Question

In this problem we'd like to solve the boundary value problem Ə x = 4 Ə 2u

Ə t Ə x2
on the interval [0, 4] with the boundary conditions u(0, t) = u(4, t) = 0 for all t.
(a) Suppose h(x) is the function on the interval [0, 4] whose graph is is the piecewise linear function connecting the points (0, 0), (2, 2), and (4,0). Find the Fourier sine series of h(z): h(x) = - Σ bx (t) sin (nkx/4).
Please choose the correct option: does your answer only include odd values of k, even values k, or all values of k? bk(t) (16/(k^2pi^2)){(-1)^{(k-1)/2))
Which values of k should be included in this summation? A. Only the even values B. Only the odd values C. All values (b) Write down the solution to the boundary value problem Ə x = 4 Ə 2u
Ə t Ə x2
on the interval [0, 4] with the boundary conditions u(0, t) = u(4, t) = 0 for all t subject to the initial conditions u(a,0) = h(a). As before, please choose the correct option: does your answer only include odd values of k, even values of k, or all values of ? [infinity]
u(x, t) = Σ
k-1 Which values of k should be included in this summation? A. Only the even values B. Only the odd values C. All values 4 br(t) sin
Previous question

Answer 1

The Fourier sine series of h(x) includes only odd values of k.

Step-by-step explanation:

To find the Fourier sine series of h(x), we need to determine the values of bx(t) for different values of k.

1. For the interval [0, 2], the function h(x) is a linear function connecting the points (0, 0) and (2, 2). So, bx(t) = 0 for all values of t.
2. For the interval [2, 4], the function h(x) is a linear function connecting the points (2, 2) and (4, 0). So, bx(t) = 2 for all values of t.

In the Fourier sine series, only odd values of k are included. So, the answer is: the Fourier sine series of h(x) includes only odd values of k.

Answer 2

The Fourier sine series of the given function h(x) and the solution to the boundary value problem will include only odd values of k because of the piecewise function's symmetry and the Dirichlet boundary conditions.

Step-by-step explanation:

In this problem, we are dealing with the boundary value problem for the heat equation on the interval [0, 4] with Dirichlet boundary conditions. We are asked to find the Fourier sine series of a given piecewise function and then write down the solution to the boundary value problem subject to these initial conditions.

For part (a), the Fourier sine series for the function h(x) defined on the interval [0, 4] will include only odd values of k. This is because sine functions with odd multiples in their arguments are odd functions about the midpoint of the given domain, matching the symmetry of h(x).

The coefficients bk(t) in the Fourier sine series expression provided in the question include a term ((-1)^{(k-1)/2}), which indeed reinforces that only odd values of k should contribute to the series because for even k, the exponent becomes a fraction and the expression is not defined.

For part (b), the solution for the boundary value problem will use a similar approach because the heat equation preserves the symmetry of the initial conditions over time. Therefore, the solution will also involve only odd values of k.