School of Mathematics and Statistics

MAST20029 Engineering Mathematics, Semester 1, 2016 Assignment 2 and Cover Sheet

Student Name Student Number

Tutor’s Name Tutorial Day/Time

Submit your assignment to your tutor’s MAST20029 assignment box before 5pm on Tuesday 3rd May.

This assignment is worth 5% of your final MAST20029 mark.

You must attach this cover sheet to your assignment.

Note:

• Full working must be shown in your analytical solutions.

• Assignments must be neatly handwritten in blue or black pen. Diagrams can be drawn in pencil.

• For the MATLAB questions, include a printout of all MATLAB code and outputs. This must be printed from within MATLAB, or must be a screen shot showing your work and the MATLAB Command window heading. You must include your name and student number in a comment in your code.

1. (a) Use MATLAB to compute the Laplace transform of f(t) = tsin(at) where a ?R is a constant. (b) Use Laplace transforms to solve the initial value problem

x00 + 9x = 2cos3t x(0) = 0, x0(0) = 1.

Hint: Your result from (a) may be useful.

(c) Using the s-shifting theorem, compute the inverse Laplace transform of

Questions continue on next page...

School of Mathematics and Statistics

MAST20029 Engineering Mathematics, Semester 1, 2016 Assignment 2 and Cover Sheet

2. The spread of a non-lethal airborne virus through a human population can be modelled by the SIR model. Let x(t) be the number of individuals, in millions, in the population who have not yet caught the virus at time t (the ‘susceptibles’), and y(t) be the number of individuals, in millions, in the population who are infected with the virus at time t (the ‘infectives’). Then the spread of the virus can be modelled by the system of differential equations

where N is the total initial population size, and ß,? and µ are constants describing the virus’s infectivity, the rate of recovery from the disease, and the per-capita birth & death rate, respectively. If we assume a population size of 4 million, and take ß = 2, ? = 1 and µ = 1, we obtain the equations

(?)

(a) Show that the system (?) has exactly two critical points, (4,0) and .

(b) Find the linearization of the system about the critical point (4,0).

(c) Use MATLAB to find the general solution of the linearized system about (4,0). Hence determine the type and stability of the critical point of the linearized system.

(d) The linearized system about the critical point is

(†)

where (you do not need to prove this).

Find the general solution of this linearized system using eigenvalues and eigenvectors.

(e) Draw by hand a phase portrait of the linearized system from (d). Be sure to include the following, and explain your reasoning to justify your conclusions:

• the straight-line orbits, if any;

• reasoning to justify the direction and behaviour of the orbits;

• at least 2 typical orbits;

• the slopes of the orbits as they cross the co-ordinate axes.

(f) Classify the type and stability of the critical point of the linearized system from ( d ).

(g) What can you conclude from your answers above about the non-linear system (?)? Explain.

(h) Use PPLANE to produce a plot of the phase portrait of the non-linear system (?), showing the behaviour of orbits around all critical points. Your printout should show the phase plane as well as the ODEs used to produce it.

(i) Interpret your results from parts (g) and (h) in terms of the virus’s spread through the population.

End of assignment.

MAST20029 Engineering Mathematics, Semester 1, 2016 Assignment 2 and Cover Sheet

Student Name Student Number

Tutor’s Name Tutorial Day/Time

Submit your assignment to your tutor’s MAST20029 assignment box before 5pm on Tuesday 3rd May.

This assignment is worth 5% of your final MAST20029 mark.

You must attach this cover sheet to your assignment.

Note:

• Full working must be shown in your analytical solutions.

• Assignments must be neatly handwritten in blue or black pen. Diagrams can be drawn in pencil.

• For the MATLAB questions, include a printout of all MATLAB code and outputs. This must be printed from within MATLAB, or must be a screen shot showing your work and the MATLAB Command window heading. You must include your name and student number in a comment in your code.

1. (a) Use MATLAB to compute the Laplace transform of f(t) = tsin(at) where a ?R is a constant. (b) Use Laplace transforms to solve the initial value problem

x00 + 9x = 2cos3t x(0) = 0, x0(0) = 1.

Hint: Your result from (a) may be useful.

(c) Using the s-shifting theorem, compute the inverse Laplace transform of

Questions continue on next page...

School of Mathematics and Statistics

MAST20029 Engineering Mathematics, Semester 1, 2016 Assignment 2 and Cover Sheet

2. The spread of a non-lethal airborne virus through a human population can be modelled by the SIR model. Let x(t) be the number of individuals, in millions, in the population who have not yet caught the virus at time t (the ‘susceptibles’), and y(t) be the number of individuals, in millions, in the population who are infected with the virus at time t (the ‘infectives’). Then the spread of the virus can be modelled by the system of differential equations

where N is the total initial population size, and ß,? and µ are constants describing the virus’s infectivity, the rate of recovery from the disease, and the per-capita birth & death rate, respectively. If we assume a population size of 4 million, and take ß = 2, ? = 1 and µ = 1, we obtain the equations

(?)

(a) Show that the system (?) has exactly two critical points, (4,0) and .

(b) Find the linearization of the system about the critical point (4,0).

(c) Use MATLAB to find the general solution of the linearized system about (4,0). Hence determine the type and stability of the critical point of the linearized system.

(d) The linearized system about the critical point is

(†)

where (you do not need to prove this).

Find the general solution of this linearized system using eigenvalues and eigenvectors.

(e) Draw by hand a phase portrait of the linearized system from (d). Be sure to include the following, and explain your reasoning to justify your conclusions:

• the straight-line orbits, if any;

• reasoning to justify the direction and behaviour of the orbits;

• at least 2 typical orbits;

• the slopes of the orbits as they cross the co-ordinate axes.

(f) Classify the type and stability of the critical point of the linearized system from ( d ).

(g) What can you conclude from your answers above about the non-linear system (?)? Explain.

(h) Use PPLANE to produce a plot of the phase portrait of the non-linear system (?), showing the behaviour of orbits around all critical points. Your printout should show the phase plane as well as the ODEs used to produce it.

(i) Interpret your results from parts (g) and (h) in terms of the virus’s spread through the population.

End of assignment.

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