Bounded Domain - Partial Differential Equations - Solved Exam, Exams of Differential Equations

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MATH348 - November 21, 2011 name: Ke. | Exam II - 50 Points x In order to receive full credit, SHOW ALL YOUR WORK. Full credit will be given only if all reasoning and work is provided. When applicable, please enclose your final answers in boxes. 1. (10 Points) Conceptual Questions. For the following questions, assume that we are considering the physical problem on a bounded domain, z € (0,7). (a) Write down the heat and wave equations and any initial conditions needed for unique solutions. For each, what does the unknown function u measure? Os et Tes es Set Se Ux) = fog Ulx,01= fo) Uy 9) = Ab) ay Os Kemp at UL moans disp la comack ee. (xa) ee 1. Cad. {b} Suppose we are given the boundary conditions u,(0, t) = 0 and u;(7,t) = 0 for each problem. Explain the physical meaning of each boundary condition for both the heat equation and wave equation. K : K eee 3s oe By Tsubakion Ste KA : Heat Reoerse or wee ok Wowvwe_ (wb) = an \aterbee ULITL) DS Same ee ey ; *S {c) The following graph gives the only nonzero initial configuration for each problem. Describe and for draw the associated heat and wave dynamics. If there is an equilibrium state then be sure to state it. f(a) = ule, 0) r= 7 Heat: Trion te whe ae. os & aloud My, whe ANE ce Zqu- Ac wees comets due +o ae ae poms will ee wy ae =a Miter Ww “we Pauls voy tern.