ArticlePDF Available

Abstract and Figures

A compact and accurate solution method is provided for problems whose infinite power series solution diverges and/or whose series coefficients are only known up to a finite order. The method only requires that either the power series solution or some truncation of the power series solution be available and that some asymptotic behavior of the solution is known away from the series' expansion point. Here, we formalize the method of asymptotic approximants that has found recent success in its application to thermodynamic virial series where only a few to (at most) a dozen series coefficients are typically known. We demonstrate how asymptotic approximants may be constructed using simple recurrence relations, obtained through the use of a few known rules of series manipulation. The result is an approximant that bridges two asymptotic regions of the unknown exact solution, while maintaining accuracy in-between. A general algorithm is provided to construct such approximants. To demonstrate the versatility of the 1 method, approximants are constructed for three nonlinear problems relevant to mathematical physics: the Sakiadis boundary layer, the Blasius boundary layer, and the Flierl-Petviashvili monopole. The power series solution to each of these problems is underspecified since, in the absence of numerical simulation, one lower-order coefficient is not known; consequently, higher-order coefficients that depend recur-sively on this coefficient are also unknown. The constructed approx-imants are capable of predicting this unknown coefficient as well as other important properties inherent to each problem. The approxi-mants lead to new benchmark values for the Sakiadis boundary layer and agree with recent numerical values for properties of the Blasius boundary layer and Flierl-Petviashvili monopole.
Content may be subject to copyright.
On the summation of divergent, truncated,
and underspecified power series
via asymptotic approximants
Nathaniel S. Barlow1, Christopher R. Stanton1, Nicole Hill2,
Steven J. Weinstein2, and Allyssa G. Cio3.
1School of Mathematical Sciences,2Department of Chemical Engineering,
3Department of Industrial Engineering
Rochester Institute of Technology, Rochester, NY 14623, USA
This article has been accepted for publication in the Quarterly Journal of
Mechanics and Applied Mathematics Published by Oxford University Press.
The final published version can be found here: https://academic.oup.com/
qjmam/article/2895119/On-the-Summation-of-Divergent-Truncated-and
Abstract
A compact and accurate solution method is provided for problems
whose infinite power series solution diverges and/or whose series coef-
ficients are only known up to a finite order. The method only requires
that either the power series solution or some truncation of the power
series solution be available and that some asymptotic behavior of the
solution is known away from the series’ expansion point. Here, we
formalize the method of asymptotic approximants that has found re-
cent success in its application to thermodynamic virial series where
only a few to (at most) a dozen series coefficients are typically known.
We demonstrate how asymptotic approximants may be constructed
using simple recurrence relations, obtained through the use of a few
known rules of series manipulation. The result is an approximant that
bridges two asymptotic regions of the unknown exact solution, while
maintaining accuracy in-between. A general algorithm is provided to
construct such approximants. To demonstrate the versatility of the
1
method, approximants are constructed for three nonlinear problems
relevant to mathematical physics: the Sakiadis boundary layer, the
Blasius boundary layer, and the Flierl-Petviashvili monopole. The
power series solution to each of these problems is underspecified since,
in the absence of numerical simulation, one lower-order coefficient is
not known; consequently, higher-order coefficients that depend recur-
sively on this coefficient are also unknown. The constructed approx-
imants are capable of predicting this unknown coefficient as well as
other important properties inherent to each problem. The approxi-
mants lead to new benchmark values for the Sakiadis boundary layer
and agree with recent numerical values for properties of the Blasius
boundary layer and Flierl-Petviashvili monopole.
1 Introduction
Power series arise in virtually all applications of mathematical physics. The
utility of such series is evident in the construction of approximations (e.g.
finite differences) or as a rigorously determined solution to a problem. Lim-
itations generally inherent to power series solutions often inhibit their di-
rect use, and they are more often useful in the implementation of numerical
schemes. For instance, a Taylor series representation of an unknown function
may not converge, as it may have a finite radius of convergence arising from
singularities (often complex) in the function it represents [1]. Even when
singularities are not a concern, higher-order terms of the series may be ex-
ceedingly difficult to compute [2], which is especially problematic if the series
converges slowly.
Several techniques have been put forward to analytically continue and
to accelerate convergence of divergent or slowly converging series; see, for
example [3] (ch. 8), [4] (chs. 14, 19, 20), [5], [6], [7], [8], [9], [10]. One
of the more well-known techniques is the Pad´e approximant method [11],
which has similarities with the method of asymptotic approximants described
herein. A Pad´e approximant (referred to as a Pad´e) is the quotient of two
polynomials, with Ntotal polynomial coefficients (distributed between the
numerator and denominator), chosen such that the Taylor expansion of the
Pad´e reproduces the power series of interest up to Nterms. This involves
solving an algebraic system with Nunknowns (the Pad´e coefficients). After
its construction, one may use the Pad´e in lieu of the original series. The Nth-
order Pad´e and Nth-order truncated original series will follow one another
2
near the expansion point. Ideally, the Pad´e will then continue on beyond
the radius of convergence (if one is present) and in general represent the
actual solution better than the truncated series. If chosen judiciously, a Pad´e
sequence (specified by a fixed difference in order between the numerator and
denominator) may converge rapidly to the correct solution as Nis increased,
whereas the original series may converge slowly or diverge. In the case of
a divergent series, the Pad´e approximates the singularity that is presumed
to be responsible for divergence [12], and in doing so enables an accurate
summation of the truncated series.
One drawback of the Pad´e method is that it is not always clear before-
hand which sequence is best suited for a given problem [12]. If one has a
power series alone, it is difficult to choose a correct Pad´e form (see [13, 14, 15]
for example). However, if the power series arises from a physical problem, it
is likely that some additional conditions or context may be gleaned. These
conditions may be conveniently available, or may need to be independently
derived. For instance, if one has a power series representation of f(x) and
it is known that fasymptotically approaches a constant value as x→ ∞,
a “symmetric” Pad´e sequence of equal order in the numerator and denom-
inator will contain Pad´es that all preserve this asymptotic condition, and
this sequence will uniformly converge towards the correct solution as Nis
increased. Baker and Gammel [11] recognized this important result and went
further to state: If a series for f(x) is known and fCxpas x (pbeing
an integer), the uniformly convergent Pad´e sequence is the one with a differ-
ence of pbetween the numerator and denominator order, hence preserving
the x→ ∞ behavior as Nis increased.
Baker & Gammel’s statement may be extended to asymptotic behaviors
beyond integer power laws using approximants other than Pad´es [16, 17, 18].
An example of such an approximant is found in the review by Frost and
Harper [10], used to find an approximate solution for the drag coefficient
on a sphere in fluid-filled tube; the approximant they used incorporates the
correct non-integer power-law asymptotic behavior. The present work may
be considered an extension of the perspective given in [10]- namely the con-
struction of approximants that are asymptotically consistent with known
behavior in the vicinity of the domain boundaries while maintaining accu-
racy in-between. In this paper, we aim to formalize this approach by defining
asymptotic approximants in general:
3
Definition 1.1 Given a power series representation of some function f(x):
f=
X
n=0
an(xx0)n,(1)
and an asymptotic behavior
fCfa(x)as xxa,(2)
where Cis a constant, an asymptotic approximant is any function fA(x)that
may be expressed analytically in closed form and that satisfies the following
three properties:
1. The N-term Taylor expansion of fAabout x0is identical to the N-term
truncation of (1).
2. lim
xxa
(fA/fa) = constant for any N.
3. The sequence of approximants converges for increasing N.
Choosing an approximant that satisfies the above definition will lead to a
uniformly convergent sequence as Nis increased that preserves the correct
asymptotic behavior. Note that in the above definition, faneed not be exact;
in fact, only the leading order is typically needed to construct an adequate
approximant. We seek approximants whose unknown coefficients can be gen-
erated with ease. We do not wish for the construction of approximants to
undergo more floating-point operations than performed in an accurate nu-
merical solution. Most of the approximant coefficients calculated here may be
obtained recursively using a few simple series relations listed in Appendix A.
This is in contrast with Pad´es and their extensions, which often require the
inversion of a linear system [3, 4, 10]. As shall be apparent, a theme of this
work is to find/impose the simplest asymptotic approximant form possible
while maintaining the desired accuracy and precision.
Approximants are closed form functions that may be analytically differen-
tiated to obtain auxiliary properties without the loss of accuracy that occurs
with discretized solutions. Also, physically relevant properties may be cast
as unknowns within an approximant, and the approximant can be used as a
predictor for such properties [19, 16, 17, 18]. This feature of approximants
provides a significant problem-solving advantage.
4
The paper is organized as follows. In Section 2, we review recent ap-
plications of asymptotic approximants towards the truncated virial series
of thermodynamics. The analysis is recast here as part of a new unified
framework that only requires the use of a few simple series relations. In
Section 3, an algorithm is provided to construct asymptotic approximants
for problems in general. The method is then applied to three nonlinear
boundary value problems in Section 4. In Section 4.1, the Sakiadis boundary
layer problem is solved using two types of asymptotic approximants - one
which is both accurate and simple in form, and another which matches its
higher-order asymptotic behavior and is capable of predicting the wall-shear
coefficient and other properties to a higher precision than reported before.
In Section 4.2, the Blasius boundary layer problem is solved using a sim-
ple asymptotic approximant, which is also capable of predicting important
quantities such as the wall-shear coefficient. In Section 4.3, an asymptotic
approximant is used to solve the Flierl-Petviashvili monopole problem and
predict relevant properties, which are shown to agree with newly generated
numerical results. Key findings are summarized in Section 5.
2 Relevant background and results from virial-
based approximants
In this section, we review the recent application of asymptotic approximants
towards problems in thermodynamics. In doing the review, simplifications
and details not previously disclosed are elucidated here and also incorporated
into the algorithm of Section 3.
The virial series is an equation of state formulated as a series expansion
about the ideal gas limit [20] (ρ0), and is expressed as
P=kT
N
X
n=1
Bn(T)ρn, B1= 1 (3)
where Pis the pressure, ρis the number density, kis the Boltzmann constant,
and Tis the temperature. The virial coefficients Bnare functions of Tas
well as other physical parameters describing a given intermolecular potential.
Specifically, the nth virial coefficient is defined as an integral over the position
of nmolecules [21]. In the absence of an exact equation of state, the virial
series represents fluid properties at low density more accurately than any
5
other theoretical or computational method, since it is effectively the Taylor
series of the exact solution about ρ=0. A barrier to its usage is that the
number of integrals appearing for each viral coefficient increases rapidly and
nonlinearly with the order of the coefficient. Consequently, significant effort is
spent to develop efficient algorithms to compute higher-order coefficients [22,
23]. As an indication of the difficulty involved, only twelve virial coefficients
are currently known (to within some precision) for the hard-sphere model
fluid [23]. For more realistic model fluids, significantly fewer coefficients are
known.
