In Chapters 14 and 15 we discussed ordinary differential equations and their solutions. These are equations that contain a dependent variable y, which is a function of a single variable x, and derivatives of y with respect to x. In this chapter, we extend the discussion to similar equations that involve functions of two or more variables x₁, x₂, …, x_n. These are called partial differential equations (PDEs) because the functional form analogous to (14.2) in general contains partial differentials with respect to several variables, including mixed derivatives. If we consider a function u of just two variables x₁ = x and x₂ = y, then examples of partial differential equations are

(16.1a)

(16.1b)

and

(16.1c)

where f(x, y) is an arbitrary function of x and y.

By analogy with the definitions in Chapter 14, the degree of the equation is defined as the power to which the highest order derivative is raised after the equation is rationalised, if necessary; and the order of a partial differential equation is the order of the highest derivative in the equation. Thus (16.1a) is first-order and (16.1b) and (16.1c) are second-order equations. In addition, all three equations are linear, in the sense that they contain only u and its derivatives to first degree; products between them are absent. Non-linear equations, such as

will not be discussed.

A linear equation is said to be homogeneous if each term contains either the dependent variable or one of its derivatives, so that if u is a solution, so is λu, where λ is a constant. Thus (16.1a) and (16.1c) are homogeneous, while (16.1b) is inhomogeneous. An important property of linear homogeneous equations is that, if u₁ and u₂ are solutions, then so is any arbitrary linear combination of them (αu₁ + βu₂), where α and β are arbitrary constants. This is called the superposition principle and is widely used in finding the general solution of some PDEs using a method known as ‘separation of variables’ that we will discuss in Sections 16.2 and 16.3.

One important difference between PDEs and ODEs needs to be mentioned. As discussed in Chapter 14, a linear combination of two or more solutions for an nth order linear homogeneous ODE will in general have n arbitrary constants, which can only be determined by suitable boundary conditions, for example, the value of the function y(x) can be specified for n values of x to determine the constants. The solution of a PDE, however, usually contains a number of arbitrary functions. For example, it is easily verified by direct substitution that the simple first-order PDE

has solutions of the form u(x, y) = y f(x), where f(x) is an arbitrary function of x. Similarly, the simple second-order equation

(16.2a)

has a general solution of the form

(16.2b)

where f(x) and g(y) are arbitrary differentiable functions, as may be found by successive integration with respect to x (at fixed y) and y (at fixed x). However, unlike the analogous situation for ODEs, it does not in general follow that an nth order PDE always contains n arbitrary functions, but it is true for linear equations with constant coefficients, which is the class of equations discussed in the rest of this chapter. As for ODEs, these arbitrary functions must be determined by imposing boundary conditions. In this case, these will have to take the form of specifying u along a continuum of points, for example along a line in the (x, y) or (x, t) plane, but the appropriate form of the boundary conditions depends on the type of PDE. This will be discussed in more detail in Section 16.6.

16.1 Some important PDEs in physics

Partial differential equations are in general far more difficult to solve than ordinary differential equations, and except for certain special types of equation, no general method of solution exists. In this chapter we will concentrate on discussing some specific second-order equations, since these include many of the most important equations in physics. These include the following, where u is a finite physical quantity that can depend on three space co-ordinates x, y, z and the time t.

1. The wave equation

   (16.3a)

   In mechanics, u could be a displacement of a vibrating medium, or in electromagnetism, the component of an electromagnetic wave, etc., and is the speed of propagation of the associated wave.

2. The diffusion equation

   (16.4)

   This describes the diffusion of material particles, where ρ is the density of diffusing particles and the constant κ is called the diffusivity. It also describes heat conduction in a region that contains no heat sources or sinks, where it is referred to as the heat conduction equation. In this case, u is the temperature and κ is called the thermal diffusivity. It is given by , where k is the thermal conductivity, s is the specific heat capacity, and ρ is the density of the material.

3. Laplace's equation

   (16.5)

   Here u could be, for example, a steady-state temperature distribution, or the electrostatic or gravitational potential in free space.

