MATHEMATICAL AND COMPUTER MODELING IN SCIENCE AND ENGINEERING · PDF fileMathematical and Computer Modeling in Science and ... Modeling in Science and Engineering 3 ... • The interplay

Sergey Pankratov, TUM 2003
Mathematical and Computer Modeling in Science and Engineering
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MATHEMATICAL AND COMPUTER MODELING IN
SCIENCE AND ENGINEERINGSergey Pankratov
Lehrstuhl fr Ingenieuranwendungen in derInformatik

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Contents Introduction Part 1. Classical deterministic systems
Chapter 1. Methodological principles of modeling (5 -76) Chapter 2. Mathematical methods for modeling (77 -132) Chapter 3. Computational techniques (133 -172) Chapter 4. Case studies (173 -334) Literature (335 -338)
Part 2. Quantum modeling Part 3. Stochastic modeling Appendices

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Introduction This course contains examples of the application of selected
mathematical methods, numerical techniques and computer algebra manipulations to mathematical models frequently encountered in various branches of science and engineering
Most of the models have been imported from physics and applied for scientific and engineering problems.
One may use the term compumatical methods stressing the merge of analytical and computer techniques.
Some algorithms are presented in their general form, and the main programming languages to be used throughout the course are MATLAB and Maple.
The dominant concept exploited throughout Part 1 of the course is the notion of local equilibrium. Mathematical modeling of complex systems far from equilibrium is mostly reduced to irreversible nonlinear equations. Being trained in such approach allows one to model a great lot of situations in science and engineering

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Part 1Classical deterministic systems

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Chapter1. Methodological Principles of Modeling
In this chapter basic modeling principles are outlined and types of models are briefly described. The relationship of
mathematical modeling to other disciplines is stressed

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The emphasis in this part is placed on dynamical systems. This is due to the fact that change is the most interesting aspect of models
In modern mathematical modeling, various mathematical concepts and methods are used, e.g. differential equations, phase spaces and flows, manifolds, maps, tensor analysis and differential geometry, variational techniques, Lie groups, ergodic theory, stochastics, etc. In fact, all of them have grown from the practical needs and only recently acquired the high-brow axiomatic form that makes their direct application so difficult
As far as computer modeling as a subset of mathematical modelingis concerned, it deals with another class of problems (algorithms, numerics, object-orientation, languages, etc.)
The aim of this course is to bridge together computational methods and basic ideas proved to be fruitful in science and engineering
Computer as a modeling tool

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Training or education? One can probably design a good interactive program without any
knowledge of analytical techniques or geometric transformations However, for modeling problems, it would be difficult in such case to get outside the prescribed set of computer tricks. This reflects a usual dichotomy between education and training
The present course has been produced with some educational purposes in mind, i.e. encompassing the material supplementary to computer-oriented training. For instance, discussion of such issues as symmetry or irreversibility is traditionally far from computer science, but needed for simulation of complex processes
Possibly, not everything contained in the slides that follow will be immediately needed (e.g. for the exam). To facilitate sorting out the material serving the utilitarian purposes, the slides covering purely theoretical concepts are designated by the symbol

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Notes on exercises Exercises to the course are subdivided into two parts. The first part
includes questions and problems needed to understand theoreticalconcepts. These questions/problems (denoted c##) are scattered throughout the text/slides. It is recommended not to skip them
The second part consists of exercises largely independent from the course text/slides. They are usually more complicated then thoseencountered in the text. One can find both these problems and their solutions in the Web site corresponding to the course (denoted ##)
Usually, exercises belonging to the second part represent some real situation to be modeled. Modeling can be performed in a variety of ways, with different aspects of the situation being emphasized and different levels of difficulty standing out. This is a typical case in mathematical modeling, implying that the situation to be modeledis not necessarily uniquely described in mathematical terms
Use of computers is indispensable for the majority of exercises

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What is mathematical modeling? Replacement of an object studied by its image a mathematical
model. A model is a simplification of reality, with irrelevant details being ignored
In a mathematical model the explored system and its attributes are represented by mathematical variables, activities are represented by functions and relationships by equations
Quasistatic models display the relationships between the system attributes close to equilibrium (e.g. the national economy models); dynamic models describe the variation of attributes as functions of time (e.g. spread of a disease)
Modeling stages: 1) theoretical, 2) algorithmic, 3) software, 4) computer implementation, 5) interpretation of results
Mathematical modeling is a synthetic discipline (superposition of mathematics, physics, computer science, engineering, biology, economics, sociology, ..). Enrichment due to interdisciplinarity

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Basic types of mathematical models
Qualitative vs. quantitative Discreet vs. continuous Analytical vs. numerical Deterministic vs. random Microscopic vs. macroscopic First principles vs. phenomenology In practice, these pure types are interpolated The ultimate difficulty: there cannot be a single recipe how to
build a model For every phenomena many possible levels of description One always has to make a choice: the art of modelingSee also: C. Zenger, Lectures on Scientific Computing,
http://www5.in.tum.de/lehre/vorlesungen/sci_comp/

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Important principles of model building Models should not contradict fundamental laws of nature (e.g. the
number of particles or mass should be conserved the continuity equation)
Testing (validation) of models against basic laws of nature Symmetry should be taken into account Scaling can be exploited to reduce the complexity To use analogies (e.g. chemical reactions and competition models) Universality: different objects are described by the same model
(e.g. vibrations of the body of a car and the passage of signalsthrough electrical filters)
Hierarchical principle: each model may incorporate submodels a step-like refinement
Modularity and reusability From problem to method, not vice versa

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Basic properties of mathematical models Causality: models based on dynamical systems are causal i.e. the
effect cannot precede the cause and the response cannot appear before the input signal is applied. Causality is a result of human experience - non-causal systems would allow us to get the signals from the future or to influence the past.
Causality is closely connected with time-reversal non-invariance (arrow of time). The time-invariance requires that direct and time-reversed processes should be identical and have equal probabilities. Most of mathematical models corresponding to real-life processes are time non-invariant (in distinction to mechanical models).
A mathematical model is not uniquely determined by investigated object or situation. Selection of the model is dictated by accuracy requirements. Examples - a table: rectangular or not, ballistics: influence of the atmosphere, atom: finite dimensions of nucleus,military planning: regular armies (linear), partisans (nonlinear).

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Basic models in physics Physical systems can be roughly separated into two classes:
particles and fields, these are two basic modelsRoughly because there is an overlap, e.g. particles as field sources The main difference in the number of degrees of freedom Any physical system consisting of a finite number N of particles
has only finite number of degrees of freedom n3NThe number of degrees of freedom is defined as dimensionality of the configuration space of a physical system, e.g.

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MATHEMATICAL AND COMPUTER MODELING IN SCIENCE AND ENGINEERING · PDF fileMathematical and Computer Modeling in Science and ... Modeling in Science and Engineering 3 ... • The interplay