Further Maths - Numerical Methods


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 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?


\[y _ {r + 1} = y _ r + \left( \frac{\text{d}y}{\text{d}x} \right)h\]
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.

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?


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

2021-12-10

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.

#####

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

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}

\[<details class="flashcard" id="bb52e9e755504a728622da0d69e3d2ef"> <summary class="flashcard-front"><p>What’s a nice way of answering numerical method questions?</p> </summary> <hr> <div class="flashcard-back"> <p>Using a table.</p> </div> </details> <details class="flashcard" id="1025d45e68d54d88bc4ba708b3ea5fb6"> <summary class="flashcard-front"><p>What assumption are you making for approximating solutions to second order differential equations?</p> </summary> <hr> <div class="flashcard-back"> \[(\frac{\text{d}^2y}{\text{d}x^2}) _ {0} \approx \frac{\left(\frac{\text{d}x}{\text{d}y}\right) _ 0 - \left(\frac{\text{d}x}{\text{d}y}\right) _ {-1}}{h}\] </div> </details> <details class="flashcard" id="e0ea3d09b0b04c6aa41fb86518bae335"> <summary class="flashcard-front">\[(\frac{\text{d}^2y}{\text{d}x^2}) _ {0} \approx \frac{\left(\frac{\text{d}x}{\text{d}y}\right) _ 0 - \left(\frac{\text{d}x}{\text{d}y}\right) _ {-1}}{h}\] <p>What is this assumption in terms of $y _ 1$, $y _ 0$ and $y _ {-1}$?</p> </summary> <hr> <div class="flashcard-back"> \[(\frac{\text{d}^2y}{\text{d}x^2}) _ {0} \approx \frac{y _ 1 - 2y _ 0 + y _ {-1}}{h^2}\] </div> </details> ##### 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}))

$$

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}?


\[y _ {r + 1} \approx 2y _ 4 - y _ {r-1} + h^2(\frac{\text{d}^2y}{\text{d}x^2}) _ {r}\]

2022-04-24

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.




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