Simple Game
Simple Game
Draw a house on a paper
2
Simple Game
90% of the people have drawn a house like this.
3
Question
How many of your Houses are like this?
Only 10% people raised their hands
4
Rapid Fire Round Quiz
5
Rapid Fire Round Quiz
What does it represent?
6
Rapid Fire Round Quiz
What does it represent?
A = lb
6
Rapid Fire Round Quiz
What does it represent?
A = lb
100% Area of a Rectangle
6
Rapid Fire Round Quiz
What does it represent?
7
Rapid Fire Round Quiz
What does it represent?
W = mg
7
Rapid Fire Round Quiz
What does it represent?
W = mg
90% Weight = Mass × Gravity
7
Rapid Fire Round Quiz
What does it represent?
8
Rapid Fire Round Quiz
What does it represent?
F = ma
8
Rapid Fire Round Quiz
What does it represent?
F = ma
90% Newton’s Second Law or Force=Mass × Acceleration
8
Rapid Fire Round Quiz
What does it represent?
9
Rapid Fire Round Quiz
What does it represent?
y = mx
9
Rapid Fire Round Quiz
What does it represent?
y = mx
100% Equation of a Straight line passing through the origin
9
Rapid Fire Round Quiz
What does it represent?
10
Rapid Fire Round Quiz
What does it represent?
z = xy
10
Rapid Fire Round Quiz
What does it represent?
z = xy
100% Complete Silence
10
Rapid Fire Round Quiz
What does it represent?
z = xy
100% Complete Silence
Hyperbolic Paraboloid
10
Rapid Fire Round Quiz
What does it represent?
11
Rapid Fire Round Quiz
What does it represent?
A = πr2
11
Rapid Fire Round Quiz
What does it represent?
A = πr2
100% Area of a Circle
11
Rapid Fire Round Quiz
What does it represent?
12
Rapid Fire Round Quiz
What does it represent?
E = mc2
12
Rapid Fire Round Quiz
What does it represent?
E = mc2
100% Einstein Equation/Theory of Relativity
12
Rapid Fire Round Quiz
What does it represent?
13
Rapid Fire Round Quiz
What does it represent?
y = ax2
13
Rapid Fire Round Quiz
What does it represent?
y = ax2
80% Equation of Parabola
13
Rapid Fire Round Quiz
What does it represent?
14
Rapid Fire Round Quiz
What does it represent?
z = xy 2
14
Rapid Fire Round Quiz
What does it represent?
z = xy 2
100% Complete Silence
14
Rapid Fire Round Quiz
What does it represent?
z = xy 2
100% Complete Silence
14
Rapid Fire Round Quiz
What does it represent?
15
Rapid Fire Round Quiz
What does it represent?
a2 + b2 = c2
15
Rapid Fire Round Quiz
What does it represent?
a2 + b2 = c2
100% Pythagoras Theorem
15
Rapid Fire Round Quiz
What does it represent?
a2 + b2 = c2
100% Pythagoras Theorem
15
Rapid Fire Round Quiz
What does it represent?
16
Rapid Fire Round Quiz
What does it represent?
x2 + y 2 = r 2
16
Rapid Fire Round Quiz
What does it represent?
x2 + y 2 = r 2
100% Equation of Circle
16
Rapid Fire Round Quiz
What does it represent?
17
Rapid Fire Round Quiz
What does it represent?
x2 + y 2 = z 2
17
Rapid Fire Round Quiz
What does it represent?
x2 + y 2 = z 2
80% Some 3D Equation
17
Rapid Fire Round Quiz
What does it represent?
x2 + y 2 = z 2
80% Some 3D Equation
17
Rapid Fire Round Quiz
What does it represent?
18
Rapid Fire Round Quiz
What does it represent?
3
x2 + 94 y 2 + z 2 − 1 − x2 z 3 −
9 2 3
y z
80
=0
18
Rapid Fire Round Quiz
What does it represent?
