This document discusses Fourier series and their application to solving partial differential equations that model transport phenomena. Specifically, it provides: 1) An overview of Fourier series and their use in expanding periodic functions as an infinite sum of sines and cosines. 2) An example of using Fourier series to solve the one-dimensional transient heat conduction equation, resulting in an infinite series solution for the dimensionless temperature. 3) Details on the steps of the Fourier series solution method, including separating variables, applying boundary conditions, and determining coefficients to obtain the series solution.

1. UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA - UNAD.
ESPECIALIZACIÓN EN INGENIERÍA DE PROCESOS DE ALIMENTOS Y BIOMATERIALES
MÉTODOS MATEMÁTICOS
SEGUNDA UNIDAD: FORMULACIÓN INTEGRAL Y DIFERENCIAL
CAPÍTULO CUATRO: FUNCIONES ESPECIALES.
LECCIÓN DOCE.
SERIES DE FOURIER.
Un caso especial y muy utilizado de la expansión de funciones ortogonales en series son las “series de Fourier”.
En un intervalo simétrico –L,L, la representación de las series de Fourier de una función f(x) se define como:
f § nSx · f § nSx ·
f( x) ¦ A n cos¨ ¸ ¦ B n sen¨ ¸
n 0 © L ¹ n 0 © L ¹
Donde n es un entero positivo.

2. Si la ecuación presentada se multiplica por cos nSx L dx y el resultado se integra entre –L y L, se pueden

3. determinar los coeficientes An. De manera similar, utilizando sen nSx L dx se obtienen lo coeficientes Bn:
1 L 1 L nSx
A0 ³ f( x)dx ; An ³ f( x) cos dx
2L L L L L
1 L nSx
B0 0 ; Bn ³ f( x )sen dx
L L L
Aunque la expansión en series de Fourier de la función f(x) dependa solo de términos de seno o de coseno, o de
ambos, si depende de si la función es regular, impar o ninguna de las dos.
Una función regular, es una donde f(x)=f(-x). x 2 , cos nx, xsenmx u un número puro son funciones regulares ( o
pares). Si una función es regular:
1L 2L nSx
A0 ³ f( x)dx ; An ³ f( x) cos dx ; B0 0
L0 L0 L
Entonces se puede decir que:
nSx § nSx ·
f( x)sen¨
f( x)sen ¸
L © L ¹
Consecuentemente, en la evaluación de los coeficientes An, la integral desde –L hasta 0 se suma a la integral de 0
a L, mientras que en la evaluación de los términos Bn estas integrales se cancelan. Esto se puede mostrar de
manera formal a continuación:
Sea f(x) una función regular (o par), entonces:
1 L 1 ªL 0 º
A0 ³ f( x )dx « ³ f( x )dx ³ f( x )dx »
2L ¬ 0
2 L L L ¼
1 ªL L º
A0 « ³ f( x )dx ³ f( x )dx»
2L ¬ 0 0 ¼
1L
A0 ³ f( x )dx
L0
Los coeficientes An y Bn se obtienen de la misma manera.
1

4. UNIVERSIDAD NACIONAL ABIERTA Y A DISTANCIA - UNAD.
ESPECIALIZACIÓN EN INGENIERÍA DE PROCESOS DE ALIMENTOS Y BIOMATERIALES
MÉTODOS MATEMÁTICOS
Una función impar, es una en la que f(x)=-f(-x). x , x 3 , sen( nx), x cos(nx) son funciones impares.
Si f(x) es impar:
A0 0
An 0
2L nSx
Bn ³ f( x )sen dx
L0 L
Si solo el intervalo 0 a L es de interés, f(x) puede expandirse solo en series senoidales o solo en series
cosenoidales, y esto puede realizarse aunque la función sea par o impar. Cuando este procedimiento se realiza la
expansión en términos de se senos genera una función impar que, en general, no representará a f(x) fuera del
intervalo o a L. y la expresión en términos de cosenos, genera una función par que de igual manera no
representa a f(x) fuera del intervalo.
Cuando el intervalo de despliegue de las series incluye al infinito, y si la función es ortogonal, se puede llegauar
en el despliegue a uno de los siguientes casos:
* Integral del seno de Fourier:
2 ªf f º
F( x) « ³ senax

5. ³ F( v)sen(av)dvda»
S ¬0
0xf
0 ¼
* Integral de coseno de Fourier:
2 ªf f º
F( x) « ³ cosax

