Open Access

Nonlinearities distribution Laplace transform-homotopy perturbation method

  • Uriel Filobello-Nino1,
  • Hector Vazquez-Leal1Email author,
  • Brahim Benhammouda2,
  • Luis Hernandez-Martinez3,
  • Claudio Hoyos-Reyes1,
  • Jose Antonio Agustin Perez-Sesma1,
  • Victor Manuel Jimenez-Fernandez1,
  • Domitilo Pereyra-Diaz1,
  • Antonio Marin-Hernandez4,
  • Alejandro Diaz-Sanchez3,
  • Jesus Huerta-Chua5 and
  • Juan Cervantes-Perez1
SpringerPlus20143:594

https://doi.org/10.1186/2193-1801-3-594

Received: 13 August 2014

Accepted: 24 September 2014

Published: 9 October 2014

Abstract

This article proposes non-linearities distribution Laplace transform-homotopy perturbation method (NDLT-HPM) to find approximate solutions for linear and nonlinear differential equations with finite boundary conditions. We will see that the method is particularly relevant in case of equations with nonhomogeneous non-polynomial terms. Comparing figures between approximate and exact solutions we show the effectiveness of the proposed method.

Keywords

Homotopy perturbation methodNonlinear differential equationApproximate solutionsLaplace transformLaplace transform homotopy perturbation methodFinite boundary conditions

Introduction

Laplace Transform (L.T.) (or operational calculus) has played an important role in mathematics (Murray 1988), not only for its theoretical interest, but also because such method allows to solve, in a simpler fashion, many problems in science and engineering, in comparison with other mathematical techniques (Murray 1988). In particular the L.T. is useful for solving ODES with constant coefficients, and initial conditions, but also can be used to solve some cases of differential equations with variable coefficients and partial differential equations (Murray 1988). On the other hand, applications of L.T. for nonlinear ordinary differential equations mainly focus to find approximate solutions, thus in reference (Aminikhan & Hemmatnezhad 2012) was reported a combination of Homotopy Perturbation Method (HPM) and L.T. method (LT-HPM), in order to obtain highly accurate solutions for these equations. However, just as with L.T; LT-HPM method has been used mainly to find solutions to problems with initial conditions (Aminikhan & Hemmatnezhad 2012; Aminikhah 2012), because it is directly related with them. Therefore (Filobello-Nino et al. 2013) presented successfully, the application of LT-HPM, in the search for approximate solutions for nonlinear problems with Dirichlet, mixed and Neumann boundary conditions defined on finite intervals. This paper introduces a modification of LT-HPM, the Non-linearities Distribution Laplace Transform-Homotopy Perturbation Method (NDLT-HPM), which will show better results for the case of linear and non-linear differential equations with non polynomial nonhomogeneous terms. The case of equations with boundary conditions on infinite intervals, has been studied in some articles, and corresponds often to problems defined on semi-infinite ranges (Aminikhah 2011; Khan et al. 2011). However the methods of solving these problems, are different from those presented in this paper (Filobello-Nino et al. 2013). As it is widely known, the importance of research on nonlinear differential equations is that many phenomena, practical or theoretical, are of nonlinear nature. In recent years, several methods focused to find approximate solutions to nonlinear differential equations, as an alternative to classical methods, have been reported, such those based on: variational approaches (Assas 2007; He 2007; Kazemnia et al. 2008; Noorzad et al. 2008), tanh method (Evans & Raslan 2005), exp-function (Xu 2007; Mahmoudi et al. 2008), Adomian’s Decomposition Method (ADM) (Adomian 1988; Babolian & Biazar 2002; Kooch & Abadyan 2012; Kooch & Abadyan 2011; Vanani et al. 2011; Chowdhury 2011; Elias et al. 2000), parameter expansion (Zhang & Xu 2007), HPM (Aminikhan & Hemmatnezhad 2012; Aminikhah 2012; Filobello-Nino et al. 2013; Aminikhah 2011; Khan et al. 2011; Marinca & Herisanu 2011; He 1998; He 1999; He 2006a; Vazquez-Leal et al. 2014; Belendez et al. 2009; He 2000; El-Shaed 2005; He 2006b; Vazquez-Leal et al. 2012a; Ganji et al. 2008; Fereidon et al. 2010; Sharma & Methi 2011; Biazar & Ghanbari 2012; Biazar & Eslami 2012; Araghi & Sotoodeh 2012; Araghi & Rezapour 2011; Bayat et al. 2014; Bayat et al. 2013; Vazquez-Leal et al. 2012b; Vazquez-Leal et al. 2012c; Filobello-Niño et al. 2012; Biazar & Aminikhan 2009; Biazar & Ghazvini 2009; Filobello-Nino et al. 2012; Khan & Qingbiao 2011; Madani et al. 2011; Ji Huan 2006; Feng et al. 2007; Mirmoradia et al. 2009; Vazquez-Leal et al. 2012; Vazquez-Leal et al 2013), Homotopy Analysis Method (HAM) (Rashidi et al. 2012a; Rashidi et al. 2012b; Patel et al. 2012; Hassana & El-Tawil 2011), and perturbation method (Filobello-Nino et al. 2013; Holmes 1995; Filobello-Niño et al. 2013b; Filobello-Nino et al. 2014) among many others. Also, a few exact solutions to nonlinear differential equations have been reported occasionally (Filobello-Niño et al. 2013a).

