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### VBA Assessment 1 – Due 7 March

Hopefully I have designed something in which you can take your time, get things right, get some good marks, and show off what you can do. You are advised to complete VBA Lab 3 first.

I recommend watching the Week 6 lectures, but perhaps it might suit students to complete VBA Assessment 1 before looking at Week 6 Exercises. Do not let VBA Assessment slide very late as you will need time to prepare for Written Assessment 1. Really you should be looking to complete VBA Assessment 1 a few days before the due date. There is no lab in Week 6 in order to give you more time.

### Written Assessment 1 – Week 7

25% Written Assessment 1, based on Weeks 1-5, so everything up to p.73.

It will be a one hour assessment, but I am going to give ye 15 minutes grace, as well as 15 minutes to upload. The test will run therefore from 09.30 to 11.00, Tuesday 9 March. It is open book — you can use your manual, any Canvas materials, as well as Excel/VBA.

40% of the marks will be numerical methods (Euler, TTT, Heun); finding approximations like p.34, Q.1-3, p.37, Q.1-3, p.48, Q.1-4 (non-Excel parts of Q.4), p.60, Q.4-7.

30% of the marks will be numerical analysis; understanding these methods and their errors like p.34, Q. 4, p.38, Q.4-6. p.49, Q.5-8

15% will be neither; like p.17, Q. 1-2, p.40, Q.1-3, p.60, Q.1-3

15% will be Runge-Kutta, like p.72, Q.1-2.

Academic Dishonesty will not be accepted and suspected breaches, such as communication with others during the assessment, will be pursued in line with this policy (Links to an external site.)

## Week 6

### Lectures

There are 61 minutes of lectures. You should schedule about an hour and a half to watch them and take the notes in your manual. You might need this extra time above 61 minutes because you will want to pause me. You should also take note of any confusions you have to ask about in the regular Q & A.

### Lab 4

If you have not yet had a chance to look at Lab 4, but have questions, you can submit to Lab 4 Second Chance.

### Theory Exercises and Q & A

p.82, Q. 1-3

Q & A to ask about Theory Exercises or anything else every Tuesday 12.30 (waiting room open 12.25).

## Week 7

There will not be much in Week 7, just a short lecture on Goal Seek for boundary value problems.

## Assessment

This is provisional and subject to change.

1. Week 6, 25% First VBA Assessment, Based (roughly) on Weeks 1-4
2. Week 7, 25 % In-Class Written Test, Based (roughly) on Weeks 1-5
3. Week 11, 25% Second VBA Assessment, Based (roughly) on Weeks 6-9
4. Week 12, 25% Written Assessment(s), Based on Weeks 6-11

## Study

Study should consist of

• doing exercises from the notes
• completing VBA exercises

## Student Resources

Please see Student Resources  for information on the Academic Learning Centre, etc..

### VBA Assessment 1 – Due Week 6 (7 March)

Hopefully I have designed something in which you can take your time, get things right, get some good marks, and show off what you can do. You are advised to complete VBA Lab 3 first.

I recommend watching the Week 5 lectures, but perhaps it might suit students to complete VBA Assessment 1 before looking at Lab 4 and Week 5 Exercises. Do not let VBA Assessment slide very late as you will need time to prepare for Written Assessment 1. Really you should be looking to complete VBA Assessment 1 a few days before the due date. There will be no lab in Week 6 in order to give you more time.

### Written Assessment 1 – Week 7

25% Written Assessment 1, based on Weeks 1-5, so everything up to p.73.

It will be a one hour assessment, but I am going to give ye 15 minutes grace, as well as 15 minutes to upload. The test will run therefore from 09.30 to 11.00, Tuesday 9 March. It is open book — you can use your manual, any Canvas materials, as well as Excel/VBA.

40% of the marks will be numerical methods (Euler, TTT, Heun); finding approximations like p.34, Q.1-3, p.37, Q.1-3, p.48, Q.1-4 (non-Excel parts of Q.4), p.60, Q.4-7.