4. Poisson's equation

   (16.6)

   The quantity u describes the same quantities as in Laplace's equation, but now in a region containing an appropriate source ρ(r). For the electrostatic potential u = φ, Poisson's equation takes the form (12.60), so that the force is proportional to the electric charge density; for the gravitational potential ψ, (12.58) and (12.59) give

   

   where ρ_m(r) is the mass density and G is Newton's gravitational constant.

5. Schrödinger's equation

   (16.7)

   This is the equation of motion for a particle of mass m in a potential V(r) in non-relativistic quantum mechanics, where ℏ is Planck's constant h divided by 2π. The quantity u is the Schrödinger wave function and is usually complex.

   An important feature of all these equations is that the spatial derivatives enter via the scalar differential operator

   

   which reflects the rotational invariance of the laws of physics. In particular, they do not include mixed partial derivatives of the forms , etc. Because of this, and subject to the forms of ρ(r) and V(r), these equations can be solved by the method of separation of variables, in which the problem is converted into one of solving a set of ordinary differential equations, each in a single variable. This method is discussed in the next two sections; other methods of solution will be treated in later sections.

16.2 Separation of variables: Cartesian co-ordinates

In this method, given a PDE for a function u(x, y, z, t), we seek solutions of the form

(16.8)

in which u is given as the product of functions of single variables. Here we have used the convention of denoting the single-variable functions by an upper-case letter and its argument by the corresponding lower-case letter. There could also be common parameters that occur in each function, but each depends on only one independent variable. We then try to rewrite the original PDE in the form of four separate ODEs for each of the functions X, Y, Z, T using a procedure to be explained below. If this is possible, then the original PDE is said to be separable, and one can seek solutions to the individual ODEs using the methods discussed in Chapters 14 and 15. If it is not possible, then the equation is not separable, and other methods must be used.

If a function can be written in the form (16.8), it is also said to be separable. Thus xy²z³tan (at) is separable in all four variables, and is said to be completely separable; whereas (x² + y^{− 2})z³sin (θt) is only separable in z and t and is therefore partially separable; but xz + ayt cannot be separated in any variable and so is inseparable. Obviously the individual solutions initially found by separation of variables are, by construction, completely separable functions. However, this is not as restrictive as it might seem, since, as we shall see, the method leads to many such solutions, usually an infinite number, and the general solution, which is rarely separable, can usually be constructed from them. This is particularly easy in the important case of linear homogeneous equations, when any linear combination of a given set of solutions is itself a solution, and for such equations it is also known as the Fourier method of solution.

In this section, we shall introduce the method using Cartesian co-ordinates, leaving the important but more complicated case of polar co-ordinates until Section 16.3. In both cases, we will use the physical equations introduced in the previous section to do this, applying boundary conditions and constraints appropriate to the function u being a physical quantity.

16.2.1 The wave equation in one spatial dimension

To illustrate the general method, we will solve the wave equation (16.3a) for a single spatial variable x, when it reduces to

(16.3b)

This equation could describe, for example, a vibrating string undergoing small transverse displacements u(x, t), where the wave velocity is . For definiteness, we will assume the string is clamped at x = 0 and x = L, so that the boundary conditions include the constraints

(16.9)

at all times t.

To solve (16.3b), we assume a separable form

(16.10)

which when substituted into (16.3b) gives,

(16.11)

Here the primes indicate differentiation with respect to the single variable appropriate to the symbol for the function, that is, and . If we now divide (16.11) by u = XT, we have

(16.12)

This equation is of a very special form, where each term that appears is a function of only one variable; is a function of x only and is a function of t only. It can therefore only be satisfied if each term is equal to the same constant, called the separation constant, which will be denoted by − k² for later convenience. Thus we can write

(16.13a)

and

(16.13b)

where .

