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## Week 10

We looked at finite differences for the Heat Equation. This completes the examinable written material.

In VBA we implemented same.

## Week 11 — 2nd 20% VBA Assessment

In the 09:00 class we will have a revision session, geared towards the 20% VBA Assessment 2. This will look at the 20% VBA Assessment 2 Tutorial Sheet It might therefore be a good idea to go through this before next week.

In the 12:00 class we will have a revision session, geared towards the 40% Written Assessment 2. This will look at the 40% Written Assessment 2 Tutorial Sheet It might therefore be a good idea to go through this before next week.

Formulae will be provided in the VBA 2 Assessment.

To understand how your student numbers generate constants (see below) see this VBA Test 2 from last year (do not read this as a sample – it included e.g. the Heat Equation which you will not be examined on (in VBA) and the Laplace’s Equation might be slightly simpler than what ye will have).

See last week’s Weekly Summary for the Format of the VBA Assessment 2.

## Week 9

We finished talking about Laplace’s Equation and started talking about the Heat Equation.

In VBA we looked at finite difference methods for Laplace’s Equation. This completes the examinable VBA material. The Heat Equation that we cover in Week 10 will not be examinable.

## Week 10

We will look at finite differences for the Heat Equation. This completes the examinable written material.

In VBA we will implement same.

## VBA Assessment 1 – Results

Have been emailed.

## Written Assessment 1 – Results

I would hope to have these with ye early next week.

## Week 8

We continued our look at Finite Differences for differential equations, including Laplace’s Equation.

In VBA we looked at the shooting method (and Goal Seek) and finite differences for boundary value problems.

## Week 9

We will continue our look at Laplace’s Equation.

In VBA we will look at Laplace’s Equation.

## Assessment

The following is the proposed assessment schedule:

2. Week 7, 20 % In-Class Written Test, More Info in Week 5
3. Week 11, 20% Second VBA Assessment, More Info in Week 9

## Study

Study should consist of

• doing exercises from the notes
• completing VBA exercises

## VBA Assessment 1 – Results

I would hope to have these with ye by the end of Week 8.

## Written Assessment 1 – Results

I would hope to have these with ye by the end of Week 8

## Week 7

In the first lecture you sat your first written assignment.

In the second lecture we looked more at boundary value problems (in particular the Shooting Method and Goal Seek). We started talking about Finite Differences.

In VBA we looked at Runge-Kutta methods.

## Written Assessment 1

Written Assessment 1 takes place Tuesday 13 March at 09:00 in the usual lecture venue.

Here is a copy of last year’s assessment. This should give you an idea of the length and format but not what questions are coming up. There are far more things I could examine.

Roughly, everything up to p. 57 is examinable. More specifically:

### Maclaurin/Taylor Series

Examples 1 & 2 on p. 17; Q. 1-2 on p.19

### ODEs in Engineering

p.22, Example, Q. 1-4

### Euler Method

p.29, Examples 1-4; p.38, Q. 1-6, 8-9; p.48, Q. 1, 5(a)

### Three Term Taylor Method

p.35, Example; p. 36, Examples 1-2; p.38, Q. 6-7, 10-14; p.48, Q. 3

### Heun’s Method

p.43, Examples 1-2; p.48, Q. 2, 4, 5(b)

### Second Order Differential Equations

p.52, Example; p.54, Example; p. 56, Q. 1-14 (some repetition here).

## VBA Assessment 1

VBA Assessment 1 will take place in Week 6, (6 & 9 March) in your usual lab time. You will not be allowed any resources other than the library of code (p.124) and formulae (p.123 parts 1 and 2) at the end of the assessment (both are provided on the assessment paper). More information in last week’s weekly summary.

## VBA Assessment 1

VBA Assessment 1 will take place in Week 6, (6 & 9 March) in your usual lab time. You will not be allowed any resources other than the library of code (p.124) and formulae (p.123 parts 1 and 2) at the end of the assessment (both are provided on the assessment paper). More information in last week’s weekly summary.

## Written Assessment 1

Written Assessment 1 takes place Tuesday 13 March at 09:00 in the usual lecture venue.

Here is a copy of last year’s assessment. This should give you an idea of the length and format but not what questions are coming up. There are far more things I could examine.

Roughly, everything up to p. 57 is examinable. More specifically:

### Maclaurin/Taylor Series

Examples 1 & 2 on p. 17; Q. 1-2 on p.19

### ODEs in Engineering

p.22, Example, Q. 1-4

### Euler Method

p.29, Examples 1-4; p.38, Q. 1-6, 8-9; p.48, Q. 1, 5(a)

### Three Term Taylor Method

p.35, Example; p. 36, Examples 1-2; p.38, Q. 6-7, 10-14; p.48, Q. 3

### Heun’s Method

p.43, Examples 1-2; p.48, Q. 2, 4, 5(b)

### Second Order Differential Equations

p.52, Example; p.54, Example; p. 56, Q. 1-14 (some repetition here).

