13.1 Partial Differential Equations

Many complex physical, geometrical, and stochastic (probabilistic) phenomena are described by partial differential equations (PDEs), involving two or more independent variables. They are obviously much more difficult to solve than ordinary differential equations (ODEs). Some applications, including weather prediction, econometric models, fluid dynamics, and nuclear engineering, might involve simultaneous PDEs with large numbers of variables. Such problems are best tackled by powerful supercomputers. We will be content to consider a few representative second-order PDEs for which analytic solutions are possible.

Scalar and vector fields which depend on more than one independent variable, which we write in the notation , etc., are very often obtained as solutions to PDEs. Some classic equations of mathematical physics which we will consider are the wave equation, the heat equation, Laplace’s equation, Poisson’s equation, and the Schrödinger equation for some exactly solvable quantum-mechanical problems.

Maxwell’s equations lead to the wave equation for components of the electric and magnetic fields, with the general form

(13.1)

Analogous equations apply to sound and other wave phenomena. For waves in one spatial dimension, the wave equation reduces to

(13.2)

The heat equation or diffusion equation is similar to the wave equation but with a first derivative in the time variable:

(13.3)

where is a constant determined by the thermal conductivity or diffusion coefficient.

Introducing the electrostatic potential into the first of Maxwell’s equation, , we obtain Poisson’s equation

(13.4)

In a region free of electric charge, this reduces to Laplace’s equation

(13.5)

In the calculus of complex variables, we encounter the two-dimensional Laplace equation for

(13.6)

For the case of monochromatic time dependence, the solution of the wave equation has the form

(13.7)

where

(13.8)

or some linear combination of sine and cosine or complex exponentials. Substituting (13.7) into the wave equation (13.1) we find

(13.9)

where we have noted that acts only on the factor while acts only on . Since either of the functions (13.8) is a solution of the ordinary differential equation

(13.10)

Equation (13.9) reduces to Helmholtz’s equation

(13.11)

after canceling out and defining . The heat equation also reduces to Helmholtz’s equation for separable time dependence in the form

(13.12)

Recall that ordinary differential equations generally yield solutions containing one or more arbitrary constants, which can be determined from boundary conditions. In contrast, solutions of partial differential equations often contain arbitrary functions. An extreme case is the second-order equation

(13.13)

This has the solutions

(13.14)

where and are arbitrary differentiable functions. More detailed boundary conditions must be specified in order to find more specific solutions.

13.2 Separation of Variables

The simplest way to solve PDEs is to reduce a PDE with n independent variables to n independent ODEs, each depending on just one variable. This isn’t always possible but we will limit our consideration to such cases. In the preceding section, we were able to reduce the wave equation and the heat equation to the Helmholtz equation when the time dependence of was separable. Consider the Helmholtz equation in Cartesian coordinates

(13.15)

If the boundary conditions allow, the solution might be reducible to a separable function of :

(13.16)

Substituting this into (13.15), we obtain

(13.17)

Division by gives the simplified equation:

(13.18)

Now, if we solve for the term , we find this function of x alone must be equal to a function of y and z, for arbitrary values of . The only way this is possible is for the function to equal a constant, say

(13.19)

A negative constant is consistent with a periodic function , otherwise it would have exponential dependence. Analogously,

(13.20)

so that

(13.21)

We have thereby reduced Eq. (13.15) to three ordinary differential equations:

(13.22)

We solved an equation of this form in Section 8.5, for two alternative sets of boundary conditions. The preceding solution of the Helmholtz equation in Cartesian coordinates is applicable to the Schrödinger equation for the quantum-mechanical particle-in-a-box problem.

13.3 Special Functions

The designation elementary functions is usually applied to the exponential, logarithmic, and trigonometric functions and combinations formed by algebraic operations. Certain other special functions occur so frequently in applied mathematics that they acquire standardized symbols and names—often in honor of a famous mathematician. We will discuss a few examples of interest in physics, chemistry, and engineering, in particular, the special functions named after Bessel, Legendre, Hankel, Laguerre, and Hermite. We already encountered in Chapter 6 the gamma function, error function, and exponential integral. Special functions are most often solutions of second-order ODEs with variable coefficients, obtained after separation of variables in PDEs. So that you will have seen the names, here is a list of some other special functions you are likely to encounter (although not in this book): Airy functions, Beta functions, Chebyshev polynomials, Elliptic functions, Gegenbauer polynomials, Jacobi polynomials, Mathieu functions, Meijer G-functions, Parabolic cylinder functions, Theta functions, Whittaker functions.

