Mathematical Models Modelling in Mathematics and Other Aspects of Life

Mathematical Modelling It is the aim of the lecture to impart the ability to - analyse real environmental systems and describe them as adequately as possible by means of simple mathematical models with regard to interesting questions; - discuss and in some cases analytically resolve the resulting mathematical equations (especially differential equation systems); - examine complicated models by means of computer programmes and thus understand the limits of simulation and the occurrence of possible mathematical artefacts.

Introduction Is mathematics related to the real world? How is mathematics useful? Why do we have to do “Word Problems” or Story Problems?

Modeling What is a model (or a mathematical model)? –Mathematics is the language of change –The use of mathematics to describe a system’s behavior.

Why Modeling Why do we need them and what purpose do they serve? –To analyze a system to be controlled or optimized –To hypothesize about how a system works –To make predictions at parameter values and/or scales that are difficult to test

Features in Modeling What are the features of a mathematical model? –Has variables, constants, and exponents –Equations –Inequations This is the most basic description.

Process of Mathematical Modeling REAL WORLDMODEL RESULT

System: Example (1)

System: Example (2)

Examples of Models ArchitecturalChemical molecules Kepler‘s Law Feedbacks

How to Model? 1.Assume and Formulate 2.Do the Maths 3.Interpret and Evaluate 4.Improve the Model

Role of models in science In building any theory you perceive the following phases: 1. Collecting: observations, measurements: In his optimally equipped observatory Uranienborg near Copenhagen Tycho Brahe ( ) collects data on planetary movements. 2. Sorting: search for the principle of classification in the collected data: Johannes Kepler ( ) classifies Brahe's planetary orbits, three laws 3. Understanding: search for a superior principle with which to understand the empirically found order Isaac Newton ( ) shows that Kepler's laws can be explained by physical principles which are valid beyond astronomy. 4. Generalising: Can the laws be transferred to other situations? Based on the equivalence of inert and heavy mass, Albert Einstein ( ) develops his general theory of relativity. 5. Prognosis: Can the (perhaps generalised) regularities be used to predict phenomena? astronomical phenomena predicted by Einstein were observed. Finally, collecting data only makes sense if further knowledge develops out of them, either as a generalised statement, prognosis or to formulate new questions, i. e. plan new experiments. Models are needed for the application of observations.

Role of models in science In building any theory you perceive the following phases: 1. Collecting: observations, measurements: In his optimally equipped observatory Uranienborg near Copenhagen Tycho Brahe ( ) collects data on planetary movements. 2. Sorting: search for the principle of classification in the collected data: Johannes Kepler ( ) classifies Brahe's planetary orbits, three laws 3. Understanding: search for a superior principle with which to understand the empirically found order Isaac Newton ( ) shows that Kepler's laws can be explained by physical principles which are valid beyond astronomy. 4. Generalising: Can the laws be transferred to other situations? Based on the equivalence of inert and heavy mass, Albert Einstein ( ) develops his general theory of relativity. 5. Prognosis: Can the (perhaps generalised) regularities be used to predict phenomena? astronomical phenomena predicted by Einstein were observed. Finally, collecting data only makes sense if further knowledge develops out of them, either as a generalised statement, prognosis or to formulate new questions, i. e. plan new experiments. Models are needed for the application of observations.

Role of models in science In building any theory you perceive the following phases: 1. Collecting: observations, measurements: In his optimally equipped observatory Uranienborg near Copenhagen Tycho Brahe ( ) collects data on planetary movements. 2. Sorting: search for the principle of classification in the collected data: Johannes Kepler ( ) classifies Brahe's planetary orbits, three laws 3. Understanding: search for a superior principle with which to understand the empirically found order Isaac Newton ( ) shows that Kepler's laws can be explained by physical principles which are valid beyond astronomy. 4. Generalising: Can the laws be transferred to other situations? Based on the equivalence of inert and heavy mass, Albert Einstein ( ) develops his general theory of relativity. 5. Prognosis: Can the (perhaps generalised) regularities be used to predict phenomena? astronomical phenomena predicted by Einstein were observed. Finally, collecting data only makes sense if further knowledge develops out of them, either as a generalised statement, prognosis or to formulate new questions, i. e. plan new experiments. Models are needed for the application of observations.

