Lecture notes (Lec 1 3)

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1 Lecture notes (Lec 1 3) What is a model? We come across various types of models in life they all represent something else in a form we can comprehend, e.g. a toy model of car or a map of a city or a road map of a city etc. It always has a purpose without purpose or aim of the study the model does not mean much, e.g. a toy prototype of car is a good for a child but not for an automobile engineer. Thus, model is a representation of given system in the language one can conceive of. It requires a translation of real world in the language one can understand. The process of developing a model is Modeling. The term real world system could refer to a physical system, a financial system, a social system, an ecological system, or any other system whose behaviours can be observed. There are many ways in which one describe a real world situation, so model can be of various types, e.g., (a) Scaled Model representation in terms of scaled down versions to predict load on a building, one uses scaled model of a building in a wind tunnel. (b) Pictorial Model representation in terms of drawings/ pictures such as a map of a city or globe (c) Conceptual Model in terms of concepts or verbal representation (d) Symbolic Model represented in terms of certain symbols, which correspond to physical features of the system. These symbols have well defined meaning and follow certain set rules. As a model is a representation of real world situation it contains less information than the real system itself. It contains only those features which are relevant from the point of view of the goal of the study. Mathematical Models are symbolic models, where the symbols are mathematical symbols/ concepts. Roughly defined, mathematical modelling is the process of constructing/ building mathematical objects (such as system of equations, a stochastic process, a geometric or algebraic structure, an algorithm or numbers) whose properties correspond in some way to a particular real world system. Why Mathematical models? There are of course many specific reasons, but most can be related in some way to the following: To gain understanding. Generally speaking, if we have a mathematical model which accurately reflects some behaviour of a real world system of interest, we can often gain improved understanding of that system through analysis of the model, e.g., blood flow in arteries or spread of an epidemic. Also, in the process of modelling, we may find out which factors are most important in the system, and how different

2 parts of the system are related. While designing a complicated equipment we may need to understand mechanism involved we need to understand lubrication mechanism of synovial joint before designing an artificial joint. To predict or simulate. Very often we wish to know what a real world system will do in the future, but it is expensive, impractical, or impossible to experiment directly with the system. Examples include nuclear reactor design, space flight, extinction of species, weather prediction and so on. To optimise some performance profit of a company To obtain response behaviour of a systems to control an epidemic what factors are important! Did we not use mathematical tools earlier? Look at some of the examples Find the height of a tower, say the Kutub Minar in New Delhi (without climbing it!). Find the mass of the Earth. Estimate the yield of rice in India from the standing crop Find the volume of blood inside the body of a person Dosage of a drug Estimate the population of the year 2500 A.D (without actually waiting till then!). So why this emphasis on Mathematical Models now? Advent of computers and development in computing skills. Use of mathematical tools to solve real world problems, which were earlier intractable. Many new areas are utilizing mathematic tools, e.g social sciences, biology, chemistry, natural sciences, etc. Thus, it is clear that much of modern science involves mathematical modelling. Scientists use mathematics to describe real phenomena, and in fact much of this activity constitutes mathematical modelling. As computers become cheaper and powerful and their use becomes more widespread, mathematical models play an increasingly important role in science. From a business perspective, it is clear that an improved ability to simulate, predict, or understand certain real world systems through mathematical modelling provides a distinct competitive advantage. Furthermore, as computing power becomes cheaper, modelling becomes an increasingly cost effective alternative to direct experimentation. How to Model? As you would have guessed by now, we encounter a variety of problems in real life, which require modelling (see the problems listed in case studies which will be discussed at some stage or the other during the course). Each of these problems is different from others and have its own distinctive feature. Therefore, there is no definite algorithm/ precise way to construct a mathematical model that will work in all situations. Modelling is sometimes viewed as an art as well as science. It involves taking whatever knowledge you may have of the system of interest and using that knowledge to create something. Since everyone has a unique way of looking at problems, different people may come up with different models for the same system. There is usually plenty of room for argument about which model is best. It is very important to understand that for any real system, there is no perfect model. One

