Scaling of Differential Equations

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

• Index