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Assessment

Considering our progress, I have decided to swap the positions of the first and second assessments. This time last year I had Section 1.2.3 completed but have (necessarily) slowed down this year. Last year Section 1.2.3 was tested both in the first (written) assessment and in the second (VBA) assessment.

It is more important that Section 1.2.3 is tested in the written component therefore the decision to switch the assessments.

Due to this change, the information in Sections 3.6 and 3.7 is now out of date.

The following is the proposed assessment schedule:

2. Week 7, 20 % In-Class Written Test, More Info in Week 5
3. Week 11, 20% Second VBA Assessment, More Info in Week 9

## VBA Assessment 1

VBA Assessment 1 will take place in Week 6, (6 & 9 March) in your usual lab time. You will not be allowed any resources other than the library of code (p.124) and formulae (p.123 parts 1 and 2) at the end of the assessment. The following is the proposed layout of the assessment:

### Q. 1: Numerical Solution of Initial Value Problem [80%]

Examples of initial value problems that might be arise include:

• Damping

$\displaystyle \frac{dv}{dt}=-\frac{\lambda}{m}v(t)$;           $v(0)=u$

• The motion of a free-falling body subject to quadratic drag:

$\displaystyle \frac{dv}{dt}=g-\frac{c}{m}v(t)^2$;           $v(0)=u$

• Newton Cooling

$\displaystyle \frac{d\theta}{dt}=-k\cdot (\theta(t)-\theta_R)$;           $\theta(0)=\theta_0$

• The charge on a capacitor

$\displaystyle \frac{dq}{dt}=\frac{E}{R}-\frac{1}{RC}q(t)$;           $q(0)=0$

Students have a choice of how to answer this problem:

• The full, 80 Marks are going for a VBA Heun’s Method implementation (like Lab 3).
• An Euler Method implementation (like Lab 2), gets a maximum of 60 Marks.

You will be asked to write a program that takes as input all the problem parameters, perhaps some initial conditions, a step-size, and a final time, and implements Heun’s Method (or Euler’s Method): similar to Exercise 1 on p. 114 and also Exercise 1 on p.109 (except perhaps implementing Heun’s Method).

If you can write programs for each of the four initial value problems above you will be in absolutely great shape for this assessment.

### Q. 2: Using your Program [20%]

You will then be asked to use your program to answer a number of questions about your model. For example, assuming Heun’s Method is used, consider the initial value problem (3.7) on p. 105.

1. Given, $v_0=0.2$, $m=3$, $\lambda=1.5$, $h=0.01$, approximate $v(0.3)$.
2. Given, $v_0=0.4$, $m=30$, $\lambda=1.5$, $h=0.1$, investigate the behaviour of $v(t)$ for large $t$.
3. Given $v_0=0.2$, $m=0.1$, $\lambda=1.5$, $h=0.5$, $T=10$, run the Heun program. Comment on the behaviour of $v(t)$. Run the same program except with $h=0.05$. Comment on the behaviour of $v(t)$.
4. Given, $v_0=0$, $m=3$, $\lambda=1.5$, $h=0.1$, $T=2$, run the Heun program. Comment on the behaviour of $v(t)$.

## Week 4

We jumped forward and looked at Heun’s Method in the 09:00 class. We went back then and looked at the Three Term Taylor Method in the afternoon. We stated in the afternoon that Heun’s Method gives the same answer as the Three Term Taylor, and without the need for implicit differentiation.

In VBA we worked on Lab 3. Those of us who did not finish the lab are advised to finish it outside class time, and are free to email me on their work if they are unsure if they are correct or not.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Week 3

We finished our work on Taylor Series, and used it to analyse the errors when using the Euler Method.

In VBA we should all have been able to finish Lab 2 (certainly up to automatic selection, p.108). Those who did so started on a Newton Cooling program. Students who have not completed these two programs (damper and Newton Cooling), are advised to work on them outside class.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Week 2

We developed the Euler Method for approximating the solution of differential equations. As we will need Taylor Series to analyse the error in this approximation — and improve Euler’s Method — we started looking at that.

In VBA we started programming the Euler Method to solve the problem of a damper.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

## Week 1

In Week 1, 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 error, accuracy & precision, iteration, convergence, meshing, error, big $\mathcal{O}$ notation, etc.

In VBA we had a quick review lab, focussing on plotting data, command buttons, message boxes, input boxes, If-statements and do-loops.