1. science
2. mathematics
3. algebra

APPROXIMATE SOLUTIONS FOR FLOW WITH A
STRETCHING BOUNDARY DUE TO PARTIAL SLIP
U. Filobello-Ninoa , H. Vazquez-Leala,∗, A. Sarmiento-Reyesb , B. Benhammoudac , V.M.
Jimenez-Fernandeza , D. Pereyra-Diaza , A. Perez-Sesmaa , J. Cervantes-Pereza , J.
Huerta-Chuad , J. Sanchez-Oreaa , A.D. Contreras-Hernandeza
a
Electronic Instrumentation and Atmospheric Sciences School, Universidad Veracruzana, Cto. Gonzalo
Aguirre Beltr´
an S/N, Xalapa, Veracruz, 91000, M´exico.
b
National Institute for Astrophysics, Optics and Electronics, Luis Enrique Erro No. 1, Sta. Maria
Tonantzintla, 72840, Puebla, Mexico.
c
Abu Dhabi Mens College, Higher Colleges of Technology, P.O. Box 25035, Abu Dhabi, United Arab
Emirates.
d
Civil Engineering School, Universidad Veracruzana, Venustiano Carranza S/N, Col. Revolucion, Poza
Rica, Veracruz, 93390, Mexico.
Abstract
In this article, the homotopy perturbation method (HPM) is coupled with versions of
Laplace-Pad´e and Pad´e methods to provide an approximate solution to the nonlinear differential equation that describes the behaviour of a flow with a stretching flat boundary due to
partial slip. Comparing results between approximate and numerical solutions, we concluded
that our results are capable to provide an accurate solution and are extremely efficient.
Keywords:
Boundary layer, fluid mechanics, homotopy perturbation method, Pad´e method, Laplace
Pad´e method, partial slip, extrusion processes, viscous.
1. Introduction
According to the classification of Prandtl, the fluid motion is divided into two regions.
The first region is near the object where the effect of friction is important and is known as
the boundary layer; while for the second type, these effects can be neglected [1,2,3]. It is
common to define the boundary layer as the region where the fluid velocity parallel to the
surface is less than 99% of the free stream velocity [1].
The boundary layer thickness δ increases from the edge along the surface on which the
fluid moves. Even for the case of a laminar flow, the exact solution of equations describing
the laminar boundary layer is very difficult to calculate and only few simple problems can
straightforward be analysed [1,3].
An interesting case is the one where a flow is induced into a viscoelastic fluid by a linearly
stretched sheet [4,52,53] (see Figure 1). Extrusion of molten polymers through a slit die for
the production of plastic sheets is an important process in polymer industry [4]. The process
∗
Corresponding author: [email protected]
1
is normally complicated from the physical point of view, because it requires significant heat
transfer between the sheet and a surrounding fluid that plays the role of a cooling medium.
An important aspect of the flow is the extensibility of the sheet which can be employed
to improve its mechanical properties along the sheet. To obtain better results is necessary
to improve the cooling rate, whereby, it is common to add some polymeric additives into
water (which is one of the most employed fluids as cooling medium) in order to have a better
control on the cooling rate. The flow due to a stretching boundary is also important in others
engineering processes of interest, such as the glass fibre drawing, and crystal growing among
many others. Detail discussion about this topic can be found in[4]. This work assumes that
the boundary conditions for the problem under study, are adequately described by Navier’s
condition, which states that the amount of relative slip is proportional to local shear stress.
Unlike what happens with fluids like water, mercury and glycerine, which do not require of
slip boundary conditions [58], there are cases where partial slip between the fluid and the
moving surface may occur. Some known cases include emulsions, as mustard and paints;
polymer solutions and clay [58].
v
u
Y
X
U=bX
Figure 1: Schematic showing a stretching boundary.
Ji-Huan He [5,6] proposed the standard HPM; it was introduced as a powerful tool to
approach several kinds of nonlinear problems. The HPM can be considered as a combination between the classical perturbation technique and the homotopy (whose origin is in the
topology), but not restricted to the limitations found in traditional perturbation methods.
