Reference no: EM131069380
1. Let Ω ⊂ R2 be the unit square: Ω = {x ∈ R2 : 0 < x1 < 1, 0 < x2 < 1}. Define F: Ω → R2 by
Verify that the divergence theorem holds for this domain Ω and this vector field F.
2. Define CN2(Ω-) = {u ∈ C2(Ω-) : x ∈ ∂Ω ⇒ ∂u/∂n(x) = 0} and
LN : C2N(Ω-) → C(Ω-)
LNu = - Δu.
Show that LN is symmetric: (LN u, v) = (u, LN v)for all u, v ∈ C2N(Ω-).
3. Solve the BVP
-Δu = f(x) in Ω,
u = 0 on ∂Ω,
where Ω is the unit square in R2 and f is the function.
4. Suppose Ω is the rectangle {x ∈ R2 : 0 < x1 < l1, 0 < x2 < l2}, and that u is a twice-continuously differentiable function defined on Ω-. Let the Fourier sine series of u be
u(x1, x2) = m=1Σ∞ n=1Σ∞amnsin(mπx1/l1)sin(nπx2/l2).
where λmn = m2π2/l12 + n2π2/l22, m, n = 1, 2, 3, ...
5. Consider an iron bar, of diameter 4cm and length 1m, with specific heat c = 0.437 J/(g K), density ρ = 7.88 g/cm3, and thermal conductivity κ = 0.836 W/(cm K). Suppose that the bar is insulated except at the ends, it is heated to a constant temperature of 5 degree Celsius, and the ends are placed in an ice bath (0 degrees Celsius). Compute the temperature (accurate to 3 digits) at the midpoint of the bar after 20 minutes.
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