3
x2 + 94 y 2 + z 2 − 1 − x2 z 3 −
9 2 3
y z
80
=0
100% Pin Drop Silence
18
Rapid Fire Round Quiz
19
Rapid Fire Round Quiz
3
x2 + 94 y 2 + z 2 − 1 − x2 z 3 −
9 2 3
y z
80
=0
19
Rapid Fire Round Quiz
3
x2 + 94 y 2 + z 2 − 1 − x2 z 3 −
9 2 3
y z
80
=0
19
Rapid Fire Round Quiz
y = mx
A = lb
F = ma
W = mg
z = xy
20
Rapid Fire Round Quiz
y = ax2
E = mc2
A = πr2
z = xy 2
21
Rapid Fire Round Quiz
x2 + y 2 = r 2
a2 + b2 = c2
x2 + y 2 = z 2
22
Quote
Mathematics is the art of giving the
same name to different things
–Henri Poincare
f (x, y, z) = 0
y = mx
A = lb
F = ma
y = ax2
A = πr2
E = mc2
2
2
x2 + y 2 = r2
a
+
b
=
c2
3
2
2 3
2
9 2 3
9 2
x + 4 y + z − 1 − x z − 80 y z = 0
W = mg
z = xy 2
x2 + y 2 = z 2
z = xy
23
Engineering Mathematics-I
Lecture 1 : Mathematical Modeling
Panchatcharam Mariappan
Assistant Professor
Department of Mathematics and Statistics
IIT Tirupati, Tirupati
23
Modelling
24
Mathematical Model
Kai Velten, Mathematical Modeling and Simulation, Introduction for Scientists
and Engineers, Wiley-VCH Verlag GmbH & Co. KGaA, 2009
25
A Few Examples
• Car/Bike is not starting
26
A Few Examples
• Car/Bike is not starting
• Headache
26
Complexity
• How complex a car is!
27
Complexity
• How complex a car is!
• How complex our body/brain is!
27
Complexity
Remarks
Scientists and Engineers: Break up the
complexity of a system and use simplified descriptions of that system
28
Model
Definition 1 (Model)
To an observer B, an object A∗ is a model of an object A to the extent that B
can use A∗ to answer questions that interest him about A.
29
Model
Definition (Model)
To an observer B, an object A∗ is a
model of an object A to the extent that
B can use A∗ to answer questions that
interest him about A.
30
Model
Definition (Model)
To an observer B, an object A∗ is a
model of an object A to the extent that
B can use A∗ to answer questions that
interest him about A.
• B: Driver
30
Model
Definition (Model)
To an observer B, an object A∗ is a
model of an object A to the extent that
B can use A∗ to answer questions that
interest him about A.
• B: Driver
• A: Car
30
Model
Definition (Model)
To an observer B, an object A∗ is a
model of an object A to the extent that
B can use A∗ to answer questions that
interest him about A.
• B: Driver
• A: Car
• A∗ : Fuel Tank/Battery
30
Model
Definition (Model)
To an observer B, an object A∗ is a
model of an object A to the extent that
B can use A∗ to answer questions that
interest him about A.
• B: Driver
• A: Car
• A∗ : Fuel Tank/Battery
30
Best/Simplest Model
Definition 2 (Best Model)
The best model is the simplest model that still serves its purpose, that is, which
is still complex enough to help us understand a system and to solve problems.
31
Modelling, Simulation,
Validation
32
Teleological Nature
Aims and Purposes
Main Purpose of Modeling and Simulation: governed by its purpose of problem
solving.