Role of models in science In building any theory you perceive the following phases: 1. Collecting: observations, measurements: In his optimally equipped observatory Uranienborg near Copenhagen Tycho Brahe ( ) collects data on planetary movements. 2. Sorting: search for the principle of classification in the collected data: Johannes Kepler ( ) classifies Brahe's planetary orbits, three laws 3. Understanding: search for a superior principle with which to understand the empirically found order Isaac Newton ( ) shows that Kepler's laws can be explained by physical principles which are valid beyond astronomy. 4. Generalising: Can the laws be transferred to other situations? Based on the equivalence of inert and heavy mass, Albert Einstein ( ) develops his general theory of relativity. 5. Prognosis: Can the (perhaps generalised) regularities be used to predict phenomena? astronomical phenomena predicted by Einstein were observed. Finally, collecting data only makes sense if further knowledge develops out of them, either as a generalised statement, prognosis or to formulate new questions, i. e. plan new experiments. Models are needed for the application of observations.

Role of models in science In building any theory you perceive the following phases: 1. Collecting: observations, measurements: In his optimally equipped observatory Uranienborg near Copenhagen Tycho Brahe ( ) collects data on planetary movements. 2. Sorting: search for the principle of classification in the collected data: Johannes Kepler ( ) classifies Brahe's planetary orbits, three laws 3. Understanding: search for a superior principle with which to understand the empirically found order Isaac Newton ( ) shows that Kepler's laws can be explained by physical principles which are valid beyond astronomy. 4. Generalising: Can the laws be transferred to other situations? Based on the equivalence of inert and heavy mass, Albert Einstein ( ) develops his general theory of relativity. 5. Prognosis: Can the (perhaps generalised) regularities be used to predict phenomena? astronomical phenomena predicted by Einstein were observed. Finally, collecting data only makes sense if further knowledge develops out of them, either as a generalised statement, prognosis or to formulate new questions, i. e. plan new experiments. Models are needed for the application of observations.

Role of models in science In building any theory you perceive the following phases: 1. Collecting: observations, measurements: In his optimally equipped observatory Uranienborg near Copenhagen Tycho Brahe ( ) collects data on planetary movements. 2. Sorting: search for the principle of classification in the collected data: Johannes Kepler ( ) classifies Brahe's planetary orbits, three laws 3. Understanding: search for a superior principle with which to understand the empirically found order Isaac Newton ( ) shows that Kepler's laws can be explained by physical principles which are valid beyond astronomy. 4. Generalising: Can the laws be transferred to other situations? Based on the equivalence of inert and heavy mass, Albert Einstein ( ) develops his general theory of relativity. 5. Prognosis: Can the (perhaps generalised) regularities be used to predict phenomena? astronomical phenomena predicted by Einstein were observed. Finally, collecting data only makes sense if further knowledge develops out of them, either as a generalised statement, prognosis or to formulate new questions, i. e. plan new experiments. Models are needed for the application of observations.

Models Models are used to express the characteristics of reality which are considered important and to neglect those which seem secondary. By these simplifications, good models allow us to obtain an easily understandable, mathematically calculable image of the real world If a model is formulated with the aid of mathematical relations, we speak of mathematical models. This lecture deals with construction and use of mathematical models in natural sciences.

Models Models are used to express the characteristics of reality which are considered important and to neglect those which seem secondary. By these simplifications, good models allow us to obtain an easily understandable, mathematically calculable image of the real world. If a model is formulated with the aid of mathematical relations, we speak of mathematical models. This lecture deals with construction and use of mathematical models in natural sciences.

Models Models are used to express the characteristics of reality which are considered important and to neglect those which seem secondary. By these simplifications, good models allow us to obtain an easily understandable, mathematically calculable image of the real world If a model is formulated with the aid of mathematical relations, we speak of mathematical models.

Walking Rates Kecepatan = Jarak/Waktu = … m/s Karena itu, jarak yang ditempuh adalah Jarak = Kecepatan (m/s). Waktu (s) = … m Contoh: Marti memiliki kecepatan: 1,2 m/s Berarti dalam 1 detik, Marti menempuh jarak = 1,2 m/s. 1 s = 1,2m Dalam 60 detik, Marti menempuh jarak = 1,2 m/s. 60 s = 72 m

Walking Rates.. Cont’d Bagaimana persamaan model dari Perjalanan Marti? d = 1,2. t Dengan d = distance, t = waktu tempuh. Jenis persamaan apakah ini? Berapakah waktu yang dibutuhkan Marti untuk menempuh 10 km?