Even with several coefficients, the virial series often converges only in a
small region near ρ= 0. Pad´e approximants have been applied to extend
this region, with a common example being the Carnahan-Starling equation of
state for hard-sphere fluids [24]. With only a finite number of terms available,
the choice of approximant (be it Pad´e or not), is important as convergence
may be accelerated by choosing an approximant with consistent asymptotic
behavior following Definition 1.1. In the following subsections, we review the
recent application of asymptotic approximants towards divergent and slowly
converging truncated virial series to construct accurate isotherms and predict
critical properties.
2.1 Soft-sphere fluids
To our knowledge, the first application of a non-Pad´e asymptotic approxi-
mant towards virial series is given in [16] and is used to describe isotherms
for soft-sphere fluids. This problem highlights the key steps in formulating
an asymptotic approximant used later in this paper, and we thus review the
technique carefully. Soft-sphere fluids [25] are defined by a molecular poten-
tial that behaves like rh, where ris the distance between molecules and h
is a “hardness” parameter. In the limit as h→ ∞, the soft-sphere molecular
potential limits to that of the hard-sphere. For soft spheres, it is typical to
rewrite (3) in terms of reduced virial coefficients ¯
Bnand reduced density ˜ρ
using the scalings provided in [16]. Equation (3) becomes
ZP
ρkT = 1 +
N
X
n=2
¯
Bn(h)˜ρn1,(4)
where Zis the compressibility factor and the coefficients ¯
Bnare now func-
tions of the hardness h. The above series diverges (see Fig. 1 for h= 4),
6
with a radius of convergence that decreases with decreasing h[16]. While (4)
accurately describes the low-density behavior, an approximant can be con-
structed by bridging (4) with the large density limit for soft-spheres:
ZC˜ρh/3as ˜ρ→ ∞.(5)
Note that if h/3 is not an integer, it is impossible to capture the above
behavior with a standard Pad´e. However, an asymptotic approximant such
as
ZA= 1 +
N1
X
n=1
An(h)˜ρn!
h/3
N1
(6a)
limits to (5) for any hand allows for an easy evaluation of its coefficients, such
that the Taylor expansion of (6a) about ˜ρ= 0 is exactly (4). An expression
for the unknown coefficients Anis found by equating the original series (4)
with the approximant form (6a) and then solving for the series in An:
1 +
N
X
n=2
¯
Bn(h)˜ρn1!N1
h/3
= 1 +
N1
X
n=1
An(h)˜ρn.
It is now clear that the Ancoefficients are the coefficients of the Taylor ex-
pansion of the left-hand side of the above expression about ˜ρ= 0. Such coef-
ficients are easily obtained using J. C. P. Miller’s formula [26] for recursively
evaluating the expansion of a series raised to any power; see equation (29) in
Appendix A. The recursion for the coefficients is
An>0=1
n
n
X
j=1 j(N1)
h/3n+j¯
Bj+1Anj, A0= 1.(6b)
Together, (6a) and (6b) provide an asymptotic approximant which converges
for all ˜ρin the fluid regime, as shown in [16] and Fig. 1 (for h=4). Note
that (6b) reduces to the formulae for A2through A10 given in the appendix
of [16], and can be used to generate higher-order approximants as higher-
order virial coefficients become available. The same procedure used above
will be applied in Section 4 to obtain asymptotic approximants for nonlinear
boundary value problems of mathematical physics.
7
0123456
˜ρ
0
20
40
60
80
100
Z
S3
A2
S7 S5
S2
S6 S4
A3-7
Figure 1: Compressibility factor Zversus reduced density ˜ρfor the r4soft-
sphere fluid. A comparison is shown between the N-term virial series (3)
(−−) labeled as SN, the corresponding approximant (6) () labeled as AN,
and data from molecular simulation [27] (). The plot spans the entire fluid
regime, as the last data point is where freezing occurs. Virial coefficients
used to generate the curves are taken from [16].
2.2 Prediction of Critical Properties
An important property of asymptotic approximants is their ability to predict
unknown quantities. Pad´e and other approximants have long been used to
predict critical properties of lattice models [28, 29, 6, 7]. With the advent of
asymptotic approximants, this approach has recently yielded accurate pre-
dictions of critical properties of model fluids [17, 18]. The prediction begins
by constructing the approximant in the same manner as the previous section.
The virial series describing P(ρ) along the critical isotherm (i.e. using
coefficients evaluated at the critical temperature T=Tc) captures low density
behavior but diverges due to a branch point singularity at the critical density
ρc[30]. The behavior of fluids near the critical point obeys a universal scaling
law [31]
P
Pc1∼ ±D
ρ
ρc1
δ
, T =Tc, ρ ρc,(7)
where Pcis the critical pressure, Dis a critical amplitude, and (for real fluids)
δis a positive non-integer critical exponent. The knowledge of a low density
series as well as an asymptotic behavior at higher density are the necessary
8
ingredients for the application of an asymptotic approximant. In particular,
the series (3) is accurate in a region where (7) does not apply and inaccurate
in the region where (7) is accurate. An approximant is provided in [17] that
captures both regions (low density and the critical region); it is given by
PA=Pc
N
X
n=0
Anρn1ρ
ρcδ
.(8a)
Following the same approach of the previous subsection, one may equate (8a)
with the virial series (3) evaluated at T=Tcand solve for the series in An:
PckTc
N
X
n=1
Bn(Tc)ρn!1ρ
ρcδ
=
N
X
n=0
Anρn
where it becomes clear that the Ancoefficients are, in fact, those in the Taylor
expansion about ρ= 0 of the left-hand side of the above expression. The An
coefficients are obtained by replacing (1 ρ/ρc)δwith its expansion about
ρ= 0 and then taking the Cauchy product (equation (27) in Appendix A)
of the two series in the left-hand side of the above expression. This leads to
Equation 4b of [17]:
An>0=PcΓ(δ+n)
n!ρn
cΓ(δ)kTc
Γ(δ)
n1
X
j=0
BnjΓ(δ+j)
ρj
cj!, A0=Pc.(8b)
If ρc,Tc, and Pcare known, one then inserts (8b) into (8a) to construct
the approximant, which captures both the low-density and critical region,
as shown in [17]. However, it is typically the case that critical properties
are poorly known and difficult to compute through molecular simulation.
Fortunately, the approximant may be used to predict these quantities. For
instance, if ρcis unknown and all other parameters are known, one may
sacrifice the highest unknown coefficient ANand instead predict ρc. This is
equivalent to setting AN=0 in (8b), leading to Equation 7 of [17],
PckTcN!
Γ(δ+N)
N
X
j=1
Γ(δ+Nj)
(Nj)! Bjρj
c= 0 (8c)
which is a polynomial in ρc. Although this predicts multiple roots for ρc,
complex or negative values may be eliminated. As one increases Nand tracks
9
the roots, typically one set of positive roots will converge to a limit point more
rapidly than others, as shown in Fig. 2a for the Lennard-Jones model fluid.
Taking the fastest converging ρcsequence in the figure and substituting it
into (8b) and (8a) for each Nleads to a set of approximant isotherms, shown
in Fig. 2b. Note in the figure that the approximant converges more rapidly
and uniformly than the virial series as Nis increased.
(a)
N
1234567
ρc
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
(b)
ρ/ρc,sim
0 0.2 0.4 0.6 0.8 1 1.2
P/Pc
0
0.2
0.4
0.6
0.8
1S1
(ideal gas)
A1
S3
S5
S7
A3,5,7
Figure 2: (a) Predictions of the critical density from (8c) for the Lennard-
Jones fluid. A dashed line indicates the value predicted from Monte-Carlo
simulations [32]. (b) Critical isotherm of the Lennard-Jones fluid. A compar-
ison is shown between the N-term virial series (3) (−−) labeled as SNand
the corresponding approximant (11) () labeled as AN. The most rapidly
converging branch of ρcroots (lowermost points) in (a) is used as an input
to the approximants shown in (b). Virial coefficients used to generate the
plots are taken from [17].
Now if both ρcand Pcare unknown, one may sacrifice the two highest
coefficients (i.e. AN=AN1= 0), leading to a system of 2 nonlinear equa-
tions to solve for ρcand Pc. Recently, this approach has been used to predict
properties and construct approximants that describe the entire critical re-
gion [18] - not just the critical isotherm. Results in Section 4 demonstrate
that this method is also useful in the prediction of quantities important to
nonlinear boundary value problems.
As mentioned in the introduction, the virtue of approximants such as (6)
and (8) is the ease with which accurate derivative and integral properties are
generated. For example, the more comprehensive critical approximant given
10
in [18] for P(ρ, T ) can be integrated with respect to ρto accurately calculate
the Helmholtz free energy, which may then be differentiated with respect
to Tto find the internal energy and specific heat. Even if these quantities
are extracted numerically after an approximant is constructed, they may be
resolved on a grid of arbitrary resolution with minimal computational expense
(only memory usage), since the approximant is an analytic expression.
3 General procedure for constructing an asymp-
totic approximant
With the aim of formalizing the method of asymptotic approximants intro-
duced in Section 2, we provide the following algorithm. This procedure is
subsequently applied to power series solutions of nonlinear boundary-value
problems in Section 4.
Algorithm 3.1 Steps for constructing an asymptotic approximant for a prob-
lem whose analytic solution f(x)is elusive:
i. Find a power series solution of the given problem and truncate to N
terms:
fS=
N
X
n=0
an(xx0)n.
Define mas the number of unknowns embedded in the coefficients an;
this value is used in step v below. Note that step i is independent of step
ii and thus their order may be interchanged.
ii. Seek the asymptotic behavior fCfaas xxa, where xa6=x0and
Cis a content. This may come from either an asymptotic expansion
about xa, a boundary condition, or some independently known limiting
behavior of f. Define pas the number of unknowns associated with the
asymptotic behavior; this value is used in step v below.
iii. Create an approximant function fAwith N+1 unknowns A0. . . AN. The
function must (a) match the correct asymptotic behavior fCfaas
xxato at least zeroth order and (b) be generalized to handle arbitrary
N; a function of an “approximant series” is well suited for this:
fA=fA N
X
n=0
An[g(x)]n!.
