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Scaling of Differential Equations
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Contents:
Scaling of Differential Equations
Preface
Dimensions and units
Fundamental concepts
Base units and dimensions
Dimensions of common physical quantities
The Buckingham Pi theorem
Absolute errors, relative errors, and units
Units and computers
Unit systems
Example on challenges arising from unit systems
PhysicalQuantity: a tool for computing with units
Parampool: user interfaces with automatic unit conversion
Pool of parameters
Fetching pool data for computing
Reading command-line options
Setting default values in a file
Specifying multiple values of input parameters
Generating a graphical user interface
Ordinary differential equation models
Exponential decay problems
Fundamental ideas of scaling
The basic model problem
The technical steps of the scaling procedure
Making software for utilizing the scaled model
Scaling a generalized problem
Variable coefficients
Scaling a cooling problem with constant temperature in the surroundings
Scaling a cooling problem with time-dependent surroundings
Scaling a nonlinear ODE
SIR ODE system for spreading of diseases
SIRV model with finite immunity
Michaelis-Menten kinetics for biochemical reactions
Vibration problems
Undamped vibrations without forcing
Undamped vibrations with constant forcing
Undamped vibrations with time-dependent forcing
Damped vibrations with forcing
Oscillating electric circuits
Exercises
Exercise 2.1: Perform unit conversion
Problem 2.2: Scale a simple formula
Exercise 2.3: Perform alternative scalings
Problem 2.4: A nonlinear ODE for vertical motion with air resistance
Exercise 2.5: Solve a decay ODE with discontinuous coefficient
Exercise 2.6: Implement a scaled model for cooling
Problem 2.7: Decay ODE with discontinuous coefficients
Exercise 2.8: Alternative scalings of a cooling model
Exercise 2.9: Projectile motion
Problem 2.10: A predator-prey model
Problem 2.11: A model for competing species
Problem 2.12: Find the period of sinusoidal signals
Problem 2.13: Oscillating mass with sliding friction
Problem 2.14: Pendulum equations
Exercise 2.15: ODEs for a binary star
Problem 2.16: Duffing’s equation
Problem 2.17: Vertical motion in a varying gravity field
Problem 2.18: A simplified Schroedinger equation
Basic partial differential equation models
The wave equation
Homogeneous Dirichlet conditions in 1D
Implementation of the scaled wave equation
Time-dependent Dirichlet condition
Velocity initial condition
Variable wave velocity and forcing
Damped wave equation
A three-dimensional wave equation problem
The diffusion equation
Homogeneous 1D diffusion equation
Generalized diffusion PDE
Jump boundary condition
Oscillating Dirichlet condition
Reaction-diffusion equations
Fisher’s equation
Nonlinear reaction-diffusion PDE
The convection-diffusion equation
Convection-diffusion without a force term
Stationary PDE
Convection-diffusion with a source term
Exercises
Problem 3.1: Stationary Couette flow
Exercise 3.2: Couette-Poiseuille flow
Exercise 3.3: Pulsatile pipeflow
Exercise 3.4: The linear cable equation
Exercise 3.5: Heat conduction with discontinuous initial condition
Problem 3.6: Scaling a welding problem
Advanced partial differential equation models
The equations of linear elasticity
The general time-dependent elasticity problem
Dimensionless stress tensor
When can the acceleration term be neglected?
The stationary elasticity problem
Quasi-static thermo-elasticity
The Navier-Stokes equations
The momentum equation without body forces
Scaling of time for low Reynolds numbers
Shear stress as pressure scale
Gravity force and the Froude number
Oscillating boundary conditions and the Strouhal number
Cavitation and the Euler number
Free surface conditions and the Weber number
Thermal convection
Forced convection
Free convection
The Grashof, Prandtl, and Eckert numbers
Heat transfer at boundaries and the Nusselt and Biot numbers
Compressible gas dynamics
The Euler equations of gas dynamics
General isentropic flow
The acoustic approximation for sound waves
Water surface waves driven by gravity
The mathematical model
Scaling
Waves in deep water
Long waves in shallow water
Two-phase porous media flow
The bidomain model in electrophysiology
The mathematical model
Scaling
An alternative
\(I_{\rm{ion}}\)
Exercises
Exercise 4.1: Comparison of vibration models for elastic structures
Exercise 4.2: A model for quasi-static poro-elasticity
Problem 4.3: Starting Couette flow
Problem 4.4: Channel flow
References
Index
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Index
