14 Chapter in Review Answers to selected odd-numbered problems begin on page ANS-37.

In Problems 1 and 2, find the steady-state temperature u(r, θ) in a circular plate of radius c if the temperature on the circumference is as given.

In Problems 3 and 4, find the steady-state temperature u(r, θ) in a semicircular plate of radius 1 if boundary conditions are as given.

  3. Find the steady-state temperature u(r, θ) in a semicircular plate of radius c if the boundaries θ = 0 and θ = π are insulated and u(c, θ) = f(θ), 0 < θ < π.
  4. Find the steady-state temperature u(r, θ) in a semicircular plate of radius c if the boundary θ = 0 is held at temperature zero, the boundary θ = π is insulated, and u(c, θ) = f(θ), 0 < θ < π.

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

  1. A one-eighth annular plate is graphed in an x y coordinate plane. The right edge of the plate coincides with the x axis. It is indicated by an arrow and is labeled u = 0 at theta = 0. The left edge of the plate is formed by the line y = x. It is indicated by an arrow and is labeled u = 0. The inner edge of the plate is indicated by an arrow and labeled u = u subscript 0 at r = 1 over 2. A thin strip of barred lines on the outer edge of the plate is labeled insulated at r = 1.

    FIGURE 14.R.1 Plate in Problem 7

  2. A portion of an annular plate plate is graphed in an x y coordinate plane such that its center coincides with the origin of the graph. The right edge of the plate coincides with the x axis. It is indicated by an arrow and is labeled u = 0 at theta = 0. The left edge of the plate is formed by a line going diagonally up to the right and is labeled u = u subscript 1 at theta = beta. The inner edge of the plate is indicated by an arrow and labeled u = 0 at r = a. The outer edge of the plate is labeled u = f(theta) at r = b.

    FIGURE 14.R.2 Plate in Problem 8

  3. If the boundary conditions for an annular plate defined by 1 < r < 2 are

    show that the steady-state temperature is

    [Hint: See Figure 14.1.4. Also, use the identity sin2 .]

  4. Find the steady-state temperature u(r, θ) in the infinite plate shown in FIGURE 14.R.3.
    A semi-circle is graphed on an x y coordinate plane such that its center coincides with the origin of the graph. The portion of the x axis to the right of the origin and left of the origin are indicated by arrows and labeled u = 0. The radius of the semi-circle is indicated by an arrow labeled I. The edge of the semi-circle is indicated by an arrow and labeled u = f(theta).

    FIGURE 14.R.3 Infinite plate in Problem 10

  5. Solve the boundary-value problem

    [Hint: See equation (12) in Section 5.3.]

  6. Suppose xk is a positive zero of J0. Show that a solution of the boundary-value problem

    is u(r, t) = u0J0(xkr) cos axkt.

  7. 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.
  8. Solve the boundary-value problem
  9. Find the steady-state temperature u(r, θ) in a sphere of unit radius if the surface is kept at

    [Hint: See Problem 22 in Exercises 12.6.]

  10. Solve the boundary-value problem

    [Hint: Proceed as in Problems 9 and 10 in Exercises 14.3, but let v(r, t) = ru(r, t). See Section 13.7.]

  11. The function u(x) = Y0(αa)J0(αx) − J0(αa)Y0(αx), a > 0 is a solution of the parametric Bessel equation

    on the interval [a, b]. If the eigenvalues λn = are defined by the positive roots of the equation

    show that the functions

    are orthogonal with respect to the weight function p(x) = x on the interval [a, b]; that is,

    [Hint: Follow the procedure on pages 702 and 703.]

  12. Use the results of Problem 17 to solve the following boundary-value problem for the temperature u(r, t) in an annular plate:

In Problems 19–22, solve the given boundary-value problem.