30% of the marks will be numerical analysis; understanding these methods and their errors like p.34, Q. 4, p.38, Q.4-6. p.49, Q.5-8

15% will be neither; like p.17, Q. 1-2, p.40, Q.1-3, p.60, Q.1-3

15% will be Runge-Kutta, like p.72, Q.1-2.

## Week 5

### Lectures

There are 82 minutes of lectures. You could schedule about 2 hours to watch them and take the notes in your manual. You might need this extra time above 82 minutes because you will want to pause me. You should also take note of any confusions you have to ask about in the regular Q & A.

### Lab 4

Lab 4 may not be up until 18 February, and can be attempted after watching the lectures above (submission not live until 20 February).

### Theory Exercises and Q & A

p.60, Q.1-7

p.72, Q.1-3

Q & A to ask about Theory Exercises or anything else every Tuesday 12.30 (waiting room open 12.25).

## Week 6

We will look at boundary value problems (in particular the Shooting Method and Goal Seek).

There will be no lab in order to give you time to complete VBA Assessment 1.

## Assessment

This is provisional and subject to change.

1. Week 6, 25% First VBA Assessment, Based (roughly) on Weeks 1-4
2. Week 7, 25 % In-Class Written Test, Based (roughly) on Weeks 1-5
3. Week 11, 25% Second VBA Assessment, Based (roughly) on Weeks 6-9
4. Week 12, 25% Written Assessment(s), Based on Weeks 6-11

## Study

Study should consist of

• doing exercises from the notes
• completing VBA exercises

## Student Resources

Please see Student Resources  for information on the Academic Learning Centre, etc..

### VBA Assessment 1 – Week 6

Hopefully I have designed something in which you can take your time, get things right, get some good marks, and show off what you can do.  Information on Written Assessment next week.

## Week 4

### Lectures

There are 84 minutes of lectures. You could schedule about 2 hours to watch them and take the notes in your manual. You might need this extra time above 84 minutes because you will want to pause me. You should also take note of any confusions you have to ask about in the regular Q & A.

The Q&A is dedicated to answering questions but do not hesitate to contact me with questions at any time. My usual modus operandi is to answer emails every morning.

### Lab

Once you have watched the lectures you should attempt VBA Lab 3.

### Theory Exercises and Q & A

p.48, Q. 1-5

p. 48, Q.5-8

Q & A to ask about Theory Exercises or anything else every Tuesday 12.30 (waiting room open 12.25).

Q&A II Slides

## Week 5

We will continue looking at second order differential equations and how to attack them numerically.

We will begin a quick study of Runge-Kutta Methods.

In VBA we will look at Lab 4, on Second Order Initial Value Problems.

## Assessment

This is provisional and subject to change.

1. Week 6, 25% First VBA Assessment, Based (roughly) on Weeks 1-4
2. Week 7, 25 % In-Class Written Test, Based (roughly) on Weeks 1-5 (notice in Week 5)
3. Week 11, 25% Second VBA Assessment, Based (roughly) on Weeks 6-9
4. Week 12, 25% Written Assessment(s), Based on Weeks 6-11

## Study

Study should consist of

• doing exercises from the notes
• completing VBA exercises

## Student Resources

Please see Student Resources  for information on the Academic Learning Centre, etc..

In theory (see Learner Workload) you are supposed to spend seven hours per week on MATH7016. My recommendation is that you watch the lecture material, then complete the lab, and if there is any time left over do the suggested theory exercises.

It is up to you to decide when you complete learning tasks and timetable yourself. There are four hours of MATH7016 slots on your timetable but remember that half an hour is taken up with the weekly Q&A. Learners should decide for themselves whether the weekly Q&A is helping their learning.

