Saniya writes a positive integer on each edge of a square. She also writes at each vertex the product of the numbers on the two edges that meet at that vertex. The sum of the numbers at the vertices is 15. What is the sum of the numbers on the edges of the square?
The length of a rectangle is one inch less than threetimes the width. Find the dimensions if the perimeteris 70 inches.
Assignment 2: BVPs/Fourier series/Parabolic equations Subject: MTH307 Due date: 9 September 2020 Value: 12% The assignment consists of 4 questions and will be marked out of 50. Task: This is a written assignment that covers material from Weeks 3 to 6, i.e. Boundary value problems, Fourier series, and parabolic differential equations, and Tutorials 4 to 6. The questions are designed to demonstrate your mastery of subject material covered in Topics 3, 4 and 5 of the study guide. For each question you will receive marks for correctness, the selection and application of the correct process or method, clear indication of your intermediate calculations, and your ability to interpret the problem and present an appropriate mathematical solution. Rationale: This assessment task will assess the following learning outcomes: · be able to numerically calculate the solutions of ordinary differential equations using various methods. · be able to adapt existing code to produce numerical solutions for differential equations. · be able to generate suitable finite difference equations from differential equations. · be able to calculate the solutions of partial differential equations using various methods. Marking criteria All questions in this assignment involve problems with a sequence of several steps. These are marked using the following criteria: Criteria Description Correctness Arithmetic, algebra and calculations are correct (except possibly with some minor rounding errors). Process/Method The indicated/ correct method is selected and carried out completely. Communication/Working You have made it clear what you have done using an appropriate mix of text, mathematical notation, neat diagrams, code excerpts and computer
The mark for each question is determined by the proportion of your solution that satisfies these criteria. A general guideline is provided below: Full marks Partial marks Zero marks Correct answer written as a clear response to the original question. Full worked solutions provided that are clear, adequate and legible and use the correct mathematical notation and reasoning, with neat diagrams, code excerpts and computer output where appropriate. Correct answer with incomplete working which lacks the appropriate notation or reasoning. OR Incorrect or partially correct answer with working that uses the correct method and appropriate notation and reasoning. Method and solution are incorrect. No working to arrive at the solution is shown. Question 1 Higher Order Differential Equations (5 marks) For a stiff pendulum with length 4.9 metres and a small amount of air resistance (ignoring earth’s rotation and the shape of the pendulum etc) the motion satisfies the following 2nd order differential equation: 𝑦 ′′ − 0.15𝑦 ′ + 2 sin 𝑦 = 0. Suppose the pendulum starts at rest at an angle of 𝑦 = 0.12. 1. Write this equation as a system of first order linear differential equations. (2 marks) 2. Use a second or higher order numerical method with step size at most ℎ = 0.5 to predict the location of the pendulum after 1 second. (2 marks) 3. Comment on the accuracy of your prediction. (1 mark) Remarks. When 𝑦 is small sin 𝑦 ≈ 𝑦. This gives the more common small angle approximation. For this question, do NOT use the small angle approximation. The purpose of this exercise is to consider accuracy when you do not know the exact solution (in this case the solution is not elementary). The solution for the “small angle approximation” can be determined, but it is complicated. You can do this question using just a calculator (just remember to use as much precision as you can on your calculator). Alternatively, you can use some of the supplied Maple programs - see Resources/Maple/Chapter2, cut the relevant parts out and paste into a new document. You will need to alter the parameters. If you are using versions with plots (which is not necessary) then you will need to change the view as appropriate. Question 2 Finite differences for ODES (15 marks in total)
Consider the following ordinary differential equation (which is a type of Euler’s equation): 𝑦 ′′𝑡 2 + 2𝑦 ′ 𝑡 − 5𝑦 = 4𝑡 Consider several accompanying conditions: IV P1: 𝑦(1) = 1, 𝑦 ′ (1) = 2; IV P2: 𝑦(1) = 1, 𝑦 ′ (1) = 5; BV P: 𝑦(1) = 1, 𝑦(2) = 3. For the following parts use a step size ℎ ≤ 0.5. If you are doing it by hand, it is recommended you use the largest permissible size (ℎ = 0.5) in all calculations below. 1. Convert the differential equation into a finite difference equation. Hints: Ignore the conditions for the moment. You can use forward, backwards or centred differences for the derivative and double derivative. Several choices will work, but it is advisable to make your selections in order to minimise truncation errors. (3 marks) 2. Use the finite difference equation from part 1 to calculate numerically for IV P1, 𝑦(2). Hint: you will need to convert the derivative condition into a finite difference equation. As before, minimise the truncation error for your choice of differences. (3 marks) 3. Use the finite difference equation from part 1 to calculate numerically for IV P2, 𝑦(2). (3 marks) 4. Use parts 2 and 3 to estimate for BV P, 𝑦(1.5). (2 marks) 5. Solve the differential equation for all three conditions analytically to check the accuracy of your numerical answers. Hint: apply the transformation 𝑥 = 𝑙𝑛(𝑡) to convert to a 2nd order linear differential equation with constant coefficients. (2 marks) 6. Is the method used here based on finite differences more or less accurate than using the Euler method? Compare them with equal step size. (2 marks) Notes. This question can be done manually by using ℎ = 0.5. This will provide practice for the exam. For this assignment this is satisfactory (even if highly inaccurate!). However, if you want reasonable accuracy, you should decrease h and use a computer program.
