Lecture 1: Basics of Mathematical Modeling

[Music] hello everyone this is dr risha diman from school of mathematics and i'm your instructor for the course mathematical modeling and simulation so first of all i would like to welcome you all and congratulate you all on your selection of this course and i assure that you are going to have fun and learn a lot of mathematics in a much more innovative and interesting manner throughout the course so without a further delay let's begin this is going to be the introductory lecture lecture one in which we are going to cover some basics of mathematical modeling so firstly we'll begin with understanding that what is the meaning of the word modeling in general and then more specifically we'll be looking at mathematical modeling and for mathematical modeling i'll be answering these three questions what it is why to do it and how to do it and at least in this first lecture i'll be definitely answering what it is and why it is and the how part will be explained in the subsequent lectures and then we'll look at some of the interesting applications of mathematical modelling followed by objectives of it although there are no well defined and fixed objectives but yet still we can discuss few of them and then we'll begin with the definition of the modeling cycle and in detail we'll be discussing the principles of mathematical modelling so let's explore all of these one by one what is modeling so to understand the modeling process we need to first understand the meaning of the word model so first question is what is a model so i would suggest you to pause the video here and think about this word that whatever comes to your mind whenever you look at the word model you read it somewhere or you hear the word model anywhere so a model is nothing but a replication of something which is happening in real so we can have few statements which can be considered as definitions of the model a model is a miniature or abstract representation of something why it is called miniature because most of the times the model is drawn on a small scale as compared to the real object a pattern of something to be made like we want to construct something but before actual construction we want to understand it through the modeling process so we'll draw its pattern or the diagrams it can be an example for imitation for example you want to learn something you want to see the how the real object or the real phenomena is happening so you will try to replicate it through modeling and it can be a representation used to visualize something so as i just told you there is no well-defined definition but all of these are different ways of understanding the word modern so before moving to the specific mathematical model i would like to give you some few examples of these models in general so the first example we are going to look at is physical model which is also known as prototype so it's clear from the name that physical means something which you can see with your eyes something which you can touch and feel and something which is nothing but a constructed copy of the modeled object for example it can be a model airplane it can be model of a solar system it can be a humanoid robot it can be model of a building or a bridge so these examples are very easy to find and this is having most of the times applications in thermodynamics fluid mechanics aeronautics and architecture obviously this is not the exhaustive list there the list is much bigger than this but this is just to give you an idea of what is the meaning of physical model next example we are going to look at is schematic model and i hope this is also clear from the name of it schematic means the word schematic we use when we want to picturize something we want to represent something through a schematic diagram or a graph so this provide means of visualizing system structure so there can be many examples of schematic type of models it can be organizational charts the process charts graphs geographical maps and the diagrams and it can also be the global positioning system so the simplest example of a schematic diagram or a schematic model is a geographical map so you want to go from one place to another and you are not familiar with the route to be followed so you'll take the help of a geographical map traditionally or you will take the help of a gps system so the gps is not going to actually show you the road but if you look at this image over here on the right hand side so gps is showing you the schematic diagram these are not the real roads but the roads represented on the form of a picture in the form of a diagram or anything like that similarly you want to understand the structure of an organization that who is the top most uh what is the top most position in a company and then what is the hierarchy of the company so there are organization charts drawn so the examples are plenty you can think of your own examples for schematic type of models and now specifically we are moving towards our actual model under discussion which is a mathematical model so so far we have learnt that whatever type of model you want to learn or understand it needs a certain language for example the schematic model needed the language of drawing it needed the language of visualization similarly a mathematical model is a representation of the behavior of real objects and phenomena in mathematical language so the important word here is mathematical language so we need to now think about this word that what do we specifically mean by mathematical language so whatever comes to your mind when you hear the word mathematics is actually mathematical language it is plus minus multiply division all those algebraic operations elementary operations square root they are all the symbolic representation of mathematics and here we want to elaborate to a bigger scale so you can imagine you have algebraic equations so an algebraic equation looks like say you have this equation to solve means you want to find the value of x so that means you are interested in solving this algebraic equation the another example can be a differential equation so all of us know how a differential equation looks like so this is a very simple example of a differential equation similarly they can be integral equations the algorithms or flow charts they are also coming in the category of mathematical models the formulas so any formula like you want to compute the roots of a quadratic equation so we know the formula is this so this formula is also a certain kind of mathematical model and the theorems and the lemma so whatever mathematical language you talk about all of these are one or other form of models so i hope now we are clear with what is a mathematical model and now we would like to move to our next question we want to understand why we do mathematical modeling what is the need of it the world is running itself why we want to understand the things from a mathematical perspective so the answer is we know that all of us have been performing mathematical modeling knowingly or unknowingly since our school days so whatever school level mathematics you have studied whatever equations and formulas you have studied ultimately they were nothing but all