## Week 3

### Lectures

There are 79 minutes of lectures. You could schedule about 2 hoursto watch them and take the notes in your manual. You might need this extra time above 79 minutes because you will want to pause me. You should also take note of any confusions you have to ask about in the regular Q & A.

The Q&A is dedicated to answering questions but do not hesitate to contact me with questions at any time. My usual modus operandi is to answer emails every morning.

### Lab

Once you have watched the lectures you should attempt VBA Lab 3.

### Theory Exercises and Q & A

p.37, Q.1-4

p.40, Q.1-3

p. 38, Q.5-6

Q & A to ask about Theory Exercises or anything else every Tuesday 12.30 (waiting room open 12.25).

## Week 4

We can avoid implicit differentiation by looking at Huen’s Method, which is an adjustment of Euler’s Method in that it uses lines.

We will also introduce second order differential equations and how to attack them numerically.

In VBA we will look at Lab 3, on Heun’s Method.

## Assessment

This is provisional and subject to change. Notice for the first assessment will be in Week 4

1. Week 6, 25% First VBA Assessment, Based (roughly) on Weeks 1-4
2. Week 7, 25 % In-Class Written Test, Based (roughly) on Weeks 1-5
3. Week 11, 25% Second VBA Assessment, Based (roughly) on Weeks 6-9
4. Week 12, 25% Written Assessment(s), Based on Weeks 6-11

## Study

Study should consist of

• doing exercises from the notes
• completing VBA exercises

## Student Resources

Please see Student Resources  for information on the Academic Learning Centre, etc..

In theory (see Learner Workload) you are supposed to spend seven hours per week on MATH7016. My recommendation is that you watch the lecture material, then complete the lab, and if there is any time left over do the suggested theory exercises.

It is up to you to decide when you complete learning tasks and timetable yourself. There are four hours of MATH7016 slots on your timetable but remember that half an hour is taken up with the weekly Q&A. Learners should decide for themselves whether the weekly Q&A is helping their learning.

## Week 2

### Lectures

There are about 82 minutes of lectures. You could schedule about 2 hours and 10 minutes to watch them and take the notes in your manual. You need this extra time above 82 minutes because you will want to pause me. You should also take note of any confusions you have to ask about in the regular Q & A (starting Tuesday 2 February)

(Last year with the strike I recorded the same stuff in a classroom)

The Q&A is dedicated to answering questions but do not hesitate to contact me with questions at any time. My usual modus operandi is to answer emails every morning.

### Lab

Once you have watched the lectures you should attempt VBA Lab 2.

### Theory Exercises and Q & A

p.34, Q.1-4

Q & A to ask about Theory Exercises or anything else every Tuesday 12.30 (waiting room open 12.25).

## Week 3

We will do more on Taylor Series and the Euler Method. When we have that done we will look at Huen’s Method.

In VBA we will finish off our Euler Method Lab.

## Assessment

I really have not yet thought about assessment but this is what I am thinking at the moment. This is provisional and subject to change.

1. Week 6, 25% First VBA Assessment, Based (roughly) on Weeks 1-4
2. Week 7, 25 % In-Class Written Test, Based (roughly) on Weeks 1-5
3. Week 11, 25% Second VBA Assessment, Based (roughly) on Weeks 6-9
4. Week 12, 25% Written Assessment(s), Based on Weeks 6-11

## Study

Study should consist of

• doing exercises from the notes
• completing VBA exercises

## Student Resources

Please see Student Resources  for information on the Academic Learning Centre, etc..

In theory (see Learner Workload) you are supposed to spend seven hours per week on MATH7016. My recommendation is that you watch the lecture material (n/a in Week 1 if you attended the live lectures), then complete the lab, and if there is any time left over do the suggested theory exercises.

## Week 1

### Lectures

We had two hours of Live Zoom. These should be recorded in “Zoom” on the left there.

By briefly looking at a number of examples (many of which we have seen before), we had a review of some central ideas from approximation theory such as approximation, measurement erroraccuracy & precisioniterationconvergencemeshingerror, etc.

