INTRODUCTION

For the remainder of this and the next chapter we shall be concerned with finding product solutions of the second-order partial differential equations

(1)

(2)

(3)

or slight variations of these equations. These classical equations of mathematical physics are known, respectively, as the one-dimensional heat equation, the one-dimensional wave equation, and Laplace’s equation in two dimensions. “One-dimensional” refers to the fact that x denotes a spatial dimension whereas t represents time; “two dimensional” in (3) means that x and y are both spatial dimensions. Laplace’s equation is abbreviated ∇²u = 0, where

is called the two-dimensional Laplacian of the function u. In three dimensions the Laplacian of u is

By comparing equations (1)–(3) with the linear second-order PDE given in Definition 13.1.1, with t playing the part of y, we see that the heat equation (1) is parabolic, the wave equation (2) is hyperbolic, and Laplace’s equation (3) is elliptic. This classification is important in Chapter 16.

Heat Equation

Equation (1) occurs in the theory of heat flow—that is, heat transferred by conduction in a rod or thin wire. The function u(x, t) is temperature. Problems in mechanical vibrations often lead to the wave equation (2). For purposes of discussion, a solution u(x, t) of (2) will represent the displacement of an idealized string. Finally, a solution u(x, y) of Laplace’s equation (3) can be interpreted as the steady-state (that is, time-independent) temperature distribution throughout a thin, two-dimensional plate.

Even though we have to make many simplifying assumptions, it is worthwhile to see how equations such as (1) and (2) arise.

Suppose a thin rod of length L has a circular cross-sectional area A and coincides with the x-axis on the interval [0, L]. See FIGURE 13.2.1. Let us suppose:

• The flow of heat within the rod takes place only in the x-direction.
• The lateral, or curved, surface of the rod is insulated; that is, no heat escapes from this surface.
• No heat is being generated within the rod by either chemical or electrical means.
• The rod is homogeneous; that is, its mass per unit volume ρ is a constant.
• The specific heat γ and thermal conductivity K of the material of the rod are constants.

To derive the partial differential equation satisfied by the temperature u(x, t), we need two empirical laws of heat conduction:

1. The quantity of heat Q in an element of mass m is

   Q = γmu, (4)

   where u is the temperature of the element.

2. The rate of heat flow Q_t through the cross section indicated in Figure 13.2.1 is proportional to the area A of the cross section and the partial derivative with respect to x of the temperature:

   Q_t = –K Au_x.(5)

Since heat flows in the direction of decreasing temperature, the minus sign in (5) is used to ensure that Q_t is positive for u_x < 0 (heat flow to the right) and negative for u_x > 0 (heat flow to the left). If the circular slice of the rod shown in Figure 13.2.1 between x and x + Δ x is very thin, then u(x, t) can be taken as the approximate temperature at each point in the interval. Now the mass of the slice is m = ρ(A Δ x), and so it follows from (4) that the quantity of heat in it is

Q = γρA Δ x u.(6)

Furthermore, when heat flows in the positive x-direction, we see from (5) that heat builds up in the slice at the net rate

–K Au_x(x, t) – [–K Au_x(x + Δ x, t)] = K A[u_x(x + Δ x, t) – u_x(x, t)].(7)

By differentiating (6) with respect to t we see that this net rate is also given by

Q_t = γρA Δx u_t .(8)

Equating (7) and (8) gives

(9)

Taking the limit of (9) as Δx → 0 finally yields (1) in the form*

It is customary to let k = K/γρ and call this positive constant the thermal diffusivity.

Wave Equation

Consider a string of length L, such as a guitar string, stretched taut between two points on the x-axis—say, x = 0 and x = L. When the string starts to vibrate, assume that the motion takes place in the xy-plane in such a manner that each point on the string moves in a direction perpendicular to the x-axis (transverse vibrations). As shown in FIGURE 13.2.2(a), let u(x, t) denote the vertical displacement of any point on the string measured from the x-axis for t > 0. We further assume:

• The string is perfectly flexible.
• The string is homogeneous; that is, its mass per unit length ρ is a constant.
• The displacements u are small compared to the length of the string.
• The slope of the curve is small at all points.
• The tension T acts tangent to the string, and its magnitude T is the same at all points.
• The tension is large compared with the force of gravity.
• No other external forces act on the string.

