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# exammit1 - MATH 152 FALL 2004 MIDTERM I Problem \$1(5 1 Let...

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MATH 152, FALL 2004: MIDTERM #I Problem \$1 (15 ) Let u ~ ( x , t) and u2(x, t) denote the solutions of the equation with initial and boundary conditions respectively u1 (x, 0) = gl (x), u1 (0, t) = fl (t), ul(L,t) = hl(t) and u2(x,0) = gz(x), uz(0,t) = f2(t), uz(L,t) = 112(t). Assume that gl I gz, fl < f2 and hl 5 h2. Prove that u1 5 u2 in the set R = [0, L] x [0, m). Hint: Let w(x, t) = ul(x, t) - u2(x,t) and prove that w(x, t) 5 0 in R. Problem #2 (zd f fiS ) This is an example of a heat problem with internal heat source for which the inaxiinum principle does not hold. Consider i ut =uz, +2(t + 1) +x(1-2) (1) u(0, t) = 0, u(1, t) = 0 u(x,O) = x(1- x) for 0 < x < 1 and t > 0. a) Verify that u(x, t) = (t + l)x(l - x) is a solution for (1). \$- b) Find the maximum M and the minimum m of the initial and boundary data. -f c ) Show that for all t > 0 the temperature distribution u(x,t) exceeds M at a certain point inside the bar [O, 11. ?i5 Problem #3 65 pt5 ) Consider the inhomogeneous problem ut = kuxx + f (5, t) (2) 4 0 , t) = g(t), u(L, t)

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exammit1 - MATH 152 FALL 2004 MIDTERM I Problem \$1(5 1 Let...

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