Indicon 2014, Pune

1. INTRODUCTION TO MATHEMATICAL
MODELLING
Prof. (Dr) Samir Kumar Das
Defence Institute of Advanced Technology (DIAT)
Girinagar, Pune 411025

2. OUTLINES OF THE PRESENTATION
DIMENSIONAL ANALYSIS
WHAT IS MATHEMATICAL MODELLING?
WHY MATHEMATICAL MODEL IS NECESSARY?
USE OF MATHEMATICAL MODEL
TYPES OF MATHEMATICAL MODEL
MATHEMATICAL MODELLING PROCESS

3. OUTLINES OF THE PRESENTATION
DISCRETE APPROACH OF COMBAT MODELS
BUCKINGHAM PI THEOREM
PREY-PREDATOR MODEL (LOTKA-VOLTERRA MODEL)
(DISCRETE MODEL , EQUILIBRIUM CONDITIONS, STABILITY ANALYSIS)
COMBAT MODELLING (LANCHESTER LAWS)
(SQUARE LAW, LINEAR LAW, PARABOLIC LAW, LOG LAW)
POPULATION MODEL ( LOGISTIC MODEL )

4. WHAT IS MATHEMATICAL MODELLING?
Representation of real world problem in mathematical
form with some simplified assumptions which helps to
understand in fundamental and quantitative way.
It is complement to theory and experiments and often to
integrate them.
Having widespread applications in all branches of
Science and Engineering & Technology, Biology,
Medicine and several other interdisciplinary areas.
2
3
1

5. WHY MATHEMATICAL MODEL IS
NECESSARY?
To perform experiments and to solve real world
problems which may be risky and expensive or time
consuming.
Emerged as a powerful, indispensable tool for studying a
variety of problems in scientific research, product and
process development and manufacturing.
Improves the quality of work and reduced changes,
errors and rework
However, mathematical model is only a complement but does not
replace theory and experimentation in scientific research.
1
2
3

6. USE OF MATHEMATICAL MODEL
Solves the real world problems and has become wide
spread due to increasing computation power and
computing methods.
Facilitated to handle large scale and complicated
problems.
Some areas where mathematical models are highly used
are : Climate modeling, Aerospace Science, Space Technology,
Manufacturing and Design, Seismology, Environment, Economics,
Material Research, Water Resource, Drug Design, Populations
Dynamics, Combat and War related problems, Medicine, Biology etc.
1
2
3

7. TYPES OF MATHEMATICAL MODEL
EMPIRICAL
MODELS
THEORETICAL
MODELS
EXPERIMENTS
OBSERVATIONS
STATISTICAL
MATHEMATICAL
COMPUTATIONAL

8. TYPES OF MATHEMATICAL PROCESS
REAL WORLD PROBLEM WORKING MODEL
MATHEMATICAL MODELRESULT / CONCLUSIONS
COMPUTATIONAL MODEL
SIMPLIFY
REPRESENT
TRANSLATESIMULATE
INTERPRET

9. FORMULATION
PROBLEM MATHEMATICAL PRARAMETERS
START
SOLUTION
EVALUTION
SATISFIED STOP
NO YES

10. TYPES OF MODELS
QUALITATIVE AND QUANTITATIVE
STATIC OR DYNAMIC
DISCRETE OR CONTINUOUS
DETERMINISTIC OR PROBABILISTIC
LINEAR OR NONLINEAR
EXPLICIT OR IMPLICIT
1
2
3
4
5
6

11. STATIC OR DYNAMIC MODEL
STATIC MODEL A static (or steady-state) model calculates
the system in equilibrium, and thus is time-invariant. A
static model cannot be changed, and one cannot enter
edit mode when static model is open for detail view.
DYNAMIC MODEL A dynamic model accounts for time-
dependent changes in the state of the system. Dynamic
models are typically represented by differential
equations.

12. DISCRETE OR CONTINUOUS MODEL
DISCRETE MODEL A discrete model treats objects as
discrete, such as the particles in a molecular model. A
clock is an example of discrete model because the clock
skips to the next event start time as the simulation
proceeds.
CONTINUOUS MODEL A continuous model represents
the objects in a continuous manner, such as the velocity
field of fluid in pipe or channels, temperatures and
electric field.