For instance, HPM method does need neither small parameter nor linearisation, just few
iterations to obtain accurate results [5,6,20-30,33-38,39,41,42,44-49,52]. The fundamentals of
HPM convergence can be found in [33,36,37].
There are other modern alternatives to find approximate solutions to the differential equations that describe some nonlinear problems such as those based on: variational approaches
[7-9, 31], tanh method [10], exp-function [11,12], Adomians decomposition method [13-18,40],
parameter expansion [19], homotopy analysis method [4,32,43], and perturbation method [50]
among many others.
To figure out how HPM method works, consider a general nonlinear equation in the form
A(u) − f (r) = 0, r ∈ Ω,
with the following boundary conditions
2
(1)
B(u, ∂u/∂n) = 0, r ∈ Γ,
(2)
where A is a general differential operator, B is a boundary operator, f (r) a known analytical
function, and Γ is the domain boundary for Ω.
Also, A can be divided into two parts L and N, where L is linear and N nonlinear; from
this last statement, (1) can be rewritten as
L(u) + N(u) − f (r) = 0.
(3)
In a broad sense, a homotopy can be constructed in the following form[5,6]
H(v, p) = (1 − p)[L(v) − L(u0 )] + p[L(v) + N(v) − f (r)] = 0,
p ∈ [0, 1], r ∈ Ω,
(4)
H(v, p) = L(v) − L(u0 ) + p[L(u0 ) + N(v) − f (r)] = 0, p ∈ [0, 1], r ∈ Ω.
(5)
or
where p is a homotopy parameter, whose values are within range of 0 and 1, u0 is the first
approximation to the solution of (3) that satisfies the boundary conditions. Assuming that
solution for (4) or (5) can be written as a power series of p
v = v0 + v1 p1 + v2 p22 + · · · .
(6)
Substituting (6) into (5) and equating identical powers for p terms, it is possible to obtain
the values for the sequence u0 , u1 , u2 , . . .
When p → 1, it yields in the approximate solution for (1) in the form
v = v0 + v1 + v2 + v3 + · · · .
(7)
Another way to build a homotopy, which is relevant for this paper, is by considering the
following general equation
L(v) + N(v) = 0,
(8)
where L(v) and N(v) are the linear and no linear operators, respectively. It is desired that
solution for L(v) = 0 describes, accurately, the original nonlinear system.
By the homotopy technique, a homotopy is constructed as follows [28]
(1 − p)L(v) + p[L(v) + N(v)] = 0.
(9)
Again, it is assumed that solution for (9) can be written in the form (6); thus, taking the
limit when p → 1, results in the approximate solution for (8).
The variation of homotopic parameter within the range [0, 1] amounts to a deformation
that begins from an initial equation with known solution until it becomes to the equation to
be solved. From a practical point of view, taking the limit p → 1, is just setting p = 1.
3
2. Pad´
e Approximant
Let u(t) be an analytical function with the Maclaurin’s expansion
u (t) =
∞
X
n=0
un tn , 0 ≤ t ≤ T .
(10)
Then the Pad´e approximant to u (t) of order [L, M] which we denote by [L/M]u (t) is defined
by [55,56,57,59]
p0 + p1 t + . . . + pL tL
[L/M]u (t) =
,
(11)
1 + q1 t + . . . + qM tM
where we considered q0 = 1, and the numerator and denominator have no common factors.
The numerator and the denominator in (11) are constructed so that u (t) and [L/M]u (t)
and their derivatives agree at t = 0 up to L + M. That is
u(t) − [L/M]u (t) = O tL+M +1 .
(12)
From (12), we have
u (t)
M
X
n=0
n
qn t −
L
X
n=0
pn tn = O tL+M +1 .
From (13), we get the following algebraic linear systems


uL q1 + . . . + uL−M +1qM = −uL+1


 uL+1 q1 + . . . + uL−M +2qM = −uL+2
..

.


 u
L+M −1 q1 + . . . + uL qM = −uL+M ,
and


p0 = u 0


 p1 = u1 + u0 q1
..

.


 p = u + u q + ...+ u q .
