Differentiaalvergelijkingen in de Aardwetenschappen

Course goals

Content

This course is a natural progression of, and builds upon, the elements of calculus and programming introduced in Wiskunde en Statistiek voor de Aardwetenschappen (GEO1-1134) and Natuurkunde en Programmeren van Aardse Processen (GEO1-1135).

The course contains 3 parts which will ultimately allow the student to solve ordinary and partial differential equations (ODEs and PDEs).

The first part is concerned with series and complex numbers, important tools for analysing and manipulating functions.

The second part introduces Fourier series, and we will see that it is a natural way to describe periodic functions. We then generalize this concept to Fourier transforms, which are a powerful tool for spectral analysis. They can be also used to solve certain differential equations very efficiently.

In the last part, we will use the previous tools (and more) to solve ordinary and the partial differential equations. Examples are the laws for heat diffusion inside the Earth, elastic wave propagation and many processes in geochemistry and fluid migration.

In each part of the course, we will make an effort to show representative examples from different branches of Geociences so that the students can form a connection between the mathematics and its practical application in Geoscience.
Planned schedule:
Week 1: Series, including: Power series of common functions, Taylor series, Binomial Series. Examples shown in lectures: Fibonacci Series – many examples in nature including e.g. ammonite curvature (paleontology) Taylor Series – Describing interatomic potential energies in mineral physics Binomial Series – modelling risks with two possible outcomes e.g. probability of landslides in a given time interval (natural hazards)
Week 2: Complex numbers
Week 3:

• Programming with Series: Students will 1. Program a series and observe the effects of truncation; 2 Represent interatomic potential energy with a Taylor series
• Simple harmonic motion, waves & periodic functions. Examples shown in lectures: – seismic waves (geophysics), sea level fluctuations (oceans & climate), paleoclimate proxies, tides.
• MID TERM EXAM 1 (SERIES & COMPLEX NUMBERS)

Week 4: Fourier Series: How to represent any periodic function with a Fourier series. Best conceptualized with musical instruments. Also nice examples from tidal water level signals.
Week 5:

• Fourier Transforms. Examples shown in lectures: extracting whole earth oscillation frequencies from large earthquakes (geophysics), FT infrared spectroscopy (mineralogy), Milankovitch cycles (oceans & (paleo)climate), sea surface waves (earth surface)
• Programming with Fourier: Students will: 1. Perform high and low pass filtering on a time series (either a seismogram or other geo-relevant time series where filtering is used) 2. Calculate the Fourier transform of a seismogram from a very large earthquake to see the resonant frequencies

Week 6:

• MID TERM EXAM 2 (FOURIER)
• ODE 1: Terminology and methods for solving 1st order ODEs. Example shown in lectures: radioactive decay (isotope geochemistry and chronology)

Week 7:

• ODE 2: Damped oscillations and 2nd order ODEs. Examples shown in lectures: how a seismometer works (geophysics), bars and bends in rivers
• Programming with ODEs: Students will model steady-state diffusion in 1D, using heat flow in the crust (tectonics) as an example
• PDE 1: Functions of several variables & partial derivatives; PDEs in geoscience: an overview, including classification of PDEs (elliptical, hyperbolic, parabolic) and where they are used in geoscience, lots of visual examples.

Week 8: PDE 2 & 3: Wave equation in 1D, Laplace's equation, Diffusion equation (in the context of heat flow) and Schrödinger equation
Week 9:

• PDE 4: Wave equation in 2D
• Programming with PDEs: Students will use python software to solve PDEs with application to a range of geocience problems, such as: 1. Solve the diffusion (heat flow) equation in 2D for cooling of the lithosphere (tectonics & geophysics); 2. Solve the diffusion equation in 2D for contamination of groundwater from a toxic spill (hydrology). 3. Solve the wave equation for propagation of seismic waves in the Earth (geophysics), propagation of tidal waves in an estuary (hydrology), or Kelvin waves in the ocean (oceans & climate).

Week 10:

• Revision class
• END TERM EXAM (ODEs and PDEs)

this course is a pre-requisite for:
GEO3-1350 = Paleomagnetism and paleogeography.
GEO3-1359 = Earth’s Heat, Dynamics, and Hazards
GEO3-1302 = Continuum mechanics and Mantle dynamics
GEO3-1312 = Introduction to Seismology