13. DETERMINISTIC OR PROBABILISTIC
(STOCHASTIC) MODEL
DETERMINISTIC MODEL A deterministic model is one in
which every set of variable states is uniquely determined
by parameters in the model and by sets of previous
states of these variables. Deterministic models describe
behaviour on the basis of some physical law.
PROBABILISTIC (STOCHASTIC) MODEL A probabilistic /
stochastic model is one where exact prediction is not
possible and randomness is present, and variable states
are not described by unique values, but rather by
probability distributions.

14. LINEAR OR NONLINEAR MODEL
LINEAR MODEL If all the operators in a mathematical
model exhibit linearity, the resulting mathematical
model is defined as linear. A linear model uses
parameters that are constant and do not vary
throughout a simulation.
NONLINEAR MODEL A nonlinear model introduces
dependent parameters that are allowed to vary
throughout the course of a simulation run, and its use
becomes necessary where interdependencies between
parameters cannot be considered.

15. EXPLICIT OR IMPLICIT MODEL
EXPLICIT MODEL Calculate the state of a system at a
using the past time from the state of the system at the
current time.
IMPLICIT MODEL Solution is obtained by solving an
equation involving both the current state of the system
and the later one which require extra computation and
could be harder to solve.

16. QUALITATIVE OR QUANTITATIVE MODEL
QUALITATIVE MODEL It is basically a conceptual model
that display visually of the important components of an
ecosystem and linkages between them. It is a
simplification of a complex system. The humans are good
at common sense with qualitative reasoning.
QUANTITATIVE MODEL Models are mathematically
focused and many times are based on complex formulas.
In addition quantitative models generally through an
input-output matrix. Quantitative modelling and
simulation give precise numerical answers.

17. DEDUCTIVE MODEL
A deductive model is a logical structure based on theory.
A single conditional statement is made and a hypothesis
(P) is stated. The conclusion (Q) is then deduced from the
statement and hypothesis. (What this model represents ?)
P Q (Conditional statement)
P (Hypothesis stated) | Q (Conclusion deducted)
Example #1
All men are mortal, Ram is man, Therefore, Ram is mortal.1
2 3
Example #2
If an angle satisfies 900<A<1800, then A is an obtuse angle,
A=1200, Therefore, A is an obtuse angle.
1
2 3

18. DEDUCTIVE MODEL
An inductive model arises from empirical findings and
generalizations from them. This is known as “Bottom-up”
approach (Qualitative). Focus on generating new theory
which is used to form hypothesis.
THEORY
HYPOTHESIS
OBSERVATION
CONFIRMATION
Deductive model is more narrow in nature and is concerned with
confirmation of hypothesis.

19. DEDUCTIVE MODEL
Deductive model is a “Top-down” approach
(Quantitative). It focus on existing theory and usually
begins with hypothesis.
OBSERVATION
PATTERN
TENTATIVE HYPOTHESIS
Inductive model is open ended and explanatory, specially at the
beginning.
THEORY

20. REAL WORLD PROBLEM FALLS IN WHICH
CATEGORY?
This is based on how much priori information is available
on the system. There are two type of models : BLACK
BOX MODEL and WHITE BOX MODEL.
BLACK BOX MODEL is a system of which there is no priori
information available.
WHITE BOX MODEL is a system where all necessary
information is available.

21. DIMENSIONAL ANALYSIS
A method with which non-dimensional can be formed
from the physical quantities occurring in any physical
problem is known as dimensional analysis.
This is a practice of checking relations among physical
quantities by identifying their dimensions.
The dimension analysis is based on the fact that a
physical law must be independent of units used to
measure the physical variables.
2
3
1

22. DIMENSIONAL ANALYSIS
The practical consequence is that any model equations
must have same dimensions on the left and right sides.
One must check before developing any mathematical
model.
4

23. DIMENSIONAL ANALYSIS
EXAMPLE
Let us take an example of heat transfer problem. We
start with the Fourier’s law of heat transfer.
Rate of heat transfer Temperature gradient
2
2
K
t x
  

 
Let us consider a uniform rod of length l with non-uniform temp.
Lying on the x-axis form x=0 to x=l. The density of the rod ( ),
specific heat (c), thermal conductivity (K) and cross-sectional area
(A) are all constant.

