MATH 467 Partial Differential Equations Exercises - Millersville ...

57. A solid cube of edge length 1 and with heat diffusivity k is initially at temperature
100 ◦ C. At time t = 0 the cube is placed in anenvironment whose constant temperature
is 0 ◦ C. Find the temperature at the center of the cube as a function of time.
58. A solid cube of edge length 1 and with heat diffusivity k is initially at temperature
100 ◦ C. Five faces of the cube are insulated. At time t = 0 the cube is placed in an
environment whose constant temperature is 0 ◦ C. Find the temperature at the center
of the cube as a function of time.
59. Solve the boundary value problem:
uxx +uyy +uzz = 0 for 0 < x < π, 0 < y < π, 0 < z < π
ux(0,y,z) = 0
ux(π,y,z) = 0
uy(x,0,z) = 0
uy(x,π,z) = 0
uz(x,y,0) = 0
uz(x,y,π) = −1+4sin 2 xcos 2 y.
60. Let f(x,t), g(y,t), and h(z,t) solve the respective heat equations
ft = kfxx
gt = kgyy
ht = khzz.
Show that u(x,y,z,t) = f(x,t)g(y,t)h(z,t) solves the partial differential equation
61. Consider the partial differential equation
ut = k(uxx +uyy +uzz).
uxx +uyy +uzz = 0
on the rectangular solid where 0 ≤ x ≤ L, 0 ≤ y ≤ M, and 0 ≤ z ≤ N. Suppose the
values of u have been specified at the eight corners of the solid. Find a solution of the
form
u(x,y,z) = axyz +bxy +cyz +dxz +ex+fy +gz +h
to the PDE.
62. Consider the partial differential equation
uxx +uyy +uzz = 0