11
This is the creative step. The goal is to capture known features of f
while leaving flexibility to continuously merge two (potentially disparate)
regions.
iv. Solve for A0. . . ANby imposing the condition that the Nth-order Taylor
expansion of fAabout x0is equal to fS. If g(x) = x, this can be accom-
plished by setting fS=fA, isolating the approximant series within fAto
one side and then Taylor expanding the other side using the relations in
Appendix A. If g(x)6=x, a matrix inversion may be required. In either
case, the resultant expressions for A0. . . ANwill be functions of the m+p
unknowns from steps i and ii.
v. Solve the algebraic system AN= 0, AN1= 0, . . . ANmp+1 = 0 for
the m+punknowns from steps i and ii. Choose the most rapidly con-
verging and physically plausible roots (for example, the physical quantity
the root represents may be real and positive). In this step, unknown co-
efficients in the approximant series are sacrificed in order to solve for
other unknowns embedded in these coefficients, while preserving the cor-
rect number of degrees of freedom.
vi. Evaluate all Ancoefficients at the values obtained in step v and substitute
these into the approximant fAfrom step iii.
vii. Ensure that fAconverges as Nis increased. As an additional metric
of the approximant’s ability to approach the correct asymptotic behav-
ior, one may plot fA/faand confirm that, for increasing N, this ratio
converges to a constant as xxa.
viii. Consistency Checks: Expand fAabout x0to Nth order; if the steps were
executed correctly this will match the first Nterms of fS. Take the limit
of fAas xxaand confirm that it agrees with fato zeroth order.
Finally, if available, verify that the approximant agrees with a full nu-
merical solution of the problem.
4 Application to nonlinear boundary value
problems
Asympotic approximants are now applied to three nonlinear boundary value
problems. In Section 4.1, we find approximant solutions to the Sakiadis
12
boundary layer problem, where we are able to take advantage of the full
asymptotic behavior, enabling us to obtain new benchmark values. In Sec-
tion 4.2, an approximant is constructed for the Blasius boundary layer prob-
lem using the leading order asymptotic behavior; this approximant confirms
existing numerically-obtained benchmark values. In Section 4.3, an approx-
imant for the Flierl-Petviashvili monopole problem is constructed using the
leading order asymptotic behavior. In this application, newly obtained nu-
merical results are confirmed and a new asymptotic constant is explored.
4.1 The Sakiadis problem
The Sakiadis problem describes steady-state developing flow created by a
moving plate in an otherwise stagnant fluid [33]. It shares the same bound-
ary layer governing equation as that of the Blasius problem (discussed in
Section 4.2), but with different boundary conditions. As shown in [33], a sim-
ilarity transform of these equations leads to the following nonlinear boundary
value problem in f(η):
2f000 +ff00 = 0 (9a)
f(0) = 0, f0(0) = 1, f 0() = 0.(9b)
As shown by Blasius [34], substituting the solution form
f=
X
n=0
anηn(10a)
into (9a) leads to the following recursion for the series coefficients
an+3 =
n
P
j=0
(j+ 1)(j+ 2) aj+2 anj
2(n+ 1)(n+ 2)(n+ 3) .(10b)
The recursion above requires knowledge of the first three coefficients in order
to generate the full series. The first two coefficients a0=f(0) and a1=f0(0)
are given by the first two boundary conditions in (9b). The third coefficient
is prescribed by the “wall shear” parameter
f00(0) κ,
and is typically obtained from a numerical solution of (9) (see [35, 36, 37]
for example). Recently, a semi-analytical fixed point method was used to
13
estimate κ; see [38] for details and a comprehensive review and comparison
of previous numerical predictions. Here, asymptotic approximants will be
used to predict κ.
From the infinite condition in (9b), it follows that the solution limits to
a constant Cat large η:
lim
η→∞ fC. (11)
Higher order asymptotic methods provided in Appendix B.1 enable the iden-
tification of an additional asymptotic constant, G, as:
lim
η→∞ (fC)eηC/2G. (12)
The constants G,C, and κwill be predicted using asymptotic approximants
in the following subsections.
4.1.1 Simple asymptotic approximant
First, we consider a simple approximant of the form
fA=CC 1 +
N
X
n=1
Anηn!1
,(13a)
which automatically satisfies both (9b) and (11). The assumed form (13a) has
the appearance of a Pad´e, but in fact it is not. Since Cis an unknown to be
predicted by the approximant, the above cannot be considered a Pad´e for the
quantity fC. Also, if the terms in (13a) were combined through a common
denominator, a specific relationship between the numerator and denominator
coefficients would be obtained. Thus (13a) cannot be considered a Pad´e for f.
We shall see below that the approximant given by (13a) contains coefficients
which are, in fact, easier to compute than those of a standard Pad´e.
The coefficients A0. . . AN,C, and κare calculated such that the N-
term Taylor expansion of (13a) about η= 0 is exactly equal to the N-term
truncation of (10). Equating (13a) with (10) and re-arranging leads to the
following
"11
C
N
X
n=0
anηn#1
= 1 +
N
X
n=1
Anηn.
Expanding the left-hand side above (utilizing equation (28) in Appendix A)
provides the following recursion for the coefficients, which are now functions
14
of κand C:
An>0=1
C
n
X
j=1
ajAnj, A0= 1.(13b)
We now sacrifice the coefficients ANand AN1to simultaneously predict κ
and C. Setting AN=AN1= 0 in (13b), we arrive at
N
X
j=1
ajANj= 0,
N1
X
j=1
ajAN1j= 0.(13c)
Once the preceding coefficients are obtained from (13b) and substituted into
the above, (13c) becomes a system of nonlinear equations in κand C, and
may be solved algebraically. The most rapidly converging κand Croots
of (13c) are given in Table 1 up to an optimal asymptotic truncation of
N= 11. That is, for N > 11, |κN+1 κN|no longer consistently decreases
as Nis increased.
We also record the value Sas the magnitude of the singularity, ηs, closest
to η=0 in the complex ηplane for the Sakiadis problem, as predicted by (13)
and listed in Table (1). This value limits the radius of convergence of the
series (10).
Lastly, Cand κare substituted into (13b) for all non-zero coefficients
(A1through AN2) to construct the approximant series in (13a). The ap-
proximant (13) is compared with the series and numerical solution of (9)
in Fig. 3. Although we are using a simple approximant form that enables
correspondingly straightforward coefficient generation, the approximant con-
verges surprisingly well, as can be seen in Fig. 3a. In Fig. 3b, we confirm
that f0and f00 are accurately obtained from the approximant, once fhas
reasonably converged (here, for N= 11). Unfortunately, our estimates for κ
and Ccease to converge for N > 11, which limits our ability to predict the
asymptotic constant Ggiven by (12). This can been seen in a plot of the
effective G
Geff = (fC)eηC/2, Geff Gas η→ ∞ (14)
versus η, which, when computed by the approximant (13), only converges
in a small region near η= 0 as shown in Fig. 3b. The reason for this poor
convergence behavior is that the approximant form (13) does not incorporate
the asymptotic correction behavior that leads to (12), and thus Geff cannot
converge uniformly.
15
Table 1: Predictions from approximants for the Sakiadis problem. For com-
parison, the numerical values of κand Cin the bottom row were computed
using the Chebfun [39] package to solve (9) via rectangular collocation with
η=replaced with a finite surrogate ηgiven in the table. Note that
the recent κestimates given in [40, 38] (reported to 7 digits) agree with the
first 7 digits of our numerical predictions and predictions from (15). The nu-
merical values of Swere obtained by inserting the numerical κand Cvalues
into (13), locating the singularity of smallest magnitude, and systematically
increasing the approximant order (up to N= 200), where the value of Shas
converged to within the reported digits.
Nκ
from (13)
C
from (13)
S
from (13)
κ
from (15)
C
from (15)
G
from (15)
5 -0.3879 2.1607 3.76 -0.464241808775 1.57791591603 -1.9379232426
7 -0.4341 1.7275 3.98 -0.445957694266 1.61001774844 -2.0928590899
9 -0.4421 1.6418 4.05 -0.443970941603 1.61533742596 -2.1255327825
11 -0.4430 1.6284 4.07 -0.443769095918 1.61603802403 -2.1306162759
13 -0.443750163304 1.61611659263 -2.1312640887
15 -0.443748473247 1.61612459984 -2.1313373822
17 -0.443748326909 1.61612536888 -2.1313450758
19 -0.443748314499 1.61612543984 -2.1313458428
21 -0.443748313462 1.61612544619 -2.1313459165
23 -0.443748313376 1.61612544675 -2.1313459234
25 -0.443748313368 1.61612544681 -2.1313459241
26 -0.443748313384 1.61612544669 -2.1313459226
27 -0.443748313370 1.61612544680 -2.1313459239
28 -0.443748313369 1.61612544680 -2.1313459240
29 -0.443748313369 1.61612544681 -2.1313459241
30 -0.443748313369 1.61612544681 -2.1313459241
numerical κ C S
from (13) η
shooting [35] -0.44374733 1.61611200 4.07217 20
quadrature [36] -0.44374831 1.61612518 4.07217 23.20512
collocation -0.44374831 1.61612545 4.07217 30
16
(a)
η
0 5 10 15 20
f
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8 S7
S9
S11 A7
A9
A11
(b)
η
0 5 10 15 20
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
A9 A11
f!
A11
f!!
A11
A7
Ge.
Figure 3: (a) Comparison between the N-term series solution (10) (−−)
labeled as SN, the corresponding approximant (13) () labeled as AN, and
the numerical solution [35] () to the Sakiadis problem (9). (b) Derivatives
of approximant (13) with N=11 and Geff given by (14) compared with the
numerical solution [35] ().
4.1.2 Asymptotic approximant with exponential corrections
We now consider an approximant,
fA=C+
N
X
n=1
AnenCη/2,(15a)
that not only matches the zeroth order infinite behavior, but also the form
of the exponential corrections from the asymptotic expansion of fas η→ ∞
provided in Appendix B.1. Note that A1in the series above is the unknown
asymptotic constant Gdefined in (12). After expanding (15a) about η= 0
and matching like terms with those of the series solution (10), we arrive at
a system of Nlinear equations,
102030··· N0
112131··· N1
122232··· N2
.
.
..
.
..
.
..
.
..
.
.
1N12N13N1··· NN1
A1
A2
A3
.
.
.
AN
=
0! a0C
1! (2/C)a1
2! (2/C)2a2
.
.
.
(N1)! (2/C)N1aN1
,
(15b)
17
where the coefficient matrix is a Vandermonde matrix with a known explicit
formula for its inverse [41]. Once the system (15b) is solved, the Anco-
efficients become functions of the unknowns Cand κ. Like the previous
approximant, we sacrifice the last two coefficients and solve the nonlinear
equations
AN(κ, C) = 0, AN1(κ, C) = 0 (15c)
to obtain values for Cand κ, provided in the 5th and 6th columns of Table 1.