The first of these equations, together with the boundary conditions X(0) = X(L) = 0, which follow from (16.9), is just the eigenvalue problem that was solved earlier in Section 15.2. There we found that solutions only exist for

(16.14)

and are

(16.15)

The corresponding values of ω are then , and the general solution to (16.13b) is therefore

(16.16)

where A_n and B_n are arbitrary constants, so that we arrive at a set of solutions

(16.17)

where n = 1, 2, … is any positive integer. Each of these solutions is a ‘normal mode’ in which u(x, t) oscillates with a single angular frequency for all x-values¹. At this point, we recall that the wave equation is linear and homogeneous, so that any linear combinations of solutions of the form (16.17) is also a solution of the original equation. The general solution is a linear combination of the normal modes, that is,

(16.18)

Finally, to determine the arbitrary constants A_n and B_n and obtain a unique solution requires imposing further boundary conditions that have to be specified according to the problem. For the case of a vibrating string, these might be that the string is released from rest at t = 0 from its initial state u(x, 0). The first of these conditions, that is,

leads to the result A_n = 0 (all n) on substituting into (16.18), so that the solution becomes

(16.19)

In addition, the coefficients B_n can be determined, given the initial displacement u(x, 0), since from (16.19)

which is just the Fourier expansion of the initial configuration of u(x, 0) extended, as an odd function, to the range − L < x < L (cf. Section 13.1.4). To invert it, we multiply both sides by and integrate from 0 to L using the orthonormality relation

(16.20)

This gives

(16.21)

and the solution (16.19) is completely determined.

16.2.2 The wave equation in three spatial dimensions

The method of Section 16.2.1 is easily extended to more than one spatial dimension. To illustrate this, we shall consider the wave equation in three dimensions with 0 < x, y, z < L, that is, for waves confined to a cubic box. We then find the form of the waves in this box by solving the equation

(16.22)

assuming that u vanishes at the walls of the box, that is,

(16.23)

In this case, substituting (16.8) into (16.22) and dividing by a = XYZT, gives

As before, since the left-hand side is independent of t and the right-hand side depends solely on t, both sides must equal a constant, which we will denote by − k². We thus obtain

(16.24a)

and

where we have transferred the term in Z to the right-hand side. Since the left-hand side of this equation is independent of Z and the right-hand side depends solely on Z, both sides again must be a constant, which we write as − k² + k²_z. We then obtain

(16.24b)

and

Taking to the right-hand side and repeating the argument then gives

(16.24c)

and

(16.24d)

where

(16.25)

We have now converted the wave equation (16.22) into four ODEs (16.24a)–(16.24d). Furthermore, (16.24b)–(16.24d) have the same form as (16.13a) with the same boundary conditions, and (16.24a) is identical to (16.13a). Solving in the same way, one finds that solutions satisfying the boundary conditions (16.23) only exist for

(16.26)

in analogy to (16.14), where n_x, n_y, n_z are positive integers; and the form of the solution is

(16.27)

in analogy to (16.17), where

(16.28)

and the arbitrary constants A and B can also depend on n_x, n_y, n_z.

The solutions (16.27) are the normal modes for waves confined to a cubic box and the general solution is a linear combination

(16.29)

These results apply to any type of wave confined to a cubic box, for example, sound waves or electromagnetic waves, and are easily generalised to any rectangular box with sides L_x, L_y, L_z.

16.2.3 The diffusion equation in one spatial dimension

We next consider the diffusion equation (16.4), which is again second order in spatial derivatives but, in contrast to the wave equation, only first order in the time derivative. To focus on this difference, we shall just consider it in one spatial dimension. The generalisation to three spatial dimensions is very similar to that discussed in detail for the wave equation and is left as an exercise for the reader.

In one spatial dimension, the diffusion equation becomes

(16.31)

and, as an example, we will find solutions subject to the boundary conditions

(16.32)

where L is a constant and f(x) is a given function that must vanish at the end points 0 and L for the boundary conditions to be consistent. We start by again assuming a solution of the separable form (16.10), and after substituting into (16.31) and dividing by u we have

Each side now has to be a constant, which we will denote by α. For α ≥ 0, it may easily be shown that the only solution consistent with the first boundary condition is u(x, t) = 0. However, for α ≡ −λ² < 0, the solutions for X(x) and T(t) are

where A, B and C are constants. The solution is therefore

(16.33)

where the constant C has been absorbed into A and B.

Using the first boundary condition with x = 0 gives

for all t, and hence A = 0. Similarly putting x = L gives

and hence only gives non-trivial solutions provided

Thus the general solution is the superposition

(16.34)

As in the corresponding result (16.19) for the wave equation, at any fixed time t the solution takes the form of a Fourier sine series, but now each term is associated with an exponential rather than oscillating time dependence.

