Numerical Heat Transfer, Part B, 38:133^156 , 2000
Copyright # 2000 Taylor & Francis
1040-7790 /00 $12.00 + .00
AN EFFIC IENT M ETH OD OF COM PUTING
EIGENVALUES IN HEAT COND UC TION
A. Haji-Sheikh
Department of Mechanical and Aerospace Engineering, The University of
Texas at Arlington, Arlington, Texas 76019-0023 , USA
J. V. Beck
Department of Mechanical Engineering, Michigan State University, East
Lansing, Michigan 48824-1226 , USA
EfŽ cient algorithms for computing eigenvalues for heat conduction problems in Cartesian
and spherical coordinates are given. Explicit approximat e relations are presented that
generally provide accurate results. When these approximate relations are followed by
a high-order Newton root-Ž nding iteration, a high degree of accuracy can be realized.
It is demonstrated that in Cartesian coordinates, eigenvalues with excellent accuracy
are obtained over the entire range of parameters.
INTROD UCTION
In this era of powerful computers and well-developed numerical methods, a
need still exists for exact solutions in heat conduction. These solutions can be used
to aid in the veri¢cation of complex numerical programs; to provide insight into
various heat conduction problems; and to aid students’ understanding of some
steady-state and transient heat transfer phenomena. In the veri¢cation of programs,
particularly important are extremely accurate solutions for two- and threedimensional transient problems. These accurate solutions can, in turn, require many
terms in the double or triple series and also accurate eigenvalues. However,
obtaining extremely accurate eigenvalues can be awkward or inef¢cient for the convective boundary condition. The objective of this article is to provide means of
obtaining these eigenvalues in a manner which itself has been veri¢ed for a wide
range of Biot numbers, a great many eigenvalues, and for several combinations
of convective conditions.
The eigenvalues for the convective boundary conditions are intended for one-,
two-, or three-dimensional Cartesian coordinates in transient cases. For steady-state
heat conduction, two- and three-dimensional cases are important. Even though multiple directions are in mind, only one direction need be considered here for the
eigenvalues, because the solutions are frequently obtained using products of
one-dimensional components or Green’s functionsösee Beck et al. [1]. Only homoReceived 22 December 1999; accepted 4 February 2000.
Address correspondence to Dr. A. Haji-Sheikh, Department of Mechanical and Aerospace
Engineering, P.O. Box 19023 UTA Station, Arlington, TX 76019-0023 , USA. E-mail: haji@mae.uta.edu
133
134
A. HAJI-SHEIKH AND J. V. BECK
NOM ENCLATUR E
Am
Bi
cn
dn
Eij
h
H
k
L
n
p
S
Xij
constants
Biot number ( ˆ hL/k)
(n ¡ 34)p
(n ¡ 12)p
function of Bi 1 and Bi 2
heat transfer coef¢cient, W/m2 K
function of n
thermal conductivity, W/m K
thickness, m
eigenvalue index 1, 2, etc.
constant
constant
Cartesian x-coordinate with boundary
conditions of the ith and jth kinds
zn
bn
gn;ij
En
zn
xij
nth eigenvalue approximation
exact nth eigenvalue
coef¢cient
deviation zn ¡ zn
nth eigenvalue, estimated
function
Subscripts
1
boundary conditions of the ¢rst kind
2
boundary conditions of the second kind
3
boundary conditions of the third kind
geneous bodies with temperature-independent properties are usually implied when
these eigenvalues are used.
Three sets of eigenconditions for Cartesian coordinates are considered, each of
which has a different set of eigenvalues, although the third condition to be considered
can include the ¢rst two. Incidentally, one of the eigenconditions also applies for a
solid sphere with a convective boundary condition at its surface. To describe the
problems, it is convenient to use some of the notation in [1]. This notation builds
on the common descriptions of the boundary conditions as being of the ¢rst kind
(that is, prescribed temperature), second kind (prescribed heat £ux), and third kind
(convective condition with a prescribed ambient temperature). A plate which has
a prescribed temperature history at x ˆ 0 and a convective boundary condition
at x ˆ L is denoted by X13. A plate which has a prescribed heat £ux at x ˆ 0
and a convective boundary condition at x ˆ L is denoted by X23. Each has a particular eigencondition in terms of a transcendental equation, which is the same
for X13 and X31 and also is the same for X23 and X32. The X13 eigencondition
and eigenvalues are nearly the same as for a solid sphere with a convective boundary
condition (which is denoted by RS03 in [1]). Actually, several other spherical cases
use the same eigencondition, namely, RS12, RS13, RS21, and RS31. The X13
eigencondition is also the same as for the X42 (or X24) case, which has boundary
conditions of the second and fourth kinds ( a high-conductivity ¢lm at the surface);
see Beck et al. [1]. The third kind of boundary condition is denoted by X33, which
has a convective boundary condition on both surfaces. The heat transfer coef¢cients
may or may not be the same on both sides. Actually, this case includes the other two
because the heat transfer coef¢cient going to zero gives the boundary condition of
the second kind and the heat transfer coef¢cient going to in¢nity gives a boundary
condition of the ¢rst kind. This X33 case is the most dif¢cult; eigenvalues have been
tabulated and simpli¢ed equations have been given in the literature for the X13
and X23 cases, but the X33 case has rarely been treated. This set of eigenvalues
can also be used for the RS22, RS23, RS32, and RS33 cases.
A brief literature review is now given. A number of ways have been proposed to
obtain numerical values for the eigenvalues for the X13, X23, and X33 cases in heat
EIGENVALUES IN HEAT CONDUCTION
135
condition. Values for the X13 and X23 cases are tabulated in many books, such as
Zucker [2]. Yovanovich [3] has proposed simple algebraic equations for the ¢rst
eigenvalues for the cases of X13, X23, and R03 (solid cylinder with a convective
boundary condition). The maximum errors are about 0.002, which is quite good
but not good enough for extremely accurate computations. No accurate equations
are given for the higher eigenvalues. Beck et al. [1] has proposed some algebraic
equations for the X13, X23, and X33 cases. These are based on series expansions.
All the eigenvalues are considered, but sometimes multiple equations are necessary
for a given case and the entire range of the Biot numbers is not covered for the
X33 case. However, a method of obtaining extremely accurate values is given
for the ranges where the approximations are valid. Another procedure for the
explicit calculation of the eigenvalues for the X23 and X13 cases is given by Leathers
and McCormick [4]. However, the computation involves numerical integration,
possibly needing on the order of 105 points.
Numerous techniques for computing eigenvalues are described in [5^7].