We looked at where ordinary differential equations come into Engineering, and we started talking about Euler’s Method.

### Lab

Where you find the time to do your VBA Lab 1 is up to you but I recommend using the two hours on your timetable. If you are serious about MATH7016 you might need to spend more than two hours.

### Theory Exercises and Q & A

p.17, Q.1-2

Q & A on these and any other MATH7016 topic next Tuesday 12.30 (waiting room open 12.25).

## Week 2

We will look at the Euler Method, and then start looking at big $\mathcal{O}$ notation, and Taylor Series.

Once I record the lectures I will put together a Week 2 announcement like this one.

## Assessment

I really have not yet thought about assessment but this is what I am thinking at the moment. This is provisional and subject to change.

1. Week 6, 25% First VBA Assessment, Based (roughly) on Weeks 1-4
2. Week 7, 25 % In-Class Written Test, Based (roughly) on Weeks 1-5
3. Week 11, 25% Second VBA Assessment, Based (roughly) on Weeks 6-9
4. Week 12, 25% Written Assessment(s), Based on Weeks 6-11

## Study

Study should consist of

• doing exercises from the notes
• completing VBA exercises

## Student Resources

Please see the Student Resources  for information on the Academic Learning Centre, etc..

I am not sure has the following observation been made:

When the Jacobi Method is used to approximate the solution of Laplace’s Equation, if the initial temperature distribution is given by $T^0(\mathbf{x})$, then the iterations $T^{\ell}(\mathbf{x})$ are also approximations to the solution, $T(\mathbf{x},t)$, of the Heat Equation, assuming the initial temperature distribution is $T^0(\mathbf{x})$.

I first considered such a thought while looking at approximations to the solution of Laplace’s Equation on a thin plate. The way I implemented the approximations was I wrote the iterations onto an Excel worksheet, and also included conditional formatting to represent the areas of hotter and colder, and the following kind of output was produced:

Let me say before I go on that this was an implementation of the Gauss-Seidel Method rather than the Jacobi Method, and furthermore the stopping rule used was the rather crude $|T^{\ell+1}_{i,j}-T^{\ell}_{i,j}|<\varepsilon$.

However, do not the iterations resemble the flow of heat from the heat source on the bottom through the plate? The aim of this post is to investigate this further. All boundaries will be assumed uninsulated to ease analysis.

## Discretisation

Consider a thin rod of length $L$. If we mesh the rod into $n$ pieces of equal length $\Delta x=L/n$, we have discretised the rod, into segments of length $\Delta x$, together with ‘nodes’ $0=x_0<\Delta x=x_1<2\Delta x=x_2<\cdots.

Suppose are interested in the temperature of the rod at a point $x\in[0,L]$, $T(x)$. We can instead consider a sampling of $T$, at the points $x_i$:

$\displaystyle T(x_i)=T(i\Delta x)=:T_i$.

Similarly we can mesh a plate of dimensions $W\times H$ into an $n\times m$ rectangular grid, with each rectangle of area $\Delta x\Delta y$, where $n\Delta x=W$ and $m\Delta y=H$, together with nodes $x_{i,j}=(i\Delta x,j\Delta y)$, and we can study the temperature of the plate at a point $\mathbf{x}\in[0,W]\times [0,H]$ by sampling at the points $x_{i,j}$:

$\displaystyle T(x_{i,j})=T(i\Delta x,j\Delta y)=:T_{i,j}$.

We can also mesh a box of dimension $W\times D\times H$ into an $n_1\times n_2\times n_2$ 3D grid, with each rectangular box of volume $\Delta x\Delta y\Delta z$, where $n_1\Delta x=W$, $n_2\Delta y=D$, and $n_3\Delta z=H$, together with nodes $x_{i,j,k}=(i\Delta x,j\Delta y,k\Delta z)$, and we can study the temperature of the box at the point $\mathbf{x}\in [0,W]\times [0,D]\times [0,H]$ by sampling at the points $x_{i,j,k}$:

$\displaystyle T(x_{i,j,k})=T(i\Delta x,j\Delta y,k\Delta z)=:T_{i,j,k}$.