L
L
L−1 1
0 L
(13)
(14)
(15)
From (14), we calculate first all the coefficients qn , 1 ≤ n ≤ M. Then, we determine the
coefficients pn , 0 ≤ n ≤ L from (15).
Note that for a fixed value of L+ M + 1, the error (12) is the smallest when the numerator
and denominator of (11) have the same degree or when the numerator has one degree higher
than the denominator.
3. Laplace-Pad´
e resummation method
Several approximate methods provide power series solutions (polynomial). Nevertheless,
sometimes, this type of solutions lacks of large domains of convergence. Therefore, LaplacePad´e resummation method [55] is used in literature to enlarge the domain of convergence of
solutions or inclusive to find exact solutions.
The Laplace-Pad´e method can be explained as follows:
4
1. First, Laplace transforma is applied to power series.
2. Next, s is substituted by 1/t in the resulting equation.
3. After that, we convert the transformed series into a meromorphic function by forming its
Pad´e approximant of order [N/M]. N and M are arbitrarily chosen, but they should be
of smaller values than the order of the power series. In this step, the Pad´e approximant
extends the domain of the truncated series solution to obtain better accuracy and
convergence.
4. Then, t is substituted by 1/s.
5. Finally, by using the inverse Laplace s transform, we obtain the exact or approximate
solution.
4. Formulation
Consider a two dimensional stretching boundary (see Figure 1). Experiments show that
the velocity of the boundary U is approximately proportional to the distance from the orifice
X [51,58], so that
U = bX.
(16)
where b is a proportionality constant.
Let (u, v) be the fluid velocities for the (X, Y ) directions, respectively. In this case the
boundary condition is adequately described by Navier’s condition, which states that the
amount of relative slip is proportional to local shear stress.
∂u
(X, 0),
(17)
∂Y
where k is a constant of proportionality and ν is the kinematic viscosity of the bulk fluid.
The relevant expressions for this case are the Navier-Stokes equations.
u(X, 0) − U = kν
uuX + vuY +
pX
− ν(uXX + uY Y ) = 0,
ρ
(18)
uvX + vvY +
pY
− ν(vXX + vY Y ) = 0,
ρ
(19)
and continuity
uX + vY = 0,
(20)
where ρ and p are density and pressure, respectively.
To solve equations (18) –(20), we have to consider boundary conditions (16) and (17),
besides that there is no lateral velocity or pressure gradient away from the stretching surface.
Next, we will show that it is possible to get an ordinary nonlinear differential equation
from (18).
5
With this end, we note that Y component of velocity is negative (v < 0) and by symmetry
arguments it only depends on Y (see Figure 1). Therefore, it is possible to define a function
y(x) ≥ 0, 0 ≤ x < ∞, such that [58].
√
v = − bνy(x),
(21)
where
x = Y
r
a
ν
(22)
From (21) and continuity equation (20) we obtain
u = bXy ′ (x),
(23)
Clearly (20) is satisfied under transformations (21), (22) and (23), while (18), adopts the
simpler form.
y ′′′ (x) − (y ′(x))2 + y(x)y ′′(x) = 0.
(24)
To deduce the boundary conditions of (24), we see from Figure 1, that v(Y = 0) = 0 and
lim u(X, Y ) = 0 and therefore
Y →∞
y(0) = 0,
y (∞) = 0,
′
(25)
(26)
Finally, substituting (16) and (23) into (17) we obtain
√
y ′(0) = k bνy ′′(0) + 1.
(27)
5. Approximate solution for a two dimensional viscous flow equation
Next, we present some solution methods to the study problem
5.1. HPM method
In this section, HPM is used to find approximate solutions for (24). Identifying the linear
part as
L = y ′′′ ,
(28)
N = −(y ′ )2 + yy ′′,
(29)
and the nonlinear as
we initiate the HPM method by constructing a homotopy based on (9), in the form
6
(1 − p)y ′′′ + p(y ′′′ − (y ′)2 + yy ′′) = 0.
(30)
Assuming that the solution has the form [5,6].
y = y0 + y1 p1 + y2 p22 + · · · .