3 always tries to improve and reach to a better model. However while modelling, one must make a trade off between accuracy, flexibility, cost. Increasing the accuracy of a model generally increases cost and decreases its flexibility. The goal of modelling process should be to obtain a sufficiently accurate and flexible model at a low cost. Note that usually in mathematics, we find very precise and explicit problems, which we are asked to solve completely. We may have to struggle to find the solution, but once we get it, we are done. This is not the case in modelling, where we encounter unclearly stated and ambiguous problems which we can never hope to solve completely! In the following we consider a general framework for the modelling process. These steps provide only a basic methodology/ broad guidelines for modelling, which are ususally followed consciously or unconsciously in modelling. However, there is no set theory of mathematical modelling. This is because no two real world problems are alike, and each new modelling exercise poses new challenges. Step 1: The starting point is the real world problem. define the problem clearly and unambiguously. The problem is then transformed into a system with a goal of study. This may require prior knowledge about the real world associated with the problem, and/or if the prior knowledge is not sufficient, then one has to design an experiment to obtain new/additional knowledge. Step 2: (System Characterization): Step 1 leads to an initial description of the problem based on prior knowledge of its behaviour. The problem as such may be very complicated and may have features which may not be relevant from the point of view of the goal. So one make some simplifications and idealizations to obtain a real world model (RWM). This involves a process of simplification and idealization known as system characterization. It is a crucial step in model building and requires a deep understanding of the physical aspects of the system. Step 3 (Mathematical Model): At this stage the system characterization is related to a mathematical formulation, which produces a mathematical model. It involves two stages, firstly selection of a suitable mathematical formulation, and then the variables of the selected formulation are related on one to one basis with the relevant features of the system. The abstract formulation is clothed in terms of physical features to give mathematical model. This step requires a strong interaction between the physical features of the system and the abstract mathematical formulation. Step 4 (Analysis): Once mathematical model is obtained, its relationship with the physical world are temporarily discarded and the mathematical formulation is solved/analysed using mathematical tools. This is done purely according to the rules of mathematics.

4 At this step, one needs to assign numerical values to various parameters of the model to obtain the model behaviour. This is done by parameter estimation using given data. Step 5 (Validation): In this step, the formulation is interpreted back in terms of the physical features of the problem to yield the behaviour of the mathematical model. The behaviour of mathematical model is then compared with that of their given problem in terms of the data of real world to determine whether the two are in reasonable agreement or not according to same predefined criterion. This is called validation. It may be pointed out here that the criterion for validation should be chosen with care. If the criterion is too stringent (i.e. it requires a very good agreement between the model behaviour and the physical world) then the resulting model will be very complex. If a less stringent model would lead to a model based on coarser system description. In general, one starts with a fairly stringent criterion and simple system characterization and mathematical formulation. Based on the degree of disagreement, either the criterion may be weakened or model be made more complex so that better agreement is achieved. Step 6 (Adequate Model): If the model passes the test of validation it is called an adequate model and process comes to an end. Otherwise, i.e. if the model does not pass the validation criterion, one needs to back track and make changes either in the description of the system (Step 2) or in the mathematical formulation itself (Step 3), and the process starts from there again. Pitfalls of Modelling In the end, there is a word of caution. It should be noted that mathematical model is only a model and not the real world problem by itself. There could be pitfalls in the mathematical model and this could be because one can make mistake at any of the above steps mentioned above. Therefore, care should be taken in using the results of the mathematical model. Real World Problem Simplifications Real World Model Test Mathematical Formulation Prediction and Validation Analysis & Interpretations Mathematical Model Fig. 1 Schematic diagram of modelling process