33
Modeling and Simulation Nature
• Definition
34
Modeling and Simulation Nature
• Definition
◦
Problem Definition to be Solved
34
Modeling and Simulation Nature
• Definition
◦
◦
Problem Definition to be Solved
A question to be Answered
34
Modeling and Simulation Nature
• Definition
◦
◦
◦
Problem Definition to be Solved
A question to be Answered
System Definition for which Answer is Required
34
Modeling and Simulation Nature
• Definition
◦
◦
◦
Problem Definition to be Solved
A question to be Answered
System Definition for which Answer is Required
• System Analysis
34
Modeling and Simulation Nature
• Definition
◦
◦
◦
Problem Definition to be Solved
A question to be Answered
System Definition for which Answer is Required
• System Analysis
◦
Identification of parts of the system related to question
34
Modeling and Simulation Nature
• Definition
◦
◦
◦
Problem Definition to be Solved
A question to be Answered
System Definition for which Answer is Required
• System Analysis
◦
Identification of parts of the system related to question
• Modeling
34
Modeling and Simulation Nature
• Definition
◦
◦
◦
Problem Definition to be Solved
A question to be Answered
System Definition for which Answer is Required
• System Analysis
◦
Identification of parts of the system related to question
• Modeling
◦
Development of a model of the system based on the results of the systems
analysis step
34
Modeling and Simulation Nature
• Definition
◦
◦
◦
Problem Definition to be Solved
A question to be Answered
System Definition for which Answer is Required
• System Analysis
◦
Identification of parts of the system related to question
• Modeling
◦
Development of a model of the system based on the results of the systems
analysis step
Definition 3 (System)
A system is an object or a collection of objects whose properties we want to
study
34
Modeling and Simulation Nature
• Simulation
35
Modeling and Simulation Nature
• Simulation
◦
Application of the model to the problem
35
Modeling and Simulation Nature
• Simulation
◦
◦
Application of the model to the problem
Derivation of a strategy to solve the problem
35
Modeling and Simulation Nature
• Simulation
◦
◦
◦
Application of the model to the problem
Derivation of a strategy to solve the problem
Answer the question
35
Modeling and Simulation Nature
• Simulation
◦
◦
◦
Application of the model to the problem
Derivation of a strategy to solve the problem
Answer the question
• Validation
35
Modeling and Simulation Nature
• Simulation
◦
◦
◦
Application of the model to the problem
Derivation of a strategy to solve the problem
Answer the question
• Validation
◦
Does strategy derived in the simulation step solve the problem or answer
the question for the real system
35
Modeling and Simulation Nature
• Simulation
◦
◦
◦
Application of the model to the problem
Derivation of a strategy to solve the problem
Answer the question
• Validation
◦
Does strategy derived in the simulation step solve the problem or answer
the question for the real system
Definition 4 (Simulation)
Simulation is the application of a model with the objective to derive strategies
that help to solve a problem or answer a question pertaining to a system.
35
Model
• System: Car
36
Model
• System: Car
• System Analysis: Identifying battery and fuel levels
36
Model
• System: Car
• System Analysis: Identifying battery and fuel levels
• Modeling:Check battery and fuel level
36
Model
•
•
•
•
System: Car
System Analysis: Identifying battery and fuel levels
Modeling:Check battery and fuel level
Simulation: Model consists of battery and fuel
36
Model
•
•
•
•
•
System: Car
System Analysis: Identifying battery and fuel levels
Modeling:Check battery and fuel level
Simulation: Model consists of battery and fuel
Validation: Apply this concept to real car
36
Model
•
•
•
•
•
•
System: Car
System Analysis: Identifying battery and fuel levels
Modeling:Check battery and fuel level
Simulation: Model consists of battery and fuel
Validation: Apply this concept to real car
Validated: If refilling the tank or battery, starts the car
36
Model
•
•
•
•
•
•
System: Car
System Analysis: Identifying battery and fuel levels
Modeling:Check battery and fuel level
Simulation: Model consists of battery and fuel
Validation: Apply this concept to real car
Validated: If refilling the tank or battery, starts the car
36
Steps Involved in Modeling
• System Analysis
37
Steps Involved in Modeling
• System Analysis
◦
Use literature, get benefit from others’ experiences.
37
Steps Involved in Modeling
• System Analysis
◦
◦
Use literature, get benefit from others’ experiences.
Discussions, meetings and consultations from various disciplines.
37
Steps Involved in Modeling
• System Analysis
◦
◦
◦
Use literature, get benefit from others’ experiences.
Discussions, meetings and consultations from various disciplines.