Now in Figure 13.2.2(b) the tensions T₁ and T₂ are tangent to the ends of the curve on the interval [x, x + Δx]. For small values of θ₁ and θ₂ the net vertical force acting on the corresponding element Δs of the string is then

T sin θ₂ – T sin θ₁ ≈ T tan θ₂ – T tan θ₁

= T [u_x(x + Δ x, t) – u_x(x, t)],^†

where T = |T₁| = |T₂|. Now ρ Δs ≈ ρ Δ x is the mass of the string on [x, x + Δ x], and so Newton’s second law gives

T [u_x(x + Δ x, t) – u_x(x, t)] = ρ Δ x u_tt

or

If the limit is taken as Δx → 0, the last equation becomes u_xx = (ρ/T)u_tt. This of course is (2) with a² = T/ρ.

Laplace’s Equation

Although we shall not present its derivation, Laplace’s equation in two and three dimensions occurs in time-independent problems involving potentials such as electrostatic, gravitational, and velocity in fluid mechanics. Moreover, a solution of Laplace’s equation can also be interpreted as a steady-state temperature distribution. As illustrated in FIGURE 13.2.3, a solution u(x, y) of (3) could represent the temperature that varies from point to point—but not with time—of a rectangular plate.

We often wish to find solutions of equations (1), (2), and (3) that satisfy certain side conditions.

Initial Conditions

Since solutions of (1) and (2) depend on time t, we can prescribe what happens at t = 0; that is, we can give initial conditions (IC). If f(x) denotes the initial temperature distribution throughout the rod in Figure 13.2.1, then a solution u(x, t) of (1) must satisfy the single initial condition u(x, 0) = f(x), 0 < x < L. On the other hand, for a vibrating string, we can specify its initial displacement (or shape) f(x) as well as its initial velocity g(x). In mathematical terms we seek a function u(x, t) satisfying (2) and the two initial conditions:

(10)

For example, the string could be plucked, as shown in FIGURE 13.2.4, and released from rest (g(x) = 0).

Boundary Conditions

The string in Figure 13.2.4 is secured to the x-axis at x = 0 and x = L for all time. We interpret this by the two boundary conditions (BC):

u(0, t) = 0,u(L, t) = 0,t > 0.

Note that in this context the function f in (10) is continuous, and consequently f(0) = 0 and f(L) = 0. In general, there are three types of boundary conditions associated with equations (1), (2), and (3). On a boundary we can specify the values of one of the following:

Here ∂u/∂n denotes the normal derivative of u (the directional derivative of u in the direction perpendicular to the boundary). A boundary condition of the first type (i) is called a Dirichlet condition, a boundary condition of the second type (ii) is called a Neumann condition, and a boundary condition of the third type (iii) is known as a Robin condition. For example, for t > 0 a typical condition at the right-hand end of the rod in Figure 13.2.1 can be

Condition (i)′ simply states that the boundary x = L is held by some means at a constant temperature u₀ for all time t > 0. Condition (ii)′ indicates that the boundary x = L is insulated. From the empirical law of heat transfer, the flux of heat across a boundary (that is, the amount of heat per unit area per unit time conducted across the boundary) is proportional to the value of the normal derivative ∂u/∂n of the temperature u. Thus when the boundary x = L is thermally insulated, no heat flows into or out of the rod and so

We can interpret (iii)′ to mean that heat is lost from the right-hand end of the rod by being in contact with a medium, such as air or water, that is held at a constant temperature. From Newton’s law of cooling, the outward flux of heat from the rod is proportional to the difference between the temperature u(L, t) at the boundary and the temperature u_m of the surrounding medium. We note that if heat is lost from the left-hand end of the rod, the boundary condition is

The change in algebraic sign is consistent with the assumption that the rod is at a higher temperature than the medium surrounding the ends so that u(0, t) > u_m and u(L, t) > u_m. At x = 0 and x = L, the slopes u_x(0, t) and u_x(L, t) must be positive and negative, respectively.