6. ³ F( v) cos(av)dvda»
S ¬0
0xf
0 ¼
* Integral de Fourier-Bessel:
ff
F( x ) ³ ³ avF( v )J p (ax )dvda 0xf ; p ! 1
00
* Integral completa de Fourier:
1 f f
F( x ) ³ ³ F( v ) cos(a( v x )dvda fxf
2 S f f
En la carpeta del capítulo cuatro ud. podrá encontrar un ejemplo de solución utilizando series de Fourier,
extraído de BRDKEY y HERSHEY, Transport Phenomena.
2

7. 654 OF TRANSPORT PHENOMENA
It is common practice to abbreviate Eqs. (13.33) and (13.34) as
(x, 0) =
(0, = 0 (13.35)
Similarly, it is desirable to transform the mass transfer equation, Eq.
using some convenient variable such as
= (13.36)
where again the transformation is useful only if the boundary conditions are
not a function of time. For one-dimensional transient mass transfer, Eq.
(13.11) becomes
(13.37)
The boundary conditions in terms of are almost identical to those in Eq.
(13.35); the variable is simply substituted for
W.2.1 Fourier Series Solution
The solution of partial differential equations using Fourier series is usually
given in an advanced mathematics course at most universities. Hence, in this
section only a typical Fourier series solution to Eq. (13.32) will be given. The
reader is referred to the several excellent texts devoted to a more complete
treatment Ml, M4,
A Fourier series may be defined as
+ + sin (13.38)
where the function f(x) is represented in terms of two periodic infinite series,
as shown in Eq. (13.38). If the function f(x) is assumed to be periodic, with
period’ 2L as shown in Fig. 13.6, then it is easily shown that
(13.39)
(13.40)
(13.41)
A function is periodic if + for all

8. UNSTEADY-STATE TRANSPORT
The power of Fourier series arises from the fact that any function may be
assumed periodic, even if it is not, by assuming that the length of the period is
the region of interest. For clarification, let us consider the following boundary
conditions:
0) = =0 =0 (13.35)
These are the simplest possible; since the transformed temperature is zero at
either end, these boundary conditions are termed “homogeneous”. Obviously,
those are not in themselves periodic; yet they may be considered as a periodic
function of period as shown in Fig. 13.7. Physically the boundary
conditions in Fig. 13.7 have no meaning for less than zero or greater than
but the mathematical assumption of periodicity allows a Fourier series
solution, as will be shown later.
Many functions likely to be encountered in our engineering problems can
be expanded in a Fourier series. There are some mathematical restrictions,
known as the Dirichlet conditions that cannot be violated. A function is
Fourier expandable if in the interval 0 2L the following are true:
1. The function f(x) is single-valued.
2. The function f(x) never becomes infinite.
3. The function f(x) has a finite number of maxima and minima.
4. The function f(x) has a finite number of discontinuities.
A practical concern that does limit the utility of Fourier series solutions is that
-4L -2L 0 2L 4L (13.35) as a periodic function.

9. 656 APPLICATIONS OF TRANSPORT PHENOMENA
the integrals in Eqs. (13.39) to (13.41) must be analytic, since and are
coefficients in an infinite series. Also, not all boundary conditions are
amenable to Fourier series solution C3, C6, Ml, M4,
The first step in the Fourier series solution of a partial differential
equation is to assume that the solution is a product of two quantities:
= (13.42)
where is a function of only and is a function of only. This assumption
can be justified only in that it leads to a solution satisfying the partial
differential equation and its boundary conditions. The assumed solution, Eq.
(13.42) is substituted into the partial differential equation of interest, Eq.
(13.32). Since is a function of only, it follows that
(13.43)
Similarly
(13.44)
Substituting the above into Eq. the result is
The variables in Eq. (13.45) are separated as follows
X ”
(13.46)
It is argued that, if is varied there is no effect on the term since
is not a function of x. Thus, the term must be independent
of x. A similar argument about varying and its effect on the term leads
to the conclusion that each side of Eq. (13.46) must be equal to a constant.
This constant shall be designated as to facilitate later forms of the
solution:
(13.47)
The constant in Eq. (13.47) must be negative in order to avoid an
exponential solution that would be inconsistent with the boundary conditions
Equation (13.47) may be decomposed into two ordinary differential
equations:
(13.48)
(13.49)

