A cornerstone of the method (and the separation of variables method) is the as-
sumption that an arbitrary function, such as the temperature field in a heat transfer problem, can be represented as a series expansion involving some set of orthogonal functions. Thus, it was assumed in Eq. (4.2-13) that a certain two-dimensional tempera- ture field, y), could be expressed as the sum of an infinite number of terms, each consisting of a basis function multiplied by a coefficient Although Example 4.2-2 illustrated the mechanics of evaluating the coefficients for one such series, it left some fundamental questions unanswered. For what kinds of functions is such a series valid? As more and more terms are evaluated, in what manner will the series converge to the function it is supposed to represent? This section addresses those questions, and it also presents a general procedure for evaluating the coefficients in such a series. The results presented here not just to series constructed using the trigonometric basis functions described in Section 4.3, but also to expansions employing any other basis
. functions derived from the general problem described in Section 4.6.
Those include Bessel functions (Section 4.7) and polynomials (Section
Generalized Fourier Series
To address the main issues, it is sufficient to consider arbitrary functions of a single variable, f (x). It is desired to represent such a function using a series of the form
where the functions a complete orthonormal sequence, as discussed in 4.3, (The starting value of has no special significance; sometimes it will be con- venient to label the first term with = Any such expansion involving
functions is referred to here as a Fourier series, and the constant coefficients are called spectral When reference is made to expansions involving spe- cific basis functions, adjectives such as "Fourier-sine," "Fourier-Bessel," or
are sometimes used.
To use a Fourier series to represent a function f ( x ) over some closed interval [a,
in for b), it is sufficient that f ( x ) be piecewise continuous in that interval [see, for example, Churchill A piecewise continuous function is one which is continuous throughout the interval, or which exhibits a finite number of jump discontinuites. At a jump discontinuity the function has different limits evaluated from the left and right, but both limits are finite. It is helpful to bear in mind that the physical processes of conduction or diffusion (and the second derivatives which are the
cussed in this section, involving (4.3-4) and boundary conditions chosen from (4.3-5), are self-adjoint.
To an orthogonal set of basis functions from a set of eigenfunctions like that generated in any of the three examples in this section, it is necessary only to ensure linear independence. Considering the eigenfunctions given by Eq. note that
-
sin(-x). Because the eigenfunctions with differ only in sign from those including negative as well as positive values of n would result in a set of functions which is not linearly independent. Accordingly, we choose to exclude the neg- ative integers. As mentioned below Eq. = is already excluded because it corresponds to the trivial solution. With regard to the eigenfunctions in Eq.note that = Consequently, the eigenfunctions for duplicate those for that once again we exclude those for In this case we must retain =
because it gives a nontrivial solution, Similar reasoning applies to the eigen- functions given by (4.3-20). For each positive root of Eq. (4.3-19) there is a "mirror image" negative root. This, coupled with the fact that involves only sines cosines, indicates that we should discard all eigenfunctions corresponding to nega- tive values of A.
For convenient reference, the sequences obtained from four common problems are given in Table These four problems include all
tions of Dirichlet and Neumann boundary conditions applied at = and x = Notice that in case IV one of the eigenfunctions is a constant, = corresponding to A =O. In each of these cases there are simple, explicit expressions for the eigenvalues. When one or more Robin (mixed) conditions is present, as in Example 4.3-3, the eigen- values must be found numerically.
With this additional information about basis functions, the reader may find it help- ful to review Example The basis functions defined by are seen now to be normalized and linearly independent versions of the eigenfunctions derived in Example Those eigenfunctions were useful because they satisfy hdmogeneous
Sequences of Functions from Certain Eigenvalue Problems in
Rectangular Coordinatesa
Case Boundary conditions Basis
the functions shown satisfy (4.3-4) in the
Representation of an Arbitrary Function Using
boundary conditions of same type as the homogeneous boundary condi- tions in the heat transfer example. Notice also that in Eq. (4.2-19) it was important for
the basis functions to satisfy Eq. (4.3-4), the differential equation of the eigenvalue problem. Finally, note the role that orthogonality played in relating to
using Eqs. (4.2-24) and (4.2-25).
4.4 REPRESENTATION OF AN ARBITRARY FUNCTION
ORTHONORMAL FUNCTIONS
A cornerstone of the method (and the separation of variables method) is the as-
sumption that an arbitrary function, such as the temperature field in a heat transfer problem, can be represented as a series expansion involving some set of orthogonal functions. Thus, it was assumed in Eq. (4.2-13) that a certain two-dimensional tempera- ture field, y), could be expressed as the sum of an infinite number of terms, each consisting of a basis function multiplied by a coefficient Although Example 4.2-2 illustrated the mechanics of evaluating the coefficients for one such series, it left some fundamental questions unanswered. For what kinds of functions is such a series valid? As more and more terms are evaluated, in what manner will the series converge to the function it is supposed to represent? This section addresses those questions, and it also presents a general procedure for evaluating the coefficients in such a series. The results presented here not just to series constructed using the trigonometric basis functions described in Section 4.3, but also to expansions employing any other basis
. functions derived from the general problem described in Section 4.6.
Those include Bessel functions (Section 4.7) and polynomials (Section
Generalized Fourier Series
To address the main issues, it is sufficient to consider arbitrary functions of a single variable, f (x). It is desired to represent such a function using a series of the form
where the functions a complete orthonormal sequence, as discussed in 4.3, (The starting value of has no special significance; sometimes it will be con- venient to label the first term with = Any such expansion involving
functions is referred to here as a Fourier series, and the constant coefficients are called spectral When reference is made to expansions involving spe- cific basis functions, adjectives such as "Fourier-sine," "Fourier-Bessel," or
are sometimes used.
To use a Fourier series to represent a function f ( x ) over some closed interval [a,
in for b), it is sufficient that f ( x ) be piecewise continuous in that interval [see, for example, Churchill A piecewise continuous function is one which is continuous throughout the interval, or which exhibits a finite number of jump discontinuites. At a jump discontinuity the function has different limits evaluated from the left and right, but both limits are finite. It is helpful to bear in mind that the physical processes of conduction or diffusion (and the second derivatives which are the
embodiments of those processes) tend to smooth the temperature or concentra- tion profile, usually ensuring that the field variable its first derivatives are continu- ous phase. Discontinuities in the field variable may exist at internal interfaces or boundaries, but such discontinuities ordinarily consist of finite jumps. Accordingly, almost conduction or diffusion problem will involve only piecewise continuous functions. A more general classification of functions which can be repre- sented (4.4-1) has been developed using linear operator theory [see and Sell (1982) and