## Finite Differences

How the temperature evolves is given by partial differential equations, expressing relationships between $T$ and its rates of change.

Keep in mind at all times:

“Any and all work, submitted at any time, will receive feedback.”

“Do not hesitate to contact me with questions at any time. My usual modus operandi is to answer all queries in the morning but sometimes I may respond sooner.”

## Week 13

### Catch Up/Revision of Lab 8 Material

The final assessment, based on Lab 8, takes place 11:00, Tuesday 12 May.

If you have not yet done so, you will undertake the learning described in Week 10. Perhaps you should also look at the theory exercises described in Week 10.

The final assessment will ask you to do the following:

Consider:

$\displaystyle k\cdot \frac{\partial^2 T}{\partial x^2}=\frac{\partial T}{\partial t}$      (1)

For one (i.e. $x_1$ or $x_2$ or $x_3$) or all (i.e. general $x_i$), use equations (2) and (3) to write (1) in the form:

$T^{\ell+1}_i=f(T_j^\ell),$

i.e. find the temperature at node $i$ at time $\ell+1\equiv(\ell+1)\cdot \Delta t$ in terms of the temperatures at the previous time $\ell\equiv \ell\cdot \Delta t$. You may take $\Delta t=0.5$.

\begin{aligned} \left.\frac{dy}{dx}\right|_{x_i}&\approx \frac{y(x_{i+1})-y(x_i)}{\Delta x}\qquad(2) \\ \left.\frac{d^2y}{dx^2}\right|_{x_i}&\approx \frac{y(x_{i+1})-2y(x_i)+y(x_{i-1})}{(\Delta x)^2}\qquad(3) \end{aligned}

so if you want to send on this work for feedback please do so.

If you submitted a Lab 8 either back before 6 April, or after, I will invite you, if necessary, to take on board the feedback I gave to that submission, and resubmit a corrected version for further feedback.

Submit work — VBA or Theory, catch-up or revision — based on Week 10 to the Lab 8 VBA/Theory Catch-up/Revision II assignment on Canvas by midnight 9 May. If you have any further questions, I am answering emails seven days a week. Email before midnight Monday 11 May to be guaranteed a response Tuesday 12 May. I cannot guarantee that I answer emails sent on Tuesday (although of course I will try).

Keep in mind at all times:

“Any and all work, submitted at any time, will receive feedback.”

“Do not hesitate to contact me with questions at any time. My usual modus operandi is to answer all queries in the morning but sometimes I may respond sooner.”

## Week 12/13

### 11:00 Tuesday 28 April, Week 12: Assessment Based on Lab 7

Students can submit work — VBA or Theory, catch-up or revision — based on Week 9 Lab 7 VBA/Theory Catch-up/Revision II assignment by today, Saturday 25 April.

If you have any further questions, I am answering emails seven days a week. Email before midnight Monday 27 April to be guaranteed a response Tuesday 28 April. I cannot guarantee that I answer emails sent on Tuesday morning (although of course I will try).

### Catch Up/Revision of Lab 8 Material

If you have not yet done so, you will undertake the learning described in Week 10.

If you feel like doing even more theory work on top of this, you will be asked to consider looking at the theory exercises described in Week 10.

If you have already conducted this learning, and submitted a Lab 8 either back before 6 April, or after, I will invite you, if necessary, to take on board the feedback I gave to that submission, and resubmit a corrected version for further feedback.

Students will be able submit work — VBA or Theory, catch-up or revision — based on Week 10 to a Lab 8 VBA/Theory Catch-up/Revision assignment on Canvas (due dates 3 May and 9 May).