Aviles-Ramos et al. [8] introduce a hybrid search algorithm based on central
differencing that can be widely used without explicit differentiation of the
transcendental equations. Recently, Stevens and Luck [9] have proposed attractive
sets of equations. These equations are attractive because they are ef¢cient and
can be made as accurate as desired, consistent with the number of signi¢cant ¢gures
used in the computation. (These sets of equations can be put in the form of
algorithms for ef¢cient programming.) The method of Stevens and Luck [9] for
the X23 case, that is, one convective boundary and one insulated surface, is modi¢ed
herein to achieve even faster convergence. For a Dirichlet boundary condition at one
surface and a convective boundary condition at the other surface, the X13 case, two
algorithms are presented in [9], one for the ¢rst eigenvalue and another for all
the others. Herein a single accurate algorithm is given covering all the eigenvalues.
These two algorithms (for the X13 and X23 cases) are properly combined for application to problems when both surfaces are subject to convective boundary conditions, the X33 case. Stevens and Luck do not consider the dif¢cult X33 case.
The main objective of this study is to investigate a robust method of ¢nding
extremely accurate eigenvalues for one-dimensional transient problems with convective boundary conditions. (As pointed out above, these same eigenvalues are used
in two- and three-dimensional problems in Cartesian coordinates and some problems in spherical coordinates.) This objective leads to modi¢cation and extension
of Stevens and Luck [9] to meet the following self-imposed goals:
1.
2.
3.
4.
The methods should provide an explicit equation with, at least, six-decimal
place accuracy, exceeding typical accuracy of tabulated information.
The method must be suf¢ciently robust that there is no numerical instability within the computer word limitation.
The basic equations must have suf¢ciently simple forms.
The methods must be capable of being expressed in terms of algorithms
that can be quickly programmed using a variety of computer languages.
To meet these goals, a preliminary function that approximates the eigenvalues
is obtained based solely on the asymptotic behavior of eigenfunctions for the X13,
X23, and X33 cases. An accurate estimate is an important initial step for a quick
136
A. HAJI-SHEIKH AND J. V. BECK
evaluation of eigenvalues. Later, the results of this study are modi¢ed to accommodate one-dimensional radial conduction in spherical bodies.
An outline of the remainder of this article is now given. The algorithm for the
X23 case is ¢rst developed. One of the most important parts of this research is
to demonstrate a methodology for ¢nding the algorithms for the eigenvalues; it
is illustrated in this section and in the next one, which addresses the X13 case.
In the following section, the results of these two sections are used to develop an
algorithm for the X33 case. Finally, the application to the RS03 (and other cases)
is discussed by introducing some minor modi¢cations.
ANALYSIS
X23 or X32 Eigenvalues
The transcendental equation for the X23 and X32 cases is
bn tan bn ¡ Bi ˆ 0
…1†
where bn is the nth eigenvalue. The eigenvalues for this case have the following
limiting conditions:
As Bi ! 0
As Bi ! 1
bn ! …n ¡ 1†p
bn ! n ¡ 12 p
which should be strictly adhered to. An algorithm based on the article of Stevens and
Luck is brie£y described here. A function that provides approximate eigenvalues, as
given in [9], is
p
zn ˆ cn ‡ x23
…2a†
4
where
and
cn ˆ …n ¡ 34†p
…2b†
Bi ¡ cn
Bi ‡ cn
…2c†
x23 ˆ
satis¢es the limiting conditions as described above. Equation (2a) has another
remarkable feature: it satis¢es the condition suggested by Eq. (1), bn ˆ Bi, when
tan bn ˆ 1. Also, according to Eqs. (2a)^2(c), when Bi ˆ cn , then zn ˆ cn ˆ Bi,
and using zn instead of bn , one can satisfy Eq. (1). However, the function zn does
not satisfy the asymptotic behavior of bn as Bi!0, described in [9]. For a set of
computed eigenvalues to be complete, it is essential to have highly accurate zn
as it approaches (n¡1)p and (n¡12)p. Equation (2a) should be modi¢ed so that
its slope of zn becomes the same as dbn =d Bi as Bi! 0. The value of dbn =dBi is computed by differentiating Bi in the eigenfunction relation, Eq. (1), with respect to
EIGENVALUES IN HEAT CONDUCTION
137
bn ; that is,
dbn
1
ˆ
dBi bn ‡ …1 ‡ Bi† tan bn
…3a†
Since bn ! …n ¡ 1†p as Bi! 0, then tan bn ˆ 0 and
dbn
1
ˆ
dBi …n ¡ 1†p
When n > 1
…3b†
Moreover,
when n ˆ 1, Eq. (1) suggests tan bn ! bn as Bi!0, and that produces
p
bn ˆ Bi. The plan is to include this asymptotic behavior of bn when Bi approaches
zero for n ˆ 1 and the limiting slope given by Eq. (3b) in the formulation of zn .
In order to maintain the form of the function x23 , as suggested in [9], and incorporate
this asymptotic behavior of bn , Eq. (2a) is modi¢ed to take the following form:
p
zn ˆ cn ‡ x23 gn;23
4
…4†
The parameter gn;23 in Eq. (4) must satisfy the following conditions:
(a) Have unit values both as Bi!0 and Bi! 1
(b) Maintain zero slope for p
zn as Bi!1
(c) Force zn to approach * Bi, when n ˆ 1, as Bi!0
(d) Provide dzn =dBi ^1/[p(n¡1)], when n > 1, as Bi!0
The factor gn;23 is selected to perform these tasks, without altering the characteristics
of Eq. (2a), as
gn;23 ˆ 1 ¡ 1:04
Bi ‡ cn ¡ p=4 Bi ‡ H…cn ¡ p=4†
¡
Bi ‡ H
Bi ‡ H
…5†
Equation (4) with gn;23 from Eq. (5) satis¢es conditions (a) and (b) unconditionally,
and it can produce in¢nite slope only for n ˆ 1. When H ˆ …p=4†2 , condition (c)
is satis¢ed; however, Eq. (5), in its present functional form, does not satisfy condition
(d) for n > 1 explicitly. Using the H value for n ˆ 1 resulted in signi¢cant improvement of the maximum deviation, bn ¡ zn , to 0.0049 in comparison with 0.0531 when
gn;23 ˆ 1. This motivated a search for an alternative functional form to be used
instead of Eq. (4) while maintaining the de¢nition of z23 , Eq. (2c).
An alternative functional form of zn is sought that would permit satisfying
condition (d). Among the possible candidates, a linear combination of two functions,
z23 and tanh(z23 )/tanh(1), is used as summarized in the equation
zn ˆ cn ‡
pgn;23 ¡1
tanh…x23 †
n x23 ‡ …1 ¡ n¡1 †
4
tanh…1†
…6†
To satisfy condition (d) using Eq. (6), the function describing the limit of dz n =dBi, as
Bi! 0, is set equal to 1/[p(n¡1)]; that is, replacing dbn =dBi in Eq. (3b) by dz n =dBi.
