SCHOOL OF MATHEMATICAL SCIENCES

SEMESTER SEMESTER 2020-2021

MTHS3001 - COMPUTERISED MATHEMATICAL METHODS IN ENGINEERING

Coursework 2

Your neat, clearly-legible solutions should be submitted electronically as a pdf file via the MTHS3001 Moodle page by 12:00 on Friday 21 May 2021. A scan of a handwritten solution is acceptable. Since this work is assessed, your submission must be entirely your own work (see the University’s policy on Academic Misconduct). Submissions made more than three weeks after the release date will receive a mark of zero. Please try to make your submission within two weeks of the release date.

1. You are given the temperature measurements in Table 1.

(a) Write down the system of equations for the unknown coefficients ci, i = 0,… , 3 of the third order polynomial p3 (t) that interpolates the 4 first measurement points, i.e. . [4 marks]

(b) Use Gaussian elimination to solve this system and find the unknown coefficients ci . You need to specify all row operations that you use. Only noting the end result will not give you full points. [9 marks]

(c) Using the results from part (a), what value do you obtain for p3 (11)? Plot the data from Table 1 together with the predictions from p3 . [5 marks]

(d) You are now asked if the polynomial that you found in part (a) provides an accurate description of the data beyond the four data points that you used to fit the polynomial. What would you say? You need to justify your answer. [1 mark]

(e) Do you think you could use the Gauss-Jacobi method to solve the linear system derived in part (a)? You need to justify your answer. [1 mark]

2. Consider the following differential equation

which describes the time evolution of caterpillars that can have devastating effects on spruce and fir forests.

(a) Use the forward Euler method with h = 1 for three initial values c(0) ∈ {0.5, 1.5, 2.5} and note down the first 4 steps. You need to show full workings. Only noting the end result will not give you full points. [4 marks]

(b) For the same initial conditions as in part (a) and h = 1, use the Euler method for an appropriate number of time steps until the rate of change  is smaller than 0.0001. What are the final values of c for the three initial conditions, and how many time steps does it take? It is convenient to modify the Matlab script that contains the Euler algorithm for this part. [6 marks]

(c) Run 200 Euler steps with h = 1 and initial conditions c(0) = 2.01 and c(0) = 1.99. What do you observe? [2 marks]

(d) Repeat part (c) for the midpoint method. It might be convenient to modify the rk4.m function to code up the midpoint method. [8 marks]