If the boundary conditions for an annular plate defined by 1 r 2 are u(1, u) sin2 u, 0u

In Problems 1 and 2, find the steady-state temperature \(u(r, \theta)\) in a circular plate of radius c if the temperature on the circumference is as given. \(u(c, \theta)=\left\{\begin{array}{ll}u_{0}, & 0<\theta<\pi \\-u_{0}, & \pi<\theta<2 \pi\end{array}\right.\)

In Problems 1 and 2, find the steady-state temperature \(u(r, \theta)\) in a circular plate of radius c if the temperature on the circumference is as given. \(u(c, \theta)=\left\{\begin{array}{lc}1, & 0<\theta<\pi / 2 \\0, & \pi / 2<\theta<3 \pi / 2 \\1, & 3 \pi / 2<\theta<2 \pi\end{array}\right.\)

In Problems 3 and 4, find the steady-state temperature \(u(r, \theta)\) in a semicircular plate of radius 1 if boundary conditions are as given. \(u(1, \theta)=u_{0}\left(\pi \theta-\theta^{2}\right)\), \(0<\theta<\pi\) u(r, 0)=0, \(u(r, \pi)=0\), 0<r<1

In Problems 3 and 4, find the steady-state temperature \(u(r, \theta)\) in a semicircular plate of radius 1 if boundary conditions are as given. \(u(1, \theta)=\sin \theta\), \(0<\theta<\pi\) u(r, 0)=0, \(u(r, \pi)=0\), 0<r<1

Find the steady-state temperature \(u(r, \theta)\) in a semicircular plate of radius c if the boundaries \(\theta=0\) and \(\theta=\pi\) are insulated and \(u(c, \theta)=f(\theta)\), \(0<\theta<\pi\).

Find the steady-state temperature \(u(r, \theta)\) in a semicircular plate of radius c if the boundary \(\theta=0\) is held at temperature zero, the boundary \(\theta=\pi\) is insulated, and \(u(c, \theta)=f(\theta)\), \(0<\theta<\pi\).

In Problems 7 and 8, find the steady-state temperature \(u(r, \theta)\) in the plate shown in the figure.

In Problems 7 and 8, find the steady-state temperature \(u(r, \theta)\) in the plate shown in the figure.

If the boundary conditions for an annular plate defined by 1 < r < 2 are \(u(1, \theta)=\sin ^{2} \theta\), \(\left. \frac{\partial u}{\partial r}\right|_{r=2}=0\), \(0<\theta<2 \pi\) show that the steady-state temperature is \(u(r, \theta)=\frac{1}{2}-\left(\frac{1}{34} r^{2}+\frac{8}{17} r^{-2}\right) \cos 2 \theta\) [Hint: See Figure 14.1.4. Also, use the identity \(\sin ^{2} \theta=\frac{1}{2}(1-\cos 2 \theta)\).]

Find the steady-state temperature \(u(r, \theta)\) in the infinite plate shown in FIGURE 14.R.3.

Suppose heat is lost from the flat surfaces of a very thin circular plate of radius 1 into a surrounding medium at temperature zero. If the linear law of heat transfer applies, the heat equation assumes the form \(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}-h u=\frac{\partial u}{\partial t}\), h>0, 0<r<1, t>0 See FIGURE 14.R.4. Find the temperature u(r, t) if the edge r = 1 is kept at temperature zero and if initially the temperature of the plate is unity throughout.

Suppose \(x_{k}\) is a positive zero of \(J_{0}\) . Show that a solution of the boundary-value problem \(a^{2}\left(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}\right)=\frac{\partial^{2} u}{\partial t^{2}}\), 0<r<1, t>0 u(1, t)=0, t>0 \(u(r, 0)=u_{0} J_{0}\left(x_{k} r\right),\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=0\), 0<r<1 is \(u(r, t)=u_{0} J_{0}\left(x_{k} r\right) \cos a x_{k} t\).

Find the steady-state temperature u(r, z) in the cylinder in Figure 14.2.5 if the lateral side is kept at temperature zero, the top z = 4 is kept at temperature 50, and the base z = 0 is insulated.

Solve the boundary-value problem \(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{\partial^{2} u}{\partial z^{2}}=0\), 0<r<1, 0<z<1 \(\left.\frac{\partial u}{\partial r}\right|_{r=1}=0\), 0<z<1 \(u(r, 0)=f(r), \quad u(r, 1)=g(r)\), 0<r<1 .

