Verfügbarkeit: Introduction to the foundations of applied mathematics

Introduction to the foundations of applied mathematics:

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Bibliographische Detailangaben
Beteilige Person:	Holmes, Mark H. (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	Dordrecht [u.a.] Springer 2009
Schriftenreihe:	Texts in applied mathematics 56
Schlagwörter:	Engineering mathematics Angewandte Mathematik Mathematisches Modell Mathematische Physik Lehrbuch
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017693572&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Umfang:	XIV, 467 S. Ill., graph. Darst.
ISBN:	9780387877495 9780387877655

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Datensatz im Suchindex

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adam_text	Contents Preface ....................................................... vii 1 Dimensional Analysis ..................................... 1 1.1 Introduction ........................................... 1 1.2 Examples of Dimensional Reduction ...................... 3 1.2.1 Maximum Height of a Projectile .................... 5 1.2.2 Drag on a Sphere ................................. 6 1.2.3 Toppling Dominoes ............................... 13 1.2.4 Endnotes........................................ 15 1.3 Theoretical Foundation .................................. 16 1.3.1 Pattern Formation ................................ 19 1.4 Similarity Variables ..................................... 22 1.5 Nondimensionalization and Scaling ....................... 25 1.5.1 Projectile Problem ............................... 26 1.5.2 Weakly Nonlinear Diffusion ........................ 30 1.5.3 Endnotes........................................ 32 Exercises .................................................. 33 2 Perturbation Methods .................................... 43 2.1 Regular Expansions ..................................... 43 2.2 How to Find a Regular Expansion ........................ 48 2.2.1 Given a Specific Function ......................... 48 2.2.2 Given an Algebraic or Transcendental Equation ...... 51 2.2.3 Given an Initial Value Problem .................... 53 2.3 Introduction to Singular Perturbations ,................... 58 2.4 Introduction to Boundary Layers ......................... 60 2.4.1 Endnotes........................................ 66 2.5 Multiple Boundary Layers ............................... 68 2.6 Multiple Scales and Two-Timing ......................... 72 Exercises .................................................. 79 Contents Kinetics .................................................. 87 3.1 Introduction ........................................... 87 3.1.1 Radioactive Decay ................................ 87 3.1.2 Predator-Prey ................................... 88 3.1.3 Epidemic Model .................................. 88 3.2 Kinetic Equations ...................................... 89 3.2.1 The Law of Mass Action .......................... 91 3.2.2 Conservation Laws ............................... 92 3.2.3 Steady-States .................................... 94 3.2.4 Examples ....................................... 94 3.2.5 End Notes .............._,,....................... 96 3.3 General Mathematical Formulation ....................... 97 3.4 Michaelis-Menten Kinetics ............................... 100 3.4.1 Numerical Solution ...............................102 3.4.2 Quasi-Steady-State Approximation .................103 3.4.3 Perturbation Approach ............................105 3.5 Assorted Applications ................................... Ill 3.5.1 Elementary and Nonelementary Reactions ........... Ill 3.5.2 Reverse Mass Action ..............................113 3.6 Steady-States and Stability ..............................114 3.6.1 Reaction Analysis ................................ 115 3.6.2 Geometric Analysis ............................... 115 3.6.3 Perturbation Analysis ............................. 118 3.7 Oscillators ............................................. 126 3.7.1 Stability ........................................ 128 Exercises ..................................................132 Diffusion ..................................................141 4.1 Introduction ...........................................141 4.2 Random Walks and Brownian Motion .....................142 4.2.1 Calculating w{m, N) ..............................145 4.2.2 Large N Approximation ...........................148 4.3 Continuous Limit .......................................149 4.3.1 What Does D Signify? ............................151 4.4 Solving the Diffusion Equation ...........................153 4.4.1 Point Source .....................................154 4.4.2 Fourier Transform ................................157 4.5 Continuum Formulation of Diffusion ......................169 4.5.1 Balance Law .....................................169 4.5.2 Fick s Law of Diffusion ............................171 4.5.3 Reaction-Diffusion Equations ......................177 4.6 Random Walks and Diffusion in Higher Dimensions .........179 4.6.1 Diffusion Equation ...............................182 4.7 Langevin Equation .....................................185 4.7.1 Properties of the Forcing ..........................188 Contents 4.7.2 Endnotes........................................194 Exercises .................................................. 194 Traffic Flow ..............................................205 5.1 Introduction ...........................................205 5.2 Continuum Variables ....................................206 5.2.1 Density .........................................207 5.2.2 Flux ............................................208 5.3 Balance Law ...........................................209 5.3.1 Velocity Formulation ..............................210 5.4 Constitutive Laws ......................................211 5.4.1 Constant Velocity ................................212 5.4.2 Linear Velocity ...................................213 5.4.3 General Velocity Formulation ......................214 5.4.4 Flux and Velocity ................................216 5.4.5 Reality Check ....................................217 5.5 Constant Velocity ......................................218 5.5.1 Characteristics ...................................221 5.6 Nonconstant Velocity ...................................225 5.6.1 Small Disturbance Approximation ..................226 5.6.2 Method of Characteristics .........................229 5.6.3 Rankine-Hugoniot Condition .......................233 5.6.4 Expansion Fan ...................................236 5.6.5 Shock Waves .....................................241 5.6.6 Return of Phantom Traffic Jams ...................245 5.6.7 Summary ........................................247 5.7 Cellular Automata Modeling .............................248 Exercises ..................................................254 Continuum Mechanics: One Spatial Dimension ...........265 6.1 Introduction ...........................................265 6.2 Coordinate Systems .....................................265 6.2.1 Material Coordinates .............................266 6.2.2 Spatial Coordinates ...............................267 6.2.3 Material Derivative ...............................270 6.2.4 End Notes .......................................272 6.3 Mathematical Tools .....................................273 6.4 Continuity Equation ....................................275 6.4.1 Material Coordinates .............................276 6.5 Momentum Equation ...................................277 6.5.1 Material Coordinates .............................279 6.6 Summary of the Equations of Motion .....................279 6.7 Steady-State Solution ...................................280 6.8 Constitutive Law for an Elastic Material ..................282 6.8.1 Derivation of Strain ..............................284 Contents 6.8.2 Material Linearity................................ 