The University of Sydney School of Mathematics and Statistics Assignment 2 MATH2065: Introduction to PDEs Semester 2, 2017 Web Page: http://http://www.maths.usyd.edu.au/u/UG/IM/MATH2065/ Lecturer: F Viera Due Thursday 26 October (Week 12) by 5:00pm via Turnitin. A single PDF file must be uploaded to Turnitin through the Learning Management System (Blackboard). ZIP files are not accepted by Turnitin. Neat handwritten solutions are acceptable; however, illegible work will not be marked. Please, make sure you receive a Turnitin Digital Receipt confirming that you have successfully uploaded your assignment. Save this receipt. 2u 2u + 2 = 0 for the temperature distribution over the x2 y annulus in R2 given by {(r, ) | 1 < r < 2, < } in the standard polar coordinates. 1. Consider Laplace's equation (a) Using the transformation between rectangular and polar coordiantes x = r cos , y = r sin , show that Laplace's equation in polar coordinates is 2 u 1 u 1 2u + + = 0. r2 r r r2 2 (b) Using separation of variables u(r, ) = R(r) G(), show that the ODE satisfied by R(r) is of the form r2 R00 + r R0 n2 R = 0, n = 0, 1, 2, 3 . . . , and calculate its solution in the two cases: (i) n = 0 and (i) n = 1, 2, 3 . . . . (c) Show that the general solution of Laplace's equation can be written in the form \u0013 \u0013 \u0012 \u0012 X X Fn Dn n n En r + n sin(n), u(r, ) = A + B ln r + Cn r + n cos(n) + r r n=1 n=1 where A, B, Cn , Dn , En , Fn are arbitrary constants (you may assume the solution for G() derived in lectures). (d) Calculate the solution if the boundary conditions are: u(1, ) = 0 and u(2, ) = 5 + cos() + sin(2). c 2017 The University of Sydney Copyright