2. (Heat type equations) (a) (15 points) Solve the initial boundary value problem ди at =K a2u 2к ди + + u(tu, дх2 х+ Хо

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2. (Heat type equations) (a) (15 points) Solve the initial boundary value problem ди at =K a2u 2к ди + + u(tu, дх2 х+ Хо дх 0<x<L, t > 0, = u(0,t) = u(L, t) = 0, u(x,0) = u(x), where k > 0, Xo > 0 are constants and the function u(t) is a bounded integrable function with support on a finite interval of t-axis. Hint: The transformation w = wexp(S* v(E)d[/2) is useful to remove the 24 term. (b) (10 points) Consider the behavior of the solution for large positive time t: Does it converge to a limit as t +00? Give the approximate form of the solution for large t. For large t find an approximation to T >0 such that the solution u(L/2,t+T) differs from u(L/2, t) by a factor of 2 ? дх