The case of Boundary Value Problems (BVPs) for nonlinear ODES includes, Michaelis Menten equation (Murray 2002; Filobello-Nino et al. 2014), that describes the kinetics of enzyme-catalyzed reactions, Gelfand’s differential equation (Filobello-Nino et al. 2013; Filobello-Niño et al. 2013b) which governing combustible gas dynamics, Troesch’s equation (Elias et al. 2000; Feng et al. 2007; Mirmoradia et al. 2009; Vazquez-Leal et al. 2012; Hassana & El-Tawil 2011; Erdogan & Ozis 2011), arising in the investigation of confinement of a plasma column by a radiation pressure, among many others.

In the same way, the theory of BVPs for linear ODES, is a well established branch of mathematics, with many applications. Between problems of interest, related to these equations, are found: The one-dimensional quantum problem, of a particle of mass m confined in a region of zero potential by an infinite potential at two points x = a and x = b (King et al. 2003), heat transfer equation (King et al. 2003), wave equation which describes for instance, transverse vibrations of a uniform stretched string between two fixed points, say x = a and x = b (Chow 1995; Zill Dennis 2012), the Laplace equation, which governs the temperature field corresponding to the steady state in a plate (Zill Dennis 2012), and so on. Generally, many problems expressed in terms of partial differential equations, give rise through method of separation of variables, to BVPs for linear ODES (Chow 1995; King et al. 2003; Zill Dennis 2012). From the above, it becomes a priority to investigate methods, to find handy analytical approximate solutions for linear and nonlinear ODES. With this end, we propose NDLT-HPM method, which as will be seen has good precision and requires a moderate computational work.

The paper is organized as follows. In Section 2, we introduce the standard HPM. Section 3, provides a basic idea of Nonlinearities Distribution Homotopy Perturbation Method (NDHPM). Section 4 introduces NDLT-HPM. Additionally Section 5 presents two cases study. Besides a discussion on the results is presented in Section 6. Finally, a brief conclusion is given in Section 7.

Standard HPM

The standard HPM was proposed by Ji Huan He, it was introduced like a powerful tool to approach various kinds of nonlinear problems. The HPM is considered as a combination of the classical perturbation technique and the homotopy (whose origin is in the topology), but not restricted to small parameters as occur with traditional perturbation methods. For example, HPM requires neither small parameter nor linearization, but only few iterations to obtain highly accurate solutions (He 1998; He 1999).

To figure out how HPM works, consider a general nonlinear equation in the form
A u f r = 0 , r Ω ,
(1)
with the following boundary conditions
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. A can be divided into two operators L and N, where L is linear and N nonlinear; so that (1) can be rewritten as
L u + N u f r = 0 .
(3)
Generally, a homotopy can be constructed as (He 1998; He 1999)
H U , p = 1 p L U L u 0 + p L U + N U f r = 0 , p 0 , 1 , r Ω ,
(4)
or
H U , p = L U L u 0 + p L u 0 + N U f r = 0 , p 0 , 1 , r Ω ,
(5)

where p is a homotopy parameter, whose values are within range of 0 and 1, u0 is the first approximation for 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.
U = v 0 + v 1 p + v 2 p 2 +
(6)

Substituting (6) into (5) and equating identical powers of p terms, there can be found values for the sequence v0, v1, v2,

When p → 1, it yields the approximate solution for (1) in the form
U = v 0 + v 1 + v 2 + v 3
(7)

Basic idea of NDHPM

(Vazquez-Leal et al. 2012b) introduced a modified version of HPM, which sometimes eases the solutions searching process for (3) and reduces the complexity of solving differential equations in terms of power series.

As first step, a homotopy of the form (Vazquez-Leal et al. 2012b) is introduced
H U , p = 1 p L U L u 0 + p L U + N U , p f r , p = 0 ,
(8)
or
H U , p = L U L u 0 + p L u 0 + N U , p f r , p = 0 , p 0 , 1 , r Ω .
(9)

It can be noticed that the homotopy function (8) is essentially the same as (4), except for the non-linear operator N and the non homogeneous function f, which contain embedded the homotopy parameter p. The standard procedure for the HPM is used in the rest of the method.

We propose that (He 1998; Vazquez-Leal et al. 2012b)
U = v 0 + v 1 p + v 2 p 2 +
(10)
When p → 1, it is expected to get an approximate solution for (3) in the form
U = v 0 + v 1 + v 2 + v 3 +
(11)

Non-linearities distribution Laplace transform-homotopy perturbation method (NDLT-HPM)

A way to introduce, NDLT-HPM is assume that NDLT-HPM follows the same steps of NDHPM until (9), next we apply Laplace transform on both sides of homotopy equation (9), to obtain
L U L u 0 + p [ L u 0 + N U , p f r , p = 0 ,
(12)

(more generally, one could substitute f(r, p) in (8) by another function g(r, p) such that, g(r, p) → f(r) when p → 1, see cases study above).