(31)
Then, substituting (31) into (30), and equating terms having identical powers of p we
obtain
p0
p1
p2
p3
p4
p5
p6
p7
y0′′′ = 0,
y1′′′ − (y0′ )2 + y0 y0′′ = 0,
y1 y0′′ − 2y0′ y1′ + y2′′′ + y0 y1′′ = 0,
2
y2′′ + y1 y1′′ − y1′ + y2 y0′′ + y3′′′ − 2y0′ y2′ = 0,
y1′′ + y0 y3′′ + y1 y2′′ − 2y0′ y3′ + y3 y0′′ + y4′′′ − 2y1′ y2′ = 0,
2
y3′′ − y2′ + y5′′′ − 2y1′ y3′ + y4 y0′′ + y2 y2′′ − 2y0′ y4′ + y0 y4′′ + y3 y1′′ = 0,
y5 y0′′ + y0 y5′′ + y6′′′ − 2y0′ y5′ + y4 y1′′ + y1 y4′′ + y2 y3′′ − 2y1′ y4′
+y3y2′′ − 2y2′ y3′ = 0,
2
: y4′′ − y3′ + y4 y2′′ + y1 y5′′ + y0 y6′′ + y6 y0′′ − 2y2′ y4′ + y5 y1′′ + y3 y3′′
−2y0′ y6′ + y7′′′ − 2y1′ y5′ = 0.
:
:
:
:
:
:
:
In order to fulfil the√boundary conditions from (24) given by (25)–(27); we find that
y0 (0) = 0, y0′ (0) = 1 + k bνa, y0′′ (0) = a, y1 (0) = 0, y1′ (0) = 0, y1′′(0) = 0, y2(0) = 0, y2′ (0) =
0, y2′′(0) = 0, y3(0) = 0, y3′ (0) = 0, y3′′(0) = 0, and so on. We have assumed the value for y ′′ (0)
as some adequate constant a, which adopts the following values for the corresponding k as
follows {k = 0, a = −1}, {k = 0.3, a = −0.701}, {k = 1, a = −0.430}, {k = 2, a = −0.284},
{k = 5, a = −0.145}, {k = 20, a = −0.0438} [53]. Thus, the results obtained from above
equations are
7
y0 =
y1 =
y2 =
y3 =
y4 =
1
x 1+
x+k a ,
2
1
1
1 2
1
2
2
1+
xk + x + k a + 2k + x a x3 ,
6
4
20
4
6
ax
1
1
1 2
2
2
1 + − xk − x + k a + 2k − x a ,
720
7
56
7
7
1 3
3 3
1 2 2
3 4
x
4
1+
xk +
x k+ x k +
x + k a4
2520
8
160
16
1760
1 2
3 2 3
3 3
3
x + 4k + x k + xk a
+
160
8
8
1 2 3
1
2
2
+ 6k + x + xk a + 4k + x a ,
16
8
8
9
13 4
29 2 3
317 5
x
1+
xk +
xk +
x
−
45360
40
440
640640
317 4
101 3 2
5
xk +
x k + k a5
+
3520
45760
87 2 2
13 3
317 4
101 3
4
+
x k + 5k + xk +
x +
x k a4
440
10
45760
1760
87 2
39 2
101 3
3
+
x k + xk +
x + 10k a3
440
20
3520
9
x
29 2 13
13
2
2
−
10k +
x + xk a + 5k + x a .
45360
440
10
40
8
(32)
(33)
(34)
(35)
(36)
y5
y6
1
589 3 3
37 2 4
419 4 2 19 5
=
1+
xk +
xk +
x k + xk
1247400
17472
208
26880
24
3431
3431
x6 + k 6 +
kx5 a6
+
19009536
1118208
37 2 3
589 3 2
3431 5 95 4
+
x k + 6k 5 +
xk +
x + xk
52
5824
1118208
24
419 4
x k a5 x11
+
13440
1
111 2 2
419 4
589 3
95 3 4
4
+
x k + 15k +
x +
x k + xk a
1247400
104
26880
5824
12
95
37
589 3 3
+ 20k 3 + xk 2 + x2 k +
x a
12
52
17472
19
37 2
95
2
2
xk +
x + 15k a +
x + 6k a x11 .