In the table, we report values up to N= 30, after which the 13th digit
oscillates without converging to a discernible limit point. The converged
values
κ=0.443748313369
and
C= 1.61612544681
obtained from approximant (15) agree with numerical predictions, and pro-
vide 5 additional digits of precision. Also, for the first time, a value of the
asymptotic constant
G=2.1313459241
is reported; convergence to this value is shown in the rightmost column of
Table 1, obtained by substituting the predictions for Cand κinto the ex-
pression for A1. Once Cand κare substituted into the expressions (arising
from the solution to (15b)) for all non-zero coefficients (A1through AN2),
the approximant series in (15a) may be constructed.
The approximant (15) is compared with the series and numerical solu-
tion of (9) in Fig. 4. The new approximant converges more rapidly than the
simpler approximant (see Fig. 3 for comparison). Also, the new approximant
captures the exponential corrections inasmuch as Geff converges into the large
ηasymptotic regime and approaches G, as shown in Fig. 4b. This behav-
ior is expected, as the correct asymptotic behavior has been incorportated
explicitly into the approximant and convergence should be uniform for large
η.
For the Sakiadis problem, a relatively simple approximant that meets the
infinite condition in (9b), such as approximant (13), is adequate for describing
fand its derivatives. However, in comparing the Geff curves of Figs. 3b and
4b, it is clear that an approximant such as (15) which includes asymptotic
corrections, is needed to predict G. As we shall see in the next section, simple
approximants can sometimes also be used to predict higher-order asymptotic
18
(a)
η
0 5 10 15 20
f
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8 S7
S9
S11 A7, 9, 11
(b)
η
0 5 10 15 20
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
f!!
A7
f!
A7
A9, 11
A7
Ge.
Figure 4: (a) Comparison between the N-term series solution (10) (−−)
labeled as SN, the corresponding approximant (15) () labeled as AN, and
the numerical solution [35] () to the Sakiadis problem (9). (b) Derivatives
of approximant (15) with N=7 and Geff given by (14) compared with the
numerical solution [35] ().
properties, provided that the approximant converges at large enough ηfor
the asymptotic behavior to be reached.
Lastly, it is interesting to note that approximant (15) contains no singu-
larities, and thus cannot be used to predict the closest singularity ηs(and
S) directly, as was done in Section 4.1.1 for approximant (13). However,
if we use either the new high precision values of κand Cfrom approxi-
mant (15) or the numerical values listed in the table as inputs to approxi-
matnt (13), the location of the singularity converges to the within the indi-
cated digits to ηs=1.211393 + 3.88781i and thus a radius of convergence
of S≡ |ηs|= 4.07217.
Although the implementation is different, the approximant (15) is similar
in style to that of the approximant of the Kidder boundary value problem
given recently in [42], where an exponential decay of the derivative at infinity
is incorporated by changing the independent variable of a Pad´e to reflect
this behavior. If convergence of this approximant for increasing order were
considered, this approach would fall within the framework of the asymptotic
approximants presented here.
The errors in the approximants of this section are given in Table 2 to en-
able comparison with other approximate treatments of the Sakiadis problem;
see for example [38].
19
Table 2: The infinity norm of the error in the Sakiadis approximants and
their 2nd derivatives (to within 1 significant digit), defined respectively as
E=||fAfnumerical||and E2=||f00
Af00
numerical||on the interval 0 η20.
NE
from (13)
E2
from (13)
E
from (15)
E2
from (15)
5 5×1016×1024×1022×102
7 1×1011×1026×1032×103
9 3×1021×1038×1042×104
11 1×1027×1049×1052×105
13 9×1062×106
15 8×1072×107
20 2×1093×1010
25 1×1011 5×1012
30 1×1011 4×1013
4.2 The Blasius problem
The Blasius problem is the archetypal boundary-layer problem found in most
undergraduate fluid mechanics books, describing the boundary layer due to
a moving fluid over a stationary flat plate [34]. The differential equation
in f(η) is the same as the Sakiadis problem (and arises after a similarity
transform is applied to the governing equations),
2f000 +ff00 = 0,(16a)
while the boundary conditions are now
f(0) = 0, f0(0) = 0, f 0() = 1.(16b)
The solution to (16) may be constructed as a power series (10) as in the
Sakiadis problem. However, one difference is that the Blasius series solution
skips every two terms, starting with the first two, which are zero as imposed
by the boundary conditions at η=0 in (16b). Again, the wall shear parameter
is defined as
f00(0) κ
which, along with the asymptotic properties [43]
lim
η→∞(fη)B(17a)
20
lim
η→∞ exp η2
4+
2f00Q, (17b)
will here be predicted by the asymptotic approximant, in a similar fashion as
done by [19] (for κusing Pad´es). Recent numerical and analytical predictions
of these quantities, including a review of previous results may be found in [44,
38, 45, 46].
The asymptotic behavior of the Blasius problem as η→ ∞ is given as
(see Appendix B.2)
fη+B+ 4Qexp[η2/4/2]
(η+B)2[1 + O(1
(η+B)2)],as η→ ∞.(18)
For comparison, the exponential correction analogous to (18) for the Sakiadis
problem is (36) in Appendix B.1. While we were able to incorporate (12)
directly in the Sakiadis approximant (15), this is not an option for the Blasius
problem since the asymptotic form (18) contains a singularity at η=Band
it is known that B < 0 [43]. To clarify this issue, note that (18) is fully valid
as η→ ∞. However, our methodology for constructing an approximant relies
on the unification of asymptotic limits such that the final form is capable of
describing all η > 0, and here is where (18) poses an issue in its ability to
inform such an approach.
While we cannot directly use the asymptotic expansion for the Blasius
problem, we can use a simple approximant that agrees asymptotically to 1st
order (i.e. with (16b) and (17a)) and then verify that it approaches the
exponential correction (17b) as η→ ∞. We use an approximant of the form
fA=η+BB 1 +
N
X
n=1
Anηn!1
,(19a)
which automatically satisfies both (16b) and (17a). As similarly discussed in
the context of the Sakiadis Approximant (13a), the above form is not a Pad´e
approximant. Firstly, Bis an unknown parameter in the approximant, and
so (19a) cannot be considered a Pad´e for fηB. Secondly, if one combines
the terms of (19a) through a common denominator, the coefficients of the
numerator will have an explicit dependence on those in the denominator,
whereas this is not the case for standard Pad´es.
The coefficients A0. . . AN,B, and κare now calculated such that the N-
term Taylor expansion of (19a) about η= 0 is exactly equal to the N-term
21
truncation of (10). Following the same procedure as in Section 4.1.1, we
arrive at the following recursion for the coefficients, which are now functions
of κand B:
An>0=1
B
n
X
j=1
˜ajAnj, A0= 1,(19b)
where ˜a1=1 and ˜aj>1=aj. The coefficients ANand AN1are now
sacrificed to simultaneously predict κand B. Setting
AN(κ, B) = 0, AN1(κ, B) = 0 (19c)
in (19b) leads to two nonlinear equations, whose most rapidly converging κ
and Broots are given in Table 3, along with the magnitude of the singularity
closest to η= 0 in the approximant, denoted by S. The table reports values
up to N= 50, beyond which convergence cannot be established in the 8th
digit. Our predictions converge to within 7 digits of the benchmark value
for κ, 6 digits for B, and 2 digits for the radius of convergence S. Again, we
determine Sas the magnitude of the closest singularity, ηs, from the origin in
the complex ηplane. Although not shown in the table, the approximant (19)
is consistent with the literature [43] in that, as Nis increased, it predicts ηs
to lie on the negative real axis.
Once Band κare substituted into (19b) for all non-zero coefficients (A1
through AN2), the approximant series in (19) may be constructed. The
approximant (19) is compared with the series and numerical solution of (16)
in Fig. 5. Like the simple Sakiadis approximant given in Section (4.1.1),
this basic Blasius approximant converges surprisingly well, as can be seen
in Fig. 5a. In Fig. 5b, we confirm that f0and f00 are accurately obtained
from the approximant, once fhas reasonably converged (here, for N= 30).
Unlike the simple Sakiadis approximant, the Blasius approximant is capable
of picking up a higher-order asymptotic quantity - namely Qdefined in (17b).
This is seen in Fig. 5c, where an effective Q, defined as
Qeff = exp η2
4+
2f00, Qeff Qas η→ ∞ (20)
is plotted versus ηfor various series and approximant order. While the ap-
proximant converges directly to a constant Qvalue at large η, it is striking
that the series solution also reaches the Qplateau prior to diverging. This
may be a result of the Blasius series having a region of convergence that over-
laps with asymptotic effects felt at relatively small η. As seen in Fig. 5c, both
22
Table 3: Predictions from the Blasius approximant (19). The infinity norm
of the error in the approximant and its 2nd derivative are recorded below (to
within 1 significant digit), defined respectively as E=||fAfnumerical||and
E2=||f00
Af00
numerical||on the interval 0 η8.8.
N κ B S E E2
5 0.12545065 -6.08950239 9.85302 3×1002×101
10 0.30018461 -2.00003122 4.47053 3×1013×102
15 0.32626056 -1.77168204 4.59756 5×1025×103
20 0.33090243 -1.73173129 4.92273 1×1021×103
25 0.33181978 -1.72323142 5.18569 2×1032×104
30 0.33200793 -1.72133641 5.43362 5×1045×105
35 0.33204717 -1.72090879 5.64460 1×1041×105
40 0.33205518 -1.72081494 5.69196 3×1052×106
45 0.33205693 -1.72079309 5.68929 5×1064×107
50 0.33205731 -1.72078801 5.68933 3×1073×108
numerical [45, 47] 0.332057336215196 -1.7207876575205 5.6900380545 0 0
the approximant and the numerical solution converge to Q0.1115, which
is consistent with the value Q=0.111483755 obtained numerically by [45] and
semi-analytically (i.e. dependent on numerical κ) in [46]1; in both references,
the value is reported well-beyond these digits, but both round to the value
above. Since the behavior (20) is not explicitly incorporated into the approx-
imant (19), the convergence of Qeff is not uniform at large η. For example,
in Fig. 5c, the behavior of the N= 20,30,40 approximants (labeled A20,
A30, A40) demonstrates the non-uniform convergence that will occur for any
order Nat some value of η.
The errors in the approximant are given in the last two columns of Ta-
ble 3 to enable comparison with other approximate treatments of the Blasius
problem [19, 43, 44, 38, 46].
The virtue of approximant (19) is that it is globally accurate from small
to large η, is able to predict κand B(i.e. does not rely on these values as
inputs), and has coefficients that are generated from a simple recursion. A
recent alternative globally accurate approximation to the Blasius solution is
provided in [46]. An attractive feature of this alternative approximation is
that it has a simpler form than the asymptotic approximant (19). However,
1Qin [46] is defined as eB2/4multiplied by the Qdefined here.
