Practice Exam.pdf - Practice Exam Math 110A Introduction to...

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Practice Exam Math 110A: Introduction to PDE Problem 1: Indicate whether each of the statements below are true or false. No justification is needed. a) If u ( x, y ) is a solution to the transport equation: 2 u x + 3 u y = 0, then u ( x, y ) must be constant along the characteristic line: 3 x - 2 y = 1. b) There are infinitely many solutions to the PDE: 2 u x + 3 u y = 0. c) The PDE: u xy - 4 x 2 y 3 = 0, is second-order, linear inhomogeneous. d) There exists a function u ( t, x ) which solves both the heat equation u t - ku xx = 0 and the wave equation u tt - u xx = 0. Problem 2: Find the specific solution to the transport equation: xyu x + (1 + y 2 ) u y = 0 , with the initial data: u ( x, 0) = x 4 . Problem 3: Consider the wave equation IVP: u tt - u xx = 0 , u (0 , x ) = 0 , u t (0 , x ) = ( 1 , | x | 6 1; 0 , | x | > 1 . (a) Calculate u (0 , t ) explicitly for all t > 0 for the solution u ( x, t ) of this problem. (b) Show that lim t + u ( t, x ) = 1 for all x R . Problem 4: Let u ( t, x ) be a (smooth) solution to the wave equation:
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