Question

Solve the following Laplace's equation in a cube, carefully explaining all steps. Separate the variables using two steps: first, separate ((x, y)) from (z), and then separate (x) from (y): (u(x,y,z) = sum_lambda Phi(x,y) h(z) = sum_lambda phi(x) psi(y)h(z)). When solving the boundary values problems (Sturm-Liouville problems) for (phi(x)) and (psi(y)), make sure to check for zero eigenvalues. Note that the single non-homogeneous boundary condition is on the z-plane, (z=0). The boundary conditions are simple enough so that all Fourier coefficient integrals are easily calculated. Many steps are very similar to the equation of vibrating membrane solved in class. ( abla^2 u = frac{partial^2 u}{partial x^2} + frac{partial^2 u}{partial y^2} + frac{partial^2 u}{partial z^2} = 0) ((0 < x < 1, 0 < y < 1, 0 < z < 1)) (u(0, y, z) = frac{partial u}{partial x}(1, y, z) = 0) (frac{partial u}{partial y}(x, 0, z) = frac{partial u}{partial y}(x, 1, z) = 0) (u(x, y, 0) = 1; u(x, y, 1) = 0)

          Solve the following Laplace's equation in a cube, carefully explaining all steps. Separate the variables using
two steps: first, separate ((x, y)) from (z), and then separate (x) from (y):
(u(x,y,z) = sum_lambda Phi(x,y) h(z) = sum_lambda phi(x) psi(y)h(z)).
When solving the boundary values problems (Sturm-Liouville problems) for (phi(x)) and (psi(y)), make sure to check
for zero eigenvalues. Note that the single non-homogeneous boundary condition is on the z-plane, (z=0). The
boundary conditions are simple enough so that all Fourier coefficient integrals are easily calculated. Many
steps are very similar to the equation of vibrating membrane solved in class.
(
abla^2 u = frac{partial^2 u}{partial x^2} + frac{partial^2 u}{partial y^2} + frac{partial^2 u}{partial z^2} = 0) ((0 < x < 1, 0 < y < 1, 0 < z < 1))
(u(0, y, z) = frac{partial u}{partial x}(1, y, z) = 0)
(frac{partial u}{partial y}(x, 0, z) = frac{partial u}{partial y}(x, 1, z) = 0)
(u(x, y, 0) = 1; u(x, y, 1) = 0)

$Solve the following Laplace's equation in a cube, carefully explaining all steps. Separate the variables using two steps: first, separate ((x, y)) from (z), and then separate (x) from (y): (u(x,y,z) = sumlambda Phi(x,y) h(z) = sumlambda phi(x) psi(y)h(z)). When solving the boundary values problems (Sturm-Liouville problems) for (phi(x)) and (psi(y)), make sure to check for zero eigenvalues. Note that the single non-homogeneous boundary condition is on the z-plane, (z=0). The boundary conditions are simple enough so that all Fourier coefficient integrals are easily calculated. Many steps are very similar to the equation of vibrating membrane solved in class. ( abla^2 u = fracpartial^2 upartial x^2 + fracpartial^2 upartial y^2 + fracpartial^2 upartial z^2 = 0) ((0 < x < 1, 0 < y < 1, 0 < z < 1)) (u(0, y, z) = fracpartial upartial x(1, y, z) = 0) (fracpartial upartial y(x, 0, z) = fracpartial upartial y(x, 1, z) = 0) (u(x, y, 0) = 1; u(x, y, 1) = 0)$

Added by Jessica P.

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