23
(a)
η
012345678
f
0
1
2
3
4
5
6
S30
A10
A20, 30, 40, 50
S20, 40, 50 S10
(b)
η
012345678
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f
A30
f′′
A30
(c)
η
012345678
Qe.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
S10
S20 S40
S50
S30
A10 A20
A50
A40
A30
Figure 5: (a) Comparison between the N-term series solution (10) (−−)
labeled as SNand the corresponding approximant (19) () labeled as AN
to the Blasius problem (16). (b) Derivatives of approximant (19) with N=30.
(c) Qeff given by (20) as predicted by the series (10) and approximant (19).
For comparison, the numerical solution from [45] is shown in all above figures
as s.
the form in [46] requires that κis known beforehand. Nevertheless, one may
directly compare the error in f00
Agiven in Table 3 with that given in figure 2
of [46].
4.3 The Flierl-Petviashvili problem
The Flierl-Petviashvili (FP) equation in similarity variables u(r) is used to
describe vortex solitons in the ocean, atmosphere [48], and Jupiter’s red
24
spot [49], and is given by
u00 +1
ru0uu2= 0 (21a)
u0(0) = 0, u()=0.(21b)
The coefficients of the power series solution
u=
X
n=0
anrn(22a)
to (21) are given by
an+2 =
an+
n
P
k=0
akank
(n+ 2)2,(22b)
which requires the specification of the first two coefficients to generate the
remaining even coefficients; all remaining odd terms are zero. The coefficient
a1=0 is known from the boundary conditions (21b) and a0is an unknown
parameter,
a0z
which, along with the asymptotic property
lim
r→∞(uerr)D(23)
will be predicted by the approximant, in a similar fashion as done by [19, 50]
(for zusing Pad´es). The property (23) above follows from an asymptotic
expansion of the FP problem as r→ ∞ (derived in Appendix B.3):
uDer
r[1 + O(1
r)],as r→ ∞.(24)
As was the case for the Blasius problem, a singularity in the asymptotic form
(here at r=0 in the above) prevents direct incorporation of (24) to construct
an approximant. However, like the Blasius approximant of Section 4.2, we
shall again use a simple approximant that agrees asymptotically to zeroth
order (i.e. with the 2nd condition of (21b)) and then verify that it approaches
the correction (24) as r→ ∞.
25
We consider a simple approximant of the form
uA=z
1 +
N
P
n=1
Anrn
,(25a)
which automatically satisfies both boundary conditions of (21), including
the condition u() = 0 not explicitly captured by (22). Note that (25a) is a
Pad´e. The simple structure of this Pad´e, however, allows for the unknowns
to be calculated without inverting a matrix. The coefficients A0. . . ANand z
in (25a) are now calculated such that the N-term Taylor expansion of (25a)
about r= 0 is exactly equal to the N-term truncation of (22). Following the
same procedure as in Section 4.1.1, we arrive at the following recursion for
the coefficients, which are now functions of z:
An>0=1
z
n
X
j=1
ajAnj, A0= 1.(25b)
The coefficients above mimic the series (22) in that it skips odd coefficients.
The unknown zin (25b) is determined by sacrificing the degree of freedom
normally used to compute AN. This is equivalent to setting AN= 0 in (25b),
leading to
N
X
j=1
ajANj= 0.(25c)
Note that since the ajcoefficients are polynomials in z, (25c) is also a polyno-
mial in z. The most rapidly converging z-roots of (25c) (for increasing N) are
given in Table 4. Once zis substituted into (25b) for all non-zero coefficients
(A1through AN1), the approximant series in (25a) may be constructed.
The magnitude of the singularity closest to r= 0 in the approximant (25) is
also given in Table 4, denoted by S. Although not shown in the table, the
approximant (25a) predicts that, within the reported digits, the singularity
appears to lie on the positive imaginary axis.
The approximant (25) is compared with the series and numerical solution
of (21) in Fig. 6. As for the simple Sakiadis approximant given in Sec-
tion 4.1.1 and the Blasius approximant of Section 4.2, the FP approximant
also converges uniformly, as can be seen in Fig. 6a. In Fig. 6b, we confirm
that u0is accurately obtained from the approximant once uhas reasonably
26
Table 4: Predictions from the Flierl-Petviashvili approximant (25a). The
numerical method for computing the values in the bottom row is described
in Appendix C. The infinity norm of the error in the approximant is also
recorded below (to within 1 significant digit), defined as E=||uAunumerical||
on the interval 0 r10.
N z S E
4 -1.5 2.82843 9×101
6 -2.14039 2.57284 3×101
8 -2.34792 2.47227 4×102
10 -2.38564 2.60878 6×103
12 -2.39117 2.59927 7×104
14 -2.39185 2.61160 2×104
16 -2.39196 2.61154 1×105
numerical -2.3919564032 2.611541077 0
converged (here, for N= 12). Like the Blasius approximant, the FP approx-
imant is capable of picking up a higher-order asymptotic quantity - here, D
defined in (23). This is seen in Fig. 6b, where an effective D, defined as
Deff =uerr, Deff Das r→ ∞ (26)
is plotted versus rfor various approximant order. As Nis increased, the
approximant Deff appears to be converging to the numerical solution, which
is in turn converging to a value of
D≈ −10.7
which may serve as a useful metric for future analytical and numerical solu-
tions of the FP problem. Again, Fig. 6 shows that convergence in Deff is not
uniform at large r, as expected by the assumed form of the approximant.
The errors in the approximant are given in the rightmost column of Ta-
ble 4 to enable comparison with other approximate treatments of the FP
equation; see for example [19].
5 Summary
In this work, we formalize a new approach to sum series. We provide both a
methodology and examples that demonstrate how asymptotic approximants
27
(a)
r
012345678910
u
-2.5
-2
-1.5
-1
-0.5
0
A6
A6
A8 S8,12,16
A8
A10,12,14,16
S6,10,14
A10,12,14,16
(b)
r
012345678910
-12
-10
-8
-6
-4
-2
0
u
A12
De.
A16
A12 A14
A10
A6 A8
Figure 6: (a) Comparison between the N-term series solution (22) (−−)
labeled as SNand the corresponding approximant (25) () labeled as AN
to the Flierl-Petviashvilli problem (21). (b) Derivatives of approximant (25)
with N=12 and Deff given by (26). For comparison, the numerical solution
(described in Appendix C) is shown in all above figures as s.
may be constructed. The key feature is that such approximants are designed
so that the asymptotic behaviors in two regions of the domain may be joined.
This approach has seen recent success in the analytic continuation of “virial”
series that describe the pressure-density-temperature dependence of various
model fluids, where no differential equation is available and usually only the
first 3 to 12 terms of these (often divergent) power series are known [16, 17,
18].
Here, we provide an additional application for asymptotically consistent
approximants - namely, a remedy for divergent power series that arise as
solutions to nonlinear ordinary differential equations. The analytic forms
provided here enable symbolic differentiation and thus allow analytical eval-
uation of the flow field (shear stress, vorticity, etc.) to any desired resolution
at low computational expense. Additionally, the approximant coefficients are
relatively straightforward to obtain, as they often only require the implemen-
tation of a few series identities, enabling them to be generated typically via
a simple recurrence relation.
Note that improvements to the forms given here are easily made, de-
pending on how many parameters are to be embedded in (or predicted by)
the approximant. Here, we use the approximant to not only find a solution,
28
but also provide the as-of-yet most accurate and precise values of the wall
shear, asymptotic constants, and singularity of smallest magnitude for the
Sakiadis boundary layer. If better estimates become available from numer-
ical (or other) techniques, these may be used as inputs to the approximant
to improve global accuracy. For the Blasius problem and Flierl-Petviashvilli
problem, more accurate values for some of these properties are, in fact, avail-
able and could have been used as inputs to the approximants; we chose
instead to predict these quantities to demonstrate that the method is self
contained and does not rely on external data.
The results presented here motivate the development and application of
asymptotic approximants to other problems where divergent, truncated, and
underspecified series arise. Considering the capabilities afforded and the ease
with which asymptotic approximants are generated, they deserve considera-
tion as an alternative to the more traditional Pad´e approximants in problems
of mathematical physics.
References
[1] E. C. Titchmarsh. The Theory of Functions. Oxford University Press,
2 edition, 1968.
[2] A. J. Masters. Virial expansions. J. Phys.: Condens. Matter,
20(283102), 2008.
[3] C. M. Bender and S. A. Orszag. Advanced Mathematical Methods for
Scientists and Engineers I: Asymptotic Methods and Perturbation The-
ory. McGraw-Hill, 1978.
[4] G. A Baker Jr. Quantative Theory of Critical Phenomena. Academic
Press, London, 1990.
[5] J. S. R. Chisholm. Generalisations of Pad´e approximants. Circuits Syst.
Signal Process, 1:279, 1982.
[6] A. J. Guttmann. Asymptotic analysis of power series expansions. In
Phase Transitions and Critical Phenomenon Vol. 13, pages 1–234. Aca-
demic Press, NY, 1989.
[7] A. J. Guttmann and I. Jensen. Series analysis. In Polygons, Polyomi-
noes, and Polycubes, pages 181–202. Springer, 2009.
29
[8] G. A. Baker Jr. and P. Graves-Morris. Pad´e approximants. Cambridge
University Press, 2 edition, 1996.
[9] I. V. Andrianov and J. Awrejcewicz. New trends in asymptotic ap-
proaches: summation and interpolation methods. Appl Mech. Rev.,
54(1), 2001.
[10] P. A. Frost and E. Y. Harper. An extended pad´e procedure for construct-
ing global approximations from asymptotic expansions: an explication
with examples. SIAM Rev., 18(1):62–91, 1976.
[11] G. A Baker Jr. and J. L. Gammel. The Pad´e approximant. J. Math.
Anal. Appl., 2:21–30, 1961.
[12] M. Van Dyke. Analysis of improvement of perturbation series. Q. J.
Mech. Appl. Math., 27(4):423–450, 1974.
[13] N. Clisby and B. M. McCoy. Ninth and tenth order virial coefficients
for hard spheres in ddimensions. J. Stat. Phys., 122(1):15–55, 2006.
[14] A. Guerrero and A. Bassi. On Pad´e approximants to virial series. J.
Chem. Phys., 129(044509), 2008.
[15] T. B. Tan, A. J. Schultz, and D. A. Kofke. Virial coefficients, equation of
state, and solid–fluid coexistence for the soft sphere model. Mol. Phys.,
109(1):123–132, 2011.