Question 3 Fourier series (15 marks in total) 1. Find the Fourier series (by using integrals) for the function 𝑓(𝑥) = { 0 − 1 < 𝑥 < 0 𝑥 2 + 1 0 < 𝑥 < 1 where 𝑓(𝑥) has period 2. (This needs to be attempted by hand, but you can use Maple to check your answer.) (8 marks) 2. Generate graphical output for the series obtained in the previous part with 2 nonzero sine terms for −1 ≤ 𝑥 ≤ 1. (2 mark) 3. Generate graphical output for the series obtained in the previous part with 10 nonzero sine terms for −1 ≤ 𝑥 ≤ 1. (1 mark) 4. Generate graphical output for the sigma-approximation series obtained in the previous part with 2 nonzero sine terms for −1 ≤ 𝑥 ≤ 1. (2 mark) 5. Generate graphical output for the sigma-approximation series obtained in the previous part with 10 nonzero sine terms and for −2 ≤ 𝑥 ≤ 2. (1 mark) 6. Comment on the effect that sigma-approximation has on the convergence of these (truncated) series. (1 marks) Question 4 Diffusion-advection equation (15 marks in total) A commonly used partial differential equation when modelling fluids is the diffusion-advection equation (also called the convection-diffusion equation). It is a combination of the heat equation and advection equation, but we will consider it in its own right here. Consider a river where the flow is a constant 𝑢 = 0.5𝑚𝑠 −1 . At one point on the river a hydrologist is going to continually add a fixed concentration of 𝐶0 = 0.10𝑘𝑔𝑚−3 of dye. The dye will flow with the water but will also diffuse according to a diffusion coefficient of 𝐷 = 0.15𝑚2 𝑠 −1 . Let 𝐶(𝑥,𝑡) be the concentration 𝑥 metres downstream from the release point and 𝑡 seconds from starting the release. Furthermore, at 𝑡 = 0 the hydrologist has spread some more dye for the first 8𝑚. A simple model for this situation is given by the diffusion-advection equation: 𝜕𝐶 𝜕𝑡 + 𝑢 𝜕𝐶 𝜕𝑥 = 𝐷 𝜕 2𝐶 𝜕�
We have initial time and spatial conditions (respectively) of: 𝐶(𝑥, 0) = 𝑓(𝑥) = { 𝐶0 (1 − 𝑥 8 ) 0 ≤ 𝑥 ≤ 8 0 𝑥 > 8 𝐶(0,𝑡) = 𝑔(𝑡) = 𝐶0 In this question we will develop a numerical method to solve this PDE. 1. Classify the diffusion-advection equation as hyperbolic/parabolic/elliptic/mixed. Explain your answer. (2 marks) 2. Construct a finite difference equation for the diffusion-advection equation. (6 marks) 3. Use your finite difference equation to predict the value at 𝑥 = 2𝑚 and 𝑡 = 2𝑠. (7 marks) Some tips on question 4.2. It is advised to use forward difference with respect to time. Due to the double derivative of 𝐶 being present, you will naturally have to sample values for (at least) 3 different 𝐶 values. This means it does not matter so much which type of difference you use for the (single) 𝐶 derivative. It is best to choose the lowest truncation error where possible. Let 𝑥𝑖 = 𝑖ℎ, 𝑡𝑗 = 𝑗𝑘 and 𝐶𝑖,𝑗 = 𝐶(𝑥𝑖 ,𝑡𝑗) where ℎ and 𝑘 are step sizes. The finite difference equation will probably be 𝐶𝑖,𝑗+1 = ⋯ You may use a computer program to help you here with these calculations. You can alternatively do them by hand instead.