The aim of the summer school is to give a broad introduction to the mathematical modeling of dynamical systems, as described by ordinary differential equations and with applications in agriculture, and to describe modern computer methods and systems for solving these differential equations either via specialized software or via appropriate modules of general-purpose problem-solving environments. At the end of the course, the students should be able to formulate simple dynamical models and to judge the accuracy and quality of a numerical computed solution.
As a dynamical system we consider a mathematical model that aims at describing the development in timeof a system or process. Dynamical systems are usually described by means of ordinary differential equations (ODEs) that relate the variation in time to the various state variables. Solving the ODEs and inspecting the computed solutions is often referred to as simulation.
Mathematical modeling is the discipline of setting up mathematical equations that describe the system being studied. In connection with dynamical systems, this involves recognizing the dynamical processes in the system and setting up the necessary ODEs - perhaps supplemented with other equations that describe other features of the model, such as constraints or linear dependencies.
Dynamical systems That appear in practical applicationswill in general become virtually impossible to treat using analytical methods, even if all the equations are known explicitly. This calls for the numerical solution of the ODEs by means of methods specialized for this purpose, such as Runge-Kutta methods. Today, algorithms for ODEs are part many software libraries, and problem solving environments such as Matlab. There is no need for writing one's own simulation software.
The main difficulty with working with "black box" software modules for solving numerical problems is that the user may have no knowledge about the features and limitations of the algorithms being used. Hence, in connection with courses in mathematical modeling by means of dynamical models, it is desirable also to teach the fundamental principles of the numerical ODE solvers underlying most current software.
Simulations of dynamical systems are used for two main purposes: to study the evolution in time of a system, and to identify model parameters in the model. The first purpose is straightforward, once the mathematical model has been set up and implemented. The second purpose arises when the aim of the simulation is to compare the simulation results with observations, in order to determine unknown or uncertain parameters in the model (i.e., we seek the model parameters that best fit the observations). These aspects should also be covered in a course on simulation.
This summer school will focus on the following main topics
The material will be illustrated with various examples from agriculture, and supplemented with guest lectures on related issues (such as stochastic differential equations and difference equations).
Throughout the course, the theory will be supplemented with exercises and computer assignments. At the end of the course, the students will work on a two-day project that involves both modeling and computing aspects.
The following key words describe the technical contents of the course.
Professor Per Christian Hansen, PhD & Dr. Techn., Dina Research School and Department of Mathematical Modelling, Technical University of Denmark.
Lecture notes, copies of overhead transparencies, assignments, and descriptions of the two-day projects. Most of this material will also be available on the Internet.
A basic understanding of mathematical modeling is required, along with some experience in using standard software systems. Background material for the participants will be available on the Internet.
Author: phd@dina.kvl.dk. Updated: 5 May 2000