Mathematical modeling

Dr. Deepak Kumar and Dr. Pooja Khurana
Department of Mathematics,
Faculty of Engineering and Technology,
Manav Rachna International Institute of Research
and Studies.
Faridabad (HR),India
Email: deepakman12@gmail.com
Mathematical Modeling

 Fundamental and quantitative way to recognize and analyze
complicated structures/systems.
 Addition to Idea and Analysis.
 Useful for situations where direct association with the real
problem is not easily achievable.
 Can be used to solve problem in sectors like financial,
transportation, energy, environment etc by making certain
assumptions and approximations.
Real World Problem
assumptions
& approximations
Mathematical Model

Now consider the following problems:
 Find the width of water
 Predict a disease outbreak/spread
 Height of a tower
 Find the mass of earth
 Predict the best possible highway route
All these above problems can be solved by doing
mathematical modeling.

Mathematical Modeling is the process that uses
mathematics to represent, analyze, construct, testing,
and developing equations, make predictions, or
otherwise provide insight into real-world phenomena. It
is a description of a system using mathematical
language. In short the process of developing a
mathematical model of any problem is termed as
mathematical modeling. Mathematical models are
used not only in the natural sciences and engineering
applications but also in the social sciences (such as
economics, psychology, sociology and political science).
Mathematical models are most extensively used by
physicists, engineers, statisticians, operations research
analysts and economists.

 Analyze the need for the model.
 List the parameters / variables which are required for the
model.
 Analyze the available relevant data.
 Analyze the governing physical principles.
❖The equation that will be used.
❖The calculations that will be made.
❖The stability and consistency of solutions.
 Analyze the parameters that can improve the model.
 Implement the model to real life problems.

Modeling World
Mathematical Form of
the Problem
Realistic World
Interpret the solution
and Validate
Formulate Model
from Text
Model
Results
Numerical/Statistical
Analysis
Solutions

1. Linear vs. Nonlinear Model:
Mathematical models are typically composed by variables,
which are reflections of quantities of interest in the
described systems, and operators that act on these
variables, which can be mathematical operators,
functions, differential operators, etc. If all the operators in
a mathematical model exhibit linearity, the resulting
mathematical model is characterized as linear.
In short: A Model is linear if
❖ Make a straight line when graphed.
❖ Have a constant rate of change.
❖ Equation can be written as ax + by=c or
y = mx + c.

Linear vs. Nonlinear: A model is which is not linear is
considered to be nonlinear otherwise. The subject of
linearity and nonlinearity is dependent on context, and
linear models may have nonlinear expressions in them.
In short: A Model is linear if
❖ Do not make a straight line when graphed.
❖ Do not have a constant rate of change.
❖ Cannot be written in the correct form.

One Sunday morning there were 14 inches of snow on
the ground. The weather warmed up. And by Tuesday
morning 2 inches had melted 2 more inches melted by
Wednesday morning. This pattern continues whole week
until no more snow was left. Write mathematical model
for the same.
Let x= days after Sunday
y= inches of snow on the ground.
Therefore; y=14-2x is the linear model for the above
problem. but when the snow is not melting at a constant
rate then it would be a non linear model.

Deterministic models are constantly offers certain output
for a given set of fixed input variables. Hence the output
continually falls with in a given specified range.
Stochastic models might also not usually offers the
identical output for a given set of enter variables,
because it incorporates some randomness. A
deterministic model means settled model. Deterministic
model are often described by differential equations.

Stochastic models are random analysis because
randomness is present, and Multiple runs are used
to estimate probability distribution.
The benefit of stochastic models are they can
predict the patterns comparable to practical
patterns. considering the fact that most of the real
structures frequently surprises us through special
outcome, this may also be due we do not
understand them completely. so the use of
deterministic strategy to find out about the real or
complicated structures are no longer really useful
for most of the time.

•Change in concept from "deterministic" (i.e., single
solution) to "stochastic" (i.e., multiple solutions related to
probability distributions).

All in all, a deterministic model produces particular
outcomes given certain inputs by the model user,
contrasting with a stochastic model which encapsulates
randomness and probabilistic events.
A computer cannot generate genuinely random numbers
because computers can only run algorithms, which are
deterministic in nature.