New data to understand the system and improve the model, and
experimental program
37
Steps Involved in Modeling
• System Analysis
◦
◦
◦
Use literature, get benefit from others’ experiences.
Discussions, meetings and consultations from various disciplines.
New data to understand the system and improve the model, and
experimental program
• Modeling and Simulation
37
Steps Involved in Modeling
• System Analysis
◦
◦
◦
Use literature, get benefit from others’ experiences.
Discussions, meetings and consultations from various disciplines.
New data to understand the system and improve the model, and
experimental program
• Modeling and Simulation
◦
Appropriate software to solve the mathematical model
37
Steps Involved in Modeling
• System Analysis
◦
◦
◦
Use literature, get benefit from others’ experiences.
Discussions, meetings and consultations from various disciplines.
New data to understand the system and improve the model, and
experimental program
• Modeling and Simulation
◦
◦
Appropriate software to solve the mathematical model
Standard software or customized software or your own software
37
Steps Involved in Modeling
• System Analysis
◦
◦
◦
Use literature, get benefit from others’ experiences.
Discussions, meetings and consultations from various disciplines.
New data to understand the system and improve the model, and
experimental program
• Modeling and Simulation
◦
◦
Appropriate software to solve the mathematical model
Standard software or customized software or your own software
• Validation
37
Steps Involved in Modeling
• System Analysis
◦
◦
◦
Use literature, get benefit from others’ experiences.
Discussions, meetings and consultations from various disciplines.
New data to understand the system and improve the model, and
experimental program
• Modeling and Simulation
◦
◦
Appropriate software to solve the mathematical model
Standard software or customized software or your own software
• Validation
◦
Compare with existing experimental data, from literature or your own
experiments, and fit data quantitatively and qualitatively.
37
Steps Involved in Modeling
• System Analysis
◦
◦
◦
Use literature, get benefit from others’ experiences.
Discussions, meetings and consultations from various disciplines.
New data to understand the system and improve the model, and
experimental program
• Modeling and Simulation
◦
◦
Appropriate software to solve the mathematical model
Standard software or customized software or your own software
• Validation
◦
Compare with existing experimental data, from literature or your own
experiments, and fit data quantitatively and qualitatively.
Remarks
Quantitative and qualitative fit may fail the validation step, if it can’t be used to
solve the problem.
37
Conceptual Model and Physical Model
Conceptual Model
• Car drawn on a paper
38
Conceptual Model and Physical Model
Conceptual Model
• Car drawn on a paper
• No Physical Reality
38
Conceptual Model and Physical Model
Conceptual Model
• Car drawn on a paper
• No Physical Reality
• All life is problem solving.
38
Conceptual Model and Physical Model
Conceptual Model
•
•
•
•
Car drawn on a paper
No Physical Reality
All life is problem solving.
Conceptual model is an
important tool to solve our
everyday problems.
38
Conceptual Model and Physical Model
Conceptual Model
•
•
•
•
Car drawn on a paper
No Physical Reality
Physical Model
• Simplified version of the
engine
All life is problem solving.
Conceptual model is an
important tool to solve our
everyday problems.
38
Conceptual Model and Physical Model
Conceptual Model
•
•
•
•
Car drawn on a paper
No Physical Reality
All life is problem solving.
Conceptual model is an
important tool to solve our
everyday problems.
