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In Problems 5-8, find the steady-state temperature $u(r, z)$ in a finite cylinder defined by $0 \le r \le 1, 0 \le z \le 1$ if the boundary conditions are as given. 5. $u(1, z) = z$, $0 < z < 1$ 6. $u(1, z) = z$, $0 < z < 1$ $\frac{\partial u}{\partial z}|_{z=0} = 0$, $0 < r < 1$ $\frac{\partial u}{\partial z}|_{z=0} = 0$, $0 < r < 1$ $\frac{\partial u}{\partial z}|_{z=1} = 0$, $0 < r < 1$ $u(r, 1) = 0$, $0 < r < 1$ 7. $u(1, z) = u_0$, $0 < z < 1$ 8. $u(1, z) = 0$, $0 < z < 1$ $u(r, 0) = 0$, $0 < r < 1$ $\frac{\partial u}{\partial z}|_{z=0} = 0$, $0 < r < 1$ $\frac{\partial u}{\partial z}|_{z=1} = 0$, $0 < r < 1$ $u(r, 1) = u_0$, $0 < r < 1$

          In Problems 5-8, find the steady-state temperature $u(r, z)$ in a
finite cylinder defined by $0 \le r \le 1, 0 \le z \le 1$ if the boundary
conditions are as given.
5. $u(1, z) = z$, $0 < z < 1$
6. $u(1, z) = z$, $0 < z < 1$
$\frac{\partial u}{\partial z}|_{z=0} = 0$, $0 < r < 1$
$\frac{\partial u}{\partial z}|_{z=0} = 0$, $0 < r < 1$
$\frac{\partial u}{\partial z}|_{z=1} = 0$, $0 < r < 1$
$u(r, 1) = 0$, $0 < r < 1$
7. $u(1, z) = u_0$, $0 < z < 1$
8. $u(1, z) = 0$, $0 < z < 1$
$u(r, 0) = 0$, $0 < r < 1$
$\frac{\partial u}{\partial z}|_{z=0} = 0$, $0 < r < 1$
$\frac{\partial u}{\partial z}|_{z=1} = 0$, $0 < r < 1$
$u(r, 1) = u_0$, $0 < r < 1$

In Problems 5-8, find the steady-state temperature u(r, z) in a
finite cylinder defined by 0 ≤ r ≤ 1, 0 ≤ z ≤ 1 if the boundary
conditions are as given.
5. u(1, z) = z, 0 < z < 1
6. u(1, z) = z, 0 < z < 1
(∂ u)/(∂ z)|z=0 = 0, 0 < r < 1
(∂ u)/(∂ z)|z=0 = 0, 0 < r < 1
(∂ u)/(∂ z)|z=1 = 0, 0 < r < 1
u(r, 1) = 0, 0 < r < 1
7. u(1, z) = u0, 0 < z < 1
8. u(1, z) = 0, 0 < z < 1
u(r, 0) = 0, 0 < r < 1
(∂ u)/(∂ z)|z=0 = 0, 0 < r < 1
(∂ u)/(∂ z)|z=1 = 0, 0 < r < 1
u(r, 1) = u0, 0 < r < 1

Added by Eric O.

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