Then H, the root of the resulting transcendental equation, is numerically computed
for a selected number of n values. The solution is sensitive to the value of
S ˆ 1:04 and the best results are obtained when S has near-unit values. The numerical computation of H begins using a constant S ˆ 1 instead of 1.04 in Eq. (5).
138
A. HAJI-SHEIKH AND J. V. BECK
The discrete data in Figure 1 are the computed values of H. A linear function,
H ^ 0:75 ‡ 1:43…n ¡ 1†
…7a†
In Figure 1 is the least-squares approximation of the discrete data. This signi¢cantly
improves the accuracy of zn , especially at small Bi values; however, it is not the
optimum value of H when n is small, e.g., n ˆ 1. Slight adjustments are made to
improve the accuracy of zn for small n values, and the equation
H ^ 0:76 ‡ 1:22…n ¡ 1†
…7b†
with S ˆ 1.04 performed well. These latter values of S and H are used for the recommended algorithm for the X23 case.
Figure 2 demonstrates the suitability of zn over a broad range of Biot numbers.
The values of bn ¡ zn in Figure 2 describe the relative error of zn . Figure 2a is for
gn;23 ˆ 1 in Eq. (4), which corresponds to zn in Eq. (2a), the Stevens and Luck
[9] equation. Next, the function zn in Eq. (6) serves as an initial estimate of the root
of Eq. (1). Figure 2b is prepared using Eq. (6) with Eq. (5), for gn;23 , H from
Eq. (7b), and S ˆ 1.04.
A modi¢ed second-order Newton method, described in the Appendix, provides
the value of zn , the ¢rst approximation for bn . The estimated difference between bn
Figure 1. Estimated values of H in the de¢nition of gn;23 , Eq. (5).
EIGENVALUES IN HEAT CONDUCTION
139
Figure 2. The bn ¡ zn deviation as a function of Bi for X23 or X32 case when n ˆ 1; 2; 5; 10; 100: (a) Eq.
(2a); (b) Eq. (6).
and zn is En , obtained from the relations
A0 ˆ Bi cos zn ‡ zn sin zn
…8a†
A2 ˆ cos zn
…8c†
A1 ˆ …1 ‡ Bi† sin zn ‡ zn cos zn
en ^ ¡
A0
A1
1‡
…A2 =A1 †…A0 =A1 †
1 ¡ 2…A2 =A1 †…A0 =A1 †
…8b†
…8d†
Equations (8a)^(8c) are the same as those in [9]. A strict second-order Newton
method formulation contains an additional ¡A0 /2 term on the right-hand side
of Eq. (8c). However, dropping ¡A0 /2 improves the results, assuming A0 ˆ 0 if
zn ˆ bn in the de¢nition of eigencondition, Eq. (1). The deviation given by Eq. (8d)
yields the approximate root of a quadratic equation, A0 ‡ A1 E ‡ A2 E2 ˆ 0. Then,
140
A. HAJI-SHEIKH AND J. V. BECK
the eigenvalue zn for the X23 case is
zn ˆ zn ‡ e n
…9†
An algorithm describing calculation of eigenvalues for X23 is given in Table 1. A
computer program can be written using these equations in the order given.
To compute zn , one calculates zn for inclusion in Eqs. (8a)^(8c) and Eq. (8d)
provides En . Figure 3 shows the accuracy of zn after one iteration. The deviations
of zn from the true eigenvalues, when using Eq. (2a), for different n and Bi values
are in Figure 3a. This is similar to the results reported in [9]. Figure 3b is for when
zn is taken from Eq. (6). Figure 3b shows the computed deviation of zn from the
exact eigenvalue bn , that is, the expected error bn ¡ zn , over a broader range of Biot
numbers. Notice that the data in Figure 3b have much lower errors than the data
in Figure 3a, by a factor of nearly 104 . This is due to a better than one-decimal-place
improvement of bn ¡ zn in Figure 2b over Figure 2a. The largest value of n appearing
in Figures 2 and 3 is 100; however, the numerical calculations were successfully
tested for n over 105 and for all Biot numbers from above 1010 to below 10¡10 . When
n > 100, the value of the maximum error stabilizes at *6£10¡8 and the peaks in
Figures 3a^3b gradually shift toward higher Bi values. The peaks are located near
the crossing points at bn ˆ zn ˆ cn . To demonstrate the accuracy of the eigenvalues
numerically, zn is computed over a small range of Biot numbers and for n ˆ 1 through
5; the results are in Table 2. The eigenvalues computed using one and two iterations
are computed with data from [2]. For one iteration, the errors in zn data agree with
deviations shown in Figure 3b.
X13 or X31 Eigenvalues
Only a minor modi¢cation of Eqs. (2a)^(2c), that is, changing 34 to 14, can provide a zn relation for the X13 case. This provides acceptable results when n ˆ 1.
Table 1. Algorithm for computing the eigenvalues
Quantities
Select n and the Biot numbers(s) and
use equations given below for
X13
X23
X33
cn
dn
x13
x23
H
gn;13
gn;23
zn
A0
A1
A2
A3
En
xn
ö
(11b)
(13a)
ö
ö
(13c)
ö
(13d)
(14a)
(14b)
(14c)
ö
(8d)
(15)
(2b)
ö
ö
(2c)
(7b)
ö
(5)
(6)
(8a)
(8b)
(8c)
ö
(8d)
(9)
(2b)
(11b)
(17b)
(18b)
(7b)
(13c)
(18c)
(20)
(21a)
(21b)
(21c)
(21d)
(21e)
(22)
EIGENVALUES IN HEAT CONDUCTION
141
Figure 3. The computed deviation, bn ¡ zn ; as a function of Bi for X23 when n ˆ 1; 2; 5; 10; 100: (a) using
zn from Eq. (2a); (b) using zn from Eq. (6).
However, without a correct value of dbn =dBi as Bi!0, when n is large, a relatively
large error shifts toward Bi ˆ 0. Since the value of dbn =dBi, as Bi! 0, cannot be
conveniently incorporated in the de¢nition of zn , an alternative scheme is used.
