Mathematical Modelling Outline

1. Mathematical Modelling
Presented By : Sanjeev Kumar Prajapati
sanjeevkumarprajapati2000@gmail.com

2. Content
Introduction
Why Mathematical Modelling
Principles of Mathematical Modelling
Advanced Mathematical Modelling
Significance in the natural sciences
Conclusion

3. What distinguishes a mathematical model from , says a
poem ,a song , a portrait or any other kind of model , is
that the mathematical model is an image or picture of
reality painted with logicals symbols instead of with
words , sounds or watercolors.
-John L.Casti

4. Introduction
Models describe our beliefs about how the world functions. In
mathematical modelling , we translate those beliefs into the language
of mathematics. This has many advantages:
1. Mathematics is a very precise language. This helps us to formulate
ideas and identify underlying assumptions.
2. All the results that mathematicians have proved over hundreds of
years are at our disposal.
3. Computers can be used to perform numerical calculations.

5. Mathematical modelling is the study of some part (or form) of some real-
life problems in mathematical terms , i.e. the conversion of a physical
situation into mathematics using some suitable conditions. Mathematical
modelling is the process of using various mathematical structures such as
graphs, equations ,diagrams, scatterplots, tree diagram and so forth-to
represent real worlds situations.
Mathematical modelling is useful for a variety of reasons . Foremost,
models represent the mathematical core of situations without extraneour
informations .The equation a=b+6, for instance , removes all of the
unnecessary words from the original statement “Ram is 6 years older
than shyam”.

6. Why Mathematical Modelling
Sometimes we solve the problems without going in to
the physical insight of the situational problems.
Situational problems need physical insight that is
introduction of physical laws and some symbols to
compare the mathematical results obtained with
practical values. To solve many problems faced by us ,
we need a technique and this is what is known as
mathematical modelling . Let us consider the following
problems :

7. 1. To find the width of a river ( particularly , when it is difficult to
cross the river).
2. To find the height of a tower ( particularly , when it is not
possible to reach the top of the tower).
3. To find the temperature at the surface of the sun.
4. To find the mass of the earth.
5. Estimate the population of India in 2022 (a person is not
allowed to wait till then).
6. To find the optimal angle in case of shot-put (by considering
the variables such as : the height of a thrower , resistance of
media , acceleration due to gravity etc.).

8. 7 To find the volume of blood inside the body of a
person (a person is not allowed to bleed
completely).
8. Estimate the yield of pulses in India from the crops
( a person is not allowed to cut it all).
9. Why heart patients are not allow to use lifts
(without knowing the physiology of a human
being).

9. Principles of Mathematical Modelling
Mathematical modelling is a principled activity and so it
has some principles behind it. These principles are
almost philosophical in nature. Some of the basic
principles of mathematical modelling are listed below
in terms of following instructions:
Identify the need for the model( for what we are
looking for )
List the parameters/variables which are required for
the model.

10. Identify the available relevant data(what is given)
Identify the circumstances that can be
applied.(assumptions)
Identify the governing physical principles.
Identify :
1. the equations that will be used.
2. the calculations that will be made.
3. the solutions which will follow.
Identify tests that can check the
1.consistency of the model
2. unity of the model

11. The above principles of mathematical
modelling lead to the following: steps for
mathematical modelling, which can be
viewed from the following diagram

13. Lets consider an example
Ex. Find the height of the given tower using
mathematical modelling.

14. Solution : Step1: Given physical situation is “to find the height of given tower”.
Step2: Let AB be the given tower (figure). Let PQ be an observer measuring the
height of the tower with his eyes at P. Let PQ=h and let height of tower be H. Let α
be the angle of elevation from the eye of the observer to the top of the tower.
PC=QB=l
tanα=AC/PC=(H-h)/l
l tanα=H-h
H=h+l tanα …………………………(1)
Note that the values of the parameters h,l and α ( using sextant) are known to
observer and so (1) gives the solution of the problem.
Step4: In case ,if the foot of the tower is not accessible ,i.e., when l is not known to
the observer , let β be the angle of deviation from P to the foot B of the tower. So
from PQB , we have,

15. tanβ=PQ/QB
tanβ=h/l
l=h/tanβ or l=hcotβ
Step 5 is not required in this situation as exact values of the parameters h , l
, α and β are known.

16. Advanced Mathematical Modelling
Mathematical modelling include the analysis of information using
statistical methods and probability ,modeling change and
mathematical relationships , mathematical decision making in finance
,and spatial and geometric modelling for decision-making. Students
will learn to become crucial consumers of the quantitative data that
surround them everyday, knowledgeable decision makers who use
logical reasoning and mathematical thinkers who can use their
quantitative skills to solve problems related to a wide range of
situations.
Mathematical modelling helps students to develop a mathematical
proficiency in a developmentally-appropriate progressions of
standards.

17. Significance in the natural sciences
Mathematical models are of great importance in the natural
sciences , particularly in Physics. Physical theories are almost
invariably expressed using mathematical models.
Throughout history , more and more accurate mathematical
models describe many everyday phenomena , but at certain
limits relativity theory and quantum mechanics must be used
;even these do not apply to all situations and need further
refinement .

18. The laws of physics are represented with simple
equations such as Newton’s laws , Maxwell’s equations
and the Schrodinger equation. These laws are such as a
basis for making mathematical models of real situations
. Many real situations are very complex and thus
modeled approximate on a computer , a model that is
computationally feasible to compute is made from the
basic laws.

19. Conclusion
Till today many mathematical models have been
developed and applied successfully to understand and
get an insight into thousands of situations. Some of the
subjects like mathematical physics , mathematical
economics , operations research , bio-mathematics
etc. are almost synonyms with mathematical
modelling.

20. Finally it can be concluded that mathematical
modelling is a good concept which helps us in various
realistic life experiment and phenomenon to convert in
to mathematical models in the form of equations ,
graphs , histogram , scatterplots , tree diagram , etc.
and then solve them.

21. It saves time , money and efforts of performing results
in laboratory with experimental tools. It has
widespread applications in every aspects of life as well
as in various branches such as engineering , medical ,
mathematical economics , medicine ,biology and
several other interdisciplinary areas. Thus it is an
advanced concept and have bright future in
upcoming time.

22. Thank you