Finally, imposing the boundary condition at t = 0 gives

Multiplying by and integrating from 0 to L using the orthonormality relation (16.20) then gives

(16.35)

which can be evaluated for any initial distribution u(x, 0) = f(x) to give the coefficients in (16.34).

16.3 Separation of variables: polar co-ordinates

In this section, we shall explore separation of variables using polar co-ordinates, rather than Cartesian co-ordinates. This is particularly useful for systems with circular, cylindrical or spherical symmetry. Throughout the discussion, which will again proceed via examples, we shall have in mind that the dependent variable is a physical quantity, and therefore will emphasise solutions that are finite and single-valued.

16.3.1 Plane-polar co-ordinates

Plane-polar co-ordinates were introduced in Section 2.2.1, where from Figure 2.4 we see that the co-ordinates (r, θ) and (r, θ + 2π) represent the same point in the plane. Hence if u(r, θ) represents a measurable quantity, it must not only be finite and continuous, but must also satisfy the boundary condition

(16.36a)

which for separable solutions

(16.37)

implies

(16.36b)

To illustrate the consequences of this, we will consider solutions of the Laplace and wave equations in two spatial dimensions.

In plane–polar co-ordinates, Laplace's equation is given by [cf. (7.29)]

(16.38a)

or equivalently,

(16.38b)

Substituting (16.37) into (16.38b) and multiplying by r² gives

(16.39)

where the left-hand side is independent of θ and the right-hand side is independent of r, so that both must be constant. Denoting the constant by m², we then have

(16.40a)

and

(16.40b)

In solving these equations, it will be convenient to treat the cases m = 0 and m ≠ 0 separately.

1. m = 0 In this case (16.40a) has the trivial solution Θ = A + Bθ, where A, B are arbitrary constants. Similarly, (16.40b) reduces to

   

   with the general solution

   

   Hence the general solution is

   

   which reduces to

   

   on imposing the boundary condition (16.36a) and absorbing the constant A into the arbitrary constants C and D.

2. m ≠ 0 In this case (16.40a) has the general solution

   

   while the corresponding general solution of (16.40b) is

   

   as is easily verified by substitution. In addition, we note that m must be an integer if the boundary condition (16.36b) is to be satisfied, so that for m ≠ 0, the separable solutions of Laplace's equation are:

   (16.41b)

   where A_m, B_m, C_m, D_m are arbitrary constants. Finally, the general solution is given by

   (16.42)

   where the various constants must be determined from boundary conditions.

The angular dependence of the separable solutions (16.41) is characteristic of the separable solutions of many other equations with circular symmetry, including the wave equation (16.3); the diffusion equation (16.4a); and the Schrödinger equation (16.7), provided the potential V(r) = V(r) is independent of angle. To illustrate this, we consider the wave equation in two spatial dimensions, which is given by

(16.43)

in planar co-ordinates. Assuming a separable solution

(16.44)

gives

after dividing by u, and hence

(16.45a)

and

(16.46)

on separating off the right-hand side and denoting the separation constant by − k². After multiplying by r², (16.46) can also be separated in a similar way to Laplace's equation to give

(16.45b)

and

(16.45c)

Here, equation (16.45b) is identical to (16.40a) and has the same solution, while on expanding the differential in (16.45c) and substituting z = kr, we obtain

which is Bessel's equation (15.53a) and has the general solution

(16.47a)

where C_m and D_m are arbitrary constants and J_m and N_m are the Bessel functions discussed in Section 15.4.1. In particular, N_m(kr) is singular at r = 0, so if we require u to be finite at r = 0, (16.47a) reduces to

(16.47b)

Finally, the general solution of (16.45) is

(16.48)

where the angular frequency . Hence the separable solutions that are finite at r = 0 are

(16.49a)

and

(16.49b)

where m = 1, 2, … must be an integer if the boundary condition (16.36b) is to be satisfied, and where C_m in (16.47b) has been absorbed into the other constants. In particular, the angular dependence of the separable solutions is the same as for Laplace's equation and is characteristic of the separable solutions of many equations with circular symmetry.

