Problem 6. (a) Consider the one-dimensional heat equation ut −uxx = 0, −∞ < x < ∞, −∞ < t < ∞. Find all solutions having

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Problem 6. (a) Consider the one-dimensional heat equation ut
−uxx = 0, −∞ < x < ∞, −∞ < t < ∞. Find all solutions
having the form of a running wave, i.e. u(x, t) = v(x − at) where a
is the wave velocity (it may be positive, negative, or zero).
(b) Using the result of part (a), and the function u(x, t) (0 ≤
x ≤ L, −∞ < t < ∞) satisfying the heat equation ut − uxx = 0
and the boundary conditions u(x, t)|x=0 = 0, u(x, t)|x=L = e^bt.
For which b such solution exists?