MATH 467 Partial Differential Equations Exercises - Millersville ...

Findthe solution to the wave equation for this initial condition. Discuss the case where
ǫ → 0 + .
46. Define new coordinates in the xy-plane by
ˆx = ax+by +f
ˆy = cx+dy +g
where a, b, c, d, f, and g are constants with ad−bc = 0. Define û(ˆx,ˆy) = u(x,y).
(a) Show that if u is C 2 , then
uxx +uyy = (a 2 +b 2 )ûˆxˆx +2(ac+bd)ûˆxˆy +(c 2 +d 2 )ûˆyˆy.
(b) Suppose that (ˆx,ˆy) are the new coordinates obtained by rotating the original axes
by some angle θ in the counterclockwise direction. Verify that a = cosθ, b = sinθ,
c = −sinθ, and d = cosθ. Show that in this case
47. Solve the boundary value problem
uxx +uyy = ûˆxˆx +ûˆyˆy.
uxx +uyy = 0 for 0 < x < π and 0 < y < π
u(x,0) = sinx
u(x,π) = sinx
u(0,y) = siny
u(π,y) = siny.
48. Find a function of the form U(x,y) = a + bx + cy + dxy such that U(0,0) = 0,
U(1,0) = 1, U(0,1) = −1, and U(1,1) = 2. Use this function to solve the following
boundary value problem.
uxx +uyy = 0 for 0 < x < 1 and 0 < y < 1
u(x,0) = 3sin(πx)+x
u(x,1) = 3x−1
u(0,y) = sin(2πy)−y
u(1,y) = y +1.
49. Solve the boundary value problem
uxx +uyy = 0 for 0 < x < π and 0 < y < π
u(x,0) = 0
u(x,π) = x(π −x)
u(0,y) = 0
u(π,y) = 0.