(1)

24. DIMENSIONAL ANALYSIS
EXAMPLE
Change of heat energy of the segment in time ( ) =
Heat in from the left side – Heat out from the right side
After rearranging
(2)
t
( , ) ( , )
x x x
c A x x t t c A x x t A t K K
x x
 
   

     
                  
( , ) ( , ) x x x
K
c x xx t t x t
t x
 
  
     
            
 
(3)
After taking the limit
2
2
k
t x
  

 
where
K
k
c
 (4)

25. DIMENSIONAL ANALYSIS
EXAMPLE
(5)
(6)
(7)
2 2 1
TLc  
 3
ML 
, 3 1
K MLT  
,
L
x
x ,
3 1 2
0 0
2 2 1 3
00 0
MLT L
O
TL T ML
K
c
k


 
  
 
    
 

0



,
0T
T
T  ,
, ,
0
0
Tt T
  

 
0
Lx x
  

 
2 2
0
2 2 2
Lx x
  

 
2
0 0
2 2
0 0
2 2 2
,
L
T T L
k
t Tx x
       
 
  
2
2T x
  

 
,

26. DIMENSIONAL ANALYSIS
ASSIGNMENT #1
2
2c c
D
t x
 

  2
2uu
t x
 

 
,
where D and are diffusion coefficient and coefficient of kinematic
viscosity respectively.

CALCULATE FOR 1-D DIFFUSION EQUATION AND 1-D FLUID
EQUATION IN DIMENSIONLESS FORM AS :
2
2
T x
  

 
THERE ARE GENERALLY THREE ACCEPTED METHODS OF DIMENSIONAL ANALYSIS :
RAYLEIGH METHOD (1904): Conceptual method expressed as a functional
relationship of some variable | BUCKINGHAM METHOD (1914): The use of
Buckingham Pi ( ) theorem as the dimensional parameters was introduced by
the Physicist Edger Buckingham in his classical paper | P. W. BRIDGMAN
METHOD (1946): Developed on pressure physics)


27. BUCKINGHAM PI THEOREM
If there are m fundamental units and n physical quantities lead to
system of m linear algebraic equations with n unknowns of the form
(10)
This can be written in the matrix form as Ay b (11)
11 1 12 2 1 1
21 1 22 2 2 2
1 1 2 2
n n
n n
m m mn n m
a y a y a y b
a y a y a y b
a y a y a y b
   
   
   
L L L
L L L
L L L L L L L L L L L L L L
L L L L L L L L L L L L L L
L L L

28. BUCKINGHAM PI THEOREM
(13)
Here A is referred as coefficient matrix of order mxn, y and
b are order nx1 and mx1 respectively. If any matrix C has at
least one determinant of order r, then the matrix C is said to
be of rank r. (14)
11 21 1
21 22 2
1 2
..............
.............
..............................
.............
n
n
m m mn
a a a
a a a
A
a a a
 
 
 
 
 
 
(12)
 T
1 2......... ny y yy   T
1 2........ mb b bb 
( )R c r

29. BUCKINGHAM PI THEOREM
In order to determine the condition for the linear system equation (10),
it is convenient to define the rank of the augmented matrix B. For the
solution of the linear system equation (10) three possible cases arise:
11 21 1 1
21 22 2 2
1 2
..............
.............
.................................
............
n
n
m m mn m
b
b
b
a a a
a a a
B
a a a
 
 
 
 
 
 
(15)
( ) ( )R A R B
( ) ( )R A R B r n  
( ) ( )R A R B r n  
CASE I :
CASE II:
CASE III:
(16)
CASE I : In this case, no solution exists.
CASE-II: In this case, a unique solution exists
CASE-III: In this case, an infinite number of solution with n-r arbitrary
unknown exists

30. BUCKINGHAM PI THEOREM
The procedure of dimensional analysis makes use of the following
assumptions:
It is possible to select m independent units (Example: m=3, ie,
length, time and mass )
There exist n quantities involved in a phenomenon whose
dimensional formulae can be expressed in terms of m
fundamental units
The dimensional quantity Q0 can be related to the independent
dimensional quantities by1 2 1, ,......... nQ Q Q 
   11 2
1 2 1 1 2 10 , ,......... , ,......... nyy y
n nQ Q Q Q Q QQ F K 
  
Where K is a non-dimensional constant and are
integer exponent
1 2 1, ......... ny y y 
(17)

31. BUCKINGHAM PI THEOREM
Equation (17) is independent of the type of units chosen and is
dimensionally Homogeneous , ie, the quantities occurring on both
sides of the equations must have same dimensions.

