Mathematical modelling

MATHEMATICAMATHEMATICALL

MODELLINGMODELLING

Mathematical modeling is an attempt to study some Mathematical modeling is an attempt to study some part of some real life problems in mathematical part of some real life problems in mathematical terms. ie; the conversion of a physical situation into terms. ie; the conversion of a physical situation into mathematics using suitable conditionsmathematics using suitable conditions

Mathematical modeling is an activity in which we Mathematical modeling is an activity in which we make model to describe the behavior of various make model to describe the behavior of various phenomenal activities of our interest in many ways phenomenal activities of our interest in many ways using words, drawings or sketches, computer using words, drawings or sketches, computer programs, mathematical formulae etc…programs, mathematical formulae etc…

Students are aware of the solution of word problems in Students are aware of the solution of word problems in arithmetic, algebra, trigonometry and linear arithmetic, algebra, trigonometry and linear programming etc…Sometimes we solve the problems programming etc…Sometimes we solve the problems without going into the physical insight of the without going into the physical insight of the situational problems. situational problems.

Situational problems need physical insight that is Situational problems need physical insight that is ‘introduction’ of physical laws and some symbols to ‘introduction’ of physical laws and some symbols to compare the mathematical results obtained with compare the mathematical results obtained with practical values. To solve many problems faced by us, practical values. To solve many problems faced by us, we need a technique and this is what is known as we need a technique and this is what is known as ‘Mathematical Modeling’.‘Mathematical Modeling’.

Let us consider the following problems :Let us consider the following problems :

• To find the width of a riverTo find the width of a river• To find the optimal angle in case of shot putTo find the optimal angle in case of shot put• To find the height of a towerTo find the height of a tower• To find the temperature at the surface of the sunTo find the temperature at the surface of the sun• Why heart patients are not allowed to use the lift?Why heart patients are not allowed to use the lift?• To find the mass of the earthTo find the mass of the earth• Estimate the yield of pulses in India from the standing cropsEstimate the yield of pulses in India from the standing crops• To find the volume of blood inside the body of a personTo find the volume of blood inside the body of a person• Estimate the population of India in 2020Estimate the population of India in 2020 All of these problems can be solved and infact have All of these problems can be solved and infact have

been solved with the help of Mathematics using Mathematical been solved with the help of Mathematics using Mathematical Modeling.Modeling.

PRINCIPLES OF MATHEMATICS MODELLINGPRINCIPLES OF MATHEMATICS MODELLING

• Identify the need for the modelIdentify the need for the model• List the parameters / variables which are required for List the parameters / variables which are required for

the modelthe model• Identify the available relevant dataIdentify the available relevant data• Identify the circumstances that can be applied Identify the circumstances that can be applied • Identify the governing physical principles Identify Identify the governing physical principles Identify (a) the equation that will be used(a) the equation that will be used (b) the calculations that will be made (b) the calculations that will be made (c) the solution which will follow(c) the solution which will follow• Identify tests that can check the Identify tests that can check the (a) consistency of the model(a) consistency of the model (b) utility of the model(b) utility of the model• Identify the parameter values that can improve the Identify the parameter values that can improve the

modelmodel

STEPS FOR MATHEMATICAL MODELLINGSTEPS FOR MATHEMATICAL MODELLING

Step 1: Identify the physical situationStep 1: Identify the physical situation

Step 2: Convert the physical situation into a Step 2: Convert the physical situation into a mathematical model by introducing parameters / mathematical model by introducing parameters / variables and using various known physical laws and variables and using various known physical laws and symbolssymbols

Step 3: Find the solution of the mathematical problemStep 3: Find the solution of the mathematical problem

Step 4: Interpret the result in terms of the original Step 4: Interpret the result in terms of the original and compare the result with observations or and compare the result with observations or experimentsexperiments

Step 5: If the result is in good agreement, then accept Step 5: If the result is in good agreement, then accept the model. Otherwise modify the hypothesis / the model. Otherwise modify the hypothesis / assumptions according to the physical situation and assumptions according to the physical situation and go to step 2go to step 2