Physical Model
• Simplified version of the
engine
• Few parts of the engine
connected to the fuel injection
process Relating our idea to
the real part of the physical
world
38
Mathematical
Modelling
39
Mathematical Model
Mathematics: A Natural Language for Modeling
40
Conceptual Model and Physical Model
Input Parameters
•
Identify list of input parameters
involved in the system
41
Conceptual Model and Physical Model
Input Parameters
•
Identify list of input parameters
involved in the system
Output Parameters
•
Identify list of Output
parameters involved in the
system
41
Conceptual Model and Physical Model
Input Parameters
•
Identify list of input parameters
involved in the system
Output Parameters
•
Identify list of Output
parameters involved in the
system
Input-Output Relation
•
Relations between input-output
41
Conceptual Model and Physical Model
Input Parameters
•
Identify list of input parameters
involved in the system
Output Parameters
•
Numerical Data
•
Collection of data from experiments,
input, output, described naturally in
mathematical terms
Identify list of Output
parameters involved in the
system
Input-Output Relation
•
•
Relations between input-output
Technology
41
Conceptual Model and Physical Model
Input Parameters
•
Identify list of input parameters
involved in the system
Numerical Data
•
Output Parameters
•
Identify list of Output
parameters involved in the
system
Input-Output Relation
•
•
Relations between input-output
Collection of data from experiments,
input, output, described naturally in
mathematical terms
y = f (x)
(y1 , y2 , · · · , ym ) = f (x1 , x2 , · · · , xn )
Initial Study
•
The system is a black box
Technology
41
Conceptual Model and Physical Model
Input Parameters
•
Identify list of input parameters
involved in the system
Numerical Data
•
Output Parameters
•
Identify list of Output
parameters involved in the
system
Input-Output Relation
•
•
Relations between input-output
Technology
Collection of data from experiments,
input, output, described naturally in
mathematical terms
y = f (x)
(y1 , y2 , · · · , ym ) = f (x1 , x2 , · · · , xn )
Initial Study
•
•
The system is a black box
Uncertainty
41
Conceptual Model and Physical Model
Input Parameters
•
Identify list of input parameters
involved in the system
Numerical Data
•
Output Parameters
•
Identify list of Output
parameters involved in the
system
Input-Output Relation
•
•
Relations between input-output
Technology
Collection of data from experiments,
input, output, described naturally in
mathematical terms
y = f (x)
(y1 , y2 , · · · , ym ) = f (x1 , x2 , · · · , xn )
Initial Study
•
•
•
The system is a black box
Uncertainty
Hypothesis: Experimenter wants to
investigate
41
Conceptual Model and Physical Model
Input Parameters
•
Identify list of input parameters
involved in the system
Numerical Data
•
Output Parameters
•
Identify list of Output
parameters involved in the
system
Input-Output Relation
•
•
Relations between input-output
Technology
Collection of data from experiments,
input, output, described naturally in
mathematical terms
y = f (x)
(y1 , y2 , · · · , ym ) = f (x1 , x2 , · · · , xn )
Initial Study
•
•
•
The system is a black box
•
Question-And-Answer Game
Uncertainty
Hypothesis: Experimenter wants to
investigate
41
Mathematical Model
There are different definitions available from different authors related to
mathematical model. To name a few
Definition 5 (Mathematical Model)
A mathematical model is a triplet (S, Q, M ) where S is a system,Q is a question
relating to S, and M is a set of mathematical statements
M = {Σ1 , Σ2 , · · · Σn }
which can be used to answer Q.
Definition 6 (Mathematical Model)
A mathematical model can be broadly defined as a formulation or equation
that expresses the essential features of a physical system or process in mathematical terms.
42
Mathematical Model
A mathematical model consists of the following elements
• Governing Equations
• Supplementary Sub-models: Defining equations and Constitutive
equations
• Constraints and Assumptions: Initial and Boundary conditions
43
Basic Group of Variables
☞ Decision variables - quantity that the decision-maker controls
☞ Input variables
☞ Output variables
☞ State variables
☞ Exo/Endogenous variables - in/dependent variable
☞ Random variables
44
State Variable
Definition 7 (State Variable)
Let (S, Q, M ) be a mathematical model.
Mathematical quantities
(s1 , s2 , · · · , sn ) which describe the state of the system S in terms of M
and which are required to answer Q are called the state variables of (S, Q, M )
45
Reduced System
Definition 8 (Reduced System)
Let s1 , s2 , · · · , sn be the state variables of a mathematical model (S, Q, M ).
Let p1 , p2 , · · · , pm be mathematical quantities (numbers, variables, functions)
which describe properties of the system S in terms of M , and which are needed
to compute the state variables. Then Sr = {p1 , p2 , · · · , pm } is the reduced
system and p1 , p2 , · · · , pm are the system parameters of (S, Q, M ).
