Acta Polytechnica Vol. 51 No. 4/2011

Rational Approximation to the Solutions of Two-Point
Boundary Value Problems

P. Amore, F. M. Fernández

Abstract

We propose a method for the treatment of two-point boundary value problems given by nonlinear ordinary differential
equations. The approach leads to sequences of roots of Hankel determinants that converge rapidly towards the unknown
parameter of the problem. We treat several problems of physical interest: the field equation determining the vortex
profile in a Ginzburg-Landau effective theory, the fixed-point equation for Wilson’s exact renormalization group, a
suitably modified Wegner-Houghton fixed point equation in the local potential approximation, and a Riccati equation.
Weconsider twomodelswhere theapproachdoesnotapply inorder to showthe limitations of ourPadé-Hankelapproach.

Keywords: nonlinear differential equations, Ginzburg-Landau, Wilson’s renormalization, Wegner-Houghton, Riccati
equation, Padé-Hankel method.

1 Introduction
Some time ago Fernández et al [1–3, 5, 6, 4, 7–9] devel-
oped a method for the accurate calculation of eigen-
functions and eigenvalues for bound states and reso-
nances of the Schrödinger equation. This approach is
based on the Taylor expansion of a regularized loga-
rithmic derivative of the eigenfunction. The physical
eigenvalue is given by a sequence of roots of Hankel
determinants constructed from the coefficients of that
series. One merit of this approach, called the Riccati-
Padé method, is the great convergence rate in most
cases and that the same equation applies to bound
states and resonances. Besides, in some cases it yields
upper and lower bounds to the eigenvalues [1].

The logarithmic derivative satisfies a Riccati
equation, and one may wonder if the method ap-
plies to other nonlinear ordinary differential equa-
tions. The purpose of this paper is to investigate
whether a kind of Padé-Hankel method may be use-
ful for two-point boundary value problems given by
nonlinear ordinary differential equations.

In Section 2 we outline the method, in Section 3
we apply it to several problems of physical interest,
and in Section 4 we discuss the relative merits of the
approach.

2 Method
It is our purpose to propose a method for the treat-
ment of two-point boundary value problems. We sup-
pose that the solution f (x) of a nonlinear ordinary
differential equation can be expanded as

f (x) = xα
∞∑

j=0

fj x
βj (1)

about x = 0, where α and β are real numbers,
and β > 0. We also assume that we can cal-
culate sufficient coefficients fj in terms of one of
them that should be determined by the boundary
condition at the other point; for example, at infin-
ity. We show several illustrative examples in sec-
tion 3.

We try a rational approximation to x−αf (x) of
the form

[M, N ](z) =

∑M
j=0 aj z

j∑N
j=0 bjz

j
. (2)

where z = xβ . The Taylor expansion of the usual
Padé approximant yields M + N + 1 coefficients of
the series (1) [11]; but in the present case we require
that the rational approximation (2) gives us one more
coefficient, that is to say, M + N + 2. If M = N + d,
N = 1, 2, . . ., d = 0, 1, . . ., this requirement leads to
the equation [1–3, 5, 6, 4, 7–9]

HdD = |fi+j+d+1|i,j=0,1,...N = 0, (3)

where D = N + 1 = 2, 3, . . . is the dimension of the
Hankel determinant HdD.

In general, equation (3) exhibits many roots and
one expects to find a sequence, for D = 2, 3, . . .
and fixed d, that converges towards the required
value of the unknown coefficient. From now on
we call it the Hankel sequence for short. If such
a convergent sequence is monotonously increasing
or decreasing we assume that it yields a lower
or upper bound, respectively. Such bounds have
been proved rigorously for some eigenvalue prob-
lems [1].

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Acta Polytechnica Vol. 51 No. 4/2011

3 Examples

In order to test the performance of the Padé-Hankel
method, in this section we consider the examples
treated by Boisseau et al [10] by means of a most
interesting algebraic approach. We first consider
the field equation determining the vortex profile in
a Ginzburg-Landau effective theory (and references
therein)

f ′′(r) +
1
r
f ′(r) +

(
1 −

n2

r2

)
f (r) − f (r)3 = 0,

r > 0.
(4)

The solution f (r) satisfies the expansion (1) with
x = r, α = n = 1, 2, . . ., and β = 2. If we substi-
tute this series into the differential equation and solve
for the coefficients fj , we obtain them in terms of the
only unknown f0 that is determined by the boundary
condition at infinity: f (r → ∞) = 1 [10] (and refer-
ences therein). The coefficients fj , and therefore the
Hankel determinant HdD , are polynomial functions of
f0. For example, for n = 1 we have