For this or the X31 case, the eigenvalues are the roots of the equation
bn cot bn ‡ Bi ˆ 0
This equation has the following limiting values:
As Bi ! 0
As Bi ! 1
bn ! …n ¡ 12†p
bn ! np
…10†
142
A. HAJI-SHEIKH AND J. V. BECK
Table 2. Calculated eigenvalue, zn , for X23 using explicit approximate formulation and comparison with
results after two and three iterations
n
1
2
3
4
5
Bi
0.1
1.0
10
100
0.1
1.0
10
100
0.1
1.0
10
100
0.1
1.0
10
100
0.1
1.0
10
100
zn , Eq. (4)
zn
Zucker [2]
After two iterations
0.31832
0.86127
1.43135
1.55541
3.17316
3.42210
4.30817
4.66492
6.30003
6.44124
7.22828
7.77558
9.43622
9.53408
10.20029
10.88711
12.57503
12.64980
13.21500
13.99889
0.311052791
0.860333589
1.428870010
1.555245130
3.173097177
3.425618473
4.305801409
4.665765142
6.299059360
6.437298159
7.228109772
7.776374078
9.435375976
9.529334370
10.200262588
10.887130102
12.574323161
12.645287193
13.214185684
13.998089735
0.31105
0.86033
ö
ö
3.17310
3.42562
ö
ö
6.29906
6.43730
ö
ö
9.43538
9.52933
ö
ö
12.57432
12.64529
ö
ö
0.311052848200298
0.860333589019380
1.428870011214077
1.555245129256167
3.173097176692870
3.425618459481728
4.305801413119223
4.665765141727248
6.299059359895646
6.437298179171947
7.228109771627249
7.776374077846953
9.435375975760847
9.529334405361963
10.20026258829591
10.88713010214771
12.57432316103787
12.64528722385664
13.21418568384292
13.99808973515508
The function
p
zn ˆ dn ‡ x13
2
…11a†
1
p
2
…11b†
Bi
Bi ‡ o
…11c†
where
dn ˆ n ¡
and
x13 ˆ
satis¢es the aforementioned limiting conditions. The variable o is a constant that
depends on n, and it is selected so that the slope dz n =dBi is the same as the slope
of the exact eigenvalue dbn =dBi as Bi!0. Accordingly, the slope dbn =dBi is computed by differentiating the eigenfunction relation, Eq. (10), with respect to bn
to obtain
dbn
1
ˆ
dBi bn ¡ …1 ‡ Bi† cot bn
…12a†
As Bi!1, bn ! np, then dbn =dBi!0 and as Bi! 0, one can set bn ˆ …n ¡ 12†p, then
EIGENVALUES IN HEAT CONDUCTION
143
cot bn ! 0, and Eq. (12a) reduces to
dbn
1
1
ˆ
ˆ
dBi bn …n ¡ 12†p
…12b†
Similarly, one can differentiate zn in Eq. (11a) with respect to Bi to obtain
dbn
p
o
p
ˆ
ˆ
ˆ
2
dBi
2
2o
…Bi ‡ o† Biˆ0
Biˆ0
…12c†
and then set p/(2o) ˆ 1/[n¡12]p or o ˆ p2 …n ¡ 12†=2 ˆ pdn /2; this reduces Eq. (11c) to
x‰13 ˆ
Bi
Bi ‡ pdn =2
…13a†
The above formulation for zn is based solely on the matching of the asymptotic
behavior of zn at small and large Biot numbers. Accordingly, to enhance the accuracy
of zn in Eq. (11a), it is reasonable to include some mid-range corrections. To reduce
the deviation bn ¡ zn within 0 µ z13 µ 1 without altering the asymptotic behavior
of zn , one can set
p
bn ¡ zn ˆ p1 x213 …1 ¡ x13 †
2
where p1 is a constant for any n. The parameter p1 was computed for a selected
number of n values so that …bn ¡ zn †2 is minimum within 0 µ x13 µ 1 and its approximate value is p1 ^ 1 ¡ 0:85=n. The modi¢ed Eq. (11a) is
p
zn ˆ dn ‡ x13 ‰1 ‡ p1 x13 …1 ¡ x13 †Š
2
…13b†
According to Eq. (13b), there will be one additional point at x13 ˆ 0:6 ‡ 0:245=n,
between x13 ˆ 0 and 1, where zn % bn . The accuracy of zn in Eq. (13a) is further
enhanced by repeating the aforementioned process while using the relation
p
bn ¡ zn ˆ p2 x213 …1 ¡ x13 †…1 ‡ x13 †…p3 ¡ x13 †
2
where p2 is a constant to be determined by p3 ^ 0:6 ‡ 0:245=n approximates the
locations where zn ˆ bn . The method of calculating p2 ^ 0:6 ¡ 0:71=n is the same
as that described for p1 . In summary, one can de¢ne a new parameter gn;13 as
gn;13 ˆ 1 ‡ x13 …1 ¡ x13 † 1 ¡
0:85
0:71
0:245
‡ …0:6 ¡
…1 ‡ x13 † 0:6 ‡
¡ x13
n
n
n
…13c†
and compute zn from the equation
p
zn ˆ dn ‡ gn;13 x13
2
…13d†
The variable zn in Eq. (13a) serves as an initial estimate of the root of Eq. (10). The
modi¢ed higher-order Newton method (see Appendix) can provide accurate results
if zn is a suitable estimate of the eigenvalues. Accordingly, one can de¢ne En as
the difference between bn and zn and estimate En from Eq. (8d) using the following
144
A. HAJI-SHEIKH AND J. V. BECK
parameters:
A0 ˆ Bi sin zn ‡ zn cos zn
…14a†
A2 ˆ sin zn ¡ 0:5A0
…14c†
A1 ˆ …1 ‡ Bi† cos zn ¡ zn sin zn
…14b†
According to Eq. (10), one can delete the ¡0.5A0 term in Eq. (14c). Unlike the X23
case, the inclusion of ¡0.5A0 in Eq. (14c) improves the accuracy of the results.
Indeed, an intermediate value such as ¡0.3A0 instead of ¡0.5A0 in Eq. (14c) further
improves the accuracy. Then, following computation of En using Eq. (8d), the
relation
zn ˆ zn ‡ e n
…15†
describes an approximate value of the eigenvalues bn . The computation procedure is
identical to that described earlier for the X23 case. A step-by-step algorithm suitable
for computer programming is in Table 1. A sample of computed zn , using Eq. (15), is
in Table 3. Comparing the last two columns, the computed zn exhibits remarkable
accuracy. Therefore, it is appropriate to study the accuracy of the computed
eigenvalues bn approximated by zn , over a large range of Biot numbers.