16.3.2 Spherical polar co-ordinates

The above discussion of plane polar co-ordinates is easily extended to the spherical polar co-ordinates (r, θ, φ) discussed in Section 11.3.1. However in this case, (r, θ, φ) and (r, θ, φ + 2π) correspond to the same point in space, so that if u is a physical quantity, we must impose the boundary condition

(16.50)

in addition to requiring that u is finite and continuous.

Again, we shall illustrate this by considering Laplace's equation (16.2). We start by using the result given in Table 12.1 for in spherical polar co-ordinates to express (16.2) as

(16.51)

and then substitute the decomposition

(16.52)

into (16.51). After dividing through by u and multiplying by r², we obtain the equation

(16.53)

The first two terms are functions of both r and θ, whereas the third term is a function of φ alone. Since the three variables are independent, this means that the third term must be a constant, and so we can set

(16.54)

The rest of the equation can then be manipulated into the form

Again we use the fact that each side is a function of a single independent variable and hence must be a constant, which we will denote by l(l + 1) for later convenience. We thus have the two equations

(16.55)

and

(16.56)

The radial equation (16.55) may be written as

and has the general solution

(16.57)

where A_l and B_l are arbitrary constants, and the possible values of l are restricted by consideration of the angular dependence, as we shall soon see.

We start with the φ-dependence. The only solutions of (16.54) compatible with the boundary conditions (16.50) are

(16.58)

Alternatively, and equivalently, we can choose a set of solutions

(16.59)

where the coefficients A_m are arbitrary constants.

We next turn to the θ dependence. On setting μ = cos θ and expanding the first term in (16.56), it becomes

(16.60)

This is the associated Legendre equation, which has solutions² that are finite for μ = cos θ = −1 (i.e. on the negative z-axis) only if l = 0, 1, 2, … and the m values are restricted to − l ≤ m ≤ l. They are called associated Legendre polynomials and denoted P^m_l(cos θ), so that we write

(16.61)

and from (16.57), (16.59) and (16.61), we find that the finite separable solutions which satisfy the boundary conditions (16.50) are

(16.62)

where l = 0, 1, 2, …, − l ≤ m ≤ l and A_lm, B_lm are arbitrary constants. The explicit forms of the first few polynomials are given in Example 15.10. For m = 0, when (16.60) reduces to Legendre's equation (15.23), they reduce to the Legendre polynomials P_l(cos θ) of order l discussed in detail in Section 15.3.1. Hence for m = 0 the solutions (16.62) reduce to

(16.63)

which are independent of the azimuthal angle φ and unchanged by rotations about the z-axis. For l = m = 0, since P₀(cos θ) = 1, (16.63) reduces to

(16.64)

which is the most general spherically symmetric solution to Laplace's equation. The most general solution, without symmetry constraints, is obtained by taking linear combinations of the separable solutions (16.63) and is

(16.65)

Finally, we note that the angular dependence of the separable solutions (16.62) is not specific to Laplace's equation, but is shared by the separable solutions of other important equations that have spherical symmetry. For example, the separable solutions of the wave equation and the diffusion equation also take the form

(16.66)

for appropriate choices of R(r) and T(t), as the reader may verify.

In practice, it is common to rewrite equations like (16.62) and (16.65) in terms of the so-called spherical harmonics defined by

(16.67a)

where the constant is chosen so that the normalisation condition

(16.67b)

is satisfied. With this convention, the first few spherical harmonics are given by

16.3.3 Cylindrical polar co-ordinates

We conclude the discussion of polar co-ordinates by considering cylindrical polar co-ordinates defined in Section 11.3.1. In this case, the co-ordinates (ρ, φ, z) and (ρ, φ + 2π, z) represent the same point, so the boundary condition corresponding to (16.50) is

(16.69)

We shall again take as our example the Laplace equation, which in cylindrical polar co-ordinates takes the form

(16.70)

where we have used the expression for given in Table 12.1. Assuming a separable solution

(16.71)

then gives

(16.72)

on substituting into (16.70) and dividing by u. In this equation, the first two terms are independent of z, while the third term depends only on z, and so must be a constant. Hence, separating the third term, we obtain

(16.73a)

and

where we have taken the separation constant to be k². The second of these equations is not separable as it stands, but multiplying through by ρ² gives

which is separable. Separating the third term and denoting the separation constant by m² then gives

(16.73b)

and

(16.73c)

where we have expanded the ρ-derivative.