Using the differential property of L.T, we have (Murray 1988)
s n U s n 1 U 0 s n 2 U 0 U n 1 0 = L u 0 pL u 0 + p N U , p + f r , p ,
(13)
or
U = 1 s n s n 1 U 0 + s n 2 U 0 + .. + U n 1 0 + L u 0 pL u 0 + p N U , p + f r , p
(14)
applying inverse Laplace transform to both sides of (14), we obtain
U = 1 1 s n s n 1 U 0 + s n 2 U 0 + .. + U n 1 0 + L u 0 pL u 0 + p N U , p + f r , p
(15)
Assuming that the solutions of (3) and f(r, p) can be expressed as a power series of p
U = n = 0 p n v n ,
(16)
f = m = 0 p m f m r .
(17)
Then substituting (16) and (17) into (15), we get
n = 0 p n ν n = 1 1 s n s n 1 U 0 + s n 2 U 0 + .. + U n 1 0 + 1 s n L u 0 pL u 0 + p N n = 0 p n ν n , p + m = 0 p m f m r ,
(18)
comparing coefficients of p, with the same power leads to
p 0 : ν 0 = 1 1 s n s n 1 U 0 + s n 2 U 0 + .. + U n 1 0 ) + L u 0 ,
p 1 : ν 1 = 1 1 s n N ν 0 L u 0 + f 0 r ,
p 2 : ν 2 = 1 1 s n N ν 0 , ν 1 , f 0 , f 1 ,
p 3 : ν 3 = 1 1 s n N ν 0 , ν 1 , ν 2 , f 0 , f 1 , f 2 , p j : ν j = 1 1 s n N ν 0 , ν 1 , ν 2 , , ν j , f 0 , f 1 , f 2 , .. f j ,
(19)
Assuming that the initial approximation has the form: U(0) = u0 = α0, U′(0) = α1,.., Un − 1(0) = αn − 1; therefore the exact solution may be obtained as follows
u = lim p 1 U = ν 0 + ν 1 + ν 2 +
(20)

LT-HPM is derived in a similar way to NDLT-HPM, the difference is that in the first case the Laplace transform applies to (5) instead of (9). From here on, takes place in essence the same procedure followed by NDLT-HPM (12), (13), (14), (15), (16), (17), (18) and (19) (Aminikhan & Hemmatnezhad 2012; Aminikhah 2012; Filobello-Nino et al. 2013; Aminikhah 2011).

Cases study

Next, NDLT-HPM, and LT-HPM are compared with the following two cases study

CASE STUDY 1

We will find an approximate solution the following nonlinear second order ordinary differential equation
d 2 y x d x 2 y 2 x e x = 0 , 0 x 1 , y 0 = 0 , y 1 = 2 .
(21)

Method 1 Employing LT-HPM

To obtain an approximate solution for (21) by applying the LTHPM method, we identify
L y = y x ,
(22)
N y = y 2 x e x ,
(23)

where prime denotes differentiation respect to x.

To solve approximately (21), first we expand the exponential term, resulting
y y 2 x 1 + x + 1 2 x 2 + .. = 0 , 0 x 1 , y 0 = 0 , y 1 = 2 .
(24)
We construct the following homotopy in accordance with (4)
1 p y y 0 + p y y 2 1 x 1 2 x 2 = 0 ,
(25)
or
y = y 0 + p y 0 + y 2 + 1 + x + x 2 2 ,
(26)

where we have kept three terms of Taylor series.

Applying Laplace transform to (26) we get
y = y 0 + p y 0 + y 2 + 1 + x + x 2 2 .
(27)
As it is explained in (Murray 1988), it is possible to rewrite (27) as
s 2 Y s s y 0 y 0 = y 0 + p y 0 + y 2 + 1 + x + x 2 2 ,
(28)

where we have defined Y(s) = (y(x)).

After applying the initial condition y(0) = 0, the last expression can be simplified as follows
s 2 Y s A = y 0 + p y 0 + y 2 + 1 + x + x 2 2 ,
(29)

where, we have defined A = y′(0).

Solving for Y(s) and applying Laplace inverse transform − 1
y x = 1 A s 2 + 1 s 2 y 0 + p y 0 + y 2 + 1 + x + x 2 2 .
(30)
Next, suppose that the solution for (30) has the form
y x = n = 0 p n ν n ,
(31)
and choosing
ν 0 x = Ax ,
(32)

as the first approximation for the solution of (21) that satisfies the condition y(0) = 0.