To demonstrate the suitability of zn , the value of bn ¡ zn is plotted in Figure 4
over a large range of Biot numbers, Bi. Without using any Newton iteration, Figure
4 yields good accuracy throughout the domain. The computed values of bn ¡ zn
are plotted in Figure 5. Figure 5 contains the deviation from the exact value for
n ˆ 1, 2, 5, 10, and 100. The solution was stable for Biot numbers between 10¡10
Table 3. Calculated eigenvalue, zn , for X13 using explicit approximate formulation and comparison with
results after two and three iterations
n
Bi
zn , Eq. (13a)
zn
Zucker [2]
After two iterations
1
0.1
1.0
10
100
0.1
1.0
10
100
0.1
1.0
10
100
0.1
1.0
10
100
0.1
1.0
10
100
1.63211
2.03036
2.85882
3.11016
4.73353
4.91387
5.75469
6.21974
7.86671
7.97955
8.70338
9.33082
11.00467
11.08625
11.69973
12.44245
14.14424
14.20798
14.73283
15.55438
1.6319945272
2.0287578382
2.8627725878
3.1104977023
4.7335118024
4.9131804394
5.7605579348
6.2210548279
7.8666927716
7.9786657124
8.7083138327
9.3317301257
11.004661096 0
11.085538406 5
11.702678081 2
12.442581015 9
14.144236840 7
14.207436725 2
14.733472342 3
15.553662970 8
1.63199
2.02876
ö
ö
4.73351
4.91318
ö
ö
7.86669
7.97867
ö
ö
11.00466
11.08554
ö
ö
14.14424
14.20744
ö
ö
1.631994527214800
2.028757838110434
2.862772587515207
3.110497702305585
4.733511802356786
4.913180439434884
5.760557932709097
6.221054827821945
7.866692771561574
7.978665712413241
8.708313830875857
9.331730125693797
11.00466109603352
11.08553840649702
11.70267808065236
12.44258101585997
14.14423684071843
14.20743672519119
14.73347234227069
15.55366297078455
2
3
4
5
EIGENVALUES IN HEAT CONDUCTION
Figure 4. The bn ¡ zn deviation as a function of Bi for X13 when n ˆ 1; 2; 5; 10; 100:
Figure 5. The bn ¡ zn deviation as a function of Bi for X13 when n ˆ 1; 2; 5; 10; 100:
145
146
A. HAJI-SHEIKH AND J. V. BECK
and 1010 , well below and above those plotted in the ¢gures. Based on data in Table 3
and data in Figure 5, Eq. (15) can provide eigenvalues with eight accurate decimal
places and the accuracy in the mid-range, where the error is the largest, satis¢es
the need of critical computations.
X33 Eigenvalues
It is remarkable that the estimated form of eigenfunctions for the X13 and X23
cases may be combined in a simple manner to construct a working relation for the
X33 case. The eigenvalues are the roots of the transcendental equation
…b2n ¡ Bi1 Bi2 † tan bn ¡ …Bi1 ‡ Bi2 †bn ˆ 0
The contribution of the X13 case is modi¢ed and will be designated as
p
E13 ˆ x13 gn;13
2
…16†
…17a†
where
Bi1 Bi2
Bi1 Bi2 ‡ 0:2 ‡ …Bi1 ‡ Bi2 †pdn =2
…17b†
pg n;23 ¡1
tanh…x23 †
n x23 ‡ …1 ¡ n¡1 †
4
tanh…1†
…18a†
Bi1 ‡ Bi2 ¡ cn
Bi1 ‡ Bi2 ‡ cn
…18b†
x13 ˆ
and gn;13 is given by Eq. (13c).
The parameter x13 is de¢ned so that E13 , Eq. (17a), reduced to px13 /2 in Eq.
(13d) as Bi1 or Bi2 becomes in¢nite. Similarly, the modi¢ed contribution of the
X23 case is
E23 ˆ
where
x23 ˆ
and
gn;23 ˆ 1 ¡ 1:04
Bi1 ‡ Bi2 ‡ Bi1 Bi2 ‡ cn ¡ p=4
Bi1 ‡ Bi2 ‡ Bi1 Bi2 ‡ H
Bi1 ‡ Bi2 ‡ Bi1 Bi2 ‡ H…cn ¡ p=4†
¡
Bi1 ‡ Bi2 ‡ Bi1 Bi2 ‡ H
…18c†
satis¢es the asymptotic behaviors of zn as Bi1 and/or Bi2 approaches zero; and the
condition bn ^(Bi 1 ‡ Bi2 ‡ Bi1 Bi2 ) 1=2 when bn ! 0 is also satis¢ed. One simple
method of ¢nding an initial estimate of the eigenvalues is to set a Taylor series
expansion of zn …x13 , x23 ), about (x13 , x23 ) equal to (0, 0), to get
zn ‡ cn ‡ E13 ‡ E23
…19†
The substitution of E13 from Eq. (17) and E23 from Eq. (18a) in Eq. (19) ensures that,
as Bi1 or Bi2 approaches in¢nity, Eq. (19) reduces to Eq. (13d). Also, as Bi1 or Bi2
approaches zero, Eq. (19) reduces to Eq. (6). Although Eq. (19) can provide accurate
EIGENVALUES IN HEAT CONDUCTION
147
results after a few iterations, the accuracy of zn over a broad range of Bi1 , Bi2 , and n
values was below the targeted accuracy of 10¡6 .
There are other methods of combining E13 and E23 to construct a zn function.
The approximate values of zn for the X13 and X23 cases, despite their high degree
of accuracy, only affect the limiting values of zn for X33. Accordingly, a simple
method is sought that provides an alternative formulation of zn for the X33 case.
This new empirical relation is
Bi1 Bi2
Bi1 Bi2
‡ …cn ‡ E23 † 1 ¡
2
Bi1 Bi2 ‡ dn
Bi1 Bi2 ‡ dn2
Bi1 Bi2
Bi1 Bi2
p
p
ˆ dn ‡ x13 ; gn;13
‡ cn ‡ x23 gn;23 1 ¡
2
2
Bi1 Bi2 ‡ dn
4
Bi1 Bi2 ‡ dn2
zn ˆ …dn ‡ E13 †
…20†
The coef¢cient dn2 in the denominator is a free numerical constant. It is evaluated by
linear regression mainly to reduce the maximum errors for different n values. It is
remarkable that the computed constant for different n values closely approximated
dn2 . Equations (19) and (20) are presented to show the method combining E13
and E23 is not unique. It is possible to construct other relations; however, the simplicity of the relation and relative accuracy of zn are the primary reasons for selecting
Eq. (20) for this numerical presentation.
Equation (20) provides an initial estimate of zn , and Figure 6 shows a
three-dimensional plot for b1 ¡ z1 . Accordingly, the reasonably good accuracy of
data in Figure 6 and a high-order Newton yield eigenvalues with reasonably good
Figure 6. The bn ¡ zn deviation as a function of Bi1 and Bi 2 when n ˆ 1 for the X33 case.