## Week 14

### 11:00 Tuesday 12 May, Week 14: Assessment Based on Lab 8

This assessment will ask you to do the following:

Consider:

$\displaystyle k\cdot \frac{\partial^2 T}{\partial x^2}=\frac{\partial T}{\partial t}$      (1)

For one (i.e. $x_1$ or $x_2$ or $x_3$) or all (i.e. general $x_i$), use equations (2) and (3) to write (1) in the form:

$T^{\ell+1}_i=f(T_j^\ell),$

i.e. find the temperature at node $i$ at time $\ell+1\equiv(\ell+1)\cdot \Delta t$ in terms of the temperatures at the previous time $\ell\equiv \ell\cdot \Delta t$. You may take $\Delta t=0.5$.

\begin{aligned} \left.\frac{dy}{dx}\right|_{x_i}&\approx \frac{y(x_{i+1})-y(x_i)}{\Delta x}\qquad(2) \\ \left.\frac{d^2y}{dx^2}\right|_{x_i}&\approx \frac{y(x_{i+1})-2y(x_i)+y(x_{i-1})}{(\Delta x)^2}\qquad(3) \end{aligned}

Keep in mind at all times:

“Any and all work, submitted at any time, will receive feedback.”

“Do not hesitate to contact me with questions at any time. My usual modus operandi is to answer all queries in the morning but sometimes I may respond sooner.”

## Week 11

### 20% VBA Assessment Based on Lab 6

Thank you to everyone for completing the assignment.

### Catch Up/Revision

You are advised to catch-up on the learning described in Week 9.

If you have already conducted this learning, and submitted a Lab 7 either back before 30 March, or after, I will invite you, if necessary, to take on board the feedback I gave to that submission, and resubmit a corrected version for further feedback. If you feel like doing even more theory work on top of this, you will be asked to consider looking at the theory exercises described in Week 9.

Students can submit work — VBA or Theory, catch-up or revision — based on Week 9 to the Lab 7 VBA/Theory Catch-up/Revision I assignment on Canvas (by Sunday 19 April), or Lab 7 VBA/Theory Catch-up/Revision II assignment by Saturday 25 April.

## Week 12/13

### Catch Up/Revision

If you have not yet done so, you will undertake the learning described in Week 10.

If you feel like doing even more theory work on top of this, you will be asked to consider looking at the theory exercises described in Week 10.

If you have already conducted this learning, and submitted a Lab 8 either back before 6 April, or after, I will invite you, if necessary, to take on board the feedback I gave to that submission, and resubmit a corrected version for further feedback.

Students will be able submit work — VBA or Theory, catch-up or revision — based on Week 10 to a Lab 8 VBA/Theory Catch-up/Revision assignment on Canvas.

I may have two due dates. Perhaps Sunday 3 May and Saturday 9 May.

## PROVISIONAL: Tuesday 12 May, Week 14: Assessment Based on Lab 8

This assessment will ask you to do the following:

Consider:

$\displaystyle k\cdot \frac{\partial^2 T}{\partial x^2}=\frac{\partial T}{\partial t}$      (1)

For one (i.e. $x_1$ or $x_2$ or $x_3$) or all (i.e. general $x_i$), use equations (2) and (3) to write (1) in the form:

$T^{\ell+1}_i=f(T_j^\ell),$

i.e. find the temperature at node $i$ at time $\ell+1\equiv(\ell+1)\cdot \Delta t$ in terms of the temperatures at the previous time $\ell\equiv \ell\cdot \Delta t$. You may take $\Delta t=0.5$.

\begin{aligned} \left.\frac{dy}{dx}\right|_{x_i}&\approx \frac{y(x_{i+1})-y(x_i)}{\Delta x}\qquad(2) \\ \left.\frac{d^2y}{dx^2}\right|_{x_i}&\approx \frac{y(x_{i+1})-2y(x_i)+y(x_{i-1})}{(\Delta x)^2}\qquad(3) \end{aligned}