INTRODUCTION

All the boundary-value problems that have been considered so far have been expressed in terms of rectangular coordinates. If, however, we wish to find temperatures in a circular disk, in a circular cylinder, or in a sphere, we would naturally try to describe the problems in polar coordinates, cylindrical coordinates, or spherical coordinates, respectively.

Because we consider only steady-state temperature problems in polar coordinates in this section, the first thing that must be done is to convert the familiar Laplace’s equation in rectangular coordinates to polar coordinates.

Laplacian in Polar Coordinates

The relationships between polar coordinates in the plane and rectangular coordinates are given by

See FIGURE 14.1.1. The first pair of equations transform polar coordinates (r, θ) into rectangular coordinates (x, y); the second pair of equations enable us to transform rectangular coordinates into polar coordinates. These equations also make it possible to convert the two-dimensional Laplacian of the function u, ∇²u = ∂²u/∂x² + ∂²u/∂y², into polar coordinates. You are encouraged to work through the details of the Chain Rule and show that

  (1)

  (2)

Adding (1) and (2) and simplifying yields the Laplacian of u in polar coordinates:

In this section we shall concentrate only on boundary-value problems involving Laplace’s equation in polar coordinates:

. (3)

Our first example is the Dirichlet problem for a circular disk. We wish to solve Laplace’s equation (3) for the steady-state temperature u(r, θ) in a circular disk or plate of radius c when the temperature of the circumference is u(c, θ) = f(θ), 0 ˂ θ ˂ 2π. See FIGURE 14.1.2. It is assumed that the two faces of the plate are insulated. This seemingly simple problem is unlike any we have encountered in the previous chapter.

EXAMPLE 1 Steady Temperatures in a Circular Plate

Solve Laplace’s equation (3) subject to u(c, θ) = f(θ), 0 ˂ θ ˂ 2π.

SOLUTION

Before attempting separation of variables we note that the single boundary condition is nonhomogeneous. In other words, there are no explicit conditions in the statement of the problem that enable us to determine either the coefficients in the solutions of the separated ODEs or the required eigenvalues. However, there are some implicit conditions.

First, our physical intuition leads us to expect that the temperature u(r, θ) should be continuous and therefore bounded inside the circle r = c. In addition, the temperature u(r, θ) should be single-valued; this means that the value of u should be the same at a specified point in the plate regardless of the polar description of that point. Since (r, θ + 2π) is an equivalent description of the point (r, θ), we must have u(r, θ) = u(r, θ + 2π). That is, u(r, θ) must be periodic in θ with period 2π. If we seek a product solution u = R(r)Θ(θ), then Θ(θ) needs to be 2π-periodic.

With all this in mind, we choose to write the separation constant in the separation of variables as λ:

The separated equations are then

(4)

(5)

We are seeking a solution of the problem

(6)

Although (6) is not a regular Sturm–Liouville problem, nonetheless the problem generates eigenvalues and eigenfunctions. The latter form an orthogonal set on the interval [0, 2π].

Of the three possible general solutions of (5),

(7)

                 (8)

                (9)

we can dismiss (8) as inherently nonperiodic unless c₁ = c₂ = 0. Similarly, solution (7) is nonperiodic unless we define c₂ = 0. The remaining constant solution Θ(θ) = c₁, c₁ # 0, can be assigned any period and so λ = 0 is an eigenvalue. Finally, solution (9) will be 2π-periodic if we take α = n, where n = 1, 2, ….* The eigenvalues of (6) are then λ₀ = 0 and λ_n = n², n = 1, 2, …. If we correspond λ₀ = 0 with n = 0, the eigenfunctions of (6) are

When λ_n = n², n = 0, 1, 2, … the solutions of the Cauchy–Euler DE (4) are

(10)

…. (11)

Now observe in (11) that r⁻ⁿ = 1/rⁿ. In either of the solutions (10) or (11), we must define c₄ = 0 in order to guarantee that the solution u is bounded at the center of the plate (which is r = 0). Thus product solutions u_n = R(r)Θ(θ) for Laplace’s equation in polar coordinates are

,

where we have replaced c₃c₁ by A₀ for n = 0 and by A_n for n = 1, 2, …; the product c₃c₂ has been replaced by B_n. The superposition principle then gives

(12)

By applying the boundary condition at r = c to the result in (12) we recognize

as an expansion of f in a full Fourier series. Consequently we can make the identifications

That is, (13)

(14)

(15)

The solution of the problem consists of the series given in (12), where the coefficients A₀, A_n, and B_n are defined in (13), (14), and (15), respectively. ≡

Observe in Example 1 that corresponding to each positive eigenvalue, λ_n = n², n = 1, 2, …, there are two different eigenfunctions—namely, cos nθ and sin nθ. In this situation the eigenvalues are sometimes called double eigenvalues.

EXAMPLE 2 Steady Temperatures in a Semicircular Plate

Find the steady-state temperature u(r, θ) in the semicircular plate shown in FIGURE 14.1.3.