148
A. HAJI-SHEIKH AND J. V. BECK
accuracy for all combinations of Bi1 , Bi2 , and n. The procedure described for the X13
and X23 cases will be repeated for this X33 study. The second-order Newton’s
method was extended to become a third-order modi¢ed Newton method and
was applied to this case. The working relations are
A0 ˆ …z2n ¡ Bi1 Bi2 † sin zn ¡ …Bi1 ‡ Bi2 †zn cos zn
A1 ˆ …2 ‡ Bi1 ‡ Bi2 †zn sin zn ‡ ‰z2n ¡ …Bi1 ‡ Bi2 † ¡ Bi1 Bi1 Š cos zn
A0
A2 ˆ …1 ‡ Bi1 ‡ Bi2 † sin zn ‡ 2zn cos zn ¡
2
2
A3 ˆ f‰6 ‡ 3…Bi1 ‡ 2† ‡ Bi1 Bi2 ¡ zn Š cos zn ¡ …6 ‡ Bi1 ‡ Bi2 †zn sin zn g=6
en ˆ
A0
A1
1‡
2
…A0 =A1 †…A2 =A1 † ¡ …A0 =A1 † …A3 =A1 †
1 ¡ 2…A0 =A1 †…A2 =A1 † ‡ 3…A0 =A1 †2 …A3 =A1 †
…21a†
…21b†
…21c†
…21d†
…21e†
where En approximates the proper root of equation A0 ‡ A1 En ‡ A2 E2n ‡ A3 E3n ˆ 0. As
for the X13 and X23 cases, the relation
zn ˆ zn ‡ e n
…22†
provides zn , the approximate eigenvalue.
Equation (22) performed well for all n values. Figure 6 is a three-dimensional
plot describing the variation of bn ¡ zn for n ˆ 1 as a function of Bi1 and Bi2 . Figure
7 is a similar three-dimensional plot for n ˆ 1, and it demonstrates the value of
bn ¡ zn that represents the error in z1 over a broad range of Bi1 and Bi2 following
Figure 7. The bn ¡ zn deviation as a function of Bi1 and Bi2 when n ˆ 1.
EIGENVALUES IN HEAT CONDUCTION
Figure 8. The b2 ¡ z2 deviation as a function of Bi1 and Bi2 for the X33 case.
Figure 9. The b2 ¡ z2 deviation as a function of Bi1 and Bi2 for the X33 case.
149
150
A. HAJI-SHEIKH AND J. V. BECK
Table 4. Calculated eigenvalue, z1 , for X33 using explicit approximate formulation and comparison with
results after two and three iterations
Bi1
Bi2
zn , Eq. (21)
z1
After two iterations
After three iterations
0.1
0.1
1.0
10
100
1.0
10
100
10
100
100
0.44420
0.92168
1.49185
1.61693
1.30592
1.88961
2.01550
2.62089
2.83147
3.07887
0.443520787 9
0.929253116 8
1.489910836 6
1.616419903 8
1.306542374 2
1.875307809 5
2.011948778 4
2.627675433 0
2.836813028 3
3.080011883 8
0.44352078788188 8
0.92925310992520 6
1.48991083663246 6
1.61641990379493 8
1.30654237418880 6
1.87530781059643 6
2.01194877843164 1
2.62767543298579 7
2.83681302827527 7
3.08001188380088 4
0.44352078788188 9
0.92925310992520 6
1.48991083663246 6
1.61641990379493 8
1.30654237418880 6
1.87530781059643 6
2.01194877843164 1
2.62767543298579 7
2.83681302827527 7
3.08001188380088 4
1.0
10
100
one iteration. It is observed from the ¢gures that the relations presented above satisfy
the asymptotic behavior of bn . Based on data plotted in Figure 7, there are better
than six accurate decimal places over a broad range of the Biot numbers when n ˆ 1.
Figure 8 demonstrates the value of bn ¡ zn as a function of Bi1 and Bi2 when n ˆ 2.
Figure 9 is a similar three-dimensional plot describing the variation of bn ¡ zn when
n ˆ 2. Figure 9 shows relatively smaller errors than those in Figure 7. Indeed, when
n > 1, Eqs. (20)^(22) yield eigenvalues with 7 or more accurate decimal places.
The error in zn is largest when n ˆ 1, and Table 4 demonstrates the accuracy of
z1 over a range of Bi1 and Bi2 . The data show excellent accuracy, especially when
the Biot number is large.
Examining the three-dimensional graphs for higher n values, up to n ˆ 104 ,
shows that the maximum error remains less than 10¡7 . Moreover, it is possible
to increase the accuracy of computed bn using a second iteration. This is demonstrated by plotting the errors in computed bn for n ˆ 1 and 2, and the results are
in Figures 10 and 11, respectively. This indicates that the accuracy of data in Figures
7 and 9 can be greatly enhanced by an additional iteration.
C ONC LUSION AND REM ARK S
Extremely accurate forward equations are given for the basic eigenconditions
in transient heat conduction for convective boundary conditionsönamely, for
the boundary conditions denoted X13 (or X31), X23 (or X32), and X33. (See Beck
et al. [1] for more description of the numbering system, which uses 1 for the temperature boundary condition, 2 for the heat £ux, and 3 for convection.) The
equations are presented in the form of algorithms that are readily programmed;
a sample program for the X33 case is in the Appendix. These same eigenconditions
are present in some spherical radial problems and also for problems with a thin
high-conductivity surface ¢lm. Very large ranges of the Biot numbers are covered.
Furthermore, this is the ¢rst article to present convenient equations for the X33
case, for two convective boundary conditions with different heat transfer coef¢cients
on either side.
When attempting to provide accurate zn for the X23 and X13 cases, it was
noticed that zn is sensitive to the choice of end values. For this reason, it is necessary
EIGENVALUES IN HEAT CONDUCTION
151
Figure 10. The deviation of computed b1 as a function of Bi1 and Bi 2 for the X33 case after two iterations.
Figure 11. The deviation of computed b2 as a function of Bi1 and Bi 2 for the X33 case after two iterations.