+
24
208
24
1
599 4 3 351 6
1693 3 4 3257 2 5
=
1+ −
xk −
xk −
xk −
xk
97297200
60928
224
26880
3360
124291 6
124291 7 7
35137 5 2
7
xk −
x x−
x a
+k −
2437120
46305280
926105600
3257 2 4 1693 3 3
35137 5
1053 5
124291 6
+ −
xk −
xk −
x k−
xk −
x
672
6720
1218560
112
46305280
3257 2 3
1
1797 4 2
6
13
6
a + −
x +
xk
x k + 21k 5
+7k −
60928
97297200
336
1693 3 2
35137 5
1797 4
5265 4 5
−
xk −
x −
x k−
xk a
4480
2437120
60928
224
599 4 1693 3
1755 3 4 13
3257 2 2
4
x k + 35k −
x −
x k−
xk a x
+ −
336
60928
6720
56
1
5265 2 3257 2
1693 3 3
3
+
35k −
xk −
x k−
x a
97297200
224
672
26880
1053
3257 2 4 13
2
+ −
xk + 21k −
x a x .
112
3360
9
(37)
(38)
y7
23
1875011
395 7
= −
1 + k8 −
x8 +
xk
9081072000
808351272960
552
392303
270805 5 3
1875011
x7 k −
x6 k 2 +
xk
−
35145707520
4792596480
68465664
1037917 4 4
5075 3 5
8393 2 6
+
a8 x15
xk +
xk +
xk
51349248
1351296
225216
1037917
8393 2 5
2765 6
23
x4 k 3 +
x k + 8k 7 +
xk
−
9081072000 808351272960
37536
552
1875011
270805 5 2
25375 3 4
xk −
x7 +
xk
+
1351296
35145707520
22821888
392303
25375 3 3 1037917 4 2
23
−
x6 k a7 x15 −
xk +
xk
2396298240
9081072000 675648
8558208
270805 5
2765 5 41965 2 4
392303
+
x k+
xk +
xk −
x6
22821888
184
75072
4792596480
6 15
23
41965 2 3
25375 3 2
6
+28k a x −
x k + 56k 5 +
xk
9081072000 56304
675648
270805 5 13825 4
1037917 4
41965 2 2 5 15
+
x +
xk +
x k+
xk ax
68465664
552
12837312
75072a
1037917 4
25375 3
13825 3
23
70k 4 +
x +
x k+
xk
−
9081072000
51349248
1351296
552
5075x3
2765xk
56k 3 2765xk 2 8393x2 k
+
+
+
+
a4 x15
+
a
184a
37536a
1351296a
552a2
23
8393 6
2
−
x + 28k a2 x15
9081072000 225216
23
395x
−
+ 8k ax15 ,
9081072000 552
(39)
and so on.
With the end to exemplify, we will consider the seventh order approximation by substituting solutions (32)–(39) into (31) and calculating the limit when p → 1.
!
7
X
(40)
y = lim
y i pi .
p→1
i=0
The above approximation is valid for all values of k ≥ 0, and corresponds to the results
reported in [52] for the cases when k = 0 and k = 20. Clearly, if higher order approximations
are considered, better accuracy is obtained but the resulting expressions could be too long.
5.2. HPM Laplce Pad´e scheme (LPHPM)
Next, we study the case k = 0, it means that y ′ (0) = 1 (see (27)). This is an interesting
case because the following exact solution was reported in [54].
10
y(x) = 1 − exp(−x).
