MATH111: Applied Mathematical Modelling I
This page contains material for the course: MATH 111
Applied Mathematical Modelling I.
This course emphasises the physical, mathematical and computational
aspects of applied mathematics in science, engineering
and industry (including financial applications). The aim of the course is
to provide students with some fundamental mathematical modelling
skills that are useful in a range of scientific and commercial
activities. Students will learn how to model a real-world problem as
an idealised mathematical system, how to solve the representative
mathematical problem and how to interpret the results.
Student learning
- Unless you are an exceptional student you should expect
to find the course difficult and time-consuming: there is not time for
all material
to be presented and reinforced in lectures.
- Students should not expect to grasp all material from the lecture
presentation. They should anticipate spending time outside the
lecture to attain the necessary level of understanding.
- The main feature distinguishing university from school is that students
are expected to learn on their own, outside the classroom.
- Ultimately, students are responsible for their own development, progress
and understanding.
- A student's failure to master something is a failure of the student, not
the educational system or the lecturer.
Your final mark in MATH111 will be determined
as follows (*). Four marks will be calculated using
scheme one (S1), scheme two (S2),
scheme three (S3) and
scheme four (S4).
Scheme | S1 |
S2 |
S3 |
S4 |
Final Exam |
60% | 50% | 50% | 50% |
Mid-session Exam |
20% | 20% | 20% | 10% |
Computer assignments |
15% | 25% | 20% | 30% |
Assignments |
5% | 5% | 10% | 10% |
- The mark for computer assignments is based upon your best
four assignments.
- Your final result will be the higher of the marks calculated
using schemes one through four.
- (*) Attendance at lab sessions, lectures and
tutorial classes may be taken into account.
- Scaling of marks is
not a standard procedure in this subject.
Note that you are not required to ``pass'' each individual
component to receive a Pass grade in MATH111. However, you
would seriously jeopardise your chances of passing this
subject if you do not aim to be successful in every component
of the assessment.
PDF file showing your: mid-session
test score (out of 40);
your maple lab score (out of 400); your
assignment mark (our of 5).
PDF file showing your mark on each
maple assignment (out of 100).
The student who receives the best mark in MATH111 is awarded the
Broadbridge Prize.
Year | Winner |
2005 | Glen Macdonald |
2006 | Andrew Holder & Oliver Towers |
2007 | Scott Harris |
- Display the information sheet for
this course.
(Last updated 1st August 2007).
I will make my over-head transparencies available.
If you miss a lecture you can download these notes and fill in
most of the blanks in your handouts by copying from the
OHTs.
- Introduction, Definitions
and Basic Concepts
- Aims.
- Overview of Discrete Systems.
- Examples of Difference Equations
- Formal Definitions.
- Revision of key ideas.
- Concept Map.
- Questions.
- First-Order Linear
Difference Equations
- Aims.
- Example of a First-Order Difference Equation:
Carp Population in a Lake.
- Solving the First-Order Difference Equation: The Case
b(n)=0.
- Solving the First-Order Difference Equation: The Case
a=1.
- Solving the First-Order Difference Equation: The General Case.
- Revision of key ideas.
- Concept map.
- Questions.
-
Applications of First-Order Difference Equations: Finance
- Aims.
- Interest.
- Loan Repayments (Amortization Scheme).
- Annuities.
- Revisionn of Key Ideas.
- Concept map.
- Questions.
- First Order
Non-Linear Difference Equations: An Introduction.
- Introduction.
- Aims.
- A linear model for population growth.
- The failure of the linear model: The need for nonlinear models.
- Cobwebbing: A graphical solution.
- Fixed points.
- The logistic equation.
- Cob-webing and fixed points: An example of putting them together.
(This is a new section to be included in the
2007 notes).
- Revision of key ideas.
- Concept map.
- Linear stability
analysis of fixed-points.
- Aims.
- Physical motivation of `stability'.
- Linear stability analysis.
- Stability of the fixed points in the logistic equation.