FLOW CHART OF MAHTEMATICAL FLOW CHART OF MAHTEMATICAL MODELLINGMODELLING

FLOW CHART OF MAHTEMATICAL FLOW CHART OF MAHTEMATICAL MODELLINGMODELLING

Physicalsituation

Mathematical modeling

Mathematicalsolution

Solution of Original problem

Accept themodel

Modifyhypothesis

Introduce

Physical Laws&symbols

Solve Interpret

In good agreement

Not in good agreement

Compare withObservation

ExamplesExamples

(i(i)) Find the height of a given tower using mathematical modeling Find the height of a given tower using mathematical modeling

Step1 : Given physical situation is “to find the height of a given Step1 : Given physical situation is “to find the height of a given tower”tower”

Step2 : Let AB be the given tower. Let PQ be an observer Step2 : Let AB be the given tower. Let PQ be an observer measuring the height of the tower with his eye at P. Let PQ=h measuring the height of the tower with his eye at P. Let PQ=h and let height of tower be H. Let x be the angle of elevation and let height of tower be H. Let x be the angle of elevation from the eye of the observer to the top of the tower.from the eye of the observer to the top of the tower.

Let l=PC=QBLet l=PC=QB

Now tan x = AC / PCNow tan x = AC / PC

= H – h / l= H – h / l

H = h + l tan x ………… (1)H = h + l tan x ………… (1)

Step3 : Note that the values of the parameters h, l, and x (using Step3 : Note that the values of the parameters h, l, and x (using secant) are known to be the observer and so (1) gives the secant) are known to be the observer and so (1) gives the solution of the problem.solution of the problem.

Step4 : In case, if the foot of the tower is not accessible. (i.e) Step4 : In case, if the foot of the tower is not accessible. (i.e) when l is not known to the observer, let y be the angle of when l is not known to the observer, let y be the angle of depression from P to the foot B of the tower. So from triangle depression from P to the foot B of the tower. So from triangle PQB, we have, PQB, we have,

tan y = PQ / QBtan y = PQ / QB

= h / l= h / l

or l = h cot yor l = h cot y

Step5 : is not required in this situation as exact values of the Step5 : is not required in this situation as exact values of the parameters h, l, x, y are known.parameters h, l, x, y are known.

(ii) Let a tank contains 1000 liters of brine which contains 250g (ii) Let a tank contains 1000 liters of brine which contains 250g of salt per liter. Brine containing 200g of salt per liter flows of salt per liter. Brine containing 200g of salt per liter flows into the tank at the rate of 25 liters per minute and the mixture into the tank at the rate of 25 liters per minute and the mixture flows out at the same rate. Assume that the mixture is kept flows out at the same rate. Assume that the mixture is kept uniform all the time by stirring what would be the amount of uniform all the time by stirring what would be the amount of salt in the tank at any time t ?salt in the tank at any time t ?

Step1 : The situation is easily identifiable.Step1 : The situation is easily identifiable.Step2 : Let y = y (t) denote the amount of salt (in kg) in the tank Step2 : Let y = y (t) denote the amount of salt (in kg) in the tank

at time t (in minutes) after the inflow, outflow starts. Further at time t (in minutes) after the inflow, outflow starts. Further assume that y is a differentiable function. assume that y is a differentiable function.

when t = 0, ie; before the inflow- outflow of the brine starts,when t = 0, ie; before the inflow- outflow of the brine starts, y = 250g * 1000 = 250kgy = 250g * 1000 = 250kg

Note that the change in y occurs due to the inflow-outflow of the Note that the change in y occurs due to the inflow-outflow of the mixture.mixture.

Now the inflow of the brine brings salt into the tank at the rate of Now the inflow of the brine brings salt into the tank at the rate of 5kg per minute (as 25 * 200g = 5kg) and the outflow of brine 5kg per minute (as 25 * 200g = 5kg) and the outflow of brine takes salt out of the tank at the rate of 25(y /1000) = y/40 kg takes salt out of the tank at the rate of 25(y /1000) = y/40 kg per minute (as at time t, the salt in the tank is y/1000 kg)per minute (as at time t, the salt in the tank is y/1000 kg)

Thus the rate of change of salt with respect to t is given byThus the rate of change of salt with respect to t is given by dy/dt = 5 – y/40dy/dt = 5 – y/40 or or dy/dt + y/40 =5 ………(1)dy/dt + y/40 =5 ………(1)This gives a mathematical model for the given problemThis gives a mathematical model for the given problem

Step3 : Equation 1 is a linear equation and can be easily solved. Step3 : Equation 1 is a linear equation and can be easily solved. The solution of (1) is given by The solution of (1) is given by

y e^ (t/40) = 200e^(t/40) + cy e^ (t/40) = 200e^(t/40) + c

or y(t) = 200 + ce^(-t/40) ……..(2)or y(t) = 200 + ce^(-t/40) ……..(2)

Where c is the constant of integrationWhere c is the constant of integration

Note that when t = 0, y = 250.Note that when t = 0, y = 250.