For a comprehensive reference on special functions, see M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, (National Bureau of Standards Applied Mathematics Series, Vol. 55,Washington, DC, 1964). This is now available online at http://dlmf.nist.gov and features graphics and hypertext links.

Following is a sampling of some “amazing” formulas involving gamma and zeta functions, two of the special functions we have already introduced. A reflection formula for gamma functions:

(13.23)

An infinite product representation of the gamma function:

(13.24)

where , the Euler-Mascheroni constant. An integral for the Riemann zeta function:

(13.25)

Euler’s relation connecting the zeta function with prime numbers, which we proved in Section 1.12:

(13.26)

For a long time, I felt intimidated by these amazing formulas involving special functions. Some resembled near miracles like “pulling rabbits out of a hat.” I could never, in a thousand years, have come up with anything as brilliant as one of these myself. I would not have been surprised to see in some derivation “and then a miracle occurs” between the next-to-last and last equations (see Sidney Harris cartoon):

Over the years, I learned to be less intimidated, assuring myself these amazing results were the product of hundreds of brilliant mathematicians using inspired guesswork, reverse engineering, and all kinds of other nefarious practices for over 300 years. Ideally, one should be fully comfortable in using these results, resisting any thoughts that we are not entitled to exploit such brilliance. Something like the habitual comfort most of us have developed in using our cars, computers, and washing machines. Without a doubt, we all “stand on the shoulders of giants.”

13.4 Leibniz’s Formula

We will later need a formula for the nth derivative of the product of two functions. This can be derived stepwise as follows:

(13.27)

Clearly, we are generating a series containing the binomial coefficients

(13.28)

and the general result is Leibniz’s formula

(13.29)

13.5 Vibration of a Circular Membrane

As our first example of a special function, we consider a two-dimensional problem: the vibration of a circular membrane such as a drumhead. The amplitude of vibration is determined by solution of the Helmholtz equation in two dimensions, most appropriately chosen as the polar coordinates . Using the scale factors and for the two-dimensional Laplacian, the Helmholtz equation can be written

(13.30)

This is subject to the boundary condition that the rim of the membrane at is fixed, so that . Also it is necessary that the amplitude be finite everywhere on the membrane, in particular, at . Assuming a separable solution , Helmholtz’s equation reduces to

(13.31)

or

(13.32)

We could complete a separation of variables by dividing by . This would imply that the function of in square brackets equals a constant, which we can write

(13.33)

This is a familiar equation, with linearly independent solutions

(13.34)

Since is an angular variable, the periodicity is required. This restricts the parameter m to integer values. Thus we obtain an ordinary differential equation for :

(13.35)

Changing the variable to , with , we can write

(13.36)

This is recognized as Bessel’s equation (8.135). Only the Bessel functions are finite at , the Neumann functions being singular there. Thus the solutions to Helmholtz’s equation are

(13.37)

The boundary condition at requires that

(13.38)

The zeros of Bessel functions are extensively tabulated. Let represent the nth root of . The first three zeros are given by:

The eigenvalues of k are given by

(13.39)

Some modes of vibration of a circular membrane, as labeled by the quantum numbers m and n, are sketched in Figure 13.1. To simplify the figure, only the sign of the wavefunction is indicated: gray for positive, white for negative. Note that modes for are twofold degenerate, corresponding to the factors and .

13.6 Bessel Functions

In Section 8.7, a series solution of Bessel’s differential equation was derived, leading to the definition of Bessel functions of the first kind:

(13.40)

This applies as well for noninteger n with the replacements . The Bessel functions are normalized in the sense that

(13.41)

Bessel functions of integer order can be determined from expansion of a generating function:

(13.42)

To see how this works, expand the product of the two exponentials:

(13.43)

The terms which contribute to are

(13.44)

which gives the expansion for . The expansion for is found from the terms proportional to and so forth.

