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# Mathematical Modeling of Physical Systems: An Introduction

## byDiran Basmadjian

### Hardcover | December 15, 2002

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Mathematical Modeling of Physical Systems provides a concise and lucid introduction to mathematical modeling for students and professionals approaching the topic for the first time. It is based on the premise that modeling is as much an art as it is a science--an art that can be mastered onlyby sustained practice. To provide that practice, the text contains approximately 100 worked examples and numerous practice problems drawn from civil and biomedical engineering, as well as from economics, physics, and chemistry. Problems range from classical examples, such as Euler's treatment of thebuckling of the strut, to contemporary topics such as silicon chip manufacturing and the dynamics of the human immunodeficiency virus (HIV). The required mathematics are confined to simple treatments of vector algebra, matrix operations, and ordinary differential equations. Both analytical andnumerical methods are explained in enough detail to function as learning tools for the beginner or as refreshers for the more informed reader. Ideal for third-year engineering, mathematics, physics, and chemistry students, Mathematical Modeling of Physical Systems will also be a welcome addition tothe libraries of practicing professionals.

### Details & Specs

Title:Mathematical Modeling of Physical Systems: An IntroductionFormat:HardcoverDimensions:368 pages, 7.52 × 9.21 × 0.91 inPublished:December 15, 2002Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0195153146

