(1) 10 points let u := u(x, t) be the temperature in a one-dimensional rod, and satisfy the following initial and boundary value problem: 4 cos x, for 0 < x < 2, t > 0, Xu(2, t) = 2, where is a constant. Denote the total thermal energy in the rod (0 < x < 2) by. Answer the following questions. (i) (2 points) what is the physical meaning of the boundary condition xu(0, t) = 0? (ii) (3 points) verify that de dt. Answer: (i) meaning the left end is insulated. (ii) de dt. 0 (uxx + x) dx = ux(2, t) ux(0, t) + 2 = 2 + 2 (iii) e(t) = 32 (iv) = 1, limit is 32. 2 (3) 10 points let f (x) be a piecewise smooth function. Denote by g(x) the fourier series of the function f (x) on the interval [ , ], that is, f (x) .