- Fixed points, their stability and bifurcations.
- Revision of key ideas.
- Questions.
- Harvesting.
- Introduction.
- Fixed harvesting.
- Proportional harvest.
- Comparison of harvesting strategies.
- Advanced Topics.
- Revision of key ideas.
- Concept map.
- First Order Difference
Equations: Concluding Thoughts.
- Review of key ideas.
- Concept map.
- Differential Equations:
Introduction, definitions and basic concepts.
- Aims.
- Overview of Continuous Systems.
- What does dy/dx mean?
- Examples of Differential Equations.
- Formal Definitions.
- Some comments on modelling.
- Revision of key ideas.
- Concep map.
- First-Order Differential
Equations.
- Aims.
- Solving the first-order differential equation: Integrable equations.
- Solving the first-order differential equation: Separable equations.
- Solving the first-order differential equation: Integrating Factors.
- Revision of key ideas.
- Concept map.
- Applications of first-order
differential equations: Lake Pollution.
- Aims.
- The lake pollution model.
- Pollutant dumped into a clean lake: Fixed flow-rate.
- Pollutant dumped into a clean lake: Seasonal flow-rate.
- Pollutant flowing into a lake.
- A degradable pollutant flowing into a lake.
- Extensions to the basic model.
- Two lakes in series.
- Rivers flowing into lake one only.
- Rivers flowing into lakes one and two.
- Revision of key ideas.
- Concept map.
- First Order
Non-Linear Differential Equations.
- Introduction.
- Aims.
- A linear model for population growth.
- The failure of the linear model: The need for nonlinear models.
- First-order Ordinary Differential Equations: Graphical Insights.
- Steady-state solutions.
- Revision of key ideas.
- Concept map.
- Linear Stability Analysis
of steady-state solutions.
- Physical motivation of `stability'.
- Linear stability analysis.
- Stability of the steady-states in the logistic equation.
- Stability of the steady-states in the Gompertz model.
- Steady-state solutions: Stability, bifurcations and
steady-state diagrams.
- Revision of key ideas.
- Concept map.
- Harvesting.
- Introduction.
- Aims.
- The logistic differential equation: A quick recap.
- Constant yield harvesting.
- Constant effort harvesting.
- Comparison of harvesting strategies.
- Revision of key ideas.
- Concept map.
- First Order Differential
Equations: Concluding Thoughts.
- Review of key ideas.
- Concept map.
- Concept maps.
- What is a concept map?
- A concept map of concept maps.
- Further Information.
- Sigma
Notation.
- What is Sigma notation?
- Questions.
- Geometric
Progression.
- What is a Geometric Progression?
- Questions.
- Taylor
series.
- Taylor series.
- Taylor series expansion.
In this section of the web-page I will try to summarise the main
ideas of each lecture.
- Week 7, Lecture 1 (Chapter 8).
- If you solve an ODE, then check that you solution is correct by:
- substituting it back into the ODE and showing that LHS=RHS.
- checking that is satisfies the initial conditions.
This is a good tip for other courses.
- Week 7, Lecture 2 (Chapter 9).
- Solving first-order differential equations,
particularly a first-order linear differential equation.
The latter will be frequently used.
- Week 7, Lecture 3 (Chapter 10).
-
- Lake pollution models are first-order linear differential
equations which we can solve using an integrating factor!
- Concentration = Mass/Volume.
- Week 8, Lectures 1 & 2 (Mid-Session Test).
- Week 8, Lecture 3 (Chapter 10).
- Lake pollution models are first-order linear differential
equations which we can solve using an integrating factor!
- It's important to be able to look at the solution of a differential
equation and determine the limit is t approaches infinity.
- Week 9, Lecture 1 & 2 (Chapter 11).
- Consider the differential equation
dx/dt =f(x).
Everything you want to know about the solution of this equation,
including how the long-term behaviour depends upon the initial
conditions, can be determined from the graph
y=f(x)
- Steady-state solutions are values of x for which
dx/dt = f(x) =0.