SOLUTION

The boundary-value problem is

Defining u = R(r)Θ(θ) and separating variables gives

and r²R″ + rR′ − λR = 0 (16)

Θ″ + λΘ = 0.(17)

The homogeneous conditions stipulated at the boundaries θ = 0 and θ = π translate into Θ(0) = 0 and Θ(π) = 0. These conditions together with equation (17) constitute a regular Sturm–Liouville problem:

(18)

This familiar problem* possesses eigenvalues λ_n = n² and eigenfunctions Θ(θ) = c₂ sin nθ, n = 1, 2, …. Also, by replacing λ by n² the solution of (16) is R(r) = c₃rⁿ + c₄r⁻ⁿ. The reasoning used in Example 1, namely, that we expect a solution u of the problem to be bounded at r = 0, prompts us to define c₄ = 0. Therefore u_n = R(r)Θ(θ) = A_nrⁿ sin nθ and

The remaining boundary condition at r = c gives the Fourier sine series

Consequently

and so

Hence the solution of the problem is given by

≡

14.1 Exercises Answers to selected odd-numbered problems begin on page ANS-36.

In Problems 1–4, find the steady-state temperature u(r, θ) in a circular plate of radius 1 if the temperature on the circumference is as given.

1. 
2. 
3. u(1, θ) = 2πθ − θ², 0 < θ <, 2π
4. u(1, θ) = θ, 0 < θ < 2π
5. If the boundaries θ = 0 and θ = π of a semicircular plate of radius 2 are insulated, we then have

   

   Find the steady-state temperature u(r, θ) if

   

   where u₀ is a constant.

6. Find the steady-state temperature u(r, θ) in a semicircular plate of radius 1 if the boundary conditions are

   

   where u₀ is a constant.

7. Find the steady-state temperature in the quarter-circular plate shown in FIGURE 14.1.4.

   

8. Find the steady-state temperature in the quarter-circular plate shown in Figure 14.1.4 if the boundaries and are insulated, and

   

9. Find the steady-state temperature in the portion of a circular plate shown in FIGURE 14.1.5.

   

10. Find the steady-state temperature in the infinite wedge-shaped plate shown in FIGURE 14.1.6. [Hint: Assume that the temperature is bounded as and as ]

    

11. Find the steady-state temperature in the plate in the form of an annulus bounded between two concentric circles of radius a and b, shown in FIGURE 14.1.7. [Hint: Proceed as in Example 1.]

    

12. If the boundary conditions for the annular plate in Figure 14.1.7 are

    

    where are constants, show that the steady-state temperature is given by

    

    [Hint: Try a solution of the form ]

13. Find the steady-state temperature in the annular plate shown in Figure 14.1.7 if the boundary conditions are

    

14. Find the steady-state temperature in the annular plate shown in Figure 14.1.7 if and

    

15. Find the steady-state temperature in the semiannular plate shown in FIGURE 14.1.8 if the boundary conditions are

    

    

16. Find the steady-state temperature in the semiannular plate shown in Figure 14.1.8 if and

    

    

    where is a constant.

17. Find the steady-state temperature in the quarter-annular plate shown in FIGURE 14.1.9.

    

18. The plate in the first quadrant shown in FIGURE 14.1.10 is one-eighth of the annular plate in Figure 14.1.7. Find the steady-state temperature

    

19. Solve the exterior Dirichlet problem for a circular disk of radius c shown in FIGURE 14.1.11. In other words, find the steady-state temperature in an infinite plate that coincides with the entire xy-plane in which a circular hole of radius c has been cut out around the origin and the temperature on the circumference of the hole is [Hint: Assume that the temperature u is bounded as ]

    

20. Consider the steady-state temperature in the semiannular plate shown in Figure 14.1.8 with and boundary conditions

    

    

    Show that in this case the choice of in (4) and (5) leads to eigenvalues and eigenfunctions. Find the steady-state temperature

Computer Lab Assignments

21. 1. Find the series solution for in Example 1 when

       

       See Problem 1.

    2. Use a CAS or a graphing utility to plot the partial sum consisting of the first five nonzero terms of the solution in part (a) for Superimpose the graphs on the same coordinate axes.
    3. Approximate the temperatures Then approximate
    4. What is the temperature at the center of the circular plate? Why is it appropriate to call this value the average temperature in the plate? [Hint: Look at the graphs in part (b) and look at the numbers in part (c).]

Discussion Problems

22. Solve the Neumann problem for a circular plate:

    

    Give the compatibility condition. [Hint: See Problem 21 of Exercises 13.5.]

23. Consider the annular plate shown in Figure 14.1.7. Discuss how the steady-state temperature can be found when the boundary conditions are

    

24. Verify that is a solution of the boundary-value problem

    

*For example, note that cos n(θ + 2π) = cos(nθ + 2nπ) = cos nθ.

*The problem in (18) is Example 2 of Section 3.9 with L = π.