects of mathematical modeling, as well as the approximations and abstractions needed for the creation of a viable model.
Chapter 1 The Process of Mathematical Modeling
1(20)
1.1 What Is Model Building?
2(2)
1.2 Modeling Framework
4(10)
1.3 Genes And Biological Reproduction
14(7)
Chapter 2 Modeling with Ordinary Differential Equations
21(70)
2.1 The Motion Of A Projectile
23(5)
2.1.1 Approximations and Simplifications
24(1)
2.1.2 Model
24(2)
2.1.3 Model Compounding
26(2)
2.2 Spring-Mass Systems
28(17)
2.2.1 Data Collection
30(1)
2.2.2 Approximations and Simplifications
30(1)
2.2.3 Mathematical Model
31(1)
2.2.4 Remarks and Refinements
32(13)
2.3 Electrical Circuits
45(8)
2.3.1 RLC Circuits
45(2)
2.3.2 Approximations
47(6)
2.4 Population Models
53(11)
2.4.1 Logistic Model
54(1)
2.4.2 Prototype Model
54(1)
2.4.3 Data and Approximations
54(2)
2.4.4 Solution of the logistic equation
56(8)
2.5 Motion In A Central Force Field
64(19)
2.5.1 Radial Coordinate System in R2
64(2)
2.5.2 Linear Pendulum
66(2)
2.5.3 Nonlinear Pendulum
68(2)
2.5.4 A Short Introduction to Elliptic Functions
70(1)
2.5.5 Motion of a Projectile on a Rotating Earth
70(1)
2.5.6 A Particle in a Central Force Field
71(1)
2.5.7 Motion of a Rocket
72(3)
2.5.8 Multistage Rockets
75(3)
2.5.9 Control of a Satellite in Orbit
78(5)
2.6 Greenhouse Effect
83(4)
2.7 Current Energy Balance of the Earth
87(4)
2.7.1 Critique of the Model
89(1)
2.7.2 Humanity and Energy
89(2)
Chapter 3 Solutions of Systems of ODEs
91(42)
3.1 Review
92(5)
3.1.1 Linear differential equations with constant coefficients
92(5)
3.2 Review Of Linear Algebra
97(4)
3.2.1 Eigenvalues and Eigenvectors
97(4)
3.3 Reformulation Of Systems Odes
101(2)
3.4 Linear Systems With Constant Coefficients
103(4)
3.5 Numerical Solution Of Initial Value Problems
107(8)
3.5.1 Euler Algorithm
108(7)
3.6 Finite Difference Approximations
115(6)
3.6.1 Extension to Higher Dimensions
120(1)
3.7 Modified Euler And Runge-Kutta Methods
121(6)
3.7.1 Modified Euler Algorithm
121(3)
3.7.2 Runge-Kutta Methods
124(3)
3.8 Boundary Value Problems
127(6)
Chapter 4 Stability Theory
133(40)
4.1 General Introduction
134(4)
4.2 Two Species Model
138(5)
4.2.1 Steady States
139(1)
4.2.2 Stability Analysis
140(3)
4.3 Basic Concepts
143(7)
4.4 Linearizable Dynamical Systems
150(4)
4.5 Linearizable Systems In Two Dimensions
154(5)
4.6 Liapounov Method
159(7)
4.7 Periodic Solutions (Limit Cycles)
166(7)
Chapter 5 Bifurcations and Chaos
173(36)
5.1 Introduction
174(1)
5.2 Bifurcations Of Co-Dimension One
175(12)
5.2.1 Trans-critical Bifurcation
176(1)
5.2.2 Saddle Point Bifurcation
177(1)
5.2.3 Pitchfork Bifurcation
178(2)
5.2.4 Subcritical Bifurcation (Hysteresis)
180(2)
5.2.5 Hopf Bifurcation
182(5)
5.3 Rossler Oscillator
187(7)
5.4 Lorenz Equations
194(3)
5.5 Nerve Models
197(2)
5.6 Miscellaneous Topics
199(7)
5.6.1 Dimension
199(4)
5.6.2 Liapunov Exponents
203(3)
5.7 Appendix A: Derivation Of Lorenz Equations
206(3)
Chapter 6 Perturbations
209(16)
6.1 Introduction
210(1)
6.2 Model Equations In Non-Dimensional Form
211(2)
6.3 Regular Perturbations
213(2)
6.4 Singular Perturbations
215(4)
6.5 Boundary Layers
219(6)
Chapter 7 Modeling with Partial Differential Equations
225(60)
7.1 The Heat (Or Diffusion) Equation
226(14)
7.1.1 Burger's Equation
234(1)
7.1.2 Similarity Solutions
235(2)
7.1.3 Stephan Problem(s)
237(3)
7.2 Modeling Wave Phenomena
240(11)
7.2.1 Nonlinear Wave Equations
244(4)
7.2.2 Riemann Invariants
248(3)
7.3 Shallow Water Waves
251(6)
7.3.1 Tsunamis
255(2)
7.4 Uniform Transmission Line