mathematical models many laws of nature are beautifully captured and explained with the help of mathematics and even if you leave aside the aesthetic sense of mathematics you will understand that there are many advantages of performing mathematical modelling for example there is no field of science and technology today which is untouched with mathematics and moreover the best way to quantify the abstract behavior of some real phenomena these words we will understand in detail when we'll go throughout the course so the quantification of some abstract behaviors abstract means something which otherwise cannot be measured or cannot be quantified but mathematical models make that possible so there are many many advantages of performing mathematical modeling and we will look at all of them so that is enough for the motivation that why we want to do mathematical modeling so now it's clear to us that mathematics is an indispensable part of the real world again i would suggest you take a pause here and try to think of at least one example where do you think that mathematics is not used at all or it is not connected with mathematics at all and i know that you are not able to find any of these such example like that so to name a few applications the technical ecological economic and other systems investigated by modern science cannot be studied adequately using regular theoretical methods so we need to have some mathematical framework for doing that direct experimentation this is a very important point to understand that direct experimentation is time consuming expensive often even dangerous or simple impossible because there are two ways of actually actually there are many ways but broadly there are two ways of understanding any real phenomena either we want to study it theoretically which i have just explained that it's difficult to construct the theoretical framework for each and every kind of real world phenomena and the other category is we want to understand it through the experimentation so here i'm telling you the limitations of performing experiments the real experiments are time consuming and expensive and in some cases it's actually impossible to perform the experiments so you can think of a real situation where it is dangerous to perform direct experiment without prior texting so even if you want to conduct experiment at a later stage but still it is much more beneficial to do the calculations or the computations mathematically prior to that so you think of launching a satellite without doing any mathematical calculations and just guessing it that this should be the velocity this should be the trajectory to be followed and so you better know the consequences i need not say that so that's just a very small example it has much more higher level of applications so that's all about answer to the question why we are going to do the mathematical modeling and now let's talk about applications so again the list is not exhaustive i cannot really list all the applications here but i have tried to cover the important ones epidemiology epidemiology is nothing but the study of epidemics in the current times if we are definitely not aware with the word epidemics then definitely it's not worth it all of us are already clear with the meaning of the word epidemic so in the current situation of kovi 19 outbreak you know how much mathematics has been helping to understand the dynamics and predicting the curves and behavior of the virus spread another application can be biological transport and vehicular traffic so here you can see a simulation of traffic uh vehicular traffic running and which you can see because to uh in later stages we will learn how to create the simulation models as well so they are also coming under the category of mathematical models and then in business you want to optimally strategies you want to decide your strategies optimally for example you have studied the course of linear programming so they are in most of the business problems you are supposed to find an optimal solution so that is also a linear programming formulation is also a kind of a mathematical model and then applications are in economics financial industry will be talking about this in our course as well engineering need not to tell you and software development so these are very few applications of mathematical modeling again i'll tell you you talk about an area and you'll definitely find an application of mathematics and the mathematical modeling there only so now let's phrase what are the objectives of mathematical modeling what are the precise objectives of mathematical modeling so the main objectives are to analyze we want to analyze something why this is happening how this is happening if this is happening then what are its effects what are its causes so that's the basic meaning of analyzing something make predictions and then based upon our analysis we try to predict just now i told you the example of this kobe 19 outbreak looking at the trends we want to make prediction or you can have another example of stock market then looking at the trends that where a particular stock is rising or falling we want to make prediction for its future value provide insight of the real world phenomena so these are the main objectives of mathematical modeling so not every model is developed to fulfill all of the objectives some models are purely constructed for the purpose of analysis some models are purely constructed for making productions a weather forecasting model can be an example here and some models are purely for this and some models are there which cover more than one objective so that will be eventually clear when we'll be studying different different examples of mathematical models and last but not the least we also want to find optimal solutions to the real world challenges this also have just explained to you so now let me come to give you the definition of modeling cycle and you will understand after this why it is called a cycle so this is a cycle because it starts from one place and ends at same place so basically it starts from here the real world we are in the real world we got to understand a phenomena so we actually move to the conceptual world so here is the cognitive activity taking place the real process we want to understand we start to imagine that it in our mind how it might be happening why it might be happening or whatever we are thinking in our mind so basically initial work of construction of model starts happening in our mind and imagination that's why it's also said that modeling is the cognitive activity so in the conceptual world we go so what is the difference between the real world and conceptual world so in conceptual world there are some things which are assumed or there are some limitations or you can say it's not a hundred percent exact representation of the real world so after conceptual world we actually construct the model physically we look at the model in front of our self and then perform the analysis of it after performing the analysis we predict or interpret the results and then these predictions we give back to the real world so here also again to understand this example you can have the example of this covin-19 outbreak