(41)
Thus, substituting k = 0 in (40) we obtain the following series solution
y(x) = −0.5x2 + x − 7.878129877 × 10−11 x19 + 1.13972148927 × 10−11 x20
+1.57691357363 × 10−10 x18 + 2.07320134446 × 10−13 x21
−1.35120811977 × 10−13 x22 + 5.87481791204 × 10−15 x23
+0.16666667x3 − 0.041666666x4 + 0.00833333333x5
−0.00138888888x6 + 0.000198412698x7 − 0.0000248015873x8
−2.755731 × 10−7 x10 + 0.000002755x9 + 2.5052108 × 10−8 x11
−2.087675 × 10−9 x12 + 1.605904 × 10−10 x13
−1.147074559 × 10−11 x14 + 7.647163731 × 10−13 x15
−4.779477332 × 10−14 x16 − 5.073836906 × 10−11 x17
(42)
As we will see (42) is accurate only for small values of x. To guarantee the validity of the
approximate solution (42) for large values of this variable, the series solution is transformed
using the Pad´e approximation and Laplace transform (see section 3). As first step, Laplace
transform [55] is applied to (42).
−1
1
9.583359 × 106 2.77283070001 × 107 1.009599 × 106
+
−
+
+
s3
s2
s20
s21
s19
8
8
7
1.51875891 × 10
1.51875891 × 10
1
1
1.0592181 × 10
−
+
+ 4− 5
+
22
23
24
s
s
s
s
s
1
1
1
1
1
0.9999999998
1
0.9999999999
1
− 13 +
+ 6 − 7 + 8 − 9 − 11 + 10 +
s
s
s
s
s
s
s12
s
s14
0.9999999997
1
1
18047
−
+ 16 − 17 − 18 ,
15
s
s
s
s
(43)
then, s is substituted by 1/x in the equation to obtain
−x3 + x2 − 9.583359 × 106 x20 + 2.77283070001 × 107 x21
+1.009599 × 106 x19 + 1.0592181 × 107 x22 − 1.51875891 × 108 x23
+x4 − x5 + x6 − x7 + x8 − x9 − x11 + x10 + 0.9999999998x12 − x13
+0.9999999999x14 − 0.9999999997x15 + x16 − x17 − 18047x18
+1.51875891 × 108 x24 .
(44)
Following Laplace-Pad´e scheme, Pad´e approximant [6/6] is applied to obtain
x2
,
1+x
here x is substituted by 1/s, the result is
11
(45)
1
,
s(s + 1)
(46)
Finally, by means of the inverse Laplace transform applied to (46), we obtain the exact
solution (41) for (24).
5.3. HPM Pad´e scheme (PHPM)
Another way to recover lost information from the truncated series (40), is by means of
applying the Pad´e approximant [7/7]. For the particular case when k = 0 (see (42)), we
obtain the following rational approximation for (24) (using dummy variables a1 and b1)
a1 = 1.15625115625 × 10−7 x7 + 0.000174825174825x5
+0.0320512820513x3 + x,
b1 = 1 + 0.1153846153x2 + 0.001456876456x4 + 0.5x + 0.01602564102x3
+0.00008741258741x5 + 0.00000323750323x6
+5.781255781 × 10−8 x7 ,
therefore, the final result is
y(x) =
a1
b1
(47)
6. Discussion
Figure 3 and Figure 5 show a comparison between Runge Kutta 4 (RK4) numerical solution for different values of k, and approximations given by Laplace Pad´e and Pad´e methods
applied to (40). From Figure 4 and Figure 6 can be noticed that the relative error obtained
by our approximations were low. As a matter of fact, the largest error for LPHPM is 0.006 when k = 5, and for PHPM is -0.0250 when k = 0. In a broad sense, if higher order
approximations are considered, better the accuracy of LP-HPM and PHPM methods. In
particular, is noteworthy that Laplace Pad´e method allows recovering the exact solution (41)
when k = 0 from the truncated series (40). This contrasts with the Figure 2, which shows
the comparison between RK4 and HPM approximations given by (40) for k = 0, 0.3, 1, 2, 5
and k = 20. It is evident that HPM series is accurate only for a restricted domain of values
for the independent variable x. Also it is worth to compare the cumbersome approximation
(42) for k = 0, with the handy expressions (41) and (47) obtained by LHPM and PHPM
respectively. In [52] HPM was employed to solve (24) with a good approximation for a restricted domain of values for x and small values for k; while in [53], the same equation was
solved using perturbation method (PM) for small values of k. Indeed is known that PM
provides, in general, better results for small values of the perturbation parameter. Unlike the
LPHPM and PHPM schemes were employed to obtain accurate solutions for different values
of k; having low relative errors for x values within the range 0 ≤ x ≤ 30, as shown in Figure
4 and Figure 6.
