Methods of Mathematical Modelling Continuous Systems and Differential Equations 1st edition by Thomas Witelski, Mark Bowen ISBN 3319230429 9783319230429

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ISBN 10: 3319230429
ISBN 13: 9783319230429
Author: Thomas Witelski, Mark Bowen

This book presents mathematical modelling and the integrated process of formulating sets of equations to describe real-world problems. It describes methods for obtaining solutions of challenging differential equations stemming from problems in areas such as chemical reactions, population dynamics, mechanical systems, and fluid mechanics. Chapters 1 to 4 cover essential topics in ordinary differential equations, transport equations and the calculus of variations that are important for formulating models. Chapters 5 to 11 then develop more advanced techniques including similarity solutions, matched asymptotic expansions, multiple scale analysis, long-wave models, and fast/slow dynamical systems. Methods of Mathematical Modelling will be useful for advanced undergraduate or beginning graduate students in applied mathematics, engineering and other applied sciences.

Methods of Mathematical Modelling Continuous Systems and Differential Equations 1st Table of contents:

Part I Formulation of Models

1 Rate Equations

1.1 The Motion of Particles

1.2 Chemical Reaction Kinetics

1.3 Ecological and Biological Models

1.4 One-Dimensional Phase-Line Dynamics

1.5 Two-Dimensional Phase Plane Analysis

1.5.1 Nullclines

1.6 Further Directions

1.7 Exercises

2 Transport Equations

2.1 The Reynolds Transport Theorem

2.2 Deriving Conservation Laws

2.3 The Linear Advection Equation

2.4 Systems of Linear Advection Equations

2.5 The Method of Characteristics

2.6 Shocks in Quasilinear Equations

2.7 Further Directions

2.8 Exercises

3 Variational Principles

3.1 Review and Generalisation from Calculus

3.1.1 Functionals

3.2 General Approach and Basic Examples

3.2.1 The Simple Shortest Curve Problem

3.2.2 The Classic Euler–Lagrange Problem

3.3 The Variational Formation of Classical Mechanics

3.3.1 Motion with Multiple Degrees of Freedom

3.4 The Influence of Boundary Conditions

3.4.1 Problems with a Free Boundary

3.4.2 Problems with a Variable Endpoint

3.5 Optimisation with Constraints

3.5.1 Review of Lagrange Multipliers

3.6 Integral Constraints: Isoperimetric Problems

3.7 Geometric Constraints: Holonomic Problems

3.8 Differential Equation Constraints: Optimal Control

3.9 Further Directions

3.10 Exercises

4 Dimensional Scaling Analysis

4.1 Dimensional Quantities

4.1.1 The SI System of Base Units

4.2 Dimensional Homogeneity

4.3 The Process of Nondimensionalisation

4.3.1 Projectile Motion

4.3.2 Terminal Velocity of a Falling Sphere in a Fluid

4.3.3 The Burgers Equation

4.4 Further Applications of Dimensional Analysis

4.4.1 Projectile Motion (Revisited)

4.4.2 Closed Curves in the Plane

4.5 The Buckingham Pi Theorem

4.5.1 Mathematical Consequences

4.5.2 Application to the Quadratic Equation

4.6 Further Directions

4.7 Exercises

Part II Solution Techniques

5 Self-Similar Scaling Solutions of Differential Equations

5.1 Finding Scaling-Invariant Symmetries

5.2 Determining the Form of the Similarity Solution

5.3 Solving for the Similarity Function

5.4 Further Comments on Self-Similar Solutions

5.5 Similarity Solutions of the Heat Equation

5.5.1 Source-Type Similarity Solutions

5.5.2 The Boltzmann Similarity Solution

5.6 A Nonlinear Diffusion Equation

5.7 Further Directions

5.8 Exercises

6 Perturbation Methods

6.1 Asymptotic Analysis: Concepts and Notation

6.2 Asymptotic Expansions

6.2.1 Divergence of Asymptotic Expansions

6.3 The Calculation of Asymptotic Expansions

6.3.1 The Expansion Method

6.3.2 The Iteration Method

6.3.3 Further Examples

6.4 A Regular Expansion for a Solution of an ODE Problem

6.5 Singular Perturbation Problems

6.5.1 Rescaling to Obtain Singular Solutions

6.6 Further Directions

6.7 Exercises

7 Boundary Layer Theory

7.1 Observing Boundary Layer Structure in Solutions

7.2 Asymptotics of the Outer and Inner Solutions

7.3 Constructing Boundary Layer Solutions

7.3.1 The Outer Solution

7.3.2 The Distinguished Limits

7.3.3 The Inner Solution

7.3.4 Asymptotic Matching

7.3.5 The Composite Solution

7.4 Further Examples

7.5 Further Directions

7.6 Exercises

8 Long-Wave Asymptotics for PDE Problems

8.1 The Classic Separation of Variables Solution

8.2 The Dirichlet Problem on a Slender Rectangle

8.3 The Insulated Wire

8.4 The Nonuniform Insulated Wire

8.5 Further Directions

8.6 Exercises

9 Weakly-Nonlinear Oscillators

9.1 Review of Solutions of the Linear Problem

9.2 The Failure of Direct Regular Expansions

9.3 Poincare–Lindstedt Expansions

9.4 The Method of Multiple Time-Scales

9.5 Further Directions

9.6 Exercises

10 Fast/slow Dynamical Systems

10.1 Strongly-Nonlinear Oscillators: The van der Pol Equation

10.2 Complex Chemical Reactions: The Michaelis-Menten Model

10.3 Further Directions

10.4 Exercises

11 Reduced Models for PDE Problems

11.1 The Method of Moments

11.2 Turing Instability and Pattern Formation

11.3 Taylor Dispersion and Enhanced Diffusion

11.4 Further Directions

11.5 Exercises

Part III Case Studies

12 Modelling in Applied Fluid Dynamics

12.1 Lubrication Theory

12.2 Dynamics of an Air Bearing Slider

12.3 Rivulets in a Wedge Geometry

12.3.1 Imbibition in a Vertical Wedge

12.3.2 Draining in a Vertical Wedge

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Tags: Thomas Witelski, Mark Bowen, Mathematical Modelling, Continuous Systems