that firstly we observe okay the outbreak is happening the virus is spreading its infection then we try to think in our mind okay how this might be happening this is coming from china this is coming from there and that this whatever might be the reason for it and then we try to understand it mathematically so we construct a model for it we will study in our course some epidemic models and there will be talking about these models in detail so after construction of model we get some answers okay this is going to rise this is going to decrease this is going to happen so those results are mathematical and then to make the predictions understandable to the layman or to the real world we interpret the result back in the language of real world so the mathematics again gets translated to the language of the real world so that completes the cycle of the process which is called math modeling so that's the meaning of this modeling cycle and that's why it's called a cycle because we started the real world and we end at the real world if we do not complete the cycle then the process is considered incomplete and now we'll move towards the understanding the principles of mathematical modeling although you will see that each model is a different kind is it is distinguishable from the other one it is having a different characteristics but still there are some commonalities in all the mathematical models and which are known as the principles so you can understand these are the fundamental rules which we have to follow in the process of mathematical modeling so basically there are four major steps the first is we have to identify the object or the system which we want to model so here basically what we are going to look at is we are going to redefine our purpose why we want to understand like i explained you that there are many objectives of mathematical modeling we should be clear with what particular objective i am performing the mathematical modeling why you are constructing this model you want to analyze it you want to use it for prediction or you want to do something else with it so that clarity is very important here that why we are doing so and then what we are trying to achieve that's basically the same thing what i've just explained you so that means here after identifying the real life object we want to define our purpose so once we have thought that so then we will follow the process of mathematical modeling so briefly i have just explained you the process of mathematical modeling through the modding cycle but we'll be talking about this process in more detail in our next lecture so here right now you just assume that some process is happening some number of steps are happening here to go towards the next step so after doing this process of modeling complete we get to actually look at the models in which we have variables and the parameters and also we have certain assumptions with us most of the times in 99 of the mathematical models we always have assumptions because we want to understand the complex behavior of real world but in the very beginning step we directly cannot make a very complicated model of the real world phenomena we have to make a simplistic version of that and for that i have to lay down certain assumptions so that's the first step we have to define some assumptions for the variables and parameters we want to define that which parameters effect we want to see again take the example of this virus spread so you want to understand how the virus is spreading so it will depend upon what kind of study you want to conduct you want to study the effect of temperature or you want to study the effect of gender bias you want to study the effect of age groups so whatever parameter effect you want to study on that particular model that you will be redefining here so the model is constructed up to this step and then what we are going to do is we will solve this model because this is that stage where you already have some maths into the picture you have some equation you have some expression in front of you so depending upon what it is differential equation integral equation algebraic equation system of equations so it can be anything we will solve it so the question arises how to solve it so there are two ways either we can solve it analytically or we can solve it numerically again we'll be talking about the solution techniques depending upon what type of model we are studying so i'll be just not going to detail at this stage right now so whatever way you solve it you have the solution at hand so once you get the solution then we are going to get the predictions that what information this model is giving us so we will interpret our mathematical results in the language of real world will give some outcome that okay our model is telling that the virus is going to spread its effect with the this much rate or that much rate or whatever is a mathematical outcome so we will interpret our results and then after giving the results we cannot just stop there we have to see whether the result is coming out to be valid or not so there are two possibilities here so firstly we need to perform the step of validation so the process of validation is further depending upon how you are doing the validation or what particular model is with you some parts of sometimes the validation is done through experimentation that okay you have the experimentation data you have the experimental data you verify the outcomes of your model with the experimental data so if they matches so that means your model is working right and sometimes the validation is done intuitively intuitively means you need to understand the laws of physics of the model there that whether the physics of the model is right in the real world whatever should happen is happening in the model also or not for example say in some population model the final answer of population is turning out to be negative so obviously a negative population is not a well defined quantity it's mathematically possible but in real life there is no sense of negative population so that means something was wrong with your model so after the validation step there will be two possibilities if it is valid then we accept the model but if it is not valid then what is going to happen so that means we know that we have to repeat the steps we have to go back to either this step where we defined the assumptions and the parameters or in some cases we may also have to go back to the first step we have to redefine the purpose of modeling so some loophole we have to identify and complete the process so that completely covers all the principles of mathematical modelling so in the next lecture we are going to look at the process of mathematical modelling in detail thank you

Channel: Dr. Maths

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Keywords: Math modeling, modeling, mathematical modelling, simulation, umc511, introduction of mathematical models, basics of models, principles of modeling, basics of mathematical modeling, aaplications of mathematics, applied mathematics, differential equations, uses of mathematics in other fields, the modeling cycle

Id: df5EK1P6Ph0

Length: 25min 44sec (1544 seconds)

Published: Sun Jul 19 2020