46
Classification of Mathematical Modeling
System
☛ Physical-Conceptual
☛ Natural-Technical
☛ Stochastic-Deterministic
☛ Continuous-Discrete
☛ Dimension
☛ Fields of Application
47
Classification of Mathematical Modeling
Questions
✐ Phenomenological-Mechanistic
✐ Stationary-Unstationary
✐ Lumped-Distributed
✐ Direct-Inverse
✐ Research-Management
✐ Speculation-Design
✐ Scale
48
Classification of Mathematical Modeling
Mathematical Statements
➶ Linear-Nonlinear
➶ Analytical-Numerical
➶ Autonomous-Nonautonomous
➶ Continuous-Discrete
➶ Difference Equations
➶ Differential Equations
➶ Integral Equations
➶ Algebraic Equations
49
Black Box to White Box
50
Don’t
Don’ts
1. Don’t Believe that the model is reality
2. Don’t extrapolate the region of fit
3. Don’t distort reality to fit the model
4. Don’t retain a discredited model
5. Don’t fall in love with your model
51
Summary
In general, the governing equations are written as a functional relationship
between input and output parameters. That is
(y1 , y2 , · · · , ym ) = f (x1 , x2 , · · · , xn )
DV
= f (IV, P, F )
where yi ’s and xj ’s denote input and output parameters respectively. DV and
IV respectively denote dependent and independent variables. P and F
denote parameters and forcing functions respectively.
52
Cancer Treatments
53
Why Cancer?
•
•
•
•
•
•
•
Due to liquor drinking and smoking
Change of life style
Heredity or Genetic Disorders
Excessive exposure to sun
Environment exposures
Working under toxic chemicals
Exposure to a few viruses
54
Types of Cancer?
•
•
•
•
•
•
•
Liver
Lungs
Kidney
Brain
Skin
Breast
and so on...
55
MICT-RFA, Liver Cancer
• Minimally Invasive Cancer Treatment
• Recommended by Interventional Radiologists (IRs)
◦
Under treatment
■
◦
Recurrence of Tumour
Over treatment
■
Kills unnecessary good cells ⇒ Death
• RFA is most Common
◦
Requires more experienced IRs
• RFA Procedure
◦
◦
Locoregional treatment
Use RFA needle (umbrella shaped) tp induce heat energy
56
Video of RFA Treatment
RFA Treatment
57
Needles
58
Needles
59
(S,Q,M)
System:
• Entire Body, Clinical Environment of RFA Treatment
Questions:
1. Can we create a simple model for RFA Treatment?
(a) Can we create liver, vessel, tumour model?
(b) Can we produce the same power produced by the needle virtually?
(c) Can we implement the same protocol followed by doctors?
2. Can we predict the temperature profile before the treatment?
3. Can we predict the cell deaths?
60
(S,Q,M)
Mathematical Statements For:
1. Heat transfer from RFA needle to the liver
2. Power generation or Heat Source from needle
3. Temperature controlled power supply
4. Cell death due to temperature
61
Input-Output System
Input-Output System
Input
1. CT Scan Images
2. Blood Perfusion
Output:
• Liver Geometry,
• Tumour Geometry
5. Needle location on image
• Vessel Geometry
• Needle Geometry
• Temperature around tumour
6. Power output from machine
• Cell States around tumour
3. Body Temperature
4. Protocol
62
Clinical Model
63
Clinical Workflow
64
Device-Specific Parameters
65
Patient-Specific Parameters
• Blood Perfusion (Tissue, Tumour, TACE)
• Specific Heat Capacity
• Thermal Conductivity
66
Mathematical Model
67
Heat Transfer
• A discipline of thermal engineering
• Conversion, exchange of heat between physical systems
68
Heat Transfer
•
•
•
•
Heat flow in a solid rod with circular cross section A
Perfectly insulated rod. No heat escapes radially
Heat flow only along the axis of symmetry of the rod
Initial temperature = room temperature
69
Heat Transfer
•
•
•
•
•
•
At one end temperature raised suddenly
Heat flow from hot to cold
δx: thickness of a section through the rod located at the point x
Some of the heat absorbed by the rod
Heat flow at x ̸= Heat flow at x + δx
Conservation Energy
Rate of change of heat content = net rate of heat conducted in and
out of section
70
Heat Transfer: Assumptions
Rate of change of heat content = net rate of heat conducted in and
out of section
• Assume no heat production inside the rod or heat loss from the
surface of the rod.
