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Ordinary Differential Equations and Models-Mathematical Modeling and Simulation-Assignment, Exercises of Mathematical Modeling and Simulation

Jaypee University of Information TechnologyMathematical Modeling and Simulation

This assignment is for Mathematical Modeling and Simulation assigned by Dr. Raima Ullal at Jaypee University of Information Technology. Its main pints are: Mass, Spring, System, ODE, Homogeneous, Model, Analytical, Solution, Initial, Conditions

Typology: Exercises

2011/2012

Uploaded on 07/03/2012

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Mathematical Modeling & Simulation
HOME WORK # 4
Due Date: Two week from date of distribution in the class.
Question No. 1
A general model for mass spring system is second order ODE based linear and non-
homogeneous model.
Let us consider a system of mass-spring with following data:
Mass, m = 1kg; Damping constant, C = 2; Spring constant , k = 5; and
R(t) = exp(-0.5t) + cost4t.
(a) Find analytical solution for initial conditions: y(0) = 0, y’(0) = 1. Then develop a
patch diagram for the model and run the simulations on MATLAB/SIMULINK.
(b) Plot the y(t) for three different initial conditions. What is effect of these initial
conditions?
(c) Find if there is any saturation value.
Question No. 2
Consider a model based on ODE:
If initial conditions are y(0) = 1, y’(0) = 0.
a) Then first find analytical solution. What kind of roots you had for the
characteristic equation. Comment on their nature. Then simulate results using
MATLAB/SIMULINK.
b) Change initial conditions and find effect. Plot all graphs on same paper.
c) Final plot dy/dt versus y (that is on y-axis plot dy/dt and on x-axis y(t)) and
comment on this graph.
Question No. 3
Consider the spreading of a highly communicable disease on an isolated island with
population size N. A portion of the population travels abroad and returns to the island
infected with the disease. You would like to predict the number of people X who will
have been infected by some time t. Consider the following model, where k > 0 is
constant:
dX/dt = kX(N – X)
(a) List two major assumptions implicit in the preceding model. How reasonable are
your assumptions?
(b) Graph dX/dt versus X after writing a Matlab program for different k values.
)(
2
2tRky
dt
dy
C
dt
yd
m
02.0
yyy
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Download Ordinary Differential Equations and Models-Mathematical Modeling and Simulation-Assignment and more Exercises Mathematical Modeling and Simulation in PDF only on Docsity!

Mathematical Modeling & Simulation HOME WORK # 4

Due Date: Two week from date of distribution in the class.

Question No. 1 A general model for mass spring system is second order ODE based linear and non- homogeneous model.

Let us consider a system of mass-spring with following data: Mass, m = 1kg; Damping constant, C = 2; Spring constant , k = 5; and R(t) = exp(-0.5t) + cost4t. (a) Find analytical solution for initial conditions: y(0) = 0, y’(0) = 1. Then develop a patch diagram for the model and run the simulations on MATLAB/SIMULINK. (b) Plot the y(t) for three different initial conditions. What is effect of these initial conditions? (c) Find if there is any saturation value.

Question No. 2 Consider a model based on ODE:

If initial conditions are y(0) = 1, y’(0) = 0. a) Then first find analytical solution. What kind of roots you had for the characteristic equation. Comment on their nature. Then simulate results using MATLAB/SIMULINK. b) Change initial conditions and find effect. Plot all graphs on same paper. c) Final plot dy/dt versus y (that is on y-axis plot dy/dt and on x-axis y(t)) and comment on this graph.

Question No. 3 Consider the spreading of a highly communicable disease on an isolated island with population size N. A portion of the population travels abroad and returns to the island infected with the disease. You would like to predict the number of people X who will have been infected by some time t. Consider the following model, where k > 0 is constant: dX/dt = kX(N – X)

(a) List two major assumptions implicit in the preceding model. How reasonable are your assumptions? (b) Graph dX/dt versus X after writing a Matlab program for different k values.

2 ky R t dt

dy C dt

d y m   

y ^  0. 2 y  y  0

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(c) Select appropriate k. Graph X versus t if the initial number of infections is X 1 < N /2. Plot a graph X versus t if the initial number of infections is X 2 > N/2. Write a program in matlab, use Euler method to solve this. (d) How the results are effected by change in initial condition? Plot X vs time for various initial conditions. (e) From above part, find the limit of X as t approaches infinity. (f) Consider an island with a population of 5000. At various times during the epidemic the number of people infected was recorded as follows:

t(days) 2 6 10 X( people infected) 1887 4087 4853 Ln(X/(N – X)) -0.5 1.5 3.

Do the collected data support the given model? Use the results in above part to estimate the constants in the model, and predict the number of people who will be infected by t = 12 days.

Question No. 4 Let x denote a guerilla force and y denote a conventional force. The autonomous system is a Lanchestrian model for conventional-guerilla combat in which there are no operational loss rates and no reinforcements:

dx/dt = - gxy ; dy/dt = - bx ;

(a) Discuss the assumptions and relationships necessary to justify the model. Does the model seem reasonable? Then develop a Matlab program to solve this. Use appropriate non-negative g and b. Try various different values and justify suitable values. Plot results as a function of time and then plot y vesus x also. (b) Solve the system analytically and obtain the parabolic law : gy^2 = 2bx + M where M = g(y 0 )^2 – 2bx 0 and y 0 and x 0 are initial values. (c) What condition must be satisfied by the initial force levels x0 and y0 for the conventional (y) force to win? If the y- force does win, how many survivors will there be?

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