INTRODUCTION

In this section we continue our examination of boundary-value problems in different coordinate systems. This time we are going to consider problems involving the heat, wave, and Laplace’s equation in spherical coordinates.

Laplacian in Spherical Coordinates

As shown in FIGURE 14.3.1, a point in 3-space is described in terms of rectangular coordinates and in spherical coordinates. The rectangular coordinates x, y, and z of the point are related to its spherical coordinates r, θ, and ϕ through the equations

(1)

By using the equations in (1) it can be shown that the Laplacian ∇²u in the spherical coordinate system is

(2)

As you might imagine, problems involving (1) can be quite formidable. Consequently we shall consider only a few of the simpler problems that are independent of the azimuthal angle ϕ.

Our first example is the Dirichlet problem for a sphere.

EXAMPLE 1 Steady Temperatures in a Sphere

Find the steady-state temperature u(r, θ) in the sphere shown in FIGURE 14.3.2.

SOLUTION

The temperature is determined from

If u = R(r)Θ(θ), the partial differential equation separates as

and so r²R″ + 2rR′ − λR = 0 (3)

sin θ Θ″ + cos θ Θ′ + λ sin θ Θ = 0. (4)

After we substitute x = cos θ, 0 ≤ θ ≤ π, (4) becomes

(5)

The latter equation is a form of Legendre’s equation (see Problems 52 and 53 in Exercises 5.3). Now the only solutions of (5) that are continuous and have continuous derivatives on the closed interval [−1, 1] are the Legendre polynomials P_n(x) corresponding to λ = n(n + 1), n = 0, 1, 2, …. Thus we take the solutions of (4) to be

Θ(θ) = P_n(cos θ).

Furthermore, when λ = n(n + 1), the general solution of the Cauchy–Euler equation (3) is

R(r) = c₁rⁿ + c₂r^{−(n + 1)}.

Since we again expect u(r, θ) to be bounded at r = 0, we define c₂ = 0. Hence u_n = A_nrⁿP_n(cos θ), and

At r = c,

Therefore A_ncⁿ are the coefficients of the Fourier–Legendre series (23) of Section 12.6:

It follows that the solution is

. ≡

Spherical Symmetry

When u depends on r and t but is independent of both angles and , then the problem is said to possess spherical symmetry. The heat and wave equations are then

and (6)

See Problems 9–11 in Exercises 14.3.

14.3 Exercises Answers to selected odd-numbered problems begin on page ANS-37.

1. Solve the problem in Example 1 if

   

   Write out the first four nonzero terms of the series solution. [Hint: See Example 3, Section 12.6.] The solution can be interpreted as the potential due to a charge of 50 volts on the upper hemisphere and a grounded lower hemisphere.

2. The solution u(r, θ) in Example 1 could also be interpreted as the potential inside the sphere due to a charge distribution f(θ) on its surface. Find the potential outside the sphere.
3. Find the solution of the problem in Example 1 if f(θ) = cos θ, 0 < θ , π. [Hint: P₁(cos θ) = cos θ. Use orthogonality.]
4. Find the solution of the problem in Example 1 if f(θ) = 1 − cos 2θ, 0 < θ < π. [Hint: See Problem 18, Exercises 12.6.]
5. Find the steady-state temperature u(r, θ) within a hollow sphere a < r < b if its inner surface r = a is kept at temperature f(θ) and its outer surface r = b is kept at temperature zero. The sphere in the first octant is shown in FIGURE 14.3.3.

   

6. The steady-state temperature in a hemisphere of radius c is determined from

   

   Solve for u(r, θ). [Hint: P_n(0) = 0 only if n is odd. Also see Problem 20, Exercises 12.6.]

7. Solve Problem 6 when the base of the hemisphere is insulated; that is,

   

8. Solve Problem 6 for r > c.
9. The time-dependent temperature within a sphere of radius 1 is determined from

   

   Solve for u(r, t). [Hint: Verify that the left side of the partial differential equation can be written as (ru). Let ru(r, t) = v(r, t) + ψ(r). Use only functions that are bounded as r → 0.]

10. A uniform solid sphere of radius 1 at an initial constant temperature u₀ throughout is dropped into a large container of fluid that is kept at a constant temperature u₁ (u₁ > u₀) for all time. See FIGURE 14.3.4. Since there is heat transfer across the boundary r = 1, the temperature u(r, t) in the sphere is determined from the boundary-value problem

    

    Solve for u(r, t). [Hint: Proceed as in Problem 9.]

    

11. Solve the boundary-value problem involving spherical vibrations:

    

    [Hint: Write the left side of the partial differential equation as a² (ru). Let v(r, t) = ru(r, t).]

12. A conducting sphere of radius c is grounded and placed in a uniform electric field that has intensity E in the z-direction. The potential u(r, θ) outside the sphere is determined from the boundary-value problem

    

    Show that

    

    [Hint: Explain why θ P_n(cos θ) sin θ dθ = 0 for all nonnegative integers except n = 1. See (24) of Section 12.6.]

In Problems 13 and 14, the time-independent partial differential equation

that arises in physics is called the Helmholtz’s equation. This equation is named after the German physician and physicist Hermann Ludwig Ferdinand von Helmholtz (1821–1894). The three-dimensional Laplacian is the spherical coordinates form given in (2) of this section.

13. 1. Proceed as in Example 1 but using and the separation constant n(n + 1) to show that the radial dependence of the solution u is defined by the equation

       

    2. Now use the second separation constant m² to show that the remaining separated equations are

       

    3. Use the substitution x = cos θ to show that the second differential equation in part (b) becomes
       
14. 1. Assume that m and n are nonnegative integers. Then find a product solution of Helmholtz’s PDE using the general solution of the ODE in part (a), the general solution of the first ODE in part (b), and a particular solution of the second ODE in part (b) of Problem 13. [Hint: See Problems 41, 42(c), and 54 in Exercises 5.3.]
    2. What product solution in part (a) would be bounded at the origin?