32. BUCKINGHAM PI THEOREM
EXAMPLE
Consider the problem of freely falling body near the surface of the
Earth. If x, w, g and t represent the distance measured from the initial
height , the weight of the body, the gravitational acceleration and the
time respectively, obtain a relation of x as a function of w, g and t.
SOLUTION
Using the fundamental units of force F, length L and time T and four
physical quantities, , , and , involve three
fundamental units. Hence m=3 and n=4 as mentioned in assumptions
one and two.
0Q x 1Q w 2Q g 3Q t

33. BUCKINGHAM PI THEOREM
SOLUTION
By using the assumption three, assume a relation of the form
  31 2
, ,
yy y
w g tx F K w g t  (18)
where K is an arbitrary non-dimensional quantity. Let [ ] denote
dimensions of a quantity. Then the relation can be written (using the
assumption four) as
       1 2 3y y y
x w g t
     
2
1 3 2 31 2 220 1 0
yy y y yy y
LTF LT F T F L T  
 
1: 0F y
2: 1L y
2 3: 0 2T y y  
3 22 2y y 
0 1 2
x K w g t
2
x K g t
(19)
(20)
(21)
(22)
(23)
The constant in this case is ½ which can not obtained form dimensional
analysis

34. BUCKINGHAM PI THEOREM
ASSIGNMENT #2
Consider the problem of drag force on a body moving through a fluid.
Let D, and V be the drag force, density of the fluid, viscosity
effect of the fluid, reference length and velocity. Using fundamental
units obtain a relation of drag force as function of density, viscosity,
length and velocity.
HINTS
, ,l 
  31 2 4
, , ,
yy y y
l vD F K l v    
1 0 0
D F L T 4 2
FL T 
 2
FL T 
, ,

35. POPULATION MODEL (LOGISTIC MODEL)
Logistic model was developed by Belgian Mathematician Pierre
Verhulst (1838) who suggested that the rate may be limited, ie, it may
depend on population density.
P
kP
t



1
P
k r
K
 
  
 
, where (1)
At low population (P<<0), the population
growth rate is maximum and equal To r.
Parameter r can be interpreted as
population growth rate in the absence of
intra-species competition. 0
r
K

36. POPULATION MODEL (LOGISTIC MODEL)
... the population growth rate declines with the
population number P and reaches zero when P=K,
parameter K is the upper limit of population
growth and it is called carrying capacity. It is
usually expressed as the amount of resources
available. If the population number exceeds K,
then population growth rate becomes negative
and population numbers decline.
(2)
There are three possible model out comes:
1. Population increases and reaches a plateau (Po<K)
2. Population decreases and reaches a plateau (Po>K)
3. Population does not change (Po=K or Po=0)
0
r
K
1
P P
r
t K
  
  
  

37. DISCRETE DYNAMICAL MODEL
Behavior of the Discrete Dynamical Model near an Equilibrium
If f '(Pe) > 1, then the solutions of the discrete dynamical model
grow away from the equilibrium (monotonically). Thus, the equilibrium
is unstable.
2. If 0 < f '(Pe) < 1, then the solutions of the discrete dynamical model
approach the equilibrium (monotonically). Thus, the equilibrium is stable.
3. If -1 < f '(Pe) < 0, then the solutions of the discrete dynamical model
oscillate about the equilibrium and approach it. Thus, the equilibrium is
stable.
4. If f '(Pe) < -1, then the solutions of the discrete dynamical model
oscillate but move away from the equilibrium. Again, the equilibrium is
unstable.
1 1 n
n n n
P
P P rP
K

 
   
 

38. PREY-PREDATOR MODEL
Prey-Predator model is known as Lotka-Vloterra
model. This model was developed independently
by Alfraid J. Lotka and Vito Voterra in 1920’s
which characterized by oscillation in the
population size of both prey-predator. The prey-
predator dynamics can be written in the
simplified form by using pair of differential
equations. This describes the relation between
herbivore-plant, parasitoid-host, lions-gazelles,
birds-insects, shark-fish etc.
ALFRED LOTKA
(1880-1949)
VITO VOLTERRA
(1860-1940)

39. PREY-PREDATOR MODEL
This model makes several simplifying assumptions:
The population will grow exponentially when predator is absent1
2
3
The predator population will starve in the absence of prey
population
Predator can consume infinite number of prey
Both the populations can move randomly through a
homogeneous environment
The prey has unlimited food supply
4
5