Finding steady-state solutions are an important took in
understanding the behaviour of the differential equation.
- Week 9, Lecture 3 (Review of mid-session test).
- Week 10, Lecture 1 (Chapter 12).
- Let x* be a steady-state of the differential
equation
dx/dt = f(x)
Then its eigenvalue is given by
λ = f'(x*).
- Week 10, Lecture 2 (Chapter 12).
- A steady state solution of the differential equation
dx/dt = f(x)
is stable if its eigenvalue is negative.
It is unstable if its eigenvalue is positive.
If the eigenvalue is zero it may be either stable or unstable.
- The stability of a steady-state solution of a differential equation
dx/dt = f(x)
can be determined in two ways.
- Algebraically by determining
the sign of the eigenvalue.
- Graphically by drawing the
function y=f(x) and looking at the sign of the
function f(x) on either side of the steady-state
solution.
- Week 10, Lecture 3 (Chapter 12).
- You need to know how to draw a steady-state diagram for the
differential equation
dx/dt = f(x,λ),
remembering that stable and unstable steady-state solutions are
indicated by solid and dashed lines respectively.
- There are three kinds of bifurcation points: limit-point,
transcritical and pitchfork.
- Week 11, Lecture 1 (Chapter 13).
- If you can draw the graph
y=f(x)
and understand how to obtain the graph
y=f(x)-H, H>0
by translating the first graph down H units then you
already understand constant yield harvesting!
- Week 11, Lecture 2 (Chapter 13).
- For the logistic equation we fixed harvesting the maximum
sustainable yield is
Hcr = rK/4. This corresponds to a
limit point bifurcation on the steady-state diagram.
- Week 11, Lecture 2 (Chapter 13).
Constant effort harvesting.
- Consider the differential equation with constant effort harvesting
dx/dt = f(x) -Ex.
The yield function is given by
Y = E*x(E)
where x(E) represents the steady-state solution(s) as a
function of the effort.
- Find the value of E that maximise the yield and
hence the maximum yield.
- Week One: MAPLE Crash
Course
-
- Basics
- Graphics
- Differentiation
- Integration
- Substitution
- Week Two: Iterating a simple
difference equation
- Population of carp in a lake.
- Week Three: Marked
Assignment.
- Investigating the discrete Ricker model
- The article that you are required to read to complete this
assignment is available as an electronic reading
in the library.
- Week Five: Marked
assignment.
- Bifurcations in first-order difference equations.
.
- Download the
maple bifurcation code.
- Download the
maple Ricker code.
- The article that you are required to read to complete this
assignment is available as an electronic reading in the
library.
- Week Seven: Marked
assignment.
- Fixed harvesting in the Ricker model.
- Week Nine: Marked
assignment.
- Solving first-order autonomous differential equations in Maple.
.
- Download the
spruce budworm code.
- The article that you are required to read to complete this
assignment is available as an electronic reading in the
library.
- Week Eleven: Marked
assignment.
- SIR Endemics with and without vaccination.
- Download the maple code.
Note that the chapter number is a guideline.
What was chapter 11 in 2004 may not be the same as chapter 11 this year!
- The mid-session test contributes either 20%
(S1-S3) or 10%
(S4) of your final mark for this
course.
- Your mark on the mid-session test will be divided by two (four)
to give you a final mark out of twenty (ten).
- Appendix A: Concept maps
- There are many web pages on concept maps. One that I find
particularly useful is
www.coun.uvic.ca/learn/program/hndouts/map_ho.html.
If you are reading for a degree in biotechnology (or related areas)
you might find the following article of interest.
James E. Bailey. 1998. Mathematical Modeling and Analysis in Biochemical
Engineering: Past Accomplishments and Future Opportunities.
Biotechnology Progress, 14, 8-20.
You can access this article via the library catalogue under
e-reading/short loans for MATH111.
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Page Created: 27th June 2003.
Last Updated: 1st November 2008.