Substituting (31) and (32) into (30), we get
n = 0 p n ν n = 1 A s 2 + 1 s 2 y 0 + p y 0 + ν 0 + p ν 1 + p 2 ν 2 + .. 2 + 1 + x + x 2 2 .
(33)
Equating terms with identical powers of p, we obtain
p 0 : ν 0 x = 1 A s 2 ,
(34)
p 1 : ν 1 x = 1 1 s 2 ν 0 2 + 1 + x + x 2 2 ,
(35)
p 2 : ν 2 x = 1 1 s 2 2 ν 0 ν 1 ,
(36)
p 3 : ν 3 x = 1 1 s 2 ν 1 2 + 2 ν 0 ν 2 ,
(37)
p 4 : ν 4 x = 1 1 s 2 2 ν 0 ν 3 + 2 ν 1 ν 2 ,
(38)
From above we solve for ν0(x), ν1(x), ν2(x),.. we obtain
p 0 : ν 0 x = Ax ,
(39)
p 1 : ν 1 x = A 2 + 1 2 x 4 12 + x 3 6 + x 2 2 ,
(40)
p 2 : ν 2 x = A 252 A 2 + 1 2 x 7 + A 90 x 6 + A 20 x 5 ,
(41)
p 3 : ν 3 x = A 2 + 1 2 A 2 6048 + 1 25920 x 10 + A 2 1440 + 1 5184 x 9 + 11 A 2 3360 + 5 4032 x 8 + x 7 252 + x 6 120 ,
(42)
p 4 : ν 4 x = 2 A 2 + 1 A 2 A 2 + 1 943488 + 1 156 A 2 6048 + 1 25920 x 13 + 23 A 3 665280 + 1.4947758 × 10 5 A x 12 + 11 A 3 665280 + 1.211332371 × 10 4 A x 11 + A 2520 x 10 + A 1080 x 9 ,
(43)

and so on.

By substituting solutions (39), (40), (41), (42) and (43) into (20) results in a fourth order approximation
y x = Ax + x 2 2 + x 3 6 + A 2 + 1 2 x 4 12 + A 20 x 5 + A 90 + 1 120 x 6 + A 252 A 2 + 1 2 + 1 252 x 7 + 11 A 2 3360 + 5 4032 x 8 + A 2 1440 + 1 5184 + A 1080 x 9 + A 2 + 1 2 A 2 6048 + 1 25920 + A 2520 x 10 + 11 A 3 665280 + 1.211332371 × 10 4 A x 11 + 23 A 3 665280 + 1.4947758 × 10 5 A x 12 + 2 A 2 + 1 A 2 A 2 + 1 943488 + 1 156 A 2 6048 + 1 25920 x 13 .
(44)
In order to calculate the value of A, we require that (44) satisfies the boundary condition y(1) = 2, so that we obtain
A = 1.096310072 .
(45)

Method 2 Employing NDLT-HPM

In accordance with NDLT-HPM, we propose the following homotopy
1 p y y 0 + p y y 2 p e p x = 0 ,
(46)

we see that (46) is not exactly of the form (8), but note that g(x, p) = pe px  → e x , if p → 1.

After expanding the exponential term, we obtain
1 p y y 0 + p y y 2 p 1 + x p + 1 2 p 2 x 2 = 0 ,
(47)
or
y = y 0 + p y 0 + y 2 + p + p 2 x + p 3 x 2 2 .
(48)
Applying Laplace transform to (48), we get
y = y 0 + p y 0 + y 2 + p + p 2 x + p 3 x 2 2 ,
(49)
it is possible to rewrite (49) as
s 2 Y s s y 0 y 0 = y 0 + p y 0 + y 2 + p + p 2 x + p 3 x 2 2 ,
(50)

where we have defined Y(s) = (y(x)).

Applying the initial condition y(0) = 0, (50) can be simplified as follows
s 2 Y s A = y 0 + p y 0 + y 2 + p + p 2 x + p 3 x 2 2 ,
(51)

where, we have defined A = y′(0).

Solving for Y(s) and applying Laplace inverse transform − 1
y x = 1 A s 2 + 1 s 2 y 0 + p y 0 + y 2 + p + p 2 x + p 3 x 2 2 .
(52)
Assuming that the solution for (52) has the form
y x = n = 0 p n ν n ,
(53)
and choosing
ν 0 x = Ax ,
(54)

as the first approximation for the solution of (21) that satisfies the condition y(0) = 0.

Substituting (53) and (54) into (52), we get
n = 0 p n ν n = 1 A s 2 + 1 s 2 y 0 + p y 0 + ν 0 + p ν 1 + p 2 ν 2 + .. 2 + p + p 2 x + p 3 x 2 2 .
(55)
Equating terms with identical powers of p, we obtain
p 0 : ν 0 x = 1 A s 2 ,
(56)
p 1 : ν 1 x = 1 1 s 2 ν 0 2 ,
(57)
p 2 : ν 2 x = 1 1 s 2 2 ν 0 ν 1 + 1 ,
(58)
p 3 : ν 3 x = 1 1 s 2 ν 1 2 + 2 ν 0 ν 2 + x ,
(59)
p 4 : ν 4 x = 1 1 s 2 2 ν 0 ν 3 + 2 ν 1 ν 2 + x 2 2 ,
(60)
Solving the above equations for ν0(x), ν1(x), ν2(x) …, we obtain
p 0 : ν 0 x = Ax ,
(61)
p 1 : ν 1 x = A 2 12 x 4 ,
(62)
p 2 : ν 2 x = A 3 252 x 7 + 1 2 x 2 ,
(63)
p 3 : ν 3 x = 135 A 4 816480 x 10 + A 20 x 5 + 1 6 x 3 ,
(64)
p 4 : ν 4 x = A 5 157248 x 13 + 11 A 2 3360 x 8 + A 90 x 6 + x 4 24 ,
(65)

and so on.