Therefore; 250 = 200 + c or c = 50Therefore; 250 = 200 + c or c = 50

then (2) reduces to then (2) reduces to

y = 200 + 50 e^(-t/40) ………..(3) y = 200 + 50 e^(-t/40) ………..(3)

Or (y – 200) / 50 = e^(-t/40)Or (y – 200) / 50 = e^(-t/40)

Or e^t/40 = 500/(y-200)Or e^t/40 = 500/(y-200)

Therefore; t = 40 logTherefore; t = 40 logee 50/(y – 200) …………….(4) 50/(y – 200) …………….(4)

Step4 : Since e^(-t/40) is always positive, Step4 : Since e^(-t/40) is always positive, from (3), we conclude that y > 200 at all from (3), we conclude that y > 200 at all times.times.

Thus, the minimum amount of salt content in Thus, the minimum amount of salt content in the tank is 200kg.the tank is 200kg.

Also from (4), we conclude that t >0 if and Also from (4), we conclude that t >0 if and only if 0<y-200<50.only if 0<y-200<50.

Ie; if and only if 200<y<250Ie; if and only if 200<y<250Ie; the amount of salt content in the tank Ie; the amount of salt content in the tank

after the start of inflow and outflow of the after the start of inflow and outflow of the brine is between 200kg and 250kg.brine is between 200kg and 250kg.

LIMITATIONS OF LIMITATIONS OF MATHEMATICAL MATHEMATICAL

MODELLINGMODELLINGTill today many mathematical modeling Till today many mathematical modeling

have been developed and applied have been developed and applied successfully to understand and get an successfully to understand and get an insight into thousands of situations. insight into thousands of situations. Some of the subjects like mathematical Some of the subjects like mathematical physics, mathematical economics, physics, mathematical economics, operation research, bio-mathematics operation research, bio-mathematics etc…are almost synonyms with etc…are almost synonyms with mathematical modeling. mathematical modeling.

But there are still a large number of situations But there are still a large number of situations which are yet to be modeled. The reason which are yet to be modeled. The reason behind this is that either the situation are behind this is that either the situation are found to be very complex or the mathematical found to be very complex or the mathematical models formed are mathematically intractable.models formed are mathematically intractable.

The development of the powerful computers and The development of the powerful computers and super computers has enabled us to super computers has enabled us to mathematically model a large number of mathematically model a large number of situations (even complex situation). Due to situations (even complex situation). Due to these fast and advanced computers, it has these fast and advanced computers, it has been possible to prepare more realistic models been possible to prepare more realistic models which can obtain better agreements with which can obtain better agreements with observations.observations.

However, we do not have good guidelines However, we do not have good guidelines for choosing various parameters / for choosing various parameters / variables and also for estimating the variables and also for estimating the values of these parameters / variables values of these parameters / variables used in a mathematical model. Infact, we used in a mathematical model. Infact, we can prepare reasonably accurate models can prepare reasonably accurate models to fit any data by choosing five or six to fit any data by choosing five or six parameters / variables. We require a parameters / variables. We require a minimal number of parameters / minimal number of parameters / variables to be able to estimate them variables to be able to estimate them accurately.accurately.

Mathematical modeling of large and Mathematical modeling of large and complex situations has its own special complex situations has its own special problems. These type of situations problems. These type of situations usually occur in the study of world usually occur in the study of world models of environment, oceanography, models of environment, oceanography, pollution control etc…Mathematical pollution control etc…Mathematical modelers from all disciplines-modelers from all disciplines-mathematics, computer science, mathematics, computer science, physics, engineering, social sciences physics, engineering, social sciences etc…are involved in meeting these etc…are involved in meeting these challenges with courage.challenges with courage.

Nandini.N.LNandini.N.L

Mathematics optionalMathematics optional

Reg.no:13971010Reg.no:13971010

KUCTE KumarapuramKUCTE Kumarapuram

Thiruvananthapuram Thiruvananthapuram

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