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Applied mathematical modelling of engineering problems

Author: Natali Hritonenko; I︠U︡ P I︠A︡t︠s︡enko
Publisher: Norwell, Mass. : Kluwer Academic Publishers, 2003.
Series: Applied optimization, v. 81.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
"The subject of the book is the "know-how" of applied mathematical modelling: how to construct specific models and adjust them to a new engineering environment or more precise realistic assumptions; how to analyze models for the purpose of investigating real life phenomena; and how the models can extend our knowledge about a specific engineering process." "Two major sources of the book are the stock of classic  Read more...
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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Natali Hritonenko; I︠U︡ P I︠A︡t︠s︡enko
ISBN: 1402074840 9781402074844
OCLC Number: 52208363
Description: xxi, 286 pages : illustrations ; 25 cm.
Contents: Chapter 1. Some Basic Models of Physical Systems 1 --
1. Basic Models of Particle Dynamics 2 --
1.1 Motion of a Particle in Gravitational Field 3 --
1.1.1 Vertical Projectile Problem 4 --
1.1.2 Free Fall with Air Resistance 4 --
1.1.3 Plane Projectile Problem 5 --
1.1.4 More General Ballistic Problems 6 --
1.2 One-Dimensional Mechanical Vibrations 6 --
1.2.1 Linear Oscillator 7 --
1.2.2 Forced Linear Vibrations and Resonance 9 --
1.2.3 Nonlinear Oscillators 11 --
1.2.4 Nonlinear Vibrations and Resonance 14 --
1.2.5 Nonlinear Electrical-Mechanical Systems 16 --
2. Inverse Problems and Integral Models 19 --
2.1 Sliding Particle and Abel's Equation 20 --
2.2 Sliding Chain 22 --
2.3 Models of Computerized Tomography 24 --
2.3.1 Radon Transform 25 --
2.3.2 Inverse Scattering Problems 28 --
Chapter 2. Models of Continuum Mechanical Systems 29 --
1. Conservation Laws in One-Dimensional Medium 30 --
1.1 Eulerian and Lagrangian Coordinates 31 --
1.2 Mass Conservation 32 --
1.3 Momentum Conservation 33 --
1.4 Energy Conservation and Thermodynamics 36 --
2. Models of One-Dimensional Continuum Dynamics 39 --
2.1 Nonlinear Model of Solid Bar Dynamics 40 --
2.2 Linearized Model of Solid Bar Dynamics 41 --
2.3 Discontinuities in Linear Models 44 --
2.3.1 Analysis of Discontinuity Propagation 45 --
2.3.2 Analysis of Size of Discontinuity 47 --
2.3.3 Discontinuities Fixed in Space 49 --
2.4 Discontinuities in Nonlinear Models 50 --
2.4.1 Nonlinear Conservation Laws 51 --
2.4.2 Impact of Diffusion and Dispersion 53 --
2.5 Models of Viscoelasticity 54 --
3. Three-Dimensional Conservation Laws and Models 56 --
3.1 Mass Conservation and Continuity Equations 57 --
3.2 Momentum Conservation and Cauchy Equations 58 --
3.2.1 Conservation of Angular Momentum 59 --
3.2.2 Newtonian Viscous Fluids 60 --
3.2.3 Inviscid Fluids 61 --
3.2.4 Propagation of Sound in Space 62 --
3.2.5 Elastic Waves in Solids 63 --
3.3 Energy Balance and Thermodynamics 65 --
3.4 Heat Balance and Diffusion Processes 66 --
3.4.1 Diffusion Equation 68 --
3.4.2 Advection-Diffusion Equation 69 --
4. Applied Modelling of Water Transport and Contamination 70 --
4.1 Description of Physical Processes 71 --
4.2 Classification of Models 72 --
4.3 Three-Dimensional Model 74 --
4.3.1 Equation of Transport of the Ingredient in Solute 75 --
4.3.2 Equation of Transport of Suspended Particles 76 --
4.3.3 Equation of Ingredient Transport on Suspended Particles 77 --
4.3.4 Equations of Surface Water Dynamics 77 --
4.3.5 Equations of Adsorption and Sedimentation 78 --
4.4 Two-Dimensional Horizontal Model and Stationary Flows 79 --
4.4.1 Equation of Ingredient Transport in Dissolved Phase 80 --
4.4.2 Equation of Suspended Particles Transport 80 --
4.4.3 Equation of Ingredient Transport on Suspended Particles 81 --
4.4.4 Equations of Water Dynamics 81 --
4.4.5 Equation of Ground Deposit Contamination 82 --
4.4.6 Analysis of Stationary Flow Problem 82 --
4.4.7 About Simulation Techniques 84 --
Chapter 3. Variational Models and Structural Stability 85 --
1. Variational Principles and Models 85 --
1.1 Basic Models of Continuum Mechanics 87 --
1.1.1 Vibrations of String 87 --
1.1.2 Transverse Vibrations of Bar 88 --
1.1.3 Vibrations of Membrane 88 --
1.1.4 Vibrations of Plate 89 --
1.2 Variational Models for Spectral Problems 90 --
1.2.1 Eigenvalues and Eigenfunctions: Simplest Case 90 --
1.2.2 Raleigh Quotient and Raleigh Method 91 --
1.2.3 Eigenvalues of Bar with Variable Shape 92 --
1.2.4 Extremal Eigenvalues of Bar with Sought-For Shape 94 --
2. Variational Models of Structural Stability 96 --
2.1 Model of Buckling Rod 97 --
2.2 Model of Anti-Plane Shear Collapse in Plasticity 98 --
2.3 Model of Capillarity Stability 101 --
Chapter 4. Integral Models of Physical Systems 105 --
