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Continuous Mathematical Models 1-Mathematical Modeling and Simulation-Lecture Slides, Slides of Mathematical Modeling and Simulation

These lecture slides are delivered at The LNM Institute of Information Technology by Dr. Sham Thakur for subject of Mathematical Modeling and Simulation. Its main points are: Continuous, Mathematical, Models, First, Order, Analytical, Constant, Coefficient, Case, Population

Typology: Slides

2011/2012

Uploaded on 07/03/2012

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Modeling and Simulation
Continuous Mathematical Models
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Download Continuous Mathematical Models 1-Mathematical Modeling and Simulation-Lecture Slides and more Slides Mathematical Modeling and Simulation in PDF only on Docsity!

Lecture Slides

on

Modeling and Simulation

Continuous Mathematical Models

  • A linear first order model based on ODE has the following general form:

where f is linear in y. Examples include equations with constant coefficients,

or equations with variable coefficients:

or, more conventionally,

f( t,y) dt

dy 

y ' a(t)yb(t)

y ayb

p( t)y g(t)

dt

dy

First Order Linear Models in Biophysics

  • Consider a colony of micro-organisms reproducing

through simple cell division under ideal conditions of

unlimited food supply and total absence of any predators.

  • It is observed that the population increases q (10% every

hour).

  • Find a mathematical model that will also produce the

same results.

Example 1: A Model from Biology

  • Consider N to be number of organisms present at any time t.
  • The after Δt time they will have increased by amount ΔN
  • This change ΔN is directly proportional to N and the time interval Δt.

rN t

N  

N  Nt

rN dt

dN ^ Mathematical model

Where, r is proportionality constant. This will be the parameter of the system

Example 1: A Model from Biology

Example 1: A Model from Biology

When time is zero, we say N = N(0);

Then C = N(0). rN; dt

dN 

This value of r is in # per hour. This is termed as growth constant.

N  0  0. 1 N  0  N  0 expr  1 

N  N  0 exp rt

When time t = 1 hour, the N icreases by 10%. Let us put this in the model.

  1. 1 exp r  r ln 1. 1   0. 0953

This model is showing very important law of exponential growth.

Example 1: A Model from Biology

Let us consider N(0)=1;

rN; dt

dN 

r = 0.0953 is in # per hour.

N  N  0 exp rt

Let us model this law of exponential growth by MATLAB.

simout To Workspace

Scope 1/s Integrator

Gain

(^00 2 4 6 8 )

50

100

150

200

250

300

Number of organisms

time(hours)

using ode45 in matlab

Population gain rate:

  • Consider N to be number of organisms present at any time t.
  • The after Δt time they will have increased by amount ΔN
  • This change ΔN is directly proportional to N and the time interval Δt.

gain rate N; population^ gain rate aN

Where, a is proportionality constant.

Population loss rate:

  • It is propotional to the square of populational at that time:

loss rate N ; 2  population loss rateb N^2

Where, b is proportionality constant.

Example 2: Population of Fruit flies

Balance Equation:

  • The net rate of change of population will be equal to gain

minus loss rates.

aN bN^2
dt
dN
  Mathematical model

Where, a and b are proportionality constants.

Let us consider both of them as positive numbers.

Example 2: Population of Fruit flies

Example 1: A Model from Biology

N(0)=100;

aN bN^2 dt

dN  

a = 0.1 in # per day. b = 0.0005 in # per day.

0 20 40 60 80 100

0

50

100

150

200

250

Number of flies

time(days)

using ode45 in matlab simout To Workspace

Scope

u^2 Math Function

1/s Integrator

-0. Gain

Gain

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Variable Coefficient Case:

  • A model with linear first order ODEs and variable

coefficients may look like this:

  • The method of integrating factors is used to find

solution.

  • It involves multiplying this equation by a function ( t ),

chosen so that the resulting equation is easily

integrated.

p( t)y g(t)
dt
dy

Method of Integrating Factors

Major features/properties of the following mathematical model:

Dependent and independent variable: y and t,

Order: one;

Linearity: It is a linear equation as it has no product term:

Homogeneity: It is non-homogeneous equation with a force term ( exp( t/2 ).

Conditions: Initial conditions can be given.

Coefficients: There are constant coefficients.

Driving term type: It has analytical term as opposed to a tabular form.

Model Equation Type: It is a single ordinary differential equation based model. It

is time dependent first order deterministic model.

e y

dt

dy t

 /^2  2

Example 3: Population of germs

/ 2 2 t  y  y  e

Example 3: Analytical solution

Let us find analytical solution using Integrating Factor

Multiplying both sides by ( t ), we obtain
We will choose ( t ) so that left side is derivative of known quantity.

Consider the following, and recall product rule:

Choose ( t ) so that

/ 2 2 t y  y  e

  y

dt
d t
dt
dy
t y t
dt
d ( )

(t ) 2 (t)  (t ) e^2 t

( ) 2 (t)y e /^2 (t)
dt
dy

 t    t 

Example 3: General Solution

The red lines are direction field and the blue lines are solutions with different initial conditions.

Example 4: Population of viruses

Consider another type of viruses in experiments showing following loss

properties:

  1. They have rate of growth equal to zero.
  2. Their first loss rate is t – 5; where t is in seconds.
  3. They have another way of loss and its rate is proportional to the

population at that time and proportionality constant is about 1/5.

y t  y

Rate of change of population = rate of growth – rate of loss

Let us call population at any time t as y(t). Then

Rate of growth = 0.0 ; first rate of loss = 5 – t.

Second Rate of loss = y/.

y ^ t^ y^ Mathematical model