Homework 2, due September 29 Math 4041, Fall 2009
1. Let v(x) = x2,x∈[0, π]. Find the sine series of v. That is, compute the
coefficients of the representation of vas
v(x) =
∞
X
n=1
vnsin(nx).
2. Find the solution of
∂u
∂t −∂2u
∂x2= 0 in t > 0,0< x < π
u(0, t)=0, u(π, t) = 0 for t > 0
u(x, 0) = x2.
3. Find the solution of
∂u
∂t −∂2u
∂x2= 0 in t > 0,0< x < π
u(0, t)=0,∂u
∂x (π, t) = 0 for t > 0
u(x, 0) = x−π/2.
4. Find u(x, t) periodic in xof period 2πsuch that
∂u
∂t −∂2u
∂x2= 0 in t > 0, x ∈R
u(x, 0) = sin x.
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