[16] N. S. Barlow, A. J. Schultz, S. J. Weinstein, and D. A. Kofke. An
asymptotically consistent approximant method with application to soft-
and hard-sphere fluids. J. Chem. Phys., 137:204102, 2012.
[17] N. S. Barlow, A. J. Schultz, S. J. Weinstein, and D. A. Kofke. Criti-
cal isotherms from virial series using asymptotically consistent approx-
imants. AIChE J., 60(9):3336–3349, 2014.
[18] N. S. Barlow, A. J. Schultz, S. J. Weinstein, and D. A. Kofke. Commu-
nication: Analytic continutation of the virial series through the critical
point using parametric approximants. J. Chem. Phys., 143:071103:1–5,
2015.
30
[19] J. P. Boyd. Pad´e approximant algorithm for solving nonlinear ordinary
differential equation boundary value problems on an unbounded domain.
Comp. Phys., 11:299–303, 1997.
[20] E. A. Mason and T. H. Spurling. The Virial Equation of State. Pergam-
mon Press, Oxford, 1969.
[21] D. A. McQuarrie. Statistical Mechanics. University Science Books, 2000.
[22] J. K. Singh and D. A. Kofke. Mayer sampling: Calculation of clus-
ter integrals using free-energy perturbation methods. Phys. Rev. Lett.,
92:220601, 2004.
[23] Richard J Wheatley. Calculation of High-Order Virial Coefficients
with Applications to Hard and Soft Spheres. Physical Review Letters,
110(20):200601, May 2013.
[24] N. F. Carnahan and K. E. Starling. Equation of state for nonattracting
rigid spheres. J. Chem. Phys., 51(2):635–636, 1969.
[25] J.-P. Hansen and I. R. McDonald. Theory of Simple Liquids. Academic
Press, London, 3rd edition, 2006.
[26] P. Henrici. Automatic computations with power series. JACM, 3:10–15,
1956.
[27] F. J. Rogers and D. A. Young. New, thermodynamically consistent,
integral equation for simple fluids. Phys. Rev. A, 30(2):999–1007, 1984.
[28] G. A Baker Jr. Application of the Pad´e approximant method to the
investigation of some magnetic properties of the Ising model. Phys.
Rev., 124(3):768–774, 1961.
[29] C. J. Thompson. Mathematical Statistical Mechanics, chapter 6.5: Nu-
merical Analysis of the Three-dimensional Ising Model. Princeton Uni-
versity Press, New Jersey, 1972.
[30] M. E. Fisher. Notes, definitions, and formulas for critical point singu-
larities. In M. S. Green and J. V. Sengers, editors, Critical phenomena,
Proceedings of a conference held in Washington, pages 21–26, April 1965.
31
[31] H Behnejad, J. V. Sengers, and M. A. Anisimov. Thermodynamic be-
haviour of fluids near critical points. In Applied Thermodynamics of
Fluids, pages 321–367. Royal Society of Chemistry, 2010.
[32] J. P´erez-Pellitero, P. Ungerer, G. Orkoulas, and A. Mackie. Critical
point estimation of the Lennard-Jones pure fluid and binary mixtures.
J. Chem. Phys., 125:054515, 2006.
[33] B. C. Sakiadis. Boundary-layer behavior on continuous solid surfaces:
II the boundary layer on a continuous flat surface. AlChE J., 7:221–225,
1961.
[34] H. Blasius. Grenzschichten in flussigkeiten mit kleiner reibung.
Zeitschrift fur Mathematik und Physik, 56:1–37, 1908.
[35] R. Cortell. Numerical comparisons of Blasius and Sakiadis flows.
MATEMATIKA, 26(2):187–196, 2010.
[36] S. A. Eftekhari and A. A. Jafari. Numerical solution of general boundary
layer problems by the method of differential quadrature. Sci. Iran. B,
20(4):1278–1301, 2013.
[37] R. Fazio. The iterative transformation method for the Sakiadis problem.
Comp. Fluids, 106:196–200, 2015.
[38] D Xu and X. Guo. Application of fixed point method to obtain semi-
analytical solution to Blasius flow and its variation. Applied Mathematics
and Computation, 224:791–802, 2013.
[39] T. A. Driscoll, N. Hale, and L. N. Trefethen, editors. Chebfun Guide.
Pafnuty Publications, Oxford, 2014.
[40] H. I. Andersson and J. B. Aarseth. Sakiadis flow with variable fluid
properties revisited. I. J. Eng. Sci., 45(2-8):554–561, 2007.
[41] N. Macon and A. Spitzbart. Inverses of Vandermonde matrices. Am.
Math. Monthly, 65(2):95–100, 1958.
[42] R. Iacono and J. P. Boyd. The Kidder equation: uxx +2xux/1αu =
0. Stud. Appl. Math., 135:63–85, 2014.
32
[43] J. P. Boyd. The Blasius function in the complex plane. Exper. Math.,
8(4):381–394, 1999.
[44] B. Yun. Intuitive approach to the approximate analytical solution for
the Blasius problem. Applied Mathematics and Computation, 215:3489–
3494, 2010.
[45] S. Anil Lal and Neeraj Paul M. An accurate taylors series solution with
high radius of convergence for the Blasius function and parameters of
asymptotic variation. JAFM, 7(4):557–564, 2014.
[46] R. Iacono and J. P. Boyd. Simple analytic approximations for the blasius
problem. Physica D, 310:72–78, 2015.
[47] J. P. Boyd. The Blasius function: Computations before computers, the
value of tricks, undergraduate projects, and open research problems.
SIAM Rev., 50(4):791–804, 2008.
[48] G. R. Flierl. Baroclinic solitary waves with radial symmetry. Dyn.
Atmos. Oceans, 3:15–38, 1979.
[49] V. I. Petviashvili. Red spot of jupiter and the drift soliton in a plasma.
JETP Lett., 32:619–622, 1981.
[50] A. M. Wazwaz. The Volterra integro-differential forms of the singular
Flierl-Petviashvili and the Lane-Emden equations with boundary con-
ditions. Rom. J. Phys., 58:685–693, 2013.
[51] R. V. Churchill. Complex Variables. McGraw-Hill, 1948.
[52] M. Abramowitz and I Stegun. Handbook of Mathematical Functions,
page 298. Dover, 1972.
[53] P. R. Turner. Guide to scientific computing. CRC press, 2 edition, 2001.
[54] M. Maleki, I. Hashim, and S. Abbasbandy. Analysis of IVPs and BVPs
on semi-infinite domains via collocation methods. Journal of Applied
Mathematics, 2012, 2012.
[55] C. Domb and M. F. Sykes. On the susceptibility of a ferromagnetic
above the curie point. Proc. Roy. Soc. London A, 240(1221):214–228,
1957.
33
A Useful Series Formulae
The following relations may be used to develop a recursion for the coefficients
of an asymptotic approximant, allowing one to avoid solving an algebraic
system. The first relation is the well-known Cauchy product of two series [51]:
N
X
n=0
anxn
N
X
n=0
Anxn=
N
X
n=0 n
X
j=0
ajAnj!xn.(27)
Setting both sides of (27) equal to one and rearranging, the recursion leads
to a representation for the N-term expansion of the reciprocal of a series:
N
X
n=0
anxn!1
=
N
X
n=0
Anxn,(28a)
where
An>0=1
a0
n
X
j=1
ajAnj, A0=1
a0
.(28b)
The generalization of (28) for the N-term expansion of a series raised to any
real power sis given by J. C. P. Miller’s formula [26]:
N
X
n=0
anxn!s
=
N
X
n=0
Anxn,(29a)
where
An>0=1
n a0
n
X
j=1
(js n+j)ajAnj, A0= (a0)s.(29b)
B Asymptotic Expansions
B.1 Sakiadis Problem
We wish to determine the η→ ∞ asymptotic behavior of the solution f(η)
to the Sakiadis equation (9). For purposes of the analysis, the infinite con-
straint (11) alone suffices. We write:
f=C+h(η),with h0 as η→ ∞ (30)
34
where Cis a constant and h(η) is a function to be determined. The form (30)
is substituted into equation (9a) to obtain
2h000 + (C+h)h00 = 0.(31)
The equation (31) may be simplified by noting that terms quadratic in hare
small relative to other terms as η→ ∞, i.e.:
2h000 +Ch00 = 0,as η→ ∞.(32)
The solution of this equation is
hGe/2,as η→ ∞,(33)
where Gis a constant. Higher order corrections can be obtained by assuming
the form
hGe/2+D(η),as η→ ∞,(34)
and applying the method of dominant balance [3] when substituted into (31),
which leads to
D(η)G2
4Ce ,as η→ ∞.(35)
The asymptotic behavior is written concisely by combining equation (30), (34),
and (35) to yield:
fC+Ge/2+G2
4Ce+O(e3C η/2),as η→ ∞.(36)
The above process for obtaining corrections may be repeated and leads to a
series of exponentials with arguments nCη/2 for n= 1,2,3,4. . . .
B.2 Blasius Problem
We wish to determine the η→ ∞ asymptotic behavior of the solution f(η)
to the Blasius problem (16). For purposes of the analysis, the infinite con-
straint (17a) alone suffices. We write:
f=η+B+g(η),with g0 as η→ ∞,(37)
where Bis an unknown constant and g(η) is a function to be determined.
The form (37) is substituted into (16) to obtain
2g000 + (η+B+g)g00 = 0.(38)
35
The equation (38) may be simplified by noting that terms quadratic in gare
small relative to other terms as η→ ∞, i.e.:
2g000 + (η+B)g00 = 0,as η→ ∞.(39)
The solution of this equation is
g00 Qe[(η2+2)/4] ,as η→ ∞,(40)
where Qis an unknown constant. We can then write:
g0Qe(B2/4) Zη
e[(ξ+B2)/4], as η→ ∞ (41)
and after application of integration by parts, we obtain:
g0∼ −2Qe(B2/4) e[(η+B2)/4]
η+B[1 + O(1
(η+B)2)],as η→ ∞.(42)
The process can then be repeated to find an expression for g, i.e.:
g2Qe(B2/4) Zη
e[(ξ+B2)/4]
ξ+Bdξ, as η→ ∞,(43)
from which integration by parts yields
g4Qe(B2/4) e[(η+B2)/4]
(η+B)2[1 + O(1
(η+B)2)],as η→ ∞,(44)
which can be combined with the original assumed form (37) to obtain the
asymptotic behavior:
fη+B+ 4Qexp[η2/4/2]
(η+B)2[1 + O(1
(η+B)2)],as η→ ∞.(45)
B.3 Flierl-Petviashvili Problem
We wish to determine the r→ ∞ asymptotic behavior of the solution u(r)
to the Flierl-Petviashvili problem (21). For purposes of the analysis, the
infinite constraint in (21b) alone suffices. Since u0 as r→ ∞, it is seen
by inspection of (21) that u2<< u as r→ ∞, leading to the linear equation
r2u00 +ru r2u= 0,as r→ ∞.(46)
36
The solution of (46) is a modified Bessel function of zeroth order [52](p.
