• Site
  • Scaling of Differential Equations
  • Preface
  • Dimensions and units
    • Fundamental concepts
      • Base units and dimensions
      • Dimensions of common physical quantities
        • Prefixes for units
      • 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
        • Example: Population dynamics
        • Example: Decay of pressure with altitude
      • The technical steps of the scaling procedure
        • Step 1: Identify independent and dependent variables
        • Step 2: Make independent and dependent variables dimensionless
        • Step 3: Derive the model involving only dimensionless variables
        • Step 4: Make each term dimensionless
        • Step 5: Estimate the scales
      • Making software for utilizing the scaled model
        • Software for the original unscaled problem
        • Software for the scaled problem
        • Implementation with joblib
      • Scaling a generalized problem
        • Exact solution
        • Theory
        • Software
      • Variable coefficients
      • Scaling a cooling problem with constant temperature in the surroundings
        • Exact solution
        • Scaling
        • Software
        • Alternative scaling
      • Scaling a cooling problem with time-dependent surroundings
        • Exact solution
        • Scaling
        • Software
        • Discussion of the time scale
      • Scaling a nonlinear ODE
        • Scaling
        • Alternative scaling
      • SIR ODE system for spreading of diseases
        • Scaling
        • Software
        • Alternative scaling
      • SIRV model with finite immunity
      • Michaelis-Menten kinetics for biochemical reactions
        • Classical analysis
        • Dimensionless ODE system
        • Determining scales
        • Conservation equations
        • Analysis of the scaled system
    • Vibration problems
      • Undamped vibrations without forcing
        • The first technical steps of scaling
        • The exact solution
        • Discussion of the displacement scale
        • Discussion of the time scale
        • The dimensionless solution
        • Alternative displacement scale
        • About frequency and dimensions
      • Undamped vibrations with constant forcing
      • Undamped vibrations with time-dependent forcing
        • Investigating scales via analytical solutions
        • The displacement and time scales
        • Finding the displacement scale from the differential equation
        • Scaling with free vibrations as time scale
        • Software
        • Choice of \(u_c\) close to resonance
        • Unit size of all terms in the ODE
        • Choice of \(u_c\) when \(\psi\gg\omega\)
        • Displacement scale based on \(I\)
      • Damped vibrations with forcing
        • The exact solution
        • Choosing scales
        • Choice of \(u_c\) at resonance
        • Choice of \(u_c\) when \(\omega\gg\psi\)
        • Choice of \(u_c\) when \(\omega\ll\psi\)
        • Software
      • 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
        • Remarks
      • 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
        • Remarks
  • Basic partial differential equation models
    • The wave equation
      • Homogeneous Dirichlet conditions in 1D
      • Implementation of the scaled wave equation
        • Waves on a string
        • Detecting an already computed case
      • Time-dependent Dirichlet condition
        • Scaling
        • Software
        • Specific case
      • Velocity initial condition
        • Analytical insight
        • Scaling
        • Nonzero initial shape
      • Variable wave velocity and forcing
        • Non-dimensionalization
        • Choosing the time scale
        • Choosing the spatial scale
        • Scaling the velocity initial condition
      • Damped wave equation
      • A three-dimensional wave equation problem
    • The diffusion equation
      • Homogeneous 1D diffusion equation
        • Choosing the time scale
        • Analytical insight
        • Choosing other scales
      • Generalized diffusion PDE
      • Jump boundary condition
      • Oscillating Dirichlet condition
        • Scaling issues
        • Exact solution
        • Time and length scales
        • The scaled problem
        • Simulations
    • Reaction-diffusion equations
      • Fisher’s equation
        • Balance of all terms
        • Fixed length scale
      • 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
        • Remarks
      • 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
        • Remarks
      • Problem 3.6: Scaling a welding problem
  • Advanced partial differential equation models
    • The equations of linear elasticity
      • The general time-dependent elasticity problem
        • Software
      • Dimensionless stress tensor
      • When can the acceleration term be neglected?
        • S waves
        • P waves
        • Time-varying load
      • The stationary elasticity problem
        • Scaling of the PDE
        • Remark on the characteristic displacement
        • Scaling of displacement boundary conditions
        • Scaling of traction boundary conditions
      • Quasi-static thermo-elasticity
    • The Navier-Stokes equations
      • The momentum equation without body forces
        • Scaling
        • Dimensonless PDEs and the Reynolds number
      • 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
        • Governing equations
        • Heating by viscous effects
        • Relation between density and temperature
        • The Boussinesq approximation
        • Scaling
      • The Grashof, Prandtl, and Eckert numbers
        • Interpretations of the Grashof number
      • Heat transfer at boundaries and the Nusselt and Biot numbers
    • Compressible gas dynamics
      • The Euler equations of gas dynamics
      • General isentropic flow
        • Elimination of the pressure
      • The acoustic approximation for sound waves
        • Wave nature of isentropic flow with small perturbations
        • Basic scaling for small wave perturbations
    • 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
        • Remarks
    • References
• Page
  • Scaling of Differential Equations
  • Index
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