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Computational Mathematics

Attach one pdf for your submission: in Q1 – 6 show your working and include this in the pdf submission. For Q7 write this as a short report.

For Q7 write this as a short report.

This includes an investigation about percolation and how to calculate percolation properties. Do some reading around this.

A useful web tool is: xsources.github.io/sitepercol.html

- Solve the following set of equations by converting them into a matrix notation and using either the Gauss Jordan or matrix inversion method

5𝑥1 − 𝑥2 + 2𝑥3 = 3

2𝑥1 + 4𝑥2 + 𝑥3 = 8

𝑥1 + 3𝑥2 − 3𝑥3 = 2

(10 marks)

In an electrical circuit, a component at 𝑡 = 0 has a charge 𝑄0 that then discharges exponentially with the remaining charge 𝑄 proportional to the initial charge. Show that (i) time taken for the amount to become 𝑄0/2 is 𝑇 where 𝑄 =

𝑄0𝑒−𝑙𝑛2 𝑡/𝑇. This is known as the ½ life. (ii) What is the time taken for the charge to reduce to 𝑄0/20?

- Solve the equation

(𝑥 − 2) 𝑑𝑦 − 𝑦 = 3, 𝑦(4) = 10.

𝑑𝑥

(10 marks)

- Find the general solution of

𝑑𝑦 + 3𝑦 = 𝑒4𝑥.

𝑑𝑥

- Solve this equation and find the particular solution

(1 + 𝑥2) 𝑑𝑦 + 3𝑥𝑦 = 5𝑥

𝑑𝑥

(10 marks)

where 𝑦 = 2 when 𝑥 = 1.

(10 marks)

The electrical resistance 𝑅 of a particular component is Normally distributed with mean 60 ohms and variance 25 ohms2. Find (i) Pr{𝑅 ≤ 65}, (ii) Pr{𝑅 > 52}; (iii) Pr{50 < 𝑅 ≤ 62}. (iv) It is required to reduce Pr{𝑅 ≤ 65} to 0.1 by changing the mean of the distribution while keeping the variance at 25 ohms2, what is the value of the mean required?

- One form of stochastic system is how an event percolates from one side to the other of a medium that contains inherent random properties. The route to cross is governed by a level of randomness 𝑝 at each spatial point and so the ability to cross from one side to the other occurs at a critical probability level 𝑝𝑐. These systems can correspond to the spread of a fire in a forest, an electrical discharge through an insulator or the way water seeps down through a mixture of sand and rock. They all exhibit an equivalent stochastic percolation property that occurs at 𝑝𝑐. Your task here is to evaluate the algorithm for the seeping of water through rocks with the python code py (repeating this analysis to create an ensemble). The algorithm to evaluate the percolation property is: a drop of liquid starts in the middle of the top layer (row 1). It then moves according to the following four options, where options with lower numbers have higher precedence.

- If space directly below is sand, move
- If space below and to the left is sand, move
- If space below and to the right is sand, move
- If the space directly to the right is sand, move

If none of these moves can be made, the drop of liquid is stuck.

In this code you can vary a) the size of the sand and rock system (𝑁 × 𝑁), b) the probability level of randomness 𝑝 that represents the proportion of impenetrable rock and porous sand – that is arranged randomly and c) the number of realizations prep from which to evaluate the statistical ensemble properties.

- For 𝑁 = 100 establish where the critical percolation level occurs 𝑝𝑐 by varying

𝑝 and use python graphics to show the 2d system. Use differing numbers of realizations.

(10 marks)

- Assess how 𝑝𝑐 varies for 𝑁 = {10, 50, 100, 200, 400} and prep =

{100, 500, 1000, 2000, 4000}. In the large 𝑁 limit it is expected that 𝑝𝑐 will give a constant ‘universal’ value, discuss the performance of this algorithm in this context.

(10 marks)

- More general percolation algorithms are needed to evaluate the percolation threshold 𝑝𝑐 for forest fires, electrical discharge and more widely. Suggest an algorithm you could use to find 𝑝𝑐 in these cases, and have a go at implementing it. Read about other percolation theory applications. Explain in 400 to 500 what their main features are (hint: what types of system do they represent) and the type of applications. Include 1 figure or table from parts 7(i) or (ii) to illustrate

(20 marks – including 10 marks for the originality of ideas)

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