By substituting solutions (61), (62), (63), (64) and (65) into (20) results in a fourth order approximation
y x = Ax + x 2 2 + x 3 6 + A 2 + 1 / 2 12 x 4 + A 20 x 5 + A 90 x 6 + A 3 252 x 7 + 11 A 2 3360 x 8 + A 4 6048 x 10 + A 5 157248 x 13 .
(66)
In order to calculate the value of A, we require that (66) satisfies the boundary condition y(1) = 2, so that we obtain
A = 1.111131377 .
(67)

Case study 2

We will find an approximate solution for the following linear third order ordinary differential equation with variable coefficients.
d 3 y x d x 3 x y x + x 2 sin x = 0 , 0 x 1 , y 0 = 0 , y 0 = 1 y 1 = 2 .
(68)

Method 1 Employing LT-HPM

To obtain a solution for (68) by applying the LT-HPM method, we identify
L y = y x ,
(69)
N y = x y x + x 2 sin x ,
(70)

where prime denotes differentiation respect to x.

To solve approximately (68), first we expand the trigonometric term, resulting
y x y x + x 2 x 1 6 x 3 + .. = 0 , 0 x 1 , y 0 = 0 , y 0 = 1 y 1 = 2 .
(71)
We construct the following homotopy in accordance with (4)
1 p y y 0 + p y x y + x 3 x 5 6 = 0 ,
(72)

where we have kept just two terms of Taylor series,

or
y = y 0 + p y 0 + x y x 3 + x 5 6 .
(73)
Applying Laplace transform to (73), we get
y = y 0 + p y 0 + x y x 3 + x 5 6 .
(74)
In accordance with (Murray 1988), it is possible to rewrite (74) as
s 3 Y s s 2 y 0 s y 0 y 0 = y 0 + p y 0 + x y x 3 + x 2 5 .
(75)
Applying the initial conditions y(0) = 0 and y′(0) = 1, (75) adopts the following form
s 3 Y s s A = y 0 + p y 0 + x y x 3 + x 2 5 ,
(76)

where, we have defined A = y″(0).

Solving for Y(s) and applying Laplace inverse transform − 1
y x = 1 A s 3 + 1 s 2 + 1 s 3 y 0 + p y 0 + x y x 3 + x 5 6 .
(77)
Assuming that the solution for (77) has the form
y x = n = 0 p n ν n ,
(78)
and choosing
ν 0 x = A 2 x 2 + x ,
(79)

let be the first approximation for the solution of (68) that satisfies the initial conditions y(0) = 0 and y′(0) = 1.

Substituting (78) and (79) into (77), we get
n = 0 p n ν n = 1 A s 3 + 1 s 2 + 1 s 3 y 0 + p y 0 + x ν 0 + p ν 1 + p 2 ν 2 + .. x 3 + x 5 6 .
(80)
Equating terms with identical powers of p, we obtain
p 0 : ν 0 x = 1 1 s 2 + A s 3 ,
(81)
p 1 : ν 1 x = 1 1 s 3 x ν 0 x 3 + x 5 6 ,
(82)
p 2 : ν 2 x = 1 1 s 3 x ν 1 ,
(83)
p 3 : ν 3 x = 1 1 s 3 x ν 2 ,
(84)
p 4 : ν 4 x = 1 1 s 3 x ν 3 ,
(85)
From above we solve for ν0(x), ν1(x), ν2(x) …, we obtain
p 0 : ν 0 x = A 2 x 2 + x ,
(86)
p 1 : ν 1 x = A / 2 1 120 x 6 + x 5 60 + x 8 2016 ,
(87)
p 2 : ν 2 x = A / 2 1 86400 x 10 + x 9 30240 + x 12 2661120 ,
(88)
p 3 : ν 3 x = x 13 51891840 + A / 2 1 188697600 x 14 + x 16 8941363200 ,
(89)
p 4 : ν 4 x = 4.723248187 × 10 12 x 17 + 1.082411043 × 10 12 A / 2 1 x 18 + 1.635084351 × 10 14 x 20 ,
(90)

and so on.