286 6.8.3 End Notes.......................................289 6.9 Morphological Basis for Deformation......................290 6.9.1 Metals..........................................290 6.9.2 Elastomers......................................293 6.10 Restrictions on Constitutive Laws ........................294 6.10.1 Frame-Indifference ................................295 6.10.2 Entropy Inequality ...............................298 6.10.3 Hyperelasticity ...................................302 Exercises ..................................................304 Elastic and Viscoelastic Materials .........................311 7.1 Linear Elasticity .......................................311 7.1.1 Method of Characteristics .........................313 7.1.2 Laplace Transform ................................316 7.1.3 Geometric Linearity ..............................327 7.2 Viscoelasticity .........................................328 7.2.1 Mass, Spring, Dashpot Systems ....................329 7.2.2 Equations of Motion ..............................331 7.2.3 Integral Formulation ..............................335 7.2.4 Generalized Relaxation Functions ..................337 7.2.5 Solving Viscoelastic Problems ......................338 Exercises ..................................................342 Continuum Mechanics: Three Spatial Dimensions .........351 8.1 Introduction ...........................................351 8.2 Material and Spatial Coordinates .........................352 8.2.1 Deformation Gradient .............................353 8.3 Material Derivative .....................................356 8.4 Mathematical Tools .....................................358 8.4.1 General Balance Law .............................361 8.5 Continuity Equation ....................................362 8.5.1 Incompressibility .................................362 8.6 Linear Momentum Equation .............................363 8.6.1 Stress Tensor .................................... 364 8.6.2 Differential Form of Equation ...................... 367 8.7 Angular Momentum .................................... 367 8.8 Summary of the Equations of Motion ..................... 368 8.9 Constitutive Laws ...................................... 368 8.9.1 Representation Theorem and Invariants .............372 8.10 Newtonian Fluid ....................................... 374 8.10.1 Pressure .........................................374 8.10.2 Viscous Stress ...................................375 8.11 Equations of Motion for a Viscous Fluid ...................378 8.11.1 Incompressibility .................................379 Contents xiii 8.11.2 Boundary and Initial Conditions ...................380 8.12 Material Equations of Motion ............................383 8.12.1 Frame-Indifference ................................385 8.12.2 Elastic Solid .....................................387 8.12.3 Linear Elasticity .................................389 8.13 Energy Equation .......................................390 8.13.1 Incompressible Viscous Fluid ......................391 8.13.2 Elasticity ........................................391 Exercises ..................................................394 9 Fluids ....................................................403 9.1 Newtonian Fluids .......................................403 9.2 Steady Flow ...........................................404 9.2.1 Plane Couette Flow ...............................405 9.2.2 Poiseuille Flow ...................................408 9.3 Vorticity ..............................................411 9.3.1 Vortex Motion ...................................412 9.4 Irrotational Flow .......................................414 9.4.1 Potential Flow ...................................417 9.5 Ideal Fluid ............................................419 9.5.1 Circulation and Vorticity ..........................420 9.5.2 Potential Flow ...................................423 9.5.3 End Notes .......................................426 9.6 Boundary Layers .......................................427 9.6.1 Impulsive Plate ..................................427 9.6.2 Blasius Boundary Layer ...........................429 Exercises ..................................................434 A Taylor s Theorem .........................................441 A.I Single Variable .........................................441 A.2 Two Variables .........................................441 A.3 Multivariable Versions ..................................442 В Fourier Analysis ..........................................445 B.I Fourier Series ..........................................445 B.2 Fourier Transform ......................................447 С Stochastic Differential Equations .........................449 D Identities .................................................451 D.I Trace .................................................451 D.2 Determinant ...........................................451 D.3 Vector Calculus ........................................452 xiv Contents E Equations for a Newtonian Fluid .........................453 E.I Cartesian Coordinates ..................................453 E.2 Cylindrical Coordinates .................................453 References ....................................................455 Index .........................................................463
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spellingShingle	Holmes, Mark H. Introduction to the foundations of applied mathematics Texts in applied mathematics Engineering mathematics Angewandte Mathematik (DE-588)4142443-8 gnd Mathematisches Modell (DE-588)4114528-8 gnd Mathematische Physik (DE-588)4037952-8 gnd
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title	Introduction to the foundations of applied mathematics
title_auth	Introduction to the foundations of applied mathematics
title_exact_search	Introduction to the foundations of applied mathematics
title_full	Introduction to the foundations of applied mathematics Mark H. Holmes
title_fullStr	Introduction to the foundations of applied mathematics Mark H. Holmes
title_full_unstemmed	Introduction to the foundations of applied mathematics Mark H. Holmes
title_short	Introduction to the foundations of applied mathematics
title_sort	introduction to the foundations of applied mathematics
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