Further Maths - Numerical Methods

Created: December 08, 2021

See Also

Try out an interactive visualisation of Euler’s method here: Euler’s method.

Flashcards

2021-12-08

How could you summarise Euler’s method for solving first-order differential equations?

Start with some point on the curve and then follow the direction of the curve.

If a gradient is given by $\frac{\text{d}y}{\text{d}x}$, how much would you increase the $y$-coordinate for a step size of $h$?
\[y_1 = y_0 + \frac{\text{d}y}{\text{d}x} h\]

What does using the assumption

\[\left( \frac{\text{d}y}{\text{d}x} \right) _ 0 \approx \frac{y _ 1 - y _ 0}{h}\]

mean you should do in a question?

Use Euler’s method.

What assumption are you making for Euler’s method?
\[\left( \frac{\text{d}y}{\text{d}x} \right)_0 \approx \frac{y_1 - y_0}{h}\]

What is the formula for $y _ {r + 1}$ given $\frac{\text{d}y}{\text{d}x}$ and $y _ r$ using Euler’s method?
\[y_{r + 1} = y_r + \left( \frac{\text{d}y}{\text{d}x} \right)h\]

If you were asked to use $5$ iterations to approximate a solution at $x = 2$ given the point $(1, 2)$ using Euler’s method, what would your step size be?
\[h = 0.2\]

What does using the assumption

\[\left( \frac{\text{d}y}{\text{d}x} \right) _ 0 \approx \frac{y _ 1 - y{-1}}{2h}\]

mean you should do in a question?

Use the midpoint method.

What is the improvement on Euler’s method called?

The midpoint method.

What assumption are you making for the midpoint method?
\[\left( \frac{\text{d}y}{\text{d}x} \right)_0 \approx \frac{y_1 - y{-1}}{2h}\]

What is the formula for $y _ {r+1}$ in terms of $y _ r$ and $y _ {r-1}$ using the midpoint method?
\[y_{r+1} \approx y_{r-1} + 2h\left(\frac{\text{d}y}{\text{d}x}\right)_{y_r}\]

What do you often need to do in order to use the midpoint method and given one point?

Use Euler’s method to find the next point.

What’s a nice way of answering numerical method questions?

Using a table.

2021-12-10

What assumption are you making for approximating solutions to second order differential equations?
\[(\frac{\text{d}^2y}{\text{d}x^2})_{0} \approx \frac{\left(\frac{\text{d}x}{\text{d}y}\right)_0 - \left(\frac{\text{d}x}{\text{d}y}\right)_{-1}}{h}\]
\[(\frac{\text{d}^2y}{\text{d}x^2}) _ {0} \approx \frac{\left(\frac{\text{d}x}{\text{d}y}\right) _ 0 - \left(\frac{\text{d}x}{\text{d}y}\right) _ {-1}}{h}\]

What is this assumption in terms of $y _ 1$, $y _ 0$ and $y _ {-1}$?
\[(\frac{\text{d}^2y}{\text{d}x^2})_{0} \approx \frac{y_1 - 2y_0 + y_{-1}}{h^2}\]

What’s the formula for $y _ {r+1}$ in terms of $y _ r$ and $y _ {r-1}$ using the approximation

$$(\frac{\text{d}^2y}{\text{d}x^2}){0} \approx \frac{y_1 - 2y_0 + y{-1}}{h^2}?
\[y_{r + 1} \approx 2y_4 - y_{r-1} + h^2(\frac{\text{d}^2y}{\text{d}x^2})_{r}\]

When do you have to use simultaneous equations when approximating a second-order differential equation?

When they ask you to use the midpoint method.

What is Simpson’s rule used for?

Approximating integrals.

How does Simpson’s rule work?

Splitting curves up into quadratics.

What is Simpson’s rule for $$

\int^b_a f(x) dx

$$ in terms of even values, odd values and endpoints?
\[\int^b_a f(x) dx \approx \frac{1}{3}h((\text{endpoints}) + 4(\text{odd values}) + 2(\text{even values}))\]

Why can’t you use Simpson’s rule when splitting up a curve into 7 strips/intervals?

Simpson’s rule only works with an even number of strips/intervals.

2022-04-24

What’s a stupid mnemonic for remembering Simpson’s rule?

On the third of Haugust, it’s the end 4 Mr Odd. He was 2 even.

“On the third of Haugust, it’s the end 4 Mr Odd. He was 2 even.” Can you turn this into Simpson’s rule?
\[\frac{1}{3} h (\text{end} + 4\text{odd} + 2\text{even})\]


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