Problem 1 (10pt). Let 41(x), 42(x) be two continuous functions defined on R. Assume that 41(x) = 42(x) for (x| 2 2 and 41(x) 0 and |x| 2 ct + 2. (b) Let v1 (x, t) be the solution of the heat equation I (v1 )t (x, t) - k(v1)zx(x, t) = 0 for (x, t) ER x (0, 00), VI (2, 0) = 41(20) for x E R. Let v2(x, t) be the solution of the heat equation (v2)t (z, t) - k(v2)xx(x, t) = 0 for (x, t) ER x (0, 00), V2 (2, 0) = 42(2 ) for x E R. Show that for any t > 0 and x E R, v1(x, t)