It remains to solve the three ODEs (16.73a)–(16.73c). For any k, the general solution of the first of these is

(16.74)

where A_k and B_k are arbitrary constants. If we impose the boundary conditions (16.69), the general solutions of (16.73b) are

(16.75)

where m ≥ 0 is an integer, and C_m, D_m are arbitrary constants. Finally, on setting η = kρ in (16.73c) we obtain

This is Bessel's equation (15.53), with general solutions of the form (15.70), i.e.

(16.76)

where E_m and F_m are arbitrary constants and J_m(kρ) and N_m(kρ) are Bessel functions of the first and second kind respectively. These Bessel functions were discussed in Section 15.4.1, where we saw that N_m(kρ) was singular at kρ = 0. If we require solutions that are finite at ρ = 0, we must therefore set F_m = 0 in (16.71). Hence those separable solutions (16.66) that are both finite and single-valued are

(16.77a)

and

(16.77b)

where we have absorbed C₀ and E_m into the other constants. Since Laplace's equation is homogeneous, more general solutions may then be formed by linear superposition of the solutions (16.77a) and (16.77b), where in applications the possible values of k and the various constants must be determined by boundary conditions, as we shall illustrate by an example.

*16.4 The wave equation: d'Alembert's solution

In the next three sections, we shall consider other methods of solution of PDEs, mainly applied to functions of two variables. We start with the wave equation (16.3b), which we will solve by introducing the new variables

(16.78)

On changing variables using (7.24), we obtain

and

so that the wave equation becomes

(16.79)

with the general solution [cf. (16.2a) and (16.2b)]

(16.80)

where f and g are arbitrary differentiable functions.

Equation (16.80) is the general solution of the wave equation and each of the two terms has a simple interpretation. Let us suppose

(16.81)

Then we have

for all t, so that the solution moves as illustrated in Figure 16.3 for a simple choice of u(x, 0). It represents a travelling wave moving in the positive x-direction with speed . Thus, for example, a simple harmonic wave with wavelength λ, wave number and angular frequency , can be written in the form

where A and α are arbitrary constants. Similarly, is a wave travelling in the minus x-direction and (16.80) shows that any other non-trivial solution of the wave equation may be written as a sum of a wave travelling to the right and one travelling to the left.

At this point, we digress briefly to indicate how this description can be extended to three dimensions. Denoting a point in space by its position vector r, one easily shows that the functions

(16.82)

are solutions of the three-dimensional wave equation, where the unit vector indicates a chosen direction in space. Since the equation of a plane perpendicular to the unit vector is , where c is a constant, then at any fixed time t, the functions f and g are constant over the whole plane. They are therefore plane waves travelling in the positive and negative directions, respectively. This may be more familiar if we note that a simple harmonic wave in the direction analogous to (16.80) is given by

where A and α are again constants and the wave vector .

We now return to the one-dimensional case and consider its solution subject to the initial conditions

(16.83a)

and

(16.83b)

where α(x) and β(x) are given functions. Substituting the general solution (16.80) into these boundary conditions then gives

(16.84a)

and

(16.84b)

where

and similarly for g′(x). We now integrate (16.84b) to get

where the integration constant c will depend on the arbitrarily chosen lower limit a of the integration. From this equation, together with (16.84a), we obtain

and

and hence

and

Finally, adding we find

(16.85)

This is d'Alembert's solution to the wave equation (16.3b) subject to the boundary conditions (16.83). It is unique and independent of the intermediate constants c and a introduced in its derivation. Furthermore, at any point x = x₀, u(x₀, t) is dependent only on the initial values u(x, 0) and in the range , and is independent of u and outside this range. This embodies the idea of ‘causality’ for the wave equation, since it is just the range from within which a signal emitted at t = 0 and travelling with speed can reach x₀ in a time less than or equal to t.