Table 1: Convergence of theHankel series for the connec-
tion parameters of the global vortex for n =1

D d = 0 d = 1

2 0.595 0.578

3 0.584 0.582 9

4 0.583 24 0.583 15

5 0.583 20 0.583 183

6 0.583 192 0.583 187

7 0.583 190 0.583 189 0

8 0.583 189 7 0.583 189 3

9 0.583 189 54 0.583 189 46

10 0.583 189 52 0.583 189 48

11 0.583 189 51 0.583 189 491

12 0.583 189 498 0.583 189 494

13 0.583 189 496 4 0.583 189 495 3

14 0.583 189 496 1 0.583 189 495 6

15 0.583 189 496 0 0.583 189 495 7

16 0.583 189 495 90 0.583 189 495 83

17 0.583 189 495 88 0.583 189 495 84

18 0.583 189 495 867 0.583 189 495 854

19 0.583 189 495 864 0.583 189 495 857

20 0.583 189 495 862 0.583 189 495 859 1

21 0.583 189 495 860 9 0.583 189 495 859 8

22 0.583 189 495 860 7 0.583 189 495 860 1

f1 = −
f0
8

, f2 =
f0

192
+

f30
24

,

f3 = −
f0

9216
−

5f30
576

, . . .

(5)

Tables 1 and 2 show two Hankel sequences with
d = 0 and d = 1 that converge rapidly towards
the result of the accurate shooting method [10] for
n = 1 and n = 2, respectively. We appreciate that
in the case n = 1 the sequences with d = 0 and
d = 1 give upper and lower bounds, respectively, that
tightly bracket the exact value of the unknown pa-
rameter of the theory: 0.583 189 495 860 60 < f0 <
0.583 189 495 860 61.

On the other hand, the appropriate Hankel se-
quences are oscillatory when n ≥ 2 and their rate of
convergence decreases with n. Table 3 shows the best
estimates of f0 for n = 2, 3, 4.

Table 2: Convergence of theHankel series for the connec-
tion parameters of the global vortex for n =2

D d = 0 d = 1

3 0.156 0.151

4 0.152 8 0.154

5 0.153 10 0.153 0

6 0.153 09 0.153 11

7 0.153 098 0.153 10

8 0.153 099 7 0.153 10

9 0.153 099 1 0.153 099

10 0.153 099 14 0.153 098 9

11 0.153 099 12 0.153 099 095

12 0.153 099 17 0.153 099 091

13 0.153 099 105 0.153 099 097

14 0.153 099 102 1 0.153 099 11

15 0.153 099 102 72 0.153 099 102

16 0.153 099 102 697 0.153 099 103

17 0.153 099 102 782 0.153 099 102 92

18 0.153 099 103 124 0.153 099 102 93

19 0.153 099 102 857 0.153 099 102 89

20 0.153 099 102 864 0.153 099 102 78

21 0.153 099 102 861 36 0.153 099 102 860

22 0.153 099 102 861 42 0.153 099 102 858

Table 3: Best estimates of the connection parameters of
the global vortex for n = 2,3,4 by means of Hankel se-
quences with D ≤ Dmax

n Dmax f0

2 21 0.153 099 102 86

3 21 0.026 183 420 7

4 26 0.003 327 173 4

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Acta Polytechnica Vol. 51 No. 4/2011

Our second example is the fixed-point equation
for Wilson’s exact renormalization group [10] (and
references therein)

2f ′′(x) − 4f (x)f ′(x) − 5xf ′(x) + f (x) = 0,
x > 0.

(6)

The solution to this equation can be expanded as in
equation (1) with α = 1 and β = 2. The first coeffi-
cients are

f1 =
f0
3

+
f20
3

, f2 =
7f0
60

+
f20
4

+
2f30
15

, . . . (7)

For large values of x the physical solution should be-
have as f (x) = ax1/5 + a2/(5x3/5) + . . . The Han-
kel sequences with d = 0 and d = 1 converge
towards the numerical result [10] (and references
therein) from above and below, respectively. Fi-
gure 1 displays the great rate of convergence of these
sequences as Δ = |f0(D, d = 0) − f0(D, d = 1)|,
D = 2, 3, . . ., from which we obtain the accu-
rate bounds −1.228 598 202 437 021 924 38 < f0 <
−1.228 598 202 437 021 924 37.

