MATH 257 Study Guide - Midterm Guide: Heat Equation
Document Summary
Be sure that this examination has 10 pages. Write your name on top of each page. In case of an exam disruption such as a re alarm, leave the exam papers in the room and exit quickly and quietly to a pre-designated location. Solve the heat equation ut(x, t) = uxx(x, t) for 0 x 1 and t 0 with non-homogeneous boundary conditions u(0, t) = 1, u(1, t) = 0, T > 0 and initial condition u(x, 0) = 1 + x. Consider the wave equation utt = uxx for x r, t > 0, with the initial conditions u(x, 0) = f (x), ut(x, 0) = g(x), where f (x) = 0 if x 0, x if 0 x 1, 0 if x > 1. (a) solve the solution u(x, t) using d"alembert"s formula. Hint: draw the graph of f (x), where it exists. (b) plot u(x, 0), u(x, 1) and u(x, 2) for |x| 4.