*16.5 Euler Equations

The method used in the previous section to obtain (16.80) as the general solution of the wave equation can be extended to solve any equation of the form

(16.86)

where A, B, C, are given constants and x, y are any variables, not necessarily Cartesian co-ordinates. Such equations are often called Euler's equations.⁴ To solve them we introduce the new variables

(16.87)

where λ₁ and λ₂ are constants. We then try to find values of λ₁ and λ₂ such that (16.86) reduces to an equation of the form (16.79), with a general solution

(16.88)

analogous to (16.80) for the wave equation.

To see whether this is possible, we change variables using (7.24) to obtain

Substituting these expressions into (16.86) gives

(16.89)

If we now choose λ_i (i = 1, 2) to be the roots of

(16.90)

then (16.89) reduces to

(16.91)

So, provided the term in square brackets does not vanish,

and the solution (16.88) follows by successive integrations.

The condition that the square bracket in (16.91) does not vanish is easily found by noting from Equation (2.7) that

so that

that is, the equation must be such that AC ≠ B².

If AC = B², the square bracket in (16.91) does vanish and (16.90) has only one solution, which is a repeated root given by . In this case, we choose, , λ₂ = 0, and substituting these in (16.87) we find that the first and third terms vanish and the equation reduces to

Direct integration then gives

(16.92)

where f and g are again arbitrary functions of ξ. The solution of the PDE when AC = B² is therefore

(16.93)

where f and g are arbitrary functions of ξ.

To summarise, we have to distinguish between the cases AC = B², when the general solution is given by (16.93); and AC ≠ B², when the general solution is given by (16.88), where λ₁, λ₂ are the roots of (16.90).

*16.6 Boundary conditions and uniqueness

So far, we have focussed on problems that can be solved exactly. However, it is often not possible to do this in practice, and then one must resort to numerical methods to find approximate solutions of a given PDE that satisfy specific boundary conditions. In these cases, especially, it is very useful to know in advance what boundary conditions result in a unique, stable solution of the PDE, where by stable we mean that very small changes in the boundary conditions do not lead to very large changes in the solution. Here we will simply state the main results without proof, since the derivations are often difficult.⁵

The boundary conditions take the form of information about the dependent variable u specified on a continuous boundary, which may be open, like a plane in three-dimensional space, or closed, like the surface of a sphere. The main types of boundary conditions are classified as follows:

• Dirichlet.

  The value of u is specified at each point of the boundary.

• Neumann

  The value of the normal derivative , where is the unit normal to the boundary, is specified at each point of the boundary.

• Cauchy

  The values of both u and are specified at each point of the boundary.

   The next step is to classify PDEs into three types, called elliptic, hyperbolic and parabolic. In doing so, we shall focus on second-order linear equations that contain only the derivatives

  (16.94a)

  with constant coefficients, where is replaced by ∂²u/∂x² if there is only a single spatial variable. Thus, we consider PDEs of the form

  (16.94b)

  where A, B, C and D are constants and ρ is a given function. This form includes many equations of physical interest, including all those listed in Section 6.1. They are then classified according to which of the partial derivatives (16.94a) occur, and by the relative sign of their coefficients. We consider each in turn.

• Elliptic equations

  These are defined as those containing and ∂²u/∂t² with coefficients of the same sign, that is, AB > 0; or just with no time derivatives. The latter are of most interest and include Laplace's equation, Poisson's equation and the Helmholtz equation

  (16.95)

  both with a source (ρ ≠ 0) and without (ρ = 0), where k² is a positive real constant.

   Elliptic equations have the property that if either Dirichlet or Neumann boundary conditions are applied on a closed boundary, then the equation has a unique and stable solution within the boundary. Consequently, if Cauchy boundary conditions are applied, the equation in general has no solutions, and the equation is said to be over-constrained. The closed boundary may be finite, as illustrated for the Laplace equation in Examples 16.7 and 16.8, which used Dirichlet conditions on a finite closed surface; or the surface may be at infinity, as illustrated for Poisson's equation in Example 16.6. Alternatively, if one wishes to determine the function outside a finite closed boundary, then either Dirichlet or Neumann conditions must be applied both on the finite boundary and at infinity.