ISBN - 13:9780195153149

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Table of Contents

PrefaceNotation1. Getting Started and Beyond1.1. When Not to ModelExample 1.1. The Challenger Space Shuttle DisasterExample 1.2. Loss of Blood Vessel Patency1.2. Some Initial Tools and Steps1.3. ClosureExample 1.3. Discharge of Plant Effluent into a RiverExample 1.4. Electrical Field Due to a DipoleExample 1.5. Design of a ThermocoupleExample 1.6. Newton's Law for Systems of Variable Mass: A False Start and the RemedyExample 1.7. Release of a Substance into a Flowing Fluid: Determination of a Mass Transfer CoefficientPractice Problems2. Some Mathematical Tools2.1. Vector Algebra2.1.1. Definition of a Vector2.1.2. Vector Equality2.1.3. Vector Addition and Subtraction2.1.4. Multiplication by a Scalar m2.1.5. The Scalar or Dot Product2.1.6. The Vector or Cross ProductExample 2.1. Distance of a Point from a PlaneExample 2.2. Shortest Distance Between Two LinesExample 2.3. Work as an Application of the Scalar ProductExample 2.4. Extension of the Scalar Product to n Dimensions: A Sale of StocksExample 2.5. A Simple Model Economy2.2. Matrices2.2.1. Types of Matrix2.2.2. The Echelon Form, Rank r2.2.3. Matrix Equality2.2.4. Matrix AdditionExample 2.6. Acquisition Costs2.2.5. Multiplication by a Scalar2.2.6. Matrix MultiplicationExample 2.7. The Product of Two MatricesExample 2.8. Matrix-Vector Representation of Linear Algebraic Equations2.2.7. Elementary Row OperationsExample 2.9. Application of Elementary Row Operations: Algebraic Equivalence2.2.8. Solution of Sets of Linear Algebraic Equations: Gaussian EliminationExample 2.10. An Overspecified System of Equations with a Unique SolutionExample 2.11. A Normal System of Equations with no Solutions2.3. Ordinary Differential Equations (ODEs)Example 2.12. A Population ModelExample 2.13. Newton's Law of Cooling2.3.1. Order of an ODE2.3.2. Linear and Nonlinear ODEs2.3.3. Boundary and Initial ConditionsExample 2.14. Classification of ODEs and Boundary Conditions2.3.4. Equivalent SystemsExample 2.15. Equivalence of Vibrating Mechanical Systems and an Electrical RLC Circuit2.3.5. Analytical Solution MethodsExample 2.16. Solution of NonLinear ODEs by Separation of VariablesExample 2.17. Mass on a Spring Subjected to a Sinusoidal Forcing FunctionExample 2.18. Application of Inversion ProceduresExample 2.19. The Mass-Spring System Revisited: ResonancePractice Problems3. Geometrical ConceptsExample 3.1. A Simple Geometry Problem: Crossing of a RiverExample 3.2. The Formation of Quasi Crystals and Tilings from Two Quadrilateral PolygonsExample 3.3. Charting of Market Price Dynamics: The Japanese Candlestick MethodExample 3.4. Surveying: The Join Calculation and the Triangulation IntersectionExample 3.5. The Global Positioning System (GPS)Example 3.6. The Orthocenter of a TriangleExample 3.7. Relative Velocity and the Wind TriangleExample 3.8. Interception of an AirplaneExample 3.9. Path of PursuitExample 3.10. Trilinear Coordinates: The Three-Jug ProblemExample 3.11. Inflecting Production Rates and Multiple Steady States: The van Heerden DiagramExample 3.12. Linear Programming: A Geometrical ConstructionExample 3.13. Stagewise Adsorption Purification of Liquids: The Operating DiagramExample 3.14. Supercoiled DNAPractice Problems4. The Effect of Forces4.1. IntroductionExample 4.1. The Stress-Strain Relation: Stored Strain Energy and Stress Due to the Impact of a Falling MassExample 4.2. Bending of Beams: Euler's Formula for the Buckling of a StrutExample 4.3. Electrical and Magnetic Forces: Thomson's Determination of e/mExample 4.4. Pressure of a Gas in Terms of Its Molecular Properties: Boyle's Law and the Ideal Gas Law, Velocity of Gas MoleculesExample 4.5. Path of a ProjectileExample 4.6. The Law of Universal Gravitation: Escape Velocity and Geosynchronous SatellitesExample 4.7. Fluid Forces: Bernoulli's Equation and the Continuity EquationExample 4.8. Lift Capacity of a Hot Air BalloonExample 4.9. Work and Energy: Compression of a Gas and Power Output of a BumblebeePractice Problems5. Compartmental ModelsExample 5.1. Measurement of Plasma Volume and Cardiac Output by the Dye Dilution MethodExample 5.2. The Continuous Stirred Tank Reactor (CSTR): Model and Optimum SizeExample 5.3. Modeling a Bioreactor: Monod Kinetics and the Optimum Dilution RateExample 5.4. Nonidealities in a Stirred Tank. Residence Time Distributions from Tracer ExperimentsExample 5.5. A Moving Boundary Problem: The Shrinking Core Model and the Quasi-Steady StateExample 5.6. More on Moving Boundaries: The Crystallization ProcessExample 5.7. Moving Boundaries in Medicine: Controlled-Release Drug DeliveryExample 5.8. Evaporation of a Pollutant into the AtmosphereExample 5.9. Ground Penetration from an Oil SpillExample 5.10. Concentration Variations in Stratified LayersExample 5.11. One-Compartment PharmacokineticsExample 5.12. Deposition of Platelets from Flowing BloodExample 5.13. Dynamics of the Human Immunodeficiency Virus (HIV)Practice Problems6. One-Dimensional Distributed SystemsExample 6.1. The Hypsometric FormulaeExample 6.2. Poiseuille's Equation for Laminar Flow in a PipeExample 6.3. Compressible Laminar Flow in a Horizontal PipeExample 6.4. Conduction of Heat Through Various GeometriesExample 6.5. Conduction in Systems with Heat SourcesExample 6.6. The Countercurrent Heat ExchangerExample 6.7. Diffusion and Reaction in a Catalyst Pellet: The Effectiveness FactorExample 6.8. The Heat Exchanger FinExample 6.9. Polymer Sheet Extrusion: The Uniformity IndexExample 6.10. The Streeter-Phelps River Pollution Model: The Oxygen Sag CurveExample 6.11. Conduction in a Thin Wire Carrying an Electrical CurrentExample 6.12. Electrical Potential Due to a Charged DiskExample 6.13. Production of Silicon Crystals: Getting Lost and Staging a RecoveryPractice Problems7. Some Simple NetworksExample 7.1. A Thermal Network: External Heating of a Stirred Tank and the Analogy to the Artifical Kidney (Dialysis)Example 7.2. A Chemical Reaction Network: The Radioactive Decay SeriesExample 7.3. Hydraulic NetworksExample 7.4. An Electrical Network: Hitting a Brick Wall and Going Around ItExample 7.5. A Mechanical Network: Resonance of Two Vibrating MassesExample 7.6. Application of Matrix Methods to Stoichiometric CalculationsExample 7.7. Diagnosis of a Plant Flow SheetExample 7.8. Manufacturing Costs: Use of Matrix-Vector ProductsExample 7.9. More About Electrical Circuits: The Electrical Ladder NetworksExample 7.10. Photosynthesis and Respiration of a Plant: An Electrial Analogue for the CO2 PathwayPractice Problems8. More Mathematical Tools: Dimensional Analysis and Numerical Methods8.1. Dimensional Analysis8.1.1. IntroductionExample 8.1. Time of Swing of a Simple PendulumExample 8.2. Vibration of a One-Dimensional Structure8.1.2. Systems with More Variables than Dimensions: The Buckingham p TheoremExample 8.3. Heat Transfer to a Fluid in Turbulent FlowExample 8.4. Drag on Submerged Bodies, Horsepower of a CarExample 8.5. Design of a Depth Charge8.2. Numerical Methods8.2.1. Introduction8.2.2. Numerical Software Packages8.2.3. Numerical Solution of Simultaneous Linear Algebraic Equations: Gaussian EliminationExample 8.6. The Global Positioning System Revisited: Using the MATHEMATICA Package for Gaussian Elimination8.2.4. Numerical Solution of Single Nonlinear Equations: Newton's MethodExample 8.7. Chemical Equilibrium: The Synthesis of Ammonia by the Haber Process8.2.5. Numerical Simulation of Simultaneous Nonlinear Equations: The Newton-Raphson MethodExample 8.8. More Chemical Equilibria: Producing Silicon Films by Chemical Vapor Deposition (CVD)8.2.6. Numerical Solution of Ordinary Differential Equations: The Euler and Runge-Kutta MethodsExample 8.9. The Effect of Drag on the Trajectory of an Artillery PiecePractice ProblemsIndex