374 equation 9.6.1). The leading-order behavior of this function is given by
equation 9.7.2 (p. 378) of [52] as
uDer
r[1 + O(1
r)],as r→ ∞,(47)
where Dis an unknown constant.
C Numerical Solution of the Flierl-Petviashvili
Problem
The numerical solution to the Flierl-Petviashvili equation used in Table 4 is
obtained by first applying the transformation α= 1 erto (21), leading to
(1 α)2d2u
2[(1 α) + 1 + α
ln(1 α)]du
uu2= 0
du
(0) = 0, u(1) = 0.(48)
Note that, like the original FP equation (21), the solution to (48) is not
unique. There is a trivial solution u= 0 and a non-trivial solution - we seek
the latter. The “Shooting Method” [53] (i.e. iterating on u(0) = zguesses)
is used to effectively convert (48) into an initial value problem, which is
solved via 4th order Runge-Kutta. The boundary condition u(1) = 0, which
informs the shooting iteration, is replaced with u(1 ε) = 0, where ε << 1.
Table 5 lists numerical predictions of zfor decreasing ε. For each value
of ε, the numerical αstep size used in the Runge-Kutta implementation
was successively decreased until convergence was established to within the
reported digits of the table.
Using the high precision zvalue of -2.3919564032 in Table 5, the mag-
nitude Sof the closest singularity to the origin in the complex r-plane is
computed by applying the ratio-test to (22b) and making a Domb-Sykes plot
of the ratios versus inverse coefficient order [55], shown in Fig. 7 and taken up
to n= 2000. A linear fit is then made near 1/(n+ 1)=0 with an n=inter-
cept of S= 2.611541077, where the digits have been conservatively truncated
since the residual of the linear fit is of order 1012.
37
Table 5: Predictions of u(0) zof (48), obtained numerically by shooting
to the boundary condition u(1 ε) = 0. The reported digits are within
the convergence tolerance of the shooting iteration. For comparison, the
previous benchmark values reported by [19] and [54] are z=2.3919564
and 2.391956403, respectively.
ε z
101-2.6972451986
102-2.4011647700
103-2.3921055641
104-2.3919584270
105-2.3919564287
106-2.3919564035
107-2.3919564032
108-2.3919564032
0 0.2 0.4 0.6 0.8 1
1/(n+ 1)
2.1
2.2
2.3
2.4
2.5
2.6
|an/an+2|1/2
0 0.001 0.002
2.60421
2.60471
2.60521
2.60571
2.60621
2.60671
2.60721
2.60771
2.60821
2.60871
2.60921
2.60971
2.61021
2.61071
2.61121
2.61158
Figure 7: Domb-Sykes ratio plot of the coefficients in (22b) using a0z=
2.3919564032 from Table 5. The intercept at 1/(n+ 1)=0 is 2.611541077.
38
... While both the Sakiadis and Blasius problems can be handled in similar ways numerically (e.g., shooting, transformation) [4][5][6][7][8], the difference in boundary conditions leads to different (approximate) analytical approaches [9]; and, the nonlinear nature of the equations yields distinctly different solutions. A common measure of the accuracy of any solution technique applied to either problem is the quantity κ, defined as ...
... well-known singularities lying the same distance S from the origin (see [10] and historical review therein) and are reported to lie at values η = S exp [i(2j + 1)π/3] (j=0, 1, 2) where S ≈ 5.6900380545. For the Sakiadis problem, S ≈ 4.07217, arising from singularities lying off the real-line in the left half-plane [9]. Approximate resummations are available, that bypass the original series' convergence barrier caused by these singularities for both the Sakiadis [9] and Blasius [9][10][11][12] problems. ...
... For the Sakiadis problem, S ≈ 4.07217, arising from singularities lying off the real-line in the left half-plane [9]. Approximate resummations are available, that bypass the original series' convergence barrier caused by these singularities for both the Sakiadis [9] and Blasius [9][10][11][12] problems. ...
Preprint
Full-text available
We examine the power series solutions of two classical nonlinear ordinary differential equations of fluid mechanics that are mathematically related by their large-distance asymptotic behaviors in semi-infinite domains. The first problem is that of the "Sakiadis" boundary layer over a moving flat wall, for which no exact analytic solution has been put forward. The second problem is that of a static air-liquid meniscus with surface tension that intersects a flat wall at a given contact angle and limits to a flat pool away from the wall. For the latter problem, the exact analytic solution -- given as distance from the wall as function of meniscus height -- has long been known (Batchelor, 1967). Here, we provide an explicit solution as meniscus height vs. distance from the wall to elucidate structural similarities to the Sakiadis boundary layer. Although power series solutions are readily obtainable to the governing nonlinear ordinary differential equations, we show that -- in both problems -- the series diverge due to non-physical complex or negative real-valued singularities. In both cases, these singularities can be moved by expanding in exponential gauge functions motivated by their respective large distance asymptotic behaviors to enable series convergence over their full semi-infinite domains. For the Sakiadis problem, this not only provides a convergent Taylor series (and conjectured exact) solution to the ODE, but also a means to evaluate the wall shear parameter (and other properties) to within any desired precision. Although the nature of nonlinear ODEs precludes general conclusions, our results indicate that asymptotic behaviors can be useful when proposing variable transformations to overcome power series divergence.
... Approximant techniques, such as the well-known Padé, have been utilized to this end [11]. A relatively new approach, asymptotic approximants [12], has been successfully applied to power series expansions to create highly accurate approximate solutions of nonlinear ODEs arising in many areas of mathematical physics [13][14][15][16][17]. In particular, if one knows the asymptotic behavior in the vicinity of the boundaries of the physical domain, the method of asymptotic approximants may be used to constrain analytic continuation by enforcing these behaviors at both ends, thus providing efficient and accurate analytic infinite series solutions which are in closed form when truncated. ...
... While one goal of this work is to assemble the above-described solution elements from across the centuries in order to give a full treatment of the pendulum series for direct use, another aim here is to add this example to a growing list of problems [12][13][14][15][16][17] in which power series solutions are demonstrated to be a viable solution technique for nonlinear ODEs, even in the more typical cases where singularity locations are unknown. The paper is organized as follows. ...
... after which, equating like powers oft in (12) leads to a recursive expression for the coefficients of the expansion of an exponential of an infinite series, collected with (9) as ...
Preprint
Full-text available
We provide an exact infinite power series solution that describes the trajectory of a nonlinear simple pendulum undergoing librating and rotating motion for all time. Although the series coefficients were previously given in [V. Fair'en, V. L'opez, and L. Conde, Am. J. Phys 56 (1), (1988), pp. 57-61], the series itself -- as well as the optimal location about which an expansion should be chosen to assure series convergence and maximize the domain of convergence -- was not examined, and is provided here. By virtue of its representation as an elliptic function, the pendulum function has singularities that lie off of the real axis in the complex time plane. This, in turn, imposes a radius of convergence on the physical problem in real time. By choosing the expansion point at the top of the trajectory, the power series converges all the way to the bottom of the trajectory without being affected by these singularities. In constructing the series solution, we re-derive the coefficients using an alternative approach that generalizes to other nonlinear problems of mathematical physics. Additionally, we provide an exact resummation of the pendulum series -- Motivated by the asymptotic approximant method given in [Barlow et al., Q. J. Mech. Appl. Math., 70 (1) (2017), pp. 21-48] -- that accelerates the series' convergence uniformly from the top to the bottom of the trajectory. We also provide an accelerated exact resummation of the infinite series representation for the elliptic integral used in calculating the period of a pendulum's trajectory. This allows one to preserve analyticity in the use of the period to extend the pendulum series for all time via symmetry.
... We develop a power series solution of the two-bubble Rayleigh equation that is slowly convergent over the entire physical domain and is thus an exact solution to the problem. We then employ the method of asymptotic approximants [10] to accelerate the convergence. An asymptotic approximant is defined as a closed-form expression whose expansion in one region is exact up to a specified order and whose asymptotic equivalence in another region is enforced [10]. ...
... We then employ the method of asymptotic approximants [10] to accelerate the convergence. An asymptotic approximant is defined as a closed-form expression whose expansion in one region is exact up to a specified order and whose asymptotic equivalence in another region is enforced [10]. Here, we assure that the approximant matches the exact power series solution as T approaches zero as well as the asymptotic behavior as T approaches the time of bubble collapse. ...
... Here, we assure that the approximant matches the exact power series solution as T approaches zero as well as the asymptotic behavior as T approaches the time of bubble collapse. The desirable feature of asymptotic approximants is their ability to attain uniform accuracy not only in these two regions, but also at all points in-between, as demonstrated thus far for problems in thermodynamics [11,12], astrophysics [13], fluid dynamics [10,14], and epidemiology [15,16]. ...
Preprint
Full-text available
The inertial collapse of two interacting and non-translating spherical bubbles of equal size is considered. The exact analytic solution to the nonlinear ordinary differential equation that governs the bubble radii during collapse is first obtained via a slowly converging power series. An asymptotic approximant is then constructed that accelerates convergence of the series and imposes the asymptotic collapse behavior when the radii are small. The solution generalizes the classical 1917 Rayleigh problem of single bubble collapse, as this configuration is recovered when the distance between the bubble centers far exceeds that of their radii.
... Besides providing analytic expressions for the quantities characterizing the solution of the SIR model, we derive a simple, accurate approximant that can be used in practise, and shares all relevant features with the exact solution, as we will show. This is a stark improvement compared with approaches, where the solution of the SIR model was for example expanded into a divergent but asymptotic series, 11,12 or where it had been obtained assuming inequalities that do hold only within a very limited range of SIR parameters, as we will show. ...
... We prefer to treat the inverse basic reproduction number k and ε as variables, so that R(0) and I(0) are determined by this set. Any other choice of two variables from the set k, ε, I(0), and R(0) would work equally well, as the two remaining ones are then given by Eq. (12). ...
... In that case the model cannot be used to calculate fractions at times prior t = 0, as this would under many circumstances lead to S > 1 or I / ∈ [0, 1] at times prior to the 'observation' time t = 0. Because the boundary conditions of case A must not be respected anymore in that case, one can use arbitrary initial conditions incompatible with Eq. (12). The model in this setup has therefore three independent parameters such as k, S(0) and I(0), while the remaining R(0) is given by Eq. (1). ...