257(4)
7.5 The Potential (Or Laplace) Equation
261(10)
7.5.1 Kirchoff Transformation
269(2)
7.6 The Continuity Equation
271(4)
7.7 Electromagnetism
275(10)
7.7.1 Maxwell Equations
275(1)
7.7.2 Electrostatic Fields
276(1)
7.7.3 Multipole Expansion
277(1)
7.7.4 Magnetostatic
278(1)
7.7.5 Electromagnetic Waves
279(1)
7.7.6 Electromagnetic Energy and Momentum
279(2)
7.7.7 Electromagnetic Potential
281(4)
Chapter 8 Solutions of Partial Differential Equations
285(68)
8.1 Method Of Separation Of Variables
286(42)
8.1.1 Method of Separation of Variables By Example
286(18)
8.1.2 Non Cartesian Coordinate Systems
304(15)
8.1.3 Boundary Value Problems with General Initial Conditions
319(4)
8.1.4 Boundary Value Problems with Inhomogeneous Equations
323(5)
8.2 Green's Functions
328(7)
8.3 Laplace Transform
335(6)
8.3.1 Basic Properties of the Laplace Transform
336(3)
8.3.2 Applications to the Heat Equation
339(2)
8.4 Numerical Solutions Of PDES
341(12)
8.4.1 Finite Difference Schemes
341(1)
8.4.2 Numerical Solutions for the Poisson Equation
341(2)
8.4.2.1 Other Boundary Conditions
343(3)
8.4.3 Irregular Regions
346(3)
8.4.4 Numerical Solutions for the Heat and Wave Equations
349(4)
Chapter 9 Variational Principles
353(38)
9.1 Extrema Of Functions
354(1)
9.2 Constraints And Lagrange Multipliers
355(2)
9.3 Calculus Of Variations
357(7)
9.3.1 Natural Boundary Conditions
363(1)
9.3.2 Variational Notation
363(1)
9.4 Extensions
364(2)
9.5 Applications
366(5)
9.6 Variation With Constraints
371(3)
9.7 Airplane Control; Minimum Flight Time
374(4)
9.8 Applications In Elasticity
378(2)
9.9 Rayleigh-Ritz Method
380(3)
9.10 The Finite Element Method In 2-D
383(7)
9.10.1 Geometrical Triangulations
383(1)
9.10.2 Linear Interpolation in 2-D
384(2)
9.10.3 Galerkin Formulation of FEM
386(4)
9.11 Appendix
390(1)
Chapter 10 Modeling Fluid Flow
391(52)
10.1 Strain And Stress
392(3)
10.2 Equations Of Motion For Ideal Fluid
395(3)
10.2.1 Continuity equation
395(1)
10.2.2 Euler's equations
396(2)
10.3 Navier-Stokes Equations
398(5)
10.4 Similarity And Reynolds' number
403(1)
10.5 Different Formulations Of Navier-Stokes Equations
404(3)
10.6 Convection And Boussinesq Approximation
407(3)
10.7 Complex Variables In 2-D Hydrodynamics
410(1)
10.8 Blasius Boundary Layer Equation
411(3)
10.9 Introduction To Turbulence Modeling
414(8)
10.9.1 Incompressible Turbulent Flow
416(2)
10.9.2 Modeling Eddy Viscosity
418(1)
10.9.3 k -- ε Model
419(1)
10.9.4 The Turbulent Energy Spectrum
420(2)
10.10 Stability Of Fluid Flow
422(3)
10.11 Astrophysical Applications
425(8)
10.11.1 Derivation of the Model Equations
426(3)
10.11.2 Steady State Model Equations
429(1)
10.11.3 Physical Meaning of the Functions H(ρ), S(ρ)
429(1)
10.11.4 Radial Solutions for the Steady State Model
430(3)
10.12 Appendix A - Gauss Theorem And Its Variants
433(2)
10.13 Appendix B - Poincare Inequality And Burger's Equation
435(2)
10.14 Appendix C - Gronwell Inequality
437(2)
10.15 Appendix D - The Spectrum
439(4)
Chapter 11 Modeling Geophysical Phenomena
443(14)
11.1 Atmospheric Structure
444(1)
11.2 Thermodynamics And Compressibility
445(3)
11.2.1 Thermodynamic Modeling
445(2)
11.2.2 Compressibility
447(1)
11.3 General Circulation
448(1)
11.4 Climate
449(8)
Chapter 12 Stochastic Modeling
457(16)
12.1 Introduction
458(1)
12.2 Pure Birth Process
458(4)
12.3 Kermack And Mckendrick Model
462(3)
12.4 Queuing Models
465(3)
12.5 Markov Chains
468(5)
Chapter 13 Answers to Problems
473(28)
Index 501
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