By substituting solutions (86), (87), (88), (89) and (90) into (20) results in a fourth order approximation
y x = x + A 2 x 2 + x 5 60 + A / 2 1 120 x 6 + x 8 2016 + x 9 30240 + A / 2 1 86400 x 10 + x 12 2661120 + x 13 51891840 + A / 2 1 188697600 x 14 + x 16 8941363200 + 4.723248187 × 10 12 x 17 + 1.082411043 × 10 12 A / 2 1 x 18 + 1.635084351 × 10 14 x 20 .
(91)
In order to calculate the value of A, we require that (91) satisfies the boundary condition y(1) = 2, so that we obtain
A = 1.965892301 .
(92)

Method 2 Employing NDLT-HPM

In accordance with NDLT-HPM, it is possible to propose the following homotopy (see (9))
1 p y y 0 + p y x y + x 2 pg x , p = 0 ,
(93)
where we have defined
g x , p = p 2 sin p x + x 3 p 5 p 3 6 ,
(94)
with the property
lim p 1 g x , p = sin x .
(95)
after expanding the two first terms of sin function, we obtain
1 p y y 0 + p y x y + x 2 p 3 x x 3 / 6 = 0 ,
(96)
or
y = y 0 + p y 0 + x y x 3 p 3 + p 3 x 5 6 .
(97)
Applying Laplace transform to (97) we get
y = y 0 + p y 0 + x y x 3 p 3 + p 3 x 5 6 ,
(98)
it is possible to rewrite (98) as
s 3 Y s s 2 y 0 s y 0 y 0 = y 0 + p y 0 + x y x 3 p 3 + p 3 x 6 5 ,
(99)

where once again, we have defined Y(s) = (y(x)).

Applying the initial conditions y(0) = 0, and y′(0) = 1, (99) can be simplified as follows
s 3 Y s s A = y 0 + p y 0 + x y x 3 p 3 + p 3 x 6 5 ,
(100)

where, we have defined A = y″(0).

Solving for Y(s) and applying Laplace inverse transform − 1
y x = 1 1 s 2 + A s 3 + 1 s 3 y 0 + p y 0 + x y x 3 p 3 + p 3 x 6 5 .
(101)
Next, we assume a series solution for y(x), in the form
y x = n = 0 p n ν n ,
(102)
let
ν 0 x = A 2 x 2 + x ,
(103)

be the first approximation for the solution of (68) that satisfies the initial conditions y(0) = 0 and y′(0) = 1.

Substituting (102) and (103) into (101), we get
n = 0 p n ν n = 1 A s 3 + 1 s 2 + 1 s 3 y 0 + p y 0 + x ν 0 + p ν 1 + p 2 ν 2 + .. x 3 p 3 + p 3 x 5 6 .
(104)
On comparing the coefficients of like powers of p we have
p 0 : ν 0 x = 1 1 s 2 + A s 3 ,
(105)
p 1 : ν 1 x = 1 1 s 3 x ν 0 ,
(106)
p 2 : ν 2 x = 1 1 s 3 x ν 1 ,
(107)
p 3 : ν 3 x = 1 1 s 3 x ν 2 ,
(108)
p 4 : ν 4 x = 1 1 s 3 x ν 3 x 3 + x 5 6 ,
(109)
Performing the above operations for ν0(x), ν1(x), ν2(x) …, we obtain
p 0 : ν 0 x = A 2 x 2 + x ,
(110)
p 1 : ν 1 x = x 5 60 + A 240 x 6 ,
(111)
p 2 : ν 2 x = x 9 30240 + A 172800 x 10 ,
(112)
p 3 : ν 3 x = x 13 51891840 + A 377395200 x 14 ,
(113)
p 4 : ν 4 x = x 6 120 + x 8 2016 + x 17 211718707200 + A 1847726899200 x 18 ,
(114)

and so on.

By substituting solutions (110), (111), (112), (113) and (114) into (20) and calculating the limit when p → 1, results in a fourth order approximation
y x = x + A 2 x 2 + 1 60 x 5 + A 240 1 120 x 6 + 1 2016 x 8 + 1 30240 x 9 + A 172800 x 10 + 1 51891840 x 13 + A x 14 377395200 + 1 211718707200 x 17 + A x 18 1847726899200 .
(115)
In order to calculate the value of A, we require that (115) satisfies the boundary condition y(1) = 2, resulting an equation for A, from which we obtain the following result
A = 1.965892301 .
(116)

Discussion

This work showed the accuracy of NDLT-HPM in solving ordinary differential equations with nonhomogeneous non-polynomial terms and finite boundary conditions, and it can be considered as a continuation of (Filobello-Nino et al. 2013) where in principle, LT-HPM already provided the possibility of solving problems, with the nonhomogeneities mentioned in this study (Aminikhan & Hemmatnezhad 2012; Aminikhah 2012; Filobello-Nino et al. 2013; Aminikhah 2011), but were not carried out. One way to introduce LT-HPM to this kind of problems, is directly apply the Laplace transform to the homotopy equation (5) and then following a procedure identical to that applied in (Filobello-Nino et al. 2013) (see also (12), (13), (14), (15), (16), (17), (18) and (19)), although a possible difficulty is that, the mathematical procedure becomes long and cumbersome, depending on the function (see (4)). It may even happen that, the method does not work if the Laplace transform does not exist. Another possibility, which was followed in this study is to use a few terms of the Taylor series of f. Although the Taylor expansion allowed apply the LT-HPM method, we noted that a possible drawback of this strategy is that it may not produce handy approximate solutions, containing more computational requirements. For comparison purposes, we will consider for both cases study, that the “exact” solution is computed using a scheme based on a trapezoid technique combined with a Richardson extrapolation as a build-in routine from Maple 17. Moreover, the mentioned routine was configured using an absolute error (A.E.) tolerance of 10− 12 .

