Download MATH 401 Spring 2012 Sample problems for the Final

MATH 401
Spring 2012
Sample problems for the Final
1. For the equation x3 − 7ϵx2 + ϵx − 10 find regular asymptotic expansions for its solutions
up to ϵ2 .
2. For the initial value problem
d2 y
+ 4y + ϵy 3 = 0
dt2
y(0) = 1
dy
(0) = 0
dt
(a) Find an approximate solution of the form u(t, ) = y0 (t) + y1 (t)ϵ + O(ϵ2 )
(b) Apply Poincare’s method to eliminate the secular term.
3. Solve the IBVP for the equation
ut = 2uxx , x ∈ (0, 1), t > 0
with the initial condition
u(x, 0) = 2 − 3x
and the following boundary conditions:
(a) u(0, t) = u(1, t) = 0
(b) ux (0, t) = ux (1, t) = 0
(c) ux (0, t) = u(1, t) = 0
4. Solve the following IBVP
ut = 2uxx , x ∈ (−1, 1), t > 0
u(x, 0) = 2 − 3x
u(−1, t) = u(1, t)
ux (−1, t) = ux (1, t)
5. Solve the IBVP for the equation
utt = 9uxx , x ∈ (0, 1), t > 0
with the initial conditions
u(x, 0) = 2x + 5,
and the following boundary conditions:
(a) u(0, t) = u(1, t) = 0
(b) ux (0, t) = ux (1, t) = 0
(c) ux (0, t) = u(1, t) = 0
ut (x, 0) = x
6. Solve the initial boundary value problem
uxx + uyy = 0, 0 < x < 1, 0 < y < 2
u(0, y) = u(1, y) = 0
u(x, 0) = 3 sin(2πx), u(x, 2) = sin(3πx)
7. Solve the initial value boundary problem for the heat equation in a two-dimensional
rectangular region
ut = 2∇2 u,
0 < x < 2, 0 < y < π, t > 0
u(x, y, 0) = 2xy
ux (0, y, t) = ux (2, y, t) = 0,
u(x, 0, t) = u(x, π, t) = 0.
8. Solve the initial value problem
utt = ∇2 u,
0 < r < 1, 0 < θ < π, t > 0
u(r, θ, 0) = 2r cos(2θ)
ut (r, θ, 0) = 0
u(0, θ, t) = u(1, θ, t) = 0,
uθ (r, 0, t) = uθ (x, π, t) = 0.
9. Solve the initial value problem for the heat equation
ut = uxx , 0 < x < π, t > 0,
u(x, 0) = x
u(0, t) = 3
u(π, t) = 1
10. Solve the initial value problem for the heat equation with time-dependent sources and
boundary conditions
ut = uxx + t2 sin 3x, 0 < x < π, t > 0,
u(x, 0) = x
u(0, t) = t
u(π, t) = 2t
11. Solve the initial value problem for the heat equation
ut = uxx , −∞ < x < ∞, t > 0
u(x, 0) = e−x
2 /9