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.
Fig. 1 ‐ Schematic diagram of modelling process
Real World
Problem
Real World Model
Mathematical
Model
Prediction and
Validation
Mathematical
Formulation
Simplifications
Analysis & Interpretations
Test
7. constant velocity 32.2 10 /0.329. Assuming the maximum value of
the diameter D = 0.00025 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.