0 5 10 15 20 25

D

10
-21

10
-18

10
-15

10
-12

10
-9

10
-6

10
-3

10
0

Δ

Fig. 1: Δ = |f0(D, d =0) − f0(D, d =1)| for Wilson’s
renormalization

The third example comes from a suitably modi-
fied Wegner-Houghton’s fixed point equation in the
local potential approximation [10] (and references
therein)

2f ′′(x)+[1+f ′(x)][5f (x)−xf ′(x)] = 0, x > 0. (8)

The solution satisfies the series (1) with α = 1 and
β = 2, and the first coefficients are

f1 = −
f0
3

−
f20
3

, f2 =
f0
60

+
2f20
15

+
7f30
60

, . . . (9)

On the other hand, the acceptable solution should
behave as f (x) = ax5 − 4/(3x) + . . . when x � 1.

Table 4 shows Hankel sequences with d = 0 and
d = 1 that clearly converge towards the numerical
value of f0 [10] (and references therein).

Table 4: Convergence of the Hankel sequences for the
Wegner-Houghton connection parameter

D d = 0 d = 1

3 −0.301 365 209 2 −0.419 012 931 2
4 −0.540 511 282 4 −0.469 645 717 0
5 −0.455 201 249 3 −0.460 479 692 6
6 −0.462 452 597 9 −0.461 693 582 1
7 −0.461 375 992 6 −0.461 509 171 7
8 −0.461 557 112 9 −0.461 537 339 3
9 −0.461 530 376 7 −0.461 533 153 5
10 −0.461 534 297 5 −0.461 533 816 5
11 −0.461 533 614 7 −0.461 533 704 3
12 −0.461 533 735 7 −0.461 533 722 7
13 −0.461 533 717 3 −0.461 533 719 6
14 −0.461 533 720 7 −0.461 533 720 2
15 −0.461 533 720 0 −0.461 533 720 1
16 −0.461 533 720 13 −0.461 533 720 119
17 −0.461 533 720 113 −0.461 533 720 115 7
18 −0.461 533 720 116 8 −0.461 533 720 116 3
19 −0.461 533 720 116 1 −0.461 533 720 116 2
20 −0.461 533 720 116 2

We have also applied our approach to the ordi-
nary differential equation for the spherically symmet-
ric skyrmion field [10] (and references therein) but
we could not obtain convergent Hankel sequences.
We do not yet know the reason for the failure of the
method in this case.

The present approach has earlier proved suitable
for the treatment of the Riccati equation derived from
the Schrödinger equation [1–3, 5, 6, 4, 7–9]. Consider,
for example, the following Riccati equation

f ′(x) − f (x)2 + x2 = 0, x > 0. (10)

The solution can be expanded as in equation (1) with
α = β = 1; the first coefficients are

f1 = f
2
0 , f2 = f

3
0 , f3 = −

1
3

+ f40 , . . .

There is a critical value f0c of f (0) = f0 such that
f (x) ∼ −x at large x if f (0) < f0c, f (x) develops a
singular point if f (0) > f0c, and f (x) ∼ x at large x if
f (0) = f0c. The present Padé-Hankel method yields
the value of f0c with remarkable accuracy, as shown
in Table 5. The rate of convergence of the Hankel se-
quence for this problem is considerably greater than
for the preceding ones.

If we substitute f (x) = −y′(x)/y(x) into equation
(10), then the function y(x) satisfies the Schrödinger
equation for a harmonic oscillator with zero energy

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Acta Polytechnica Vol. 51 No. 4/2011

Table 5: Convergence of theHankel sequenceswith d =0
for the Riccati equation

D f0

4 0.676 2

5 0.675 970

6 0.675 978 5

7 0.675 978 23

8 0.675 978 240 3

9 0.675 978 240 059

10 0.675 978 240 067 5

11 0.675 978 240 067 277

12 0.675 978 240 067 285 0

13 0.675 978 240 067 284 722

14 0.675 978 240 067 284 729

15 0.675 978 240 067 284 728 99

16 0.675 978 240 067 284 729 00

17 0.675 978 240 067 284 729 00

Table 6: Convergence of theHankel sequenceswith d =4
for the Thomas-Fermi equation

