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Due to this change, the information in Sections 3.6 and 3.7 is now out of date.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

## Week 5

In class we will look at higher order initial value problems and in VBA we will look at Lab 4.

## Study

Study should consist of

• doing exercises from the notes
• completing VBA exercises