5 Note that constructing a model requires: A clear picture of the goal of the modeling exercise. Exactly which aspects of the system do you wish to understand or predict, and how accurately do you need to do it? A picture of the key factors involved in the system and how they relate to each other. This often requires taking a greatly simplified view of the system, neglecting factors known to influence the system, and making assumptions which may or may not be correct. Thus, Mathematical modelling is an iterative process which involves interdisciplinary interactions. It is an art as well as a science. The art aspect deals with the intuition required at various stages, which scientific aspect deals with the precise translation and use of mathematical tools to get solution of the problem. Our process can be briefly described as follows: 1. Make general observations of phenomena, 2. Formulate a hypothesis, 3. Develop a method to test hypothesis, 4. Obtain data, 5. Test hypothesis against data, 6. Attempt to confirm or deny hypothesis. It results in asking the questions like Why? Why modelling? What is the goal? Find? List the data from real world or what more you need to know? Given? What is given list it. Assume? What assumptions can be made from the observations? How? How is the system governed by physical principles? Predict? The formulation and solve Valid? Interpret the solution and validate with real world. Verify? Test to verify is the model adequate? If not Improve. It may be noted that good models already exist for parts of the system. The goal is then to assemble these submodels" to represent the whole system of interest. Good models already exist for a different system, which can be translated or modified to apply to the system of interest. This is due to one of the greatest virtues of mathematics i.e. its generality. A general model exists which includes the system of interest as a special case, but it is very difficult to compute with or analyze the general model. The goal is then to simplify or make approximations to the general model which will still reflect the behavior of the particular system of interest.

6 Fall of a raindrop To illustrate the process, we consider the problem of a raindrop falling from a cloud at moderate height. Goal To find time the taken by a raindrop falling from a cloud at moderate height to reach the ground. Real world consists of rain drop, cloud, ground, surrounding environment. Observations: (i) The velocity of raindrop increases as the distance travelled increases. (ii) A large raindrop takes about 40 sec. to reach ground from a cloud at the height of 1024 sec. (experimental observation) Simplifications/ Idealization Raindrop is a particle falling from rest along a straight line. The variables here are: time, distance, velocity of the raindrop. Mathematical Formulation: If x(t) is the distance travelled in time t by the rain drop after its fall from cloud, then its velocity is rate of change of x with time, i.e.,. Now from the assumption we have with x = 0 at t = 0 (here k is the constant of proportionality). Solving this we get the x = 0 for all time the raindrop is not moving at all. This is not correct. So we have to correct the things. It may be pointed out that while observation is correct, the drop is falling under gravity and Galileo had observed that An object falling from moderate height under gravity gains an extra 32 ft/ sec in velocity in each second. Thus we have a modified formulation, which gives: 32 As initial velocity 0 (rain drop falls from rest), we have 32, 0 0. This gives time to cover a distance of 1024 ft is 8 sec. this much less than the observed experimental value. (However, this value has been observed for a heavy ball falling from this height under gravity). It may be noted that both rain drop and a heavy ball experience a resistance due to air through which they fall. While, this air drag can be neglected in the case of heavy ball, it is not so in case of rain drop. Therefore, model needs an improvement to consider air drag opposing the model. For this we have an empirical result Stokes law which says For a spherical droplet (with diameter D < ft) falling under gravity in motionless air, it experiences air drag which opposes the motion is proportional to velocity and is equal to 0.329x10 5 v/d 2 Thus we get, a new formulation: , 0 0, 0 0 One can easily solve this equation, however we can have some mathematical simplifications it is observed that rain drop attains 99% of its terminal velocity in a very short time (Verify it by solving the equation) and then it continues to fall with a

7 constant velocity / Assuming the maximum value of the diameter D = ft, we get the rain drop travels 1 ft in 165 sec. i.e. it is hardly moving. Again the model is not adequate for the given goal and needs to be further modify. However, it is good enough for a fog droplet. (Diameter of raindrop is larger than the given value). Ex. Look for the improved models.