152
A. HAJI-SHEIKH AND J. V. BECK
Table 5. Calculated eigenvalue, zn , for RS13 and comparison with results after two and three iterations
n
Ba
zn
zn
Zucher [2]
Second iteration
1
¡1.00
¡0.95
¡0.90
¡0.80
¡0.60
¡0.40
¡0.20
0.00
¡1.00
¡0.95
¡0.90
¡0.80
¡0.60
¡0.40
¡0.20
0.00
¡1.00
¡0.95
¡0.90
¡0.80
¡0.60
¡0.40
¡0.20
0.00
¡1.00
¡0.95
¡0.90
¡0.80
¡0.60
¡0.40
¡0.20
0.00
¡1.00
¡0.95
¡0.90
¡0.80
¡0.60
¡0.40
¡0.20
0:00
0.00000
0.39305
0.54397
0.75002
1.03511
1.24961
1.42428
1.57080
4.49362
4.50455
4.51552
4.53757
4.58181
4.62591
4.66952
4.71239
7.72518
7.73162
7.73807
7.75098
7.77685
7.80268
7.82842
7.85398
10.90408
10.90865
10.91323
10.92239
10.94073
10.95906
10.97735
10.99557
14.06617
14.06972
14.07327
14.08038
14.09459
14.10881
14.12300
14.13717
0.0000000
0.3853669
0.5422809
0.7593083
1.0527958
1.2644040
1.4320323
1.5707963
4.4934095
4.5045364
4.5156604
4.5378886
4.5821879
4.6261383
4.6695848
4.7123890
7.7252518
7.7317240
7.7381957
7.7511351
7.7769834
7.8027626
7.8284393
7.8539816
10.9041217
10.9087070
10.9132922
10.9224613
10.9407885
10.9590911
10.9773570
10.9955743
14.0661939
14.0697485
14.0733030
14.0804114
14.0946232
14.1088235
14.1230066
14.1371669
0
0.38537
0.54228
0.75931
1.05279
1.26440
1.43203
1.57080
4.49341
4.50454
4.51566
4.53789
4.58219
4.62614
4.66958
4.71239
7.72525
7.73172
7.73820
7.75114
7.77698
7.80276
7.82844
7.85398
10.90412
10.90871
10.91329
10.92246
10.94079
10.95909
10.97736
10.99557
14.06619
14.06975
14.07330
14.08041
14.09462
14.10882
14.12301
14.13717
0
0:385368098094658
0:542280885416155
0:759307689030632
1:05279429376987
1:26440357774200
1:432032236243418
1:570796326794897
4:493409457909064
4:504536385052075
4:515660437913874
4:537888582246557
4:582187926046559
4:626138290981610
4:669584780905963
4:712388980384690
7:725251836937707
7:731724026038532
7:738195664946898
7:751135101682629
7:776983426127267
7:802762575422832
7:828439313712818
7:853981633974483
10:90412165942890
10:90870704909461
10:91329224451455
10:92246127745309
10:94078851202827
10:95909110563603
10:97735698123371
10:99557428756428
14:06619391283147
14:06974851962217
14:07330303618783
14:08041143825164
14:09462320275459
14:10882348067746
14:12300659712559
14:13716694115407
2
3
4
5
a B ˆ hr =k ¡ 1.
0
to have highly accurate asymptotic values when developing zn for the X33 case. Also,
for any given n, the zn for the X13 and X23 cases depend on one Biot number,
whereas zn for the X33 case depends on two Biot numbers. This makes obtaining
a simple relation for X33 more demanding. However, for less critical applications,
one may use Eqs. (2a)^(2c) for X23, Eqs. (11a)^(11c) for X13, and Eq. (19) with
E23 ˆ px23 =4 for X33.
EIGENVALUES IN HEAT CONDUCTION
153
When using the algorithms in Table 1, the numerical accuracy for the X13 and
X23 cases exceeds the self-imposed target of accuracy, that is, an error of less than
10¡7 . Also, this target of accuracy for X33 is satis¢ed when n > 1 and marginally
satis¢ed when n ˆ 1. Therefore, the explicit relations that include one high-order
iteration provide accuracy suf¢cient for all practical applications. The procedure
to compute these eigenvalues for Cartesian coordinates is summarized in Table 1.
The equation derived to obtain approximate eigenvalues in Cartesian
coordinates applies equally to spherical coordinates; only minor modi¢cation is
needed to accomplish this task. The procedure described here applies to spherical
coordinates once gn;13 in Eq. (13c) is replaced by
gn;13 ˆ 1 ‡
p
1 ¡ 0:8
2:18835
exp ¡dn2 Bi ‡ n
x13 …1 ¡ x13 † ‡
n
p
…23a†
and Eq. (23a) is modi¢ed for inclusion in the X33 case as
gn;13 ˆ 1 ‡
1 ¡ 0:8
2:18835
Bi1 Bi2
exp ¡dn2
‡n
x13 …1 ¡ x13 † ‡
n
Bi1 ‡ Bi2
p
…23b†
These changes are needed because the Bi term in Eqs. (10) and (23a) may be replaced
by a constant B that can go from ¡1 to in¢nity (see R03 in [1], p. 468); at B ˆ ¡1,
dbn =dB ! 1. For example, using a solid sphere with radius ro , this constant is
B ˆ hro =k ¡ 1. When Bi µ 0, the computed eigenvalues in Table 5 are compared with
the tabulated data in [2]. The largest error is in the vicinity of Bi ˆ ¡1. Also, Eq. (23a)
can be used successfully when Bi > 0; however, the data accuracy using Eq. (13a) is
slightly less than those from Eq. (13c). For a sphere with inner radius ri and outer
radius ro , the parameters Bi1 and Bi2 in Eqs. (16) and (23b) may be replaced by
B1 and B2 , respectively. For the RS33 case described in [1], when the heat transfer
coef¢cients at the inner and outer surfaces are h1 and h2 , one obtains
B1 ˆ …h1 a=k ‡ 1†…b=a ¡ 1† and B1 ˆ …h2 b=k ‡ 1†…1 ¡ a=b†. All computations reported
here were performed using Mathematica [10], and a sample program for X33 is in the
Appendix.
R EFER ENCES
1. J. V. Beck, K. D. Cole, A. Haji-Sheikh, and B. Litkouhi, Heat Conduction Using Green’s
Functions, Hemisphere, Washington, DC, 1992.
2. Ruth Zucker, Elementary Transcendental Functions, in M. Abramowitz and I. Stegun
(eds.), Handbook of Mathematical Functions, National Bureau of Standards AMS 55,
p. 224, June 1964.
3. M. M. Yovanovich, Simple Explicit Expressions for Calculation of the Heisler-Grober
Charts, 1996 National Heat Transfer Conf., Houston, Texas, Paper AIAA-96-3968, 1996.
4. R. A. Leathers and N. J. McCormick, Closed-Form Solutions for Transcendental
Equations of Heat Transfer, ASME J. Heat Transfer, vol. 118, pp. 970^973 , 1996.
5. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in
Fortran. The Art of Scienti¢c Computing, Cambridge University Press, Cambridge, UK,
1994.
6. C. J. F. Ridders, IEEE Trans. Circuits Systems, vol. CAS-26, pp. 979^980, 1979.
154
A. HAJI-SHEIKH AND J. V. BECK
7. R. P. Brent, Algorithms for Minimization without Derivatives, chaps. 3, 4, Prentice-Hall,
Englewood Cliffs, NJ, 1973.
8. C. Aviles-Ramos, K. T. Harris, and A. Haji-Sheikh, A Hybrid Root Finder, Proc. Fifth
Integral Methods in Science and Engineering, Houghton, MI, 1998.
9. J. W. Stevens and R. Luck, Explicit Approximations for All Eigenvalues of 1-D Transient
Heat Conduction Equations, Heat Transfer Eng., vol. 20, no. 2, pp. 35^41, 1999.
10. S. Wolfram, The Mathematica, 3d ed., Cambridge University Press, Cambridge, UK,
1996.
11. T. R. McCalla, Introduction to Numerical Methods and Fortran Programming, Wiley,
New York, 1967.