40. PREY-PREDATOR MODEL
If there is no predator, the first assumption would imply that prey
grows exponentially. Let us consider N is prey, using Pierre Verhulst
model this can be written as
N
rN
t



0IC : (0)N N(4) (5)
The solution of equation (4) can be expressed as
0( ) rt
N t N e
Where is initial populations and r is the growth rate. Here the
number of pray would increase exponentially. Since there are
predators, which Must account for negative component of growth
rate.
(6)
N
rN aPN
t

 

Where P is number of predators and a is the attack rate. The term
shows the losses from prey population due to predation.
(7)
0N
aPN

41. PREY-PREDATOR MODEL
Now we consider predator population. If there are no food supply, the
population would die out at rate proportional to its size.
(8)
Predator mortality rate
In the presence of prey, this decline is opposed by the predator birth
rate
Where the term is the birth rate and c a constant conversion
rate of eaten prey into new predator in abundance. Combining
equation (7) and equation (9), coupled model can be obtained which
is known as Lotka-Volterra model or prey-predator model.
(9)
P
qp
t

 

q 
P
qP caNP
t

  

caNP

42. PREY-PREDATOR MODEL
Prey Model:
(10)
The Lotka-Volterra model consist of a system of linked differential
equations that can be separated from each other and can not be
solved in closed form.
Assuming N>0 and P>0
N
rN aPN
t

 

Prey Model:
P
qP caNP
t

  

EQUILIBRIUM CONDITIONS:
0
N
t



,   0r aP N  ,
r
P
a


0
P
t



,   0caN q P 
q
N
ca


(11)
(12)

43. PREY-PREDATOR MODEL
GRAPHICAL EQUILIBRIUM
Prey pop size
Predator
Pop Size
r/a
dN/dt =0
Prey (H) equilibrium
(dN/dt=0) is determined by
predator population size.
If the predator population
size is large the prey
population will go extinct.
If the predator population is
small the prey population
size increases.

44. PREY-PREDATOR MODEL
GRAPHICAL EQUILIBRIUM
Prey pop size
Predator
Pop Size
Predator (P) equilibrium
(dP/dt=0) is determined by
prey population size.
If the prey population size is
large the predator
population will increase.
If the prey population is
small the predator
population goes extinct.
q/ca
dP/dt =0

45. PREY-PREDATOR MODEL
PREY-PREDATOR INTERACTION
Prey pop size
Predator
Pop Size
The stable dynamic of predators and
prey is a cycle.
CASE – I : When Prey population is
above equilibrium and Predator
population below the equilibrium
CASE – II : When both Prey and
Predator populations are below
equilibrium
CASE – III : When Predator
population size is above the
equilibrium and Prey below
equilibrium
CASE – IV : When both Prey and
Predator populations are above the
equilibrium
q/ca
dN/dt =0
r/a
Case - II
Case - I
Case - III
Case - IV
dP/dt =0

46. PREY-PREDATOR MODEL
Three possible outcomes of interactions
The oscillations are stable
The oscillations are damped (convergent
oscillation)
The oscillations are divergent and can
lead to extinction
This model predicts cyclical
Relationship between predator
(P) and Prey (N)
Due to increase of consumption rate ,
decrease of prey takes place and
therefore aPN decreases .

47. PREY-PREDATOR MODEL
LOWINTIMACYHIGH
LOW LETHALITY HIGH
PARASITOIDSPARASITE
GRAZER PREDATOR

48. STABILITY ANALYSIS
0
N
t



  1 1
1 ,
f f
df N P dN dp
N P
 
 
 
0
P
t



  2 2
2 ,
f f
df N P dN dp
N P
 
 
 
1 1
2 2
( ) ( )
( ) ( )
N PN P
f f
r aP N r aP N
N P N P
J
f f
caN q P caN q P
N PN P   
      
       
    
       
        
Since and
P
N
Prey Zero Growth Isocline
PredatorZeroGrowth
Isocline

49. STABILITY ANALYSIS
 
r q
r a a
r aP aN a ca
J
r qcaP caN q ca ca q
a ca
 
 
    
              
             
    
0
0
q
J c
cr
 
 
 
 
The trace of J is equal to zero which is not
meeting stability conditions. Therefore,
equilibrium condition is not stable.
 