One of the many integral representations of Bessel functions is

(13.45)

An identity involving the derivative is

(13.46)

In particular, . An addition theorem for Bessel functions states that

(13.47)

Asymptotic forms for the Bessel function for small and large values of the argument are given by

(13.48)

and

(13.49)

Bessel functions of the second kind show the following limiting behavior:

(13.50)

and

(13.51)

The two kinds of Bessel functions thus have the asymptotic dependence of slowly damped cosines and sines. In analogy with Euler’s formula , we define Hankel functions of the first and second kind:

(13.52)

The asymptotic forms of the Hankel functions are given by

(13.53)

having the character of waves propagating to the right and left, respectively.

Trigonometric functions with imaginary arguments suggested the introduction of hyperbolic functions: and . Analogously, we can define modified Bessel functions (or hyperbolic Bessel functions) of the first and second kind as follows:

(13.54)

and

(13.55)

Their asymptotic forms for large x are

(13.56)

We have given just a meager sampling of formulas involving Bessel functions. Many more can be found in Abramowitz & Stegun and other references. G.N. Whittaker’s classic A Treatise on the Theory of Bessel Functions (Cambridge University Press, 1952) is a ponderous volume devoted entirely to the subject.

13.7 Laplace’s Equation in Spherical Coordinates

Laplace’s equation in spherical polar coordinates can be written

(13.57)

We consider separable solutions

(13.58)

where the functions are known as a spherical harmonics. Substitution of (13.58) into (13.57), followed by division by and multiplication by , separates the radial and angular variables:

(13.59)

This can hold true for all values of only if each of the two parts of this partial differential equation equals a constant. Let the first part equal and the second equal . The ordinary differential equation for r becomes

(13.60)

It is easily verified that the general solution is

(13.61)

Returning to the spherical harmonics, these evidently satisfy a partial differential equation in two variables:

(13.62)

Again we assume a separable solution with

(13.63)

Substituting (13.63) into (13.62), dividing by , and multiplying by , we achieve separation of variables:

(13.64)

Setting the term containing to a constant , we obtain the familiar differential equation

(13.65)

Solutions periodic in can be chosen as

(13.66)

Alternative solutions are and .

13.8 Legendre Polynomials

Separation of variables in Laplace’s equation leads to an ODE for the function :

(13.67)

Let us first consider the case and define a new independent variable

(13.68)

This transforms Eq. (13.67) to

(13.69)

which is known as Legendre’s differential equation. We can construct a solution to this linear second-order equation by exploiting Leibniz’s formula (13.29). Begin with the function

(13.70)

which is a solution of the first-order equation

(13.71)

Differentiating times, we obtain

(13.72)

where

(13.73)

This is a solution of Eq. (13.69) for

(13.74)

With a choice of constant such that , the Legendre polynomials can be defined by Rodrigues’ formula:

(13.75)

Reverting to the original variable , the first few Legendre polynomials are

(13.76)

The Legendre polynomials obey the orthonormalization relations

(13.77)

A generating function for Legendre polynomials is given by

(13.78)

An alternative generating function involves Bessel function of order zero:

(13.79)

Another remarkable connection between Legendre polynomials and Bessel functions is the relation

(13.80)

Returning to Eq. (13.67) for arbitrary values of m, the analog of Eq. (13.69) can be written

(13.81)

The solutions are readily found to be

(13.82)

known as associated Legendre functions. These reduce to the Legendre polynomials (13.75) when . Since is a polynomial of degree , is limited to the values . The associated Legendre functions obey the orthonormalization relations

(13.83)

The orthonormalized solutions to Eq. (13.67) are thus given by

(13.84)

13.9 Spherical Harmonics

Combining the above functions of and , we obtain the spherical harmonics

(13.85)

where

(13.86)

The factor is appended in applications to quantum mechanics, in accordance with the Condon and Shortley phase convention. Two important special cases are

(13.87)

and

(13.88)