The importance of providing analytical solutions, although approximate, with good accuracy is that numerical solutions only provide a qualitative idea of the problem to be solved.
12
Besides, just as it was shown in one of our cases study, a solution obtained by a numerical method like Runge-Kutta, could hide the case of an exact solution. This possibility
was successfully explored by LPHPM for case study k = 0. Finally, other reason why we
are interested in obtaining analytical approximation solutions is that numerical algorithms
could give some problems, such as numerical instabilities, oscillations, among others. This
means that the numerical solutions may not correspond to the real solution of the original
differential equation [61].
Just as it was seen with our HPM solutions, one disadvantage of approximate with polynomials is its tendency to oscillate, this gives rise that the obtained solutions diverge, especially
for the case of problems defined on open intervals. This can be attributed to the possibility
that the radius of convergence may not be sufficiently large to contain the boundaries of the
domain of study [57,60].
In order to improve the aforementioned, the use of rational functions was proposed,
through the Pad´e approximation method.
The Pad´e approximation is an extension of the Taylor Polynomial approximation but for
rational functions. When the denominator of the approximation is a zero degree polynomial
function, the Pad´e approximation is reduced to a Maclaurin Polynomial.
Pad´e-approximate extrapolation technique consists in approximating a truncated series
(such as those resulting of HPM) by a rational function, this latter extends the range of
validity of the initial polynomial. The above results are particularly relevant if the truncated
series represent the solution of a differential equation.
We observe that even though the series has a finite region of convergence, Pad´e approximant let obtain the limit of the function under study as x takes large values, if L = M (see
Figures 3 and 5) [56]. In Fact, the rational functions whose denominator and numerator have
the same values of L and M, or a degree almost identical, give rise to a results better than
those obtained by methods based on polynomial functions [60].
Finally, the convergence is uniform in any compact region in the case of Pad´e approximation, whereas truncated Mclaurin series is valid only in a near neighbourhood of zero point.
The above explains why HPM-Pad´e has larger intervals of convergence in comparison with
HPM series [56].
Such as it was explained LPHPM employ Pad´e approximant to the results of apply Laplace
transform to the truncated series of HPM, and the rational function obtained in this way
acquire the benefits of Pad´e method above mentioned.
Nevertheless, one possible advantage of LPHPM is that the application of inverse Laplace
transform at the last step of the method, could result in the exact solution of the problem.
7. Conclusions
This work showed that some nonlinear problems may be adequately approximated using
the coupling of the HPM method with Laplace Pad´e, and Pad´e methods to deal with HPM
truncated power series. For instance, the flow induced by a stretching sheet is adequately
described by our approximations given by LPHPM and PHPM (see Figure 3 through Figure
6). Figure 4 and Figure 6 illustrate the relative error and show that the proposed solutions
are highly accurate. Since that this procedure is, in principle, applicable to other similar
13
k
k
k
k
k
k
Figure 2: Fourth order Runge Kutta numerical solution for (24) (solid line) and HPM solution (40) (symbols).
k
k
k
k
k
k
Figure 3: Fourth order Runge Kutta solution for (24) (symbols) and LPHPM (solid line).
14
k
k
k
k
k
k
Figure 4: Relative error for different cases of LPHPM.
k
k
k
k
k
k
Figure 5: Fourth order Runge Kutta numerical solution for (24) (symbols) and PHPM (solid line).
15
k
k
k
k
k
k
Figure 6: Relative error for different cases of PHPM.
problems, we conclude that LPHPM and PHPM are methods with high potential in the
search for analytical approximate solutions for nonlinear problems.
Acknowledgements
We gratefully acknowledge the financial support from the National Council for Science
and Technology of Mexico (CONACyT) through grant CB-2010-01 #157024.
Disclosure Policy
The authors declare that there is no conflict of interests regarding the publication of this
paper.
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