AP PEND IX
This Appendix is concerned with using a simple high-order Newton method for
¢nding the root of the equation
F …x† ˆ 0
…A:1†
It is possible to use the ¢rst- or higher-order Newton method to ¢nd the roots of this
equation. A ¢rst-order Newton method was not suf¢ciently accurate for a one-term
approximate explicit equation. The second-order Newton with the square-root term
because unstable when the Biot number was extremely large. This modi¢ed Newton
method is used because it has nearly the accuracy of the second-order Newton
method and it is numerically stable similar to the ¢rst-order Newton. Moreover,
it can be applied to the X33 case, where it is desirable to accommodate a third-degree
polynomial. Although this method is different from Bailey’s iterative method [11],
they are conceptually similar. If x0 is an estimate of the root, h ˆ x ¡ x0 , then, using
the Taylor series for F …x† ˆ F ‰x0 ‡ …x ¡ x0 †Š ˆ F …x ‡ h†, one can write
F …x† ˆ F …x0 † ‡ F 0 …x0 †h ‡
M
ˆ
mˆ0
1 00
1
F …x0 †h2 ‡ F 000 …x0 †h3 ‡ ¢ ¢ ¢
2!
3!
Am hm ˆ 0
…A:2†
where Am ˆ F 0 …x†=m! computed at x ˆ x0 . When h is small, the ¢rst-order Newton
equation uses the ¢rst two terms in the above series to approximate h as
h0 ˆ
¡A0
A1
…A:3†
However, for a better approximation, without signi¢cant additional efforts, one can
write h ˆ h0 ‡ Dh, where Dh½h if h is already small. Substituting for h in the polynomial represented by Eq. (A.2) and deleting the terms that contain Dh with a power
of 2 or higher leads to a relation that determines Dh,
Dh ˆ ¡
ˆ
M
m
mˆ2 Am h
M
m¡1
mˆ1 mA m h
M
m
m
mˆ2 …¡1† Am …A0 =A1 †
M
m
m¡1
mˆ1 m…¡1† Am …A0 =A1 †
…A:4†
EIGENVALUES IN HEAT CONDUCTION
155
When M ˆ 1, Eq. (A.4) yields Dh ˆ 0, which corresponds to a ¢rst-order Newton
method. When M ˆ 2 it approximates a second-order Newton method, while
M ˆ 3 improves the accuracy; however, it is rarely necessary to use M > 3.
Example. As an example, consideration is given to ¢nding the roots of the
equation
F …x† ˆ x sin x ¡ B cos x ˆ 0
…A:5†
which describes the eigenvalues for the X23 case. If x0 is an estimate of the root, then
A0 ˆ F …x0 † ˆ x0 sin x0 ¡ B cos x ˆ 0
A1 ˆ F 0…x0 † ˆ …1 ‡ B† sin x0 ‡ x cos x
F 00 …x0 † …1 ‡ B† cos x0 ‡ cos x0 ¡ x0 sin x0
A2 ˆ
ˆ
2!
2!
ˆ ‰…B cos x0 ¡ x0 sin x0 † ‡ 2 cos x0 Š=2 ^ cos x0
…A:5†
…A:5†
…A:5†
Using M ˆ 2, the value of x is
x ^ x0 ¡
A0
…A2 =A1 †…A0 =A1 †
1‡
A1
1 ¡ 2…A2 =A1 †…A0 =A1 †
…A:5†
The following symbolic function in Mathematica demonstrates the code for X33
when M ˆ 3. The ¢rst two arguments of this function are Bi1 and Bi2 , while the
third argument refers to the nth eigenvalue to be computed. This function accepts
all positive numbers, including 0 and 1, for either Bi1 or Bi2 .
(*Mathematica function begins here and ends after the return statement*)
eigen[bi1^ ,bi2^ ,n^ ]: ˆ (
cn ˆ (n¡3/4)*Pi;
dn ˆ (n¡1/2)*Pi;
d ˆ 1.22*(n¡1) ‡ 0.76;
p23 ˆ 1/n;
term ˆ Limit[Limit[(Sqrt[(b1 ‡ b2 ‡ b1*b2 ‡ cn¡Pi/4)/(b1 ‡ b2 ‡ b1*b2 ‡ d)]¡
(b1 ‡ b2 ‡ b1*b2 ‡ Sqrt[d*(cn¡Pi/4)])/(b1 ‡ b2 ‡ b1*b2 ‡ d)),b1! bi1], b2!bi2];
g23 ˆ 1¡1.04*term;
x23 ˆ Limit[Limit[(b1 ‡ b2¡cn)/(b1 ‡ b2 ‡ cn),b1!bi1],b2! bi2];
x13 ˆ Limit[Limit[b1*b2/(b1*b2 ‡ 0.2 ‡ (b1 ‡ b2)*(Pi*Pi*(n ¡0.5)/2)),b1 ! bi1],b2! bi2];
g13 ˆ 1 ‡ x13*(1¡x13)(1¡0.85/n¡(0.6¡0.71/n)*(x13 ‡ 1)*(x13¡0.6¡0.25/n));
e23 ˆ g23*(p23*x23 ‡ (1¡p23)*Tanh[x23]/Tanh[1]);
e13 ˆ 2*g13*x13;
y23 ˆ cn ‡ Pi*e23/4;
y13 ˆ dn ‡ Pi*e13/4;
param ˆ Limit[Limit[b1*b2/(b1*b2 ‡ dnê2),b1! bi1],b2! bi2];
z0 ˆ (y13*param ‡ y23*(1¡param));
a0 ˆ Limit[Limit[((z0^ 2¡b1*b2)*Sin[z0] ¡(b1 ‡ b2)*z0*Cos[z0])/(b1 ‡ b2),b1! bi1],b2!
bi2];
a1 ˆ Limit[Limit[((z0^ 2¡(b1 ‡ b2)¡b1*b2)*Cos[z0] ‡ (2 ‡ b1 ‡ b2)*z0*Sin[z0])/(b1 ‡ b2),
b1! bi1],b2! bi2];
a2 ˆ Limit[Limit[((1 ‡ b1 ‡ b2)*Sin[z0] ‡ 2*z0*Cos[z0])/(b1+b2),b1! bi1], b2! bi2]-a0/2;
156
A. HAJI-SHEIKH AND J. V. BECK
a3=Limit[Limit[(((6+3*(b1+b2) +b1* b2-z0*z0)*Cos[z0]-(6+b1+b2)*z0*Sin[z0])/6)
(b1+b2),b1! bi1],b2! bi2];
e=-a0/a1-(-a3*a ^ 3+a2*a0 ^ 2*a1)/(3*a3*a0*a0*a1-2*a2*a0*a1 ^ 2+a1 ^ 4);
z1=z0+e;
Return[z1]
(*Calling the function to calculate eignevalues with 10 digit output*)
N[eigen[1,10,1],10]
Out[2]=1.87530781
N[eigen[1,01],10]
Out [4]=0.86033358 9
N[eigen[In¢nity,1,1],10]
Out[6]=2.028757838