Preprint
We revisit the Susceptible-Infectious-Recovered/Removed (SIR) model which is one of the simplest compartmental models. Many epidemological models are derivatives of this basic form. While an analytic solution to the SIR model is known in parametric form for the case of a time-independent infection rate, we derive an analytic solution for the more general case of a time-dependent infection rate, that is not limited to a certain range of parameter values. Our approach allows us to derive several exact analytic results characterizing all quantities, and moreover explicit, non-parametric, and accurate analytic approximants for the solution of the SIR model for time-independent infection rates. We relate all parameters of the SIR model to a measurable, usually reported quantity, namely the cumulated number of infected population and its first and second derivatives at an initial time t=0, where data is assumed to be available. We address the question on how well the differential rate of infections is captured by the Gauss model (GM). To this end we calculate the peak height, width, and position of the bell-shaped rate analytically. We find that the SIR is captured by the GM within a range of times, which we discuss in detail. We prove that the SIR model exhibits an asymptotic behavior at large times that is different from the logistic model, while the difference between the two models still decreases with increasing reproduction factor. This part A of our work treats the original SIR model to hold at all times, while this assumption will be released in part B. Releasing this assumption allows to formulate initial conditions incompatible with the original SIR model.
... We then develop exact solutions for these lengths as infinite series in powers of λ. Since these solutions converge poorly near λ c , we apply the method of asymptotic approximants (Barlow et al., 2017) to describe axial meniscus length uniformly over the whole range 0 ≤ λ < λ c for both configurations. We end with a discussion of implications relevant to the measurement of surface tension using a spinning bubble tensiometer. ...
... (2.41) for a spinning bubble, over the entire respective intervals 0 ≤ λ ≤ λ c . Interested readers may consult Barlow et al. (2017) and the references therein for an extensive presentation of the method applied to a wide range of problems in mathematical physics. Briefly, asymptotic approximants go beyond the well-known Padé approximants in that they incorporate asymptotic behaviors that are often singular in ways other than just poles (Bender & Orszag, 1978), thus dramatically improving the approximant's power to extend the region of convergence. ...
Article
The interface shape of a fluid in rigid body rotation about its axis and partially filling the container is often the subject of a homework problem in the first graduate fluids class. In that problem, surface tension is neglected, the interface shape is parabolic and the contact angle boundary condition is not satisfied in general. When surface tension is accounted for, the shapes exhibit much richer dependencies as a function of rotation velocity. We analyze steady interface shapes in rotating right-circular cylindrical containers under rigid body rotation in zero gravity. We pay special attention to shapes near criticality, in which the interface, or part thereof, becomes straight and parallel to the axis of rotation at certain specific rotational speeds. We examine geometries where the container is axially infinite and derive properties of their solutions. We then examine in detail two special cases of menisci in a cylindrical container: a meniscus spanning the cross-section and a meniscus forming a bubble. In each case, we develop exact solutions for the respective axial lengths as infinite series in powers of appropriate rotation parameters, and we find the respective asymptotic behaviors as the shapes approach their critical configuration. Finally, we apply the method of asymptotic approximants to yield analytical expressions for the axial lengths of the menisci over the whole range of rotation speeds. In this application, the analytical solution is employed to examine errors introduced by the assumption that the interface is a right circular cylinder; this assumption is key to the spinning bubble method used to measure surface tension.
... Since these solutions converge poorly near λ c , we apply the method of asymptotic approximants [7] to describe axial meniscus length uniformly over the whole range 0 ≤ λ < λ c for both configurations. We end with a discussion of implications relevant the measurement of surface tension using a spinning bubble tensiometer. ...
... (41) for a spinning bubble, over the entire respective intervals 0 ≤ λ ≤ λ c . Interested readers may consult Barlow et al. [7] and references therein for an extensive presentation of the method applied to a wide range of problems in mathematical physics. ...
Preprint
Full-text available
We analyze steady interface shapes in zero gravity in rotating right circular cylindrical containers under rigid body rotation. Predictions are made near criticality, in which the interface, or part thereof, becomes straight and parallel to the axis of rotation. We examine geometries where the container is axially infinite and derive properties of their solutions. We then examine in detail two special cases of menisci in a cylindrical container: a meniscus spanning the cross section; and a meniscus forming a bubble. In each case we develop exact solutions for the meniscus height and the bubble length as infinite series in powers of appropriate rotation parameters; and we find the respective asymptotic behaviors as the shapes approach their critical configuration. Finally we apply the method of asymptotic approximants to yield analytical expressions for the height of the meniscus and the length of a spinning bubble over the whole range of rotation speeds. As the spinning bubble method is commonly used to measure surface tension, the latter result has practical relevance.
... Since the infinite series solutions converge very poorly near λ c , we apply the method of asymptotic approximants [7] to describe axial length uniformly over the whole range 0 ≤ λ < λ c for both configurations. ...
... (22) (the height of a rotating meniscus) and (40) (the length of a spinning bubble) over their entire respective intervals 0 ≤ λ ≤ λ c . The method has most recently been revisited and reexamined by Barlow et al. [7], and interested readers may consult their article and references therein for an extensive presentation of the method applied to problems covering a broad range of physics. ...
Preprint
Full-text available
We analyze steady interface shapes in zero gravity in rotating right circular cylindrical containers under rigid body rotation. Predictions are made near criticality, in which the interface, or part thereof, becomes straight and parallel to the axis of rotation. We examine geometries where the container is axially infinite and derive properties of their solutions. We then examine in detail two special cases of menisci in a cylindrical container: a meniscus spanning the cross section; and a meniscus forming a bubble. In each case we develop exact solutions for the meniscus height and the bubble length as infinite series in powers of appropriate rotation parameters; and we find the respective asymptotic behaviors as the shapes approach their critical configuration. Finally we apply the method of asymptotic approximants to yield analytical expressions for the height of the meniscus and the length of a spinning bubble over the whole range of rotation speeds. As the spinning bubble method is commonly used to measure surface tension, the latter result has practical relevance.
... Besides providing analytic expressions for the quantities characterizing the solution of the SIR model, we derive a simple, accurate approximant that can be used in practise, and shares all relevant features with the exact solution, as we will show. This is a significant improvement compared with approaches, where the solution of the SIR model was for example expanded into a divergent but asymptotic series, 15,16 or where it had been obtained assuming inequalities that do hold only within a very limited range of SIR parameters, as we will show. ...
... We emphasize that this special case includes the standard case used by most analysis before that the infection and recovery rates are constants with respect to time. Equation (16) still allows us to take into account an arbitrary time-dependence of the infection rate a(t) which, of course, then is identical to the time dependence of the recovery rate µ(t) due to assumption (16). Then it is covenient to introduce for arbitrary time dependence of the infection rate a(t) the new dimensionless time variable τ with τ (0) = 0 via ...
Article
Full-text available
We revisit the susceptible-infectious-recovered/removed (SIR) model which is one of the simplest compartmental models. Many epidemological models are derivatives of this basic form. While an analytic solution to the SIR model is known in parametric form for the case of a time-independent infection rate, we derive an analytic solution for the more general case of a time-dependent infection rate, that is not limited to a certain range of parameter values. Our approach allows us to derive several exact analytic results characterizing all quantities, and moreover explicit, non-parametric, and accurate analytic approximants for the solution of the SIR model for time-independent infection rates. We relate all parameters of the SIR model to a measurable, usually reported quantity, namely the cumulated number of infected population and its first and second derivatives at an initial time t = 0, where data is assumed to be available. We address the question of how well the differential rate of infections is captured by the Gauss model (GM). To this end we calculate the peak height, width, and position of the bell-shaped rate analytically. We find that the SIR is captured by the GM within a range of times, which we discuss in detail. We prove that the SIR model exhibits an asymptotic behavior at large times that is different from the logistic model, while the difference between the two models still decreases with increasing reproduction factor. This part A of our work treats the original SIR model to hold at all times, while this assumption will be relaxed in part B. Relaxing this assumption allows us to formulate initial conditions incompatible with the original SIR model.
... First of all, we note that other researchers have considered, motivated by the ongoing COVID epidemic, the temporal aspects of the standard SIR dynamics. We mention in particular Cadoni [9] (a related, but quite involved, approach had been considered by Harko, Lobo and Mak [31] ) and Barlow and Weinstein [10] , who obtained an exact solution for the SIR equations in terms of a divergent but asymptotic series [32] ; see also [33,34] for a different approach to exact solution of SIR and SIR-type models. ...
Article
Different countries – and sometimes different regions within the same countries – have adopted different strategies in trying to contain the ongoing COVID-19 epidemic; these mix in variable parts social confinement, early detection and contact tracing. In this paper we discuss the different effects of these ingredients on the epidemic dynamics; the discussion is conducted with the help of two simple models, i.e. the classical SIR model and the recently introduced variant A-SIR (arXiv:2003.08720) which takes into account the presence of a large set of asymptomatic infectives.
Article
This paper is devoted to an overview of the basic properties of the Padé transformation and its generalizations. The merits and limitations of the described approaches are discussed. Particular attention is paid to the application of Padé approximants in the mechanics of liquids and gases. One of the disadvantages of asymptotic methods is that the standard ansatz in the form of a power series in some parameter usually does not reflect the symmetry of the original problem. The search for asymptotic ansatzes that adequately take into account this symmetry has become one of the most important problems of asymptotic analysis. The most developed technique from this point of view is the Padé approximation.
Article
Full-text available
Critical phenomena in fluid and fluid mixtures have been the subject of many theoretical and experimental studies during the past decades as has been elucidated in various reviews.1–15 The most striking result of these studies has been the discovery of critical-point universality. Universality of cr...
Article
Full-text available
The mathematical structure imposed by the thermodynamic critical point motivates an approximant that synthesizes two theoretically sound equations of state: the parametric and the virial. The former is constructed to describe the critical region, incorporating all scaling laws; the latter is an expansion about zero density, developed from molecular considerations. The approximant is shown to yield an equation of state capable of accurately describing properties over a large portion of the thermodynamic parameter space, far greater than that covered by each treatment alone.
Chapter
The purpose of these lectures is to introduce some nonperturbative renormalization group (RG) methods for the study of certain problems of statistical mechanics and quantum field theory. Our aim is a nonperturbative construction of asymtotically free quantum field theories and non-gaussian critical points.
Article
In this paper, we establish the Volterra integro-differential forms of the singular Flierl-Petviashvili equation and the singular Lane-Emden equation. We use the variational iteration method (VIM) to effectively handle any singular equation of the form identical to these equations. The Volterra integro-differential forms of the singular equations overcome the singular behaviour at the origin x = 0, do not use a variety of Lagrange multipliers, and facilitate the computational work. The Padé approximant will be used for the Flierl-Petviashvili equation that is valid in an infinite domain.