Mathematical modeling



Prof. Dr. Gerrit Lohmann

24. April- 24. July 2006: Monday 11-13, Room S3032



This lecture is about concepts of mathematical modeling, and about the kinds of techniques that are useful for modeling. It covers exact and approximate analytical techniques, numerical methods, and model inference based on observations. A focus will be on applications across a broad range of Earth System Modeling and Analysis. It discusses the general components of the modeling process and highlights the potential of modeling in practice. A part of the lecture provides case studies, with examples, exercises, and projects.


Literature:
The nature of mathematical modeling, N. Gershenfeld, Cambridge University Press, Cambridge, 2003, 344 pp. link for applications link for programs

Applied Mathematical Modeling: A Multidisciplinary Approach; von D. R. Shier, K. T. Wallenius, pp. 443, CRC PrILlc, ISBN    1584880481

Imboden, D. und Koch, S. (2003).Systemanalyse - Einf?hrung in die mathematische Modellierung nat?rlicher Systeme Springer-Verlag, Berlin Heidelberg.

Script for part of the course Mathematical modeling, G. Lohmann, 2006



Preliminary Schedule:
1) 24.4. Introduction, Aim of modelling, Linear differential equations, Laplace transformation
ppt file,
Script_VL1.doc
pdf_VL1: Laplace Transformation

2) 8.5. Types of Models, Examples: Exponential & Logistic equations, Newton's cooling, Numerical schemes
ppt file,
Script_VL2.doc,
download Excel examples
pdf_VL2-1_Numerics_ode
pdf_VL2-2_Numerics_ode

3) 15.5. Practical applications with R: Ordinary differential equations
intro to R,
R-2.2.0-win32.exe

Here are the examples:
lesson_14May_commands.R
EulerForward_1storder.R,
EulerForward_1storder_generalized.R,
EulerForward_2storder_generalized.R,
EulerForward_1storder_LogisticE.R,
EulerForward_2storder_LogisticE.R

4) 22.5. ODE continued
ppt file,
Script_VL4.doc,

5) 29.5. Examples, Stability theory, Bifurcations
Overview ppt file,
Box_inorganicCC.doc
Bifurcation ppt file,
Stability Theory: doc file


6) 12.6. Partial differential equations with applications in Earth System Science, Diffusion and advection
Bifurcation ppt file,
Skript_mathe_VL_partial1.doc
Diff_adv_solvppt file,
Stability doc file
numerics 1 pdf file

numerics 2 pdf file (not used)
numerics 3 pdf file (not used)


7) 19.6. Practical units: Partial differential equations
numerics pdf (from the last lesson)
work sheet
Diffusion_explicit.R
AdVDiffusion_leapfrog.R

8) 26.6. Master and Fokker-Planck Equation, Bownian Motion
ppt file
doc file

9) 3.7. continued: Master and Fokker-Planck Equation, Bownian Motion
ppt file
doc file
10) 10.7. Practical units: Applications for stochastic systems
work sheet
brown_oneparticle.R
2D_Diffusion.R
brown_multipleparticle.R
brown_multipleparticle_potential.R

11) 17.7. Linear time series analysis, Statistical Modelling, Significance testing
2Timeser_VL11.ppt
some background material pdf1
some background material pdf2

12) 24.7. Practical units: Spectrum, wavelet, EOF, SSA
worksheet
corexperiments.R
spectrum.R


for fun:
Practical unit (left over, not during the lesson 7): Difference equations, Logistic map and Chaos, Bifurcations
work sheet
logisticmap.R


How to get the Credit Points / Schein?
Projects with R, Practical work within the lessons
Tutor: Thomas Laepple

Oral exam: List of catchwords: repeat lessons 1)-12) !!!
Day: 4 August 2006; 8:30, Room S3032