Of course, at the ends of the rod we can specify different conditions at the same time. For example, we could have

= 0andu(L, t) = u₀, t > 0.

We note that the boundary condition in (i)′ is homogeneous if u₀ = 0; if u₀ ≠ 0, the boundary condition is nonhomogeneous. The boundary condition (ii)′ is homogeneous; (iii)′ is homogeneous if u_m = 0 and nonhomogeneous if u_m ≠ 0.

Boundary-Value Problems

Problems such as

(11)

and

(12)

are called boundary-value problems. The problem in (11) is classified as a homogeneous BVP since the partial differential equation and the boundary conditions are homogeneous.

Variations

The partial differential equations (1), (2), and (3) must be modified to take into consideration internal or external influences acting on the physical system. More general forms of the one-dimensional heat and wave equations are, respectively,

(13)

and (14)

For example, if there is heat transfer from the lateral surface of a rod into a surrounding medium that is held at a constant temperature u_m, then the heat equation (13) is

where h is a constant. In (14) the function F could represent the various forces acting on the string. For example, when external, damping, and elastic restoring forces are taken into account, (14) assumes the form

(15)

13.2 ExercisesAnswers to selected odd-numbered problems begin on page ANS-33.

In Problems 1– 6, a rod of length L coincides with the interval [0, L] on the x-axis. Set up the boundary-value problem for the temperature u(x, t).

1. The left end is held at temperature zero, and the right end is insulated. The initial temperature is f(x) throughout.
2. The left end is held at temperature u₀, and the right end is held at temperature u₁. The initial temperature is zero throughout.
3. The left end is held at temperature 100°, and there is heat transfer from the right end into the surrounding medium at temperature zero. The initial temperature is f(x) throughout.
4. There is heat transfer from the left end into a surrounding medium at temperature 20°, and the right end is insulated. The initial temperature is f(x) throughout.
5. The left end is at temperature sin(πt/L), the right end is held at zero, and there is heat transfer from the lateral surface of the rod into the surrounding medium held at temperature zero. The initial temperature is f(x) throughout.
6. The ends are insulated, and there is heat transfer from the lateral surface of the rod into the surrounding medium held at temperature 50°. The initial temperature is 100° throughout.

In Problems 7–10, a string of length L coincides with the interval [0, L] on the x-axis. Set up the boundary-value problem for the displacement u(x, t).

7. The ends are secured to the x-axis. The string is released from rest from the initial displacement x(L – x).
8. The ends are secured to the x-axis. Initially the string is undisplaced but has the initial velocity sin(πx/L).
9. The left end is secured to the x-axis, but the right end moves in a transverse manner according to sin πt. The string is released from rest from the initial displacement f(x). For t > 0 the transverse vibrations are damped with a force proportional to the instantaneous velocity.
10. The ends are secured to the x-axis, and the string is initially at rest on that axis. An external vertical force proportional to the horizontal distance from the left end acts on the string for t > 0.

In Problems 11 and 12, set up the boundary-value problem for the steady-state temperature u(x, y).

11. A thin rectangular plate coincides with the region in the xy-plane defined by 0 ≤ x ≤ 4, 0 ≤ y ≤ 2. The left end and the bottom of the plate are insulated. The top of the plate is held at temperature zero, and the right end of the plate is held at temperature f(y).
12. A semi-infinite plate coincides with the region defined by 0 ≤ x ≤ π, y ≥ 0. The left end is held at temperature e^–y, and the right end is held at temperature 100° for 0 < y ≤ 1 and temperature zero for y > 1. The bottom of the plate is held at temperature f(x).

*Recall from calculus that

^†tan θ₂ = u_x(x + Δx, t) and tan θ₁ = u_x(x, t) are equivalent expressions for slope.