Following is a listing of spherical harmonics through :

A graphical representation of these functions is given in Figure 13.2. Surfaces of constant absolute value are drawn, with intermediate shadings representing differing complex values of the functions. In quantum mechanics, spherical harmonics are eigenfunctions of orbital angular momentum operators such that

(13.89)

and

(13.90)

The spherical harmonics are an orthonormal set with respect to integration over solid angle:

(13.91)

Linear combinations of and contain the real functions:

(13.92)

These are called tesseral harmonics since they divide the surface of a sphere into tesserae—four-sided figures bounded by nodal lines of latitude and longitude. (Nodes are places where the wavefunction equals zero.) When , the spherical harmonics are real functions:

(13.93)

These correspond to zonal harmonics since their nodes are circles of latitude which divide the surface of the sphere into zones. When , the spherical harmonics reduce to

(13.94)

The corresponding real functions

(13.95)

are called sectoral harmonics. The three types of surface harmonics are shown in Figure 13.3.

13.10 Spherical Bessel Functions

Helmholtz’s equation in spherical polar coordinates can be obtained by adding a constant to Laplace’s equation (13.57):

(13.96)

As in the case of Laplace’s equation, Eq. (13.96) has separable solutions

(13.97)

Since we now know all about spherical harmonics, we can proceed directly to the radial equation:

(13.98)

Were it not for the factor 2 in the second term, this would have the form of Bessel’s equation (13.35). As in Section 13.5, we redefine the variables to and . We also rewrite as n to conform to conventional notation. The differential equation now reads

(13.99)

With the substitution , the equation reduces to

(13.100)

which is recognized as Bessel’s equation of odd-half order (if n is an integer). The linearly independent solutions are and , the latter being proportional to since the order is not an integer. The solutions to Eq. (13.99) are known as spherical Bessel functions and conventionally defined by

(13.101)

and

(13.102)

You can verify that Eq. (13.99) with has the simple solutions and . In fact, spherical Bessel functions have closed-form expressions in terms of trigonometric functions and powers of x, given by

(13.103)

Explicit formulas for the first few spherical Bessel functions follow:

(13.104)

There are also spherical analogs of the Hankel functions:

(13.105)

The first few are

(13.106)

13.11 Hermite Polynomials

The quantum-mechanical harmonic oscillator satisfies the Schrödinger equation:

(13.107)

To reduce the problem to its essentials, simplify the constants with , or alternatively, replace x by x. Correspondingly, . We must now solve a second-order differential equation with nonconstant coefficients:

(13.108)

A useful first step is to determine the asymptotic solution to this equation, giving the form of as . For sufficiently large values of , , so that the differential equation can be approximated by

(13.109)

This suggests the following manipulation:

(13.110)

Now, the first-order differential equation

(13.111)

can be solved exactly to give

(13.112)

To build in this asymptotic behavior, let

(13.113)

This reduces Eq. (13.108) to a differential equation for :

(13.114)

To construct a solution to Eq. (13.114), we begin with the function

(13.115)

which is clearly the solution of the first-order differential equation

(13.116)

Differentiating this equation times using Leibniz’s formula (13.29), we obtain

(13.117)

where

(13.118)

We find that satisfies

(13.119)

which is known as Hermite’s differential equation. The solutions in the form

(13.120)

are known as Hermite polynomials, the first few of which are enumerated below:

(13.121)

Comparing Eq. (13.119) with Eq. (13.114), we can relate the parameters

(13.122)

Referring back to the original harmonic-oscillator equation (13.107), this leads to the general formula for energy eigenvalues

(13.123)

A generating function for Hermite polynomials is given by

(13.124)

Using the generating function, we can evaluate integrals over products of Hermite polynomials, such as

(13.125)

Thus, the functions

(13.126)

form an orthonormal set with

(13.127)

13.12 Laguerre Polynomials

The quantum-mechanical problem of a particle moving in a central field is represented by a three-dimensional Schrödinger equation with a spherically symmetric potential :

(13.128)

As in the case of Helmholtz’s equation, we have separability in spherical polar coordinates: . In convenient units with , the ODE for the radial function can be written