1. Construction of Integral Models 106 --
1.1 Converting Differential Models to Integral Models 106 --
1.1.1 Initial Value Problems 107 --
1.1.2 Boundary Value Problems for Ordinary Differential Equations: Green's Function 112 --
1.1.3 Boundary Value Problems for Partial Differential Equations: Boundary Integral Equation Method 114 --
1.2 Integral Models Occurring in Physical Problems 120 --
1.2.1 Integral Model of Membrane Vibrations 120 --
1.2.2 Integral Models of Nuclear Reactors Dynamics 122 --
2. Modelling of Traffic Noise Propagation 125 --
3. Modelling of Mine Rope Dynamics 129 --
3.1 Description of Physical Process 130 --
3.2 Differential Model 131 --
3.3 Integral Model 134 --
3.4 Some Generalizations 137 --
Chapter 5. Modeling in Bioengineering 139 --
1. Models of Population Dynamics and Control 141 --
1.1 Classic Models for One Species Population 142 --
1.1.1 Malthus Model 142 --
1.1.2 Verhulst-Pearl Model 144 --
1.1.3 Population Control and Harvesting 146 --
1.2 Age-Dependent Models for One Species Population 148 --
1.2.1 Linear Integral Model (Lotka Model) 149 --
1.2.2 Linear Differential Model (Lotka-Von Foerster Model) 150 --
1.2.3 Equivalence of Integral and Differential Models 151 --
1.3 Nonlinear Age-Dependent Models with Intra-Species Competition 152 --
1.4 Models with Delay 154 --
1.5 Difference Models 155 --
1.6 Spatial Diffusion Models of Population Dynamics 159 --
1.6.1 Random Walk Models 159 --
1.6.2 Diffusion Models 163 --
2. Bifurcation Analysis for Nonlinear Integral Models 166 --
2.1 Stationary Solutions 167 --
2.2 Stability Analysis 168 --
2.2 Connection with Difference Models 177 --
2.3.1 Single Seasonal Reproduction 177 --
2.3.2 Double Seasonal Reproduction 178 --
2.3 Open Problems 180 --
Chapter 6. Models of Technological Renovation in Production Systems 183 --
1. Traditional Models of Technological Renovation 184 --
1.1 Aggregated Models of Optimal Investments 185 --
1.2 Age-Specific Models of Equipment Replacement 186 --
1.3 Statistical Models of Equipment Renewal 188 --
2. Models of Equipment Replacement Under Technological Change 189 --
2.1 Self-Organizing Market Model of Enterprise Under Technological Change 190 --
2.2 Aggregated Model with Endogenous Useful Life of Equipment 194 --
2.2.1 Integral Macroeconomic Models of Technological Renovation 195 --
2.2.2 Statement of Optimization Problem 197 --
2.3 Disaggregated Integral Model of Equipment Replacement 199 --
2.3.1 Description of Production System 199 --
2.3.2 Construction of Model 200 --
2.3.3 About Prediction Problems 202 --
2.3.4 Statement of Optimization Problem 204 --
3. Qualitative Analysis of Optimal Equipment Replacement 206 --
3.1 About Optimal Control Problems in Integral Models 206 --
3.1.1 General Statement of Optimal Control Problem 207 --
3.1.2 Necessary Conditions of Extremum 208 --
3.1.3 Lagrange Multipliers Method 211 --
3.1.4 Novelty and Common Features 214 --
3.2 Optimal Equipment Replacement in Aggregate Model 215 --
3.2.1 Structure of Aggregated Optimization Problem 216 --
3.2.2 Equation for Turnpike Regimes of Equipment Replacement 218 --
3.2.3 Infinite-Horizon Discounted Optimization 220 --
3.2.4 Finite-Horizon Optimization 221 --
3.2.5 Discussion of Results 223 --
3.3 Optimal Equipment Replacement in Disaggregated Models 225 --
3.3.1 Model with Different Lifetimes of Equipment 229 --
3.4 Open Problems 230 --
4. Mathematical Details and Proofs 232 --
1. Miscellaneous Facts of Analysis 241 --
1.1 Vector and Integral Calculus 241 --
1.1.1 Gradient, Divergence and Rotation 241 --
1.1.2 Gauss Divergence Theorem 242 --
1.1.3 Dubois-Reymond's Lemma 243 --
1.1.4 Leibniz's Formula for Derivatives 243 --
1.2 Functional Spaces 243 --
1.3 Calculus of Variations and Euler Equations 244 --
2. Mathematical Models and Equations 245 --
2.1 Classification of Mathematical Models 245 --
2.1.1 Deterministic and Stochastic Models 245 --
2.1.2 Continuous and Discrete Models 246 --
2.1.3 Linear and Nonlinear Models 247 --
2.1.4 Difference, Differential and Integral Models 248 --
2.2 Integral Dynamical Models and Volterra Integral Equations 254 --
2.2.1 Solvability of Volterra Integral Equations 254 --
2.2.2 Correctness and Stability of Volterra Integral Equations 256 --
2.2.3 Stability of Volterra Integral Equations 258 --
2.2.4 Integral Inequalities 258.
Series Title: Applied optimization, v. 81.
Responsibility: by Natali Hritonenko, Yuri Yatsenko.
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Abstract:

Discusses the 'know-how' of applied mathematical modelling. This title provides a theoretical background to guide the development of practical models and their investigation. It considers general  Read more...

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