• Hyperbolic equations

  These are defined as those containing and ∂²u/∂t² with coefficients of the opposite sign, that is, AB < 0. They may in principle also contain terms in ∂u/∂t, but in practice such terms are usually absent in physical applications. Examples of hyperbolic equations are the wave equations (16.3a) and (16.3b) and the Klein-Gordon equation.

  

  which plays an important role in relativistic quantum mechanics, where c is the speed of light and m is a particle mass.

   Hyperbolic equations have unique and stable solutions if Cauchy boundary conditions are applied on an open boundary. In physical applications, this is usually taken to correspond to a constant time, which can always be chosen to be t = 0. One thus has to specify u(r, t = 0) and the time derivative of u(r, t) at t = 0 to obtain a unique and stable solution, as illustrated in the wave equation in Section 16.2.1 [cf. (16.19)] and Example 16.2 for standing waves constrained to vanish at x = 0, L; and for travelling waves in one dimension in Section 16.4 [cf. (16.83, 85)].

• Parabolic equations

  These are defined as those containing terms in and ∂u/∂t, but not terms in ∂²u/∂t², that is, AB = 0. Examples are the diffusion equation (16.4) and the Schrödinger equation (16.7). In this case, unique and stable solutions are obtained if Dirichlet or Neumann conditions are imposed on an open boundary. This is almost always chosen to be constant time t = t₀, and unique and stable solutions for t > t₀ are obtained given either u(r, t₀) or . This is illustrated for the case of Dirichlet boundary conditions for the examples given in Section 16.2.3 and Section 16.6.1 below.

   Finally, we stress again that we have only considered equations containing the partial derivatives (16.89), since this covers many of the most important PDEs in physical applications. The discussion can, however, be extended to all linear second-order PDEs with constant coefficients.⁶

*16.6.1 Laplace transforms

In Section 14.2.4, we introduced Laplace transforms (14.44) and showed how they could be used to obtain solutions of ODEs that automatically incorporated given boundary conditions. This method can be extended in principle to PDEs in which the boundary conditions are given at an initial time t = 0, although, as in ODEs, it may be difficult to perform the inverse Laplace transform required to obtain the final solution.

To illustrate this, we shall consider the bounded solution of the diffusion equation in one spatial dimension (16.31) in the range 0 < x < ∞ for times t > 0, with boundary conditions

(16.96)

This could, for example, describe the temperature distribution due to heat flow along a long rod with one end at x = 0, if the rod is initially at zero temperature u = 0, but is in contact at t > 0 with a heat bath of constant temperature u₀ at the end x = 0, and the sides of the bar are perfectly lagged so that heat flow from the sides can be neglected.

To solve this problem, we take the Laplace transform of both sides of (16.31) with respect to time t from (14.44) and (14.45a). We then have, with an appropriate change of notation:

(16.97a)

and

(16.97b)

where we have used (16.96) to set u(x, 0) = 0 in (16.97b). Hence (16.31) becomes

(16.98)

with the general solution

where α = (p/κ)^1/2 > 0 and A and B are arbitrary constants. If u(x, t) is bounded as x → ∞, which is an obvious requirement if it represents a temperature distribution, then it follows that F(x, p) must also be bounded as x → ∞, so that A = 0. The value of B is then found by imposing the boundary condition u(0, t) = u₀, which from (16.97a) gives

Hence,

(16.99a)

and the final solution is given by the inverse transform

(16.99b)

At this point, we remind the reader that, as discussed in Section (14.2.4), finding inverse Laplace transforms is difficult and often impossible to do in closed form. One frequently has to resort to tables of such transforms like that of Table 14.1, or the more extensive tables available in the literature.⁷ In the case above, the required inverse transform can be expressed in terms of the error function

(16.100a)

and the associated complementary error function

(16.100b)

The error function is normalised so that it tends to unity as t → ∞, since (cf. Example 11.11)

The behaviour of both (16.100a) and (16.100b) is shown in Figure 16.4. The relevance of this becomes clear on taking the Laplace transform of , which can be shown to be [cf. Example 16.11]

(16.101a)

This obviously implies

(16.101b)

which, together with (16.99a) and (16.99b), gives

(16.102)

as the final solution. Hence u(x, t) → u₀ as t → ∞, but more slowly as x increases, which is what one intuitively expects if u represents the temperature of a long bar, as in this example.