Find the steady-state temperature \(u(r, \theta)\) in a sphere of unit radius if the surface is kept at \(u(1, \theta)=\left\{\begin{array}{lr}100, & 0<\theta<\pi / 2 \\-100, & \pi / 2<\theta<\pi\end{array}\right.\) [Hint: See Problem 22 in Exercises 12.6.]

Solve the boundary-value problem \(\frac{\partial^{2} u}{\partial r^{2}}+\frac{2}{r} \frac{\partial u}{\partial r}=\frac{\partial^{2} u}{\partial t^{2}}\), 0<r<1, t>0 \(\left.\frac{\partial u}{\partial r}\right|_{r=1}=0\), t>0 u(r, 0)=f(r), \(\left. \frac{\partial u}{\partial t}\right|_{t=0}=g(r)\), 0<r<1 [Hint: Proceed as in Problems 9 and 10 in Exercises 14.3, but let v(r, t) = ru(r, t). See Section 13.7.]

The function \(u(x)=Y_{0}(\alpha a) J_{0}(\alpha x)-J_{0}(\alpha a) Y_{0}(\alpha x)\), a > 0 is a solution of the parametric Bessel equation \(x^{2} \frac{d^{2} u}{d x^{2}}+x \frac{d u}{d x}+\alpha^{2} x^{2} u=0\) on the interval [a, b]. If the eigenvalues \(\lambda_{n}=\alpha_{n}^{2}\) are defined by the positive roots of the equation \(Y_{0}(\alpha a) J_{0}(\alpha b)-J_{0}(\alpha a) Y_{0}(\alpha b)=0\) show that the functions \(u_{m}(x)=Y_{0}\left(\alpha_{m} a\right) J_{0}\left(\alpha_{m} x\right)-J_{0}\left(\alpha_{m} a\right) Y_{0}\left(\alpha_{m} x\right)\) \(u_{n}(x)=Y_{0}\left(\alpha_{n} a\right) J_{0}\left(\alpha_{n} x\right)-J_{0}\left(\alpha_{n} a\right) Y_{0}\left(\alpha_{n} x\right)\) are orthogonal with respect to the weight function p(x) = x on the interval [a, b]; that is, \(\int_{a}^{b} x u_{m}(x) u_{n}(x) d x=0\), \(m \neq n\). [Hint: Follow the procedure on pages 693 and 694.]

Use the results of Problem 17 to solve the following boundary value problem for the temperature u(r, t) in an annular plate: \(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}=\frac{\partial u}{\partial t}\), a<r<b, t>0 u(a, t)=0, u(b, t)=0, t>0 u(r, 0)=f(r), a<r<b

Discuss how to solve \(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{\partial^{2} u}{\partial z^{2}}=0\), 0<r<c, 0<z<L with the boundary conditions given in FIGURE 14.R.5.

Carry out your ideas and find u(r, z) in Problem 19. [Hint: Review (11) Section 13.5.]

In Problems 21–24, solve the given boundary-value problem. \(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{\partial^{2} u}{\partial z^{2}}=0\), 0<r<1, 0<z<1 u(1, z)=100, 0<z<1 \(\left.\frac{\partial u}{\partial z}\right|_{z=0}=0\), 0<r<1 u(r, 1)=200, 0<r<1

In Problems 21–24, solve the given boundary-value problem. \(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{\partial^{2} u}{\partial z^{2}}=0\), 0<r<3, 0<z<1 \(u(3, z)=u_{0}\), 0<z<1 u(r, 0)=0, 0<r<3 u(r, 1)=0, 0<r<3

In Problems 21–24, solve the given boundary-value problem. \(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{\partial^{2} u}{\partial z^{2}}=0\), 0<r<1, z>0 u(1, z)=0, z>0 u(r, 0)=100, 0<r<1

In Problems 21–24, solve the given boundary-value problem. \(\frac{\partial^{2} u}{\partial r^{2}}+\frac{1}{r} \frac{\partial u}{\partial r}+\frac{\partial^{2} u}{\partial z^{2}}=0\), 0<r<1, z>0 u(1, z)=0, z>0 \(u(r, 0)=u_{0}\left(1-r^{2}\right)\), 0<r<1