In this study was considered the exponential and sine functions respectively and we saw that the process of getting approximate solutions by using LT-HPM, was unnecessarily long and complicated. In order to deal with the above mentioned problems, this paper introduced NDLT-HPM.

At the first place, we studied a nonlinear second order ordinary differential equation with an exponential nonhomogeneous non-polynomial term. This example, proposed the application of LT-HPM, keeping only three terms of the Taylor expansion of e x from where it was obtained the fourth order approximation (44) and although the final approximation had good accuracy (Figure 1), it is clear that the procedure of solution was cumbersome.

On the other hand, the application of NDLT-HPM to the same problem is outlined in (61), (62), (63), (64) and (65) and can be seen by inspection that exist a considerable saving of computational effort, even NDLT-HPM approximation (66) not only turned out to be clearly shorter than (44), but from the Figures 1 and 2 is scarcely less accurate.
Figure 1

Comparison between numerical solution of ( 21) and LT-HPM, NDLT-HPM approximations ( 44), ( 66).

Figure 2

Absolute Error (A.E.) between numerical solution of (21) and LT-HPM, NDLT-HPM approximations ( 44), ( 66).

Next, we found an approximate solution for the linear third-order equation of variable coefficients, (68) and although we kept only two terms of the Taylor series of sin(x), LT-HPM got a precise approximation (see Figure 3). Iterations (86), (87), (88), (89) and (90) for LT-HPM show again a long computational process compared to NDLT-HPM (110), (111), (112), (113) and (114) but the Figure 3 reveals that in fact, both methods are highly accurate, and although NDLT-HPM is handier its absolute error is again slightly less accurate.

In more precise terms, Figure 2 shows that LT-HPM, NDLT-HPM approximations (44) and (66), are accurate analytical approximate solutions for (21). The biggest absolute error (A.E) of LT-HPM and NDLT-HPM turned out to be 0.000006 and 0.000012 respectively, while from Figure 4 we conclude that the second case study got for the same methods, the values of A.E 0.004 and 0.014. In Spite of this it is noted that NDLT-HPM got a slightly small loss of accuracy with respect to LT-HPM, the comparison of computational effort for both methods leads to the conclusion that NDLT-HPM is more compact, handy and easy to compute, therefore it is an useful tool with good accuracy in the search of solutions for ODES of the type already mentioned.
Figure 3

Comparison between numerical solution of ( 68) and LT-HPM, NDLT-HPM approximations ( 91), ( 115).

Figure 4

Absolute Error (A.E.) between numerical solution of ( 68) and LT-HPM, NDLT-HPM approximations ( 91), ( 115).

Finally, we observe that the proposed homotopy formulations (46) and (93) are something different from the original propose in (9) (Vazquez-Leal et al. 2012b), which shows the richness and flexibility of NDHPM and of course of NDLT-HPM. Indeed, the mentioned homotopies were formulated in this way, with the aim that the computational work was reduced considerably, without losing a great precision in the results. Although it is possible to consider other variants of the homotopy given in (9), the key point is that, in the limit when p → 1, the homotopy equation is reduced to the differential equation to be solved.

Conclusions

In this paper NDLT-HPM was introduced as a useful strategy capable of supporting approximate methods, simplifying mathematical iterative procedure, building handy and easy computable expressions in comparison with LT-HPM, in the search for analytical approximate solutions for linear and nonlinear ordinary differential equations with finite boundary conditions, for the case of equations with nonhomogeneous non-polynomial terms. Moreover, the accuracy of the proposed approximate solutions are in good agreement with the exact solutions.

Such as it was explained, NDLT-HPM method expresses the problem of finding an approximate solution for an ordinary differential equation, in terms of solving an algebraic equation for some unknown initial condition (Filobello-Nino et al. 2013). Figure 1 through Figure 4 show how good this procedure is in the search for analytical approximate solutions with good precision, and a moderate computational effort. In addition, just as with LT-HPM, the proposed method does not need to solve several recurrence differential equations. From all the above, we conclude that NDLT-HPM method is a reliable and precise tool in practical applications.

Declarations

Acknowledgments

We gratefully acknowledge the financial support from the National Council for Science and Technology of Mexico (CONACyT) through grant CB-2010-01 #157024. The authors would like to thank Rogelio-Alejandro Callejas-Molina, and Roberto Ruiz-Gomez for their contribution to this project.