10. UNSTEADY-STATE TRANSPORT 657
The solutions to Eqs. (13.48) and (13.49) are assumed to be
= exp( (13.50)
(13.51)
Appropriate boundary conditions for a direct Fourier series solution must be
the simplest possible, i.e., homogeneous:
0) =
=0
=0 (13.35)
At this stage in the solution, there are four constants and to be
determined from the boundary conditions. Using the boundary condition at
the point = 0, Eq. (13.42) becomes
= =0 (13.52)
Since for the nontrivial case T(t) cannot be zero for all it follows that Eq.
(13.52) is true only if
=0 (13.53)
Substituting the results of Eq. (13.53) into Eq. Eq. (13.51) becomes
= + (13.54)
Since the sine of zero is zero and the cosine of zero is one, then by Eq. (13.54)
must be zero. Next, the boundary condition at the other end is applied, and
by similar reasoning
=0= sin (13.55)
Equation (13.55) equals zero only if is zero or if sin is zero or if both
are zero. However, if is zero, then is zero and our assumed solution, Eq.
(13.42) is trivial. Therefore
sin =0 (13.56)
The sine of an arbitrary angle is zero if = etc. Hence
1, 2, 3, . . (13.57)
From Eq. the constant is found to be
(13.58)
The solution as determined so far is substituted from Eqs.
(13.55) and (13.58) into Eq. (13.42):
= (13.59)

11. APPLICATIONS OF TRANSPORT PHENOMENA
where each and every j, as j goes from 1 to is a solution to the original
partial differential equation, Eq. (13.32). Thus, the total solution is the sum
over j of all possible solutions, since Eq. (13.32) is a linear partial differential
equation:
(13.60)
where the product has been replaced by
The remaining boundary condition at time zero, Eq. is used to
evaluate Applying that condition to Eq. (13.60):
0) (13.61)
Since the exponential of zero is one, Eq. (13.61) reduces to
= (13.62)
A comparison of Eq. (13.62) with the Fourier series of Eq. (13.38) shows that
must be zero for all j and
(13.63)
By substituting Eq. (13.63) into Eq. a complete solution is now
available. Note that a Fourier series solution is possible as long as the
integration in Eq. (13.63) can be performed. For the case of constant
integration of Eq. (13.63) yields
(-cos 0)] = [for odd j] (13.64)
The simplification of Eq. (13.64) resulted from the following reasoning. The
cosine of zero is one. The cosine of equals -1 for odd j and for even j.
Hence
-(cos jn) + 1 = 0 if j is even
-(cos +1=2 if j is odd (13.65)
Note that a single value of from Eq. (13.64) cannot possibly satisfy the
boundary condition of Eq. (13.33). Hence, it is argued that only the sum from
one to infinity of all possible will satisfy the boundary condition, since Eq.
(13.32) [or Eq. is a linear partial differential equation.
Combining results, the final solution to Eq. (13.32) as determined by the

12. UNSTEADY-STATE TRANSPORT 659
method of Fourier series is
Equation (13.66) expresses in terms of an infinite series the dimensionless
temperature for any and when the boundary conditions of Eq. (13.35)
are valid. However, the engineer will usually be interested in the temperature
for a particular and c or for a series of and combinations. In general a
Fourier series such as Eq. (13.38) or Eq. (13.66) converges very slowly,
especially for small Often thousands of terms are required in order to
evaluate to the required accuracy. In fact, evaluation of Eq. (13.66) on a
digital computer requires almost as much effort as a direct numerical method
(to be discussed subsequently), not including the lengthy steps required to get
Eq. (13.66).
A mass transfer problem with nonhomogeneous boundary conditions is
solved in Example 13.2.
Example A 3-in. schedule 40 pipe is 3 ft long and contains helium at
26.03 atm and 317.2 K as shown in Fig. 13.8. The ends of the pipe are
initially capped by removable partitions. At time zero, the partitions are
removed, and across each end of the pipe flows a stream of air plus helium at the
same temperature and pressure. On the left end, the stream is 90 percent air and
10 percent He (by volume) and on the right 80 percent air and 20percent He. It
may be assumed that the flow effectively maintains the helium concentration
constant at the ends. If isothermal conditions are maintained and there are no
end effects associated with the air flowing past the pipe, calculate the composition
profile (to four decimal places) after 1.2 h at space increments of Use
Fourier series. The value of is 0.7652 x
Answer. First, note that mass transfer in Fig. 13.8 occurs in the z direction only;
there is no transport in either the or e-directions. Since the previous equations
(derived with heat transfer as the example) are in terms of the direction, the .
solution to this problem arbitrarily use the direction as the direction of mass
transfer.
FIGURE
90% air, 10% He 80% air, 20% He Transient diffusion of helium
atm, 44°C 1 atm, 44°C in a pipe.