D 2f2

10 −1.588 070 9
11 −1.588 070 6
12 −1.588 071 03
13 −1.588 071 024
14 −1.588 071 022 7
15 −1.588 071 022 64
16 −1.588 071 022 609
17 −1.588 071 022 609
18 −1.588 071 022 611 6
19 −1.588 071 022 611 5
20 −1.588 071 022 611 39
21 −1.588 071 022 611 38
22 −1.588 071 022 611 37
23 −1.588 071 022 611 37
24 −1.588 071 022 611 375 6
25 −1.588 071 022 611 375 37
26 −1.588 071 022 611 375 32
27 −1.588 071 022 611 375 315 4
28 −1.588 071 022 611 375 315 2
29 −1.588 071 022 611 375 315 4
30 −1.588 071 022 611 375 313 7

on the half line: y′′(x)−x2y(x) = 0, and the problem
solved above is equivalent to finding the logarithmic
derivative at origin y′(0)/y(0) so that y(x) behaves as
exp(−x2/2) at infinity. Obviously, any approach for
linear differential equations is suitable for this prob-
lem.

Finally, we consider two examples discussed by
Bender et al [12]; the first of them is the instanton
equation

f ′′(x) + f (x) − f (x)3 = 0 (11)
with the boundary conditions f (0) = 0, f (∞) = 1.
The solution to this equation is f (x) = tanh

(
x/

√
2
)

.

The expansion of f (x) is a particular case of equation
(1) with α = 1 and β = 2; its first coefficients being

f1 = −
f0
6

,
f0
(
6f20 + 1

)
120

,

f3 = −
f0
(
66f20 + 1

)
5 040

, . . . ,

(12)

where f0 = f
′(0) is the unknown. The Hankel series

with d = 0 and d = 1 converge rapidly giving upper
and lower bounds, respectively, to the exact result
f0 = 1/

√
2.

The second example is the well known Blasius
equation [12]

2y′′′(x) + y(x)y′′(x) = 0 (13)

with the boundary conditions y(0) = y′(0) = 0,
y′(∞) = 1. The expansion of the solution in a Taylor
series about x = 0 is a particular case of equation (1)
with α = 2 and β = 3; its first coefficients are

f1 = −
f20
60

, f2 =
11f30
20160

, . . . (14)

Since, in general, fj ∝ f j+10 , then the only root of
the Hankel determinants is f0 = 0, which leads to
the trivial solution y(x) ≡ 0. We thus see another
case where the Padé-Hankel method does not apply.

4 Conclusions
We have presented a simple method for the treat-
ment of two-point boundary value problems. If there
is a suitable series for the solution about one point,
we construct a Hankel matrix with the expansion co-
efficients and obtain the physical value of the un-
determined coefficient from the roots of a sequence
of determinants. The value of this coefficient given
by a convergent Hankel sequence is exactly the one
that produces the correct asymptotic behaviour at
the other point. We cannot prove this assumption
rigorously, but it seems that if there is a convergent
sequence, it yields the correct answer. Moreover, in
some cases the Hankel sequences produce upper and
lower bounds bracketing the exact result tightly.

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Acta Polytechnica Vol. 51 No. 4/2011

The present Padé-Hankel approach is not as gen-
eral as the one proposed by Boisseau et al [10], as
we have already seen that the former does not appar-
ently apply to the skyrmion problem or to the Blasius
equation [12]. However, our procedure is much sim-
pler and more straightforward and may be a suitable
alternative for treating problems of this kind. Be-
sides, if our approach converges, it yields remarkably
accurate results, as shown in the examples above.

References

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upper and lower bounds for energy eigenvalues
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[2] Fernández, F. M.: Strong Coupling Expan-
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[3] Fernández, F. M., Guardiola, R.: Accurate
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[4] Fernández, F. M.: Resonances for a Perturbed
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[7] Fernández, F. M.: Quantization Condition for
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[8] Fernández, F. M.: Direct Calculation of Stark
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[9] Fernández, F. M.: Tunnel Resonances for One-
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[10] Boisseau, B., Forgács, P., Giacomini, H.: An
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[11] Bender, C. M., Orszag, S. A.: Advanced math-
ematical methods for scientists and engineers,
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[12] Bender, C. M., Pelster, A., Weissbach, F.:
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Paolo Amore
E-mail: paolo@cgic.ucol.mx
Facultad de Ciencias
Universidad de Colima
Bernal Dı́az del Castillo 340, Colima, Colima, Mexico

Francisco M. Fernández
E-mail: fernande@quimica.unlp.edu.ar
INIFTA (Conicet,UNLP), Diag. 113 y 64 S/N
Sucursal 4, Casilla de Correo 16, 1900 La Plata, Ar-
gentina

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