• T (x, t): temperature of the rod at position x, at time t.
• Some of the heat energy is absorbed by the rod.
• Changes in temperature at time t + δt
• Temperature difference:
T (x, t + δt) − T (x, t)
71
Heat Equation: Derivation
• Temperature difference:
T (x, t + δt) − T (x, t)
• Amount of heat required to change the temperature of the entire mass of
the cross section??
•
•
•
•
•
•
Proportional to mass of section??
Mass = density x volume
Volume = A × δx
(ξ,t)
Rate of change of heat content = cρAδx ∂T∂t
ξ ∈ (x, x + δx)
c : proportionality factor or specific heat constant depends on the metal
72
Heat Transfer: Derivation
Rate of change of heat content = net rate of heat conducted in and
out of section
• Heat flux J(x, t): The rate of heat passing through a cross-section,
per unit area, per unit time
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Fourier’s Law
• More the area is exposed ⇒ More will be the heat transferred
J ∝A
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Fourier’s Law
• More is the temperature difference ⇒ More will be the heat transferred
J ∝ dT
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Fourier’s Law
• Farther is the heat source ⇒ Lesser is the heat transferred
J∝
1
dx
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Fourier’s Law
dT
dx
dT
J = −kA
dx
∂T
J = −kA
∂x
J ∝A
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Bioheat Equation
(ξ,t)
net rate of heat conducted in
Rate of change of heat content = cρAδx ∂T∂t
and out of section
∂T (x + δx, t)
∂T (xx, t)
− kA
∂x
∂x
T (x + δx, t) ∂T (xx, t)
∂T (ξ, t)
cρAδx
= kA
−
∂t
∂x
∂x
∂T (ξ, t)
kA T (x + δx, t) ∂T (xx, t)
cρA
=
−
∂t
δx
∂x
∂x
J(x + δx, t) − J(x, t) = kA
cρ
∂T (ξ, t)
∂2T
=k 2
∂t
∂x
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Bioheat Equation
∂2T
∂T (ξ, t)
=k 2
∂t
∂x
2
∂T (ξ, t)
∂ T
∂2T
∂2T
cρ
=k
+
+
∂t
∂x2
∂y 2
∂z 2
cρ
cρ
∂T (ξ, t)
= k∆T + Qext + Qsink
∂t
79
Pennes Bioheat Equation
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Engineering
Mathematics-I
81
What is Engineering Mathematics?
• Brand new series of two courses:
◦
◦
Engineering Mathematics I: Compulsary for all 1st semester BTech
students.
Engineering Mathematics II: Compulsary for all 2nd semester BTech
Students.
• Both courses closely follow the text book “AAdvanced Engineering
Mathematics” by Erwin Kreyszig.
• Designed to introduce engineering students those areas of mathematics
that are most relevant for solving practical problems.
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What is in Engineering Mathematics I and II?
Engineering Mathematics I:
• Multivariable Calculus, and
• Ordinary differential equations.
Engineering Mathematics II:
• Linear algebra, and
• Theory of probability.
83
Instruction format
Lectures:
• Monday: 9:00 am - 9:50 am
• Tuesday: 11:00 am - 11:50 am
• Thursday: 10:00 am - 10:50 am
Tutorials:
• Friday: 2:10 pm - 3:00 pm
84
Thanks
Doubts and Suggestions
panch.m@iittp.ac.in
85
Engineering Mathematics-I
Lecture 1 : Mathematical Modeling
Panchatcharam Mariappan
Assistant Professor
Department of Mathematics and Statistics
IIT Tirupati, Tirupati
85