0 0
0 , det
0 0
q q
A Ac c
cr cr


   
       
   
   
2 2
0, ,qr qr i qr       

50. STABILITY ANALYSIS
The British theorist F. W. Lanchester (1914) developed this
theory based on World War-I, aircraft engagement to explain
why concentration of forces was useful in modern warfare.
Both the models work on the basis of attrition :
HOMOGENEOUS
1. A single scalar represents a unit’s combat power.
2. Both sides have the same weapon effectiveness.
HETEROGENEOUS
Attrition is assured by weapon type and target type and other
variability factors.

51. STABILITY ANALYSIS
HOMOGENEOUS
1. Homogeneous Model is an “Academic Model”.
2. Useful for review the ancient battles.
3. Not proper model for modern warfare.
HETEROGENEOUS
More appropriate for “Modern Battlefield”.
The following battlefield functions are sometimes combined and
can be modeled by separate algorithm/theory.
Direct Fire | Indirect Fire | Air to Ground Fire | Ground to Air
Fire | Air to Air Fire | Minefield Attrition

52. COMBAT MODELLING
(LANCHESTER LAWS)
1. Forces are homogeneous.
2. Similar Weapons Systems.
3. Which can accomplish the same effects.
Lanchester’s Combat model shows force-on-force interaction.
Blue Force (B) acts upon a Red Force (R) in accordance with
same effect ( ).
Red Force (R) acts upon a Blue Force (B) in accordance with
same effect ( ).
Blue Force Red Force
B(t) R(t)





53. COMBAT MODELLING
(LANCHESTER LAWS)
This and are attrition between forces. The variables which
represents these effects are and , are called attrition rates
as they represent the rates at which reds kill blues and blues kill
reds.
 
 
Attrition of Blue forces
dB
R
dt
  , for B>0
dR
B
dt
  , for R>0
IC: at t=0 0(0)B B 0(0)R R
(1)
(2)
Attrition of Red forces

54. COMBAT MODELLING
(LANCHESTER LAWS)
(3)
(5)
;
dB dR
dt dt
R B 
   
,
dB dR
BdB RdR
R B
 
 
   
0 0
B R
B R
BdB RdR  
0 0
2 2
2 2
B R
B R
B R
 
   
      
   
   2 2 2 2
0 0B B R R   
(6)
(7)
(4)

55. SQUARE LAW
Since the strengths of the opposing forces appear with
exponents of two, the name “Square Law” is given to that law
which the set of equations describes.
It is assumed that combat continues until one side’s unit count
reaches to zero. If this is true, then at the end of the combat ,
we define the number of units remaining as Bf and Rf
respectively, we can write

56. SQUARE LAW
0fB 
2 2
0 0fR R B


 
   
 
2 2
0 0B R 
2 2
0 0fB B R


 
   
 
2 2
0 0R B  0fR 
then
then
And
If :
If :

57. SQUARE LAW
OTHER ASSUMPTIONS
1. The forces are within weapons range of one another.
2. The effects of weapons rounds are independent.
3. Fire is uniformly distributed over the enemy targets.
4. Attrition coefficient are constant and known.
5. All of the forces are committed at the beginning and there are
no reinforcements.
These assumptions appear to restrict the applicability of Lanchester’s
Model.
Combat on today’s battlefield is very complex and very much from the
type proposed by Lanchester’s Model.
Some of these are not longer seen to be appropriate in modelling what
we term combat under modern conditions.
Although, it is not necessary that all the assumptions fit the
experimental model perfectly, some deviation is quite possible.

58. SQUARE LAW
ASSIGNMENT#3
Initial Blue Force = 2000 ( men ) =B0
Initial Red Force = 1000 ( men ) =R0
Blue Effect = = 0.002( 1/hour )
Red Effect = = 0.001( 1/hour )


HINTS
To determine who will win, each must have victory conditions.
Battle termination model, assuming both side fight to annihilation.
   0 and 0f fR t B t 
   0 and 0f fB t R t 
0
0
B
R



It can be shown that a square-law battle will be won by the blue (B)
If and only if
Who will win the battle after how many hours?
Red Wins :
Blue Wins :

59. EXTENSION OF LANCHESTER LAWS
1 w
dB B
R
dt R


 
   
 
1 w
dB B
R
dt R


 
   
 
1 w
dR R
B
dt B


 
   
 
0=> T/Tw
1/2=> FT/FTw
1=> F/Fw

60. Write to me at
samirkdas@diat.ac.in hod_am@diat.ac.in samirkumar_d@yahoo.com
THANK YOU
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