Deterministic(same output)
f(x) = b x (1 – x), b = 4, x0 =
0.5 , let us find first 10 iterates:
Stochastic(may or may not be
same output)
f(x) = bx (1 – x) + k c, b = 4, x0 =
0.5, k = 0.02, c is a Random
Variable
First Attempt Second attempt First Attempt Second attempt
0.84 0.84 0.86 0.86
0.5376 0.5376 0.4816 0.4816
0.994345 0.994345 0.998646 0.998646
0.0224922 0.0224922 0.0254096 0.0254096
0.0879454 0.0879454 0.119056 0.0990559
0.320844 0.320844 0.419526 0.356975
0.871612 0.871612 0.974096 0.938176
0.447617 0.447617 0.120932 0.232008
0.989024 0.989024 0.44523 0.732721
0.0434219 0.0434219 0.988001 0.783363

Static Model
 Static means fixed.
 Output is determined only
by the current input, reacts
instantaneously.
 Relationship does not
change.
 Relationship is represented
by an algebraic equation.
Dynamic Model
 Dynamic means change.
 Output takes time to react.
 Relationship changes with
time, depend on past
inputs and initial conditions
 Relationship is represented
by an Differential equation.
 We require future input or
past input.

A dynamical model is used to express the behavior of the
system over time whereas static model does not.
Dynamic models typically are represented with difference
equations or differential equations.

Now we have y as output and x as input as we want to
find the value of y at x=1 means the value of y at 1 is
depending upon future x(2) and past inputs x(0)
therefore it is a dynamical model. In second example if
we want to find y at x=1 we want current input therefore
it is a static model.
( ) ( )
.........(2)x n
y n e=
( ) ( ) ( )
( ) ( ) ( )
2 1 ...........(1)
1; 1 2 0
y n x n x n
for n y x x
= − −
= = −

 Discrete models do not take into account the function of
time and usually uses time-advance methods”. A
discrete model is that if the values belonging to the set
are distinct and separate. Age, Height, Shoe size change
in pocket number of books in a bag pack.
 Continuous models typically are represented with f(t)
and the changes are reflected over continuous time
intervals. A continuous model is that if the values
belonging to the given set of data can take on any value
within a finite or infinite interval. True height, true weight,
time, speed, temperature, volume etc.

 Number of rabbits in a field. (D)
 Size of cars gas tank. (C)
 Number of text messages sent today. (D)
 Number of doughnuts eaten this week. (D)
 The level of lead in water. (C)
 Number of goals scored in a soccer match. (D)
 Length of a leaf. (C)
 No of languages spoken. (D)
 No of books on a shelf. (D)
 No of people in a family. (D)

➢ Quantitative models lead to a detailed, numerical
predication about responses, it gives us information that
can be counted or expressed numerically.
❖ Can be used in calculations and statistical tests.
❖ can be represented in tabular and graphically.
❖ Can be either discrete or continuous form.
➢ Qualitative models lead to general descriptions about
the responses.
❖ It is descriptive data provides depth of understanding.
❖ Adds feel or texture to quantities findings.

Quantitative
 Level of occurrence.
 Asks how many or
how much?
 Studies events
 Objective
 Discovery and proof
 More definitive
 Describes
Qualitative
 Depth of understanding
 Asks why?
 Studies motivation
 Subjective
 Enables discovery
 Exploratory in nature
 Interprets

Lets consider an example of a cat if we say it has 4
legs and 10 pounds weight means we are talking
about quantitative data and qualitative would be that
the cat is yellow in color and its fur is soft. Another
example is of book shelf lets say we have 50 books
in the shelf and it is 50 cm tall means we are talking
about quantitative data and qualitative would be that
the books are of multi colors and has a smooth
surface.

➢ Today modeling is being used for studying a varying
array of problems. Some of these are listed below.
❖ Seismology
❖ Drug Design
❖ Climate Modeling
❖ Biology
❖ Environment
❖ Material Research
❖ Medicine
❖ Transport and Communication

Mathematical modeling

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