Expand the function f(x) = sin(27x)] in a real Fourier series valid on the interval -...

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Expand the function f(x) = sin(27x)] in a real Fourier series valid on the interval - ≤x≤ (b) Write down the series solution u(t, x) for the heat equation u, = xx for 1>0,- ≤x≤ with the initial condition u(0, x) = sin(2x)] and periodic boundary conditions u(t. -1/2) = u(t. 1/2). u.(t, -1/2) = u(t, 1/2) (i.e. the solutions are 1-periodic in x, which corresponds to the case of the heated ring). (c) What is the behavior of the solution as 1 → 00? Expand the function f(x) = sin(27x)] in a real Fourier series valid on the interval - ≤x≤ (b) Write down the series solution u(t, x) for the heat equation u, = xx for 1>0,- ≤x≤ with the initial condition u(0, x) = sin(2x)] and periodic boundary conditions u(t. -1/2) = u(t. 1/2). u.(t, -1/2) = u(t, 1/2) (i.e. the solutions are 1-periodic in x, which corresponds to the case of the heated ring). (c) What is the behavior of the solution as 1 → 00?