## Week 4

We could avoid the implicit differentiation of $k$-Term-Taylor Methods by looking at Huen’s Method, which is an adjustment of Euler’s Method in that it uses lines.

We started talking about second order differential equations.

We also looked at calculating a Maclaurin Series

In VBA we did Heun’s Method. Unfortunately the storm messed up the Lab for DME2C and they need to do the lab remotely as per the email, as well as MCQIII.

## Further Remark: Heun’s Method

There is a statement in the notes that Heun’s method is of order the second derivative. The exact situation with the Heun’s method error is fairly complicated. The errors depend not just on the nature of the solution but also the nature of the differential equation. For example, consider

• $\dfrac{dy}{dx}=2\sqrt{y(x)}$, $y(0)=1$,
• $\dfrac{dy}{dx}=\dfrac{2y(x)}{1+x}$, $y(0)=1$,
• $\dfrac{dy}{dx}=2x+2$, $y(0)=1$.

The function $y(x)=(x+1)^2$ is a solution of each of these, but only the third will yield zero errors with Heun. It turns out that the Heun’s method error depends in an important way on how the $F(x,y)$ depends on $y$; so above we have $F(x,y)=2\sqrt{y}$, then $F(x,y)=2y/(1+x)$, then $F(x,y)=2x+2$. How exactly that works we will not get into but we will say:

1. It is still true that the local Heun errors are $\mathcal{O}(h^3)$, and so
2. The global Heun error is $\mathcal{O}(h^2)$, and also
3. Heun’s method will have problems when the third derivative is large.

## 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.

## Week 6

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

## DME2C Lab 18 March

DME2C have already been impacted negatively by Storm Eunice. Rather than have them hit again with no lab on this day (for the university is closed for the bank holiday), we will have no DME2C Lab in Week 8, but two in Week 9, with the additional lab held:

• Monday, 21 March, C125.

I will also set up a remote option for those who would prefer that.

## Assessment

See Canvas for the assessment schedule. There will be more information on assessments the Friday of the week two before the assessment week (i.e. info on Friday of Week 4 about the Week 6 assessment).

## MCQ League

The standings after MCQIII will not be updated until next week.

## Study

Study should consist of

• doing exercises from the notes
• completing VBA exercises