(13.129)

We consider the electron in a hydrogen atom or hydrogenlike ion () “orbiting” around a nucleus of atomic number Z. The attractive Coulomb potential in atomic units () can be written

(13.130)

It is again useful to find asymptotic solutions to the differential equation. When the equation is approximated by

(13.131)

noting that the energy E will be negative for bound states of the hydrogenlike system. We find the asymptotic solution

(13.132)

As , Eq. (13.129) is approximated by

(13.133)

which is just Laplace’s equation in spherical coordinates. The solution finite at suggests the limiting dependence

(13.134)

We can incorporate both limiting forms by writing

(13.135)

in terms of a new variable

(13.136)

where n is a constant to be determined. The differential equation for then works out to

(13.137)

Following the strategy used to solve the Hermite and Legendre differential equations, we begin with a function

(13.138)

where is a positive integer. This satisfies the first-order differential equation

(13.139)

Differentiating this equation times using Leibniz’s formula (13.29), we obtain

(13.140)

where

(13.141)

Laguerre polynomials are defined by Rodrigues’ formula:

(13.142)

We require a generalization known as associated Laguerre polynomials, defined by

(13.143)

These are solutions of the differential equation

(13.144)

Comparing Eqs. (13.137) and (13.144), we can identify

(13.145)

where n must be a positive integer. The bound-state energy hydrogenlike eigenvalues are therefore determined:

(13.146)

with the normalized radial functions

(13.147)

The conventional definition of the constant is

(13.148)

such that

(13.149)

Laguerre and associated Laguerre polynomials can be found from the following generating functions:

(13.150)

(13.151)

13.13 Hypergeometric Functions

A geometric series is a function of whose terms constitute a geometric progression, in which the ratio of successive terms is equal. Consider, for example, with z and a, in general, being complex quantities,

(13.152)

This series converges to the value , provided that . A certain type of generalization of a geometric series is known as a hypergeometric series or hypergeometric function. This has the form of a power series in z in which the coefficients are replaced by ratios of rational functions of constants.

A rudimentary example of a hypergeometric function can be written

(13.153)

Here , known as the Pochhammer symbol (also called a rising or ascending factorial), is defined by

(13.154)

or

(13.155)

The last form is valid even if a is not an integer. For reduces to the geometric series for z. When a is a negative integer, the series terminates, for example,

(13.156)

Next consider the hypergeometric function

(13.158)

The series is convergent for finite z, provided that It is known as the confluent hypergeometric function (explanation later) or Kummer function. It is a solution of the differential equation:

(13.159)

Two important properties of the confluent geometric function are Kummer’s first formula, again for or a negative integer,

(13.160)

and Kummer’s second formula:

(13.161)

For large , the function has the asymptotic forms:

(13.162)

For certain combinations of a, c, and z, the confluent hypergeometric function reduces to some of the special functions considered earlier in this chapter. First, the rather trivial case:

(13.163)

(Even more trivial is .) Some other elementary functions are obtained from

(13.164)

and

(13.165)

The error function can be represented by

(13.166)

When , we obtain a relation for Bessel functions:

(13.167)

Another formula for a Bessel function:

(13.168)

For , the Laguerre polynomial is given simply by

(13.169)

Hermite polynomials reduce slightly to different forms for even and odd n. For ,

(13.170)

while, for ,

(13.171)

Gauss’ famous hypergeometric differential equation is given by

(13.172)

where c does not equal There exist 24 possible solutions involving transformed versions of . We will consider only the simplest one, which is what was originally called the hypergeometric function:

(13.173)

There is a huge amount of mathematical technology involving the hypergeometric function. The interested reader can consult numerous references which cover these in great detail. We will content ourselves with some simple results:

(13.174)

(13.175)

(13.176)

The Legendre polynomials can be represented by:

(13.177)

The concept of confluence involves the substitution , followed by taking the limit . This reduces the hypergeometric differential equation and the hypergeometric function to their analogs for the confluent hypergeometric function.

It is also possible to define generalized hypergeometric functions of higher order, involving more than three constants:

(13.178)