13. OF TRANSPORT PHENOMENA
Concentration is related to partial pressure by Eq. (2.37):
= (2.37)
where is the gas constant (0.082057 atm from Table C.l). The
partial pressure of species A is defined by Eq. (2.38):
PA = (2.38)
where is the mole fraction of A and is the total pressure.
Example 2.7 illustrated the method of converting partial pressures into
concentrations. Initially, the partial pressure of helium in the tube equals the total
pressure, 26.03 atm. When the partitions are removed, the of
helium at the ends of the pipe are
= = 2.603 atm
= = = atm
Inserting these into Eq. the concentrations are
= = x = 1.0 kmol
= x = 0.1 kmol (ii)
= x = 0.2 kmol
Following the nomenclature of Eq. these are
0) = 1.0 kmol
= 0.1 kmol (iii)
= 0.2 kmol
where the total length of the pipe is or 3 Although does not equal
C it is still convenient to transform to 0,:
=
0) = = 0.9 =
= =
t) = = 0.1 (vii)
where the definition of 0, is arbitrarily based on for ease of notation,
and have also been introduced in the above equations. Using the above
transformation, Eq. (13.37) still applies:
(13.37)
where the total length of the pipe is or 3 ft.
The boundary conditions in Eqs. (iv) through (vii) do not allow a solution
by Fourier series because the condition in Eq. (vii) does not equal zero. This
limitation is easily circumvented by expressing the total solution as the sum of the
unsteady-state (or transient or particular) solution 0, and the steady-state
solution 0,
= + (viii)

14. TRANSPORT 661
The steady-state solution is obtained from Fick’s law, Eq. (2.4). For
equimolar counter diffusion, it is easily shown that the ratio must be
constant at steady-state. If the ratio is constant, then the ratio
is also constant [cf. Eq. Since two points uniquely determine the equation
for a straight line, the equation for is
=0+
= = =
The boundary conditions for the transient solution are found by combining
the last two equations. First, Eq. (viii) is solved for t); then the conditions
in Eqs. (v) through (vii) are inserted into the resulting equation:
0) = 0) = 0.9 =
t) = t) =0 0=0
= = 0. =0
The solution to Eq. (13.37) subject to the above boundary conditions follows the
derivation of Eq. (13.66) from the inception, Eq. up through Eq.
which after replacing w i t h D i s
(xiii)
boundary condition of Eq. (x) is used to evaluate the remaining constant,
= = 0.9 [
A comparison of Eq. (xiv) with the Fourier series of Eq. (13.38) shows that
must be zero for all and
2L
Equation (xv) equals the sum of two integrals:
The first integral is identical to that in Eq. (13.64):
[for odd
The second integral may be located in a standard table of integrals
(xviii)
a

15. APPLICATIONS OF TRANSPORT PHENOMENA
where equals Substituting, the second integral is
The first term in the above is
( x x )
since cos = for even values of j and -1 for odd values of j. The second term
is zero since sin equals zero for all values of j:
sin 0 = sin 0=0
The value of is the combination of Eqs. (xvii) and (xx). The final
expression is obtained by the summation of all between 1 and infinity, which is
then included in Eq. (xiii):
(xxii)
,.
As a rule, these infinite series converge slowly; hence, it may be advantageous to
combine the two infinite series:
(xxiii)
Note that if Eq. (xxiii) reduces to Eq. (13.66). Inserting Eqs. (ix) and
(xxiii) into Eq. (viii) yields the final expression for the concentration or for as
a function of time and distance:
(xxiv)
where
D=
A computer program to evaluate Eq. (xxiv) at the and of interest is
given in Fig. 13.9. Equation (xxiv) is a converging infinite series, in which the
signs of the terms alternate in an irregular pattern. The best procedure is to
combine all terms of like sign, then test the magnitude to see if that combination
is less than the accuracy desired. In the case of a simple alternating converging
series, the truncation error is smaller in absolute value than the first term
neglected and is of the same sign. The results are given in Table 13.2. The last
column is an indication of how many terms are required for the infinite series to
converge.