The University of Sydney School of Mathematics and Statistics Assignment MATH3078/MATH3978/MATH4078: PDEs and Waves Semester 2, 2024 Web Page: https://canvas.sydney.edu.au/courses/60714 Lecturer: Robert Marangell • This assignment is due by 23:59 Sunday 29th September 2024. Late assignments will not be accepted. • Please submit your typeset Assignment as a single PDF document using TurnItIn in Canvas. Handwritten assignments will not be accepted. • Please show all working and present your arguments clearly. • To ensure compliance with our anonymous marking obligations, please do not include your name in any area of your assignment; only your SID should be present. Copyright© 2024 The University of Sydney 1 1. (a) Solve Laplace’s equation on the interior of the half disk H := {(x, y)|x2 + y2 ≤ 1 , y ≥ 1} with boundary conditions u(x, 0) = 0 for −1 ≤ x ≤ 1 u(x, y) = Im (z4) = 4x3y−4xy3 for x2+y2 = 1 y > 0. (b) Plot your solution using your favourite software package. (c) What is u(0, 12)? (d) What are the maximum and minimum values that the solution takes on the half disk, and at what points do they occur? 2. Consider the following function f(x) on the interval [0, 7] (plotted below). f(x) =  3x− 15 2 5 2 ≤ x ≤ 3 6− 3 2 x 3 ≤ x ≤ 9 2 3 2 x− 15 2 9 2 ≤ x ≤ 5 0, otherwise (a) Compute the Fourier sine series of the periodic extension of f(x) with period 14. Compute the exact expression for the coeffieicients (b) Write the solution to the boundary value problem (wave equation) utt = uxx u(0, t) = u(7, t) = 0 on the interval [0, 7] with initial profiles u(x, t) = f(x) and ut(x, t) = 0. (c) Use your favourite software package to make a plot of the solution from (b) at times t = 3, 4, and t = 5. Be sure to say which program you used, and what the code you used to produce the plots. 3. Consider the PDE initial boundary value problem on the interval [0, pi]. ut = uxx + 2ux + u u(0, t) = u(pi, t) = 0 with u(x, 0) = f(x). (1) (a) Solve the equation using separation of variables. (b) Now solve the equation by making an ansatz eaxw(x, t) = u(x, t), where a is a real number to be determined. Find a so that if u(x, t) solve eq. (1), then w(x, t) will solve the heat equation, with appropriately modified initial and boundary conditions. Solve this heat equation and then undo your transformation to produce the solution u to eq. (1). (c) Now either use separation of variables, or generalise the method from part (b) by making the substitution u = eax+btw for unknown constants a and b to solve 2 the following general, constant coefficient parabolic equation with homogeneous Dirichlet boundary conditions: ut = uxx + 2Aux +Bu u(0, t) = u(pi, t) = 0 with u(x, 0) = f(x). (2) Briefly discuss the the long term behaviour of the solution. Justify your answer. 4. Please list your sources for what you used to solve these problems. You can use what you like to help you with figuring out a problem, but what you write up should be your own understanding, and youneed to tell me what you used to get to this understanding. This includes each other. So for example if you found an example of a certain type on a given webpage, and used Mathematica to help you compute something, and you worked together with another student to complete the assignment, you need to document all of these facts on the sources page. You don’t need to adhere to a particular style of sourcing To maintain anonymity, please only include the SID of the students you worked on. Your assignment will be compared with the other students’ as well as what is on the web via the turnitin program. Should you have too great of a match without properly citing your sources, you may be in violation of the university’s academic honesty policy. Solution: I used Mathematica for my graphs and some of the questions, as well as the textbook by Olver, and the book Intro to PDEs by Strauss and some Chat GPT to draw inspiration for some of the questions. 3 51作业君版权所有