Authors’ Affiliations

(1)
Electronic Instrumentation and Atmospheric Sciences School, Universidad Veracruzana
(2)
Higher Colleges of Technology, Abu Dhabi Men’s College
(3)
National Institute for Astrophysics, Optics and Electronics
(4)
Department of Artificial Intelligence, Universidad Veracruzana
(5)
Civil Engineering School, Universidad Veracruzana

References

  1. Adomian G: A review of decomposition method in applied mathematics. Math Anal Appl 1988, 135: 501-544. 10.1016/0022-247X(88)90170-9View ArticleGoogle Scholar
  2. Aminikhah H: Analytical approximation to the solution of nonlinear Blasius viscous flow equation by LTNHPM. Int Sch Res Netw ISRN Math Anal 2011, 2012(957473):10. doi:10.5402/2012/957473Google Scholar
  3. Aminikhah H: The combined Laplace transform and new homotopy perturbation method for stiff systems of ODE s. Appl Math Model 2012, 36: 3638-3644. 10.1016/j.apm.2011.10.014View ArticleGoogle Scholar
  4. Aminikhan H, Hemmatnezhad M: A novel effective approach for solving nonlinear heat transfer equations. Heat Transfer Asian Res 2012, 41(6):459-466. 10.1002/htj.20411View ArticleGoogle Scholar
  5. Araghi MF, Rezapour B: Application of homotopy perturbation method to solve multidimensional schrodingers equations. Int J Math Arch (IJMA) 2011, 2: 1-6. ISSN 2229–5046Google Scholar
  6. Araghi MF, Sotoodeh M: An enhanced modified homotopy perturbation method for solving nonlinear volterra and fredholm integro-differential equation. World Applied Sciences Journal 2012, 20: 1646-1655.Google Scholar
  7. Assas LMB: Approximate solutions for the generalized K-dV- Burgers’ equation by He’s variational iteration method. Phys Scr 2007, 76: 161-164. doi:10.1088/0031-8949/76/2/008 10.1088/0031-8949/76/2/008View ArticleGoogle Scholar
  8. Babolian E, Biazar J: On the order of convergence of Adomian method. Appl Math Comput 2002, 130(2):383-387. doi:10.1016/S0096-3003(01)00103-5View ArticleGoogle Scholar
  9. Bayat M, Pakar I, Emadi A: Vibration of electrostatically actuated microbeam by means of homotopy perturbation method. Struct Eng Mech 2013, 48: 823-831. 10.12989/sem.2013.48.6.823View ArticleGoogle Scholar
  10. Bayat M, Bayat M, Pakar I: Nonlinear vibration of an electrostatically actuated microbeam. Latin Am J Solids Struct 2014, 11: 534-544. 10.1590/S1679-78252014000300009View ArticleGoogle Scholar
  11. Belendez A, Pascual C, Alvarez ML, Méndez DI, Yebra MS, Hernández A: High order analytical approximate solutions to the nonlinear pendulum by He’s homotopy method. Phys Scr 2009, 79(1):1-24. doi:10.1088/0031-8949/79/01/015009View ArticleGoogle Scholar
  12. Biazar J, Aminikhan H: Study of convergence of homotopy perturbation method for systems of partial differential equations. Comput Math Appl 2009, 58(11–12):2221-2230.View ArticleGoogle Scholar
  13. Biazar J, Eslami M: A new homotopy perturbation method for solving systems of partial differential equations. Comput Math Appl 2012, 62: 225-234.View ArticleGoogle Scholar
  14. Biazar J, Ghanbari B: The homotopy perturbation method for solving neutral functional-differential equations with proportional delays. J King Saud Univ 2012, 24: 33-37. 10.1016/j.jksus.2010.07.026View ArticleGoogle Scholar
  15. Biazar J, Ghazvini H: Convergence of the homotopy perturbation method for partial differential equations. Nonlinear Anal 2009, 10(5):2633-2640. 10.1016/j.nonrwa.2008.07.002View ArticleGoogle Scholar
  16. Chow TL: Classical Mechanics. John Wiley and Sons Inc., New York; 1995.Google Scholar
  17. Chowdhury SH: A comparison between the modified homotopy perturbation method and Adomian decomposition method for solving nonlinear heat transfer equations. J Appl Sci 2011, 11: 1416-1420. doi:10.3923/jas.2011.1416.1420View ArticleGoogle Scholar
  18. Elias D, Khuri SA, Shishen X: An algorithm for solving boundary value problems. J Comput Phys 2000, 159: 125-138. 10.1006/jcph.2000.6452View ArticleGoogle Scholar
  19. El-Shaed M: Application of He’s homotopy perturbation method to Volterra’s integro differential equation. Int J Nonlinear Sci Numerical Simul 2005, 6: 163-168.Google Scholar
  20. Erdogan U, Ozis T: A smart nonstandard finite difference scheme for second order nonlinear boundary value problems. J Comput Phys 2011, 230(17):6464-6474. 10.1016/j.jcp.2011.04.033View ArticleGoogle Scholar
  21. Evans DJ, Raslan KR: The Tanh function method for solving some important nonlinear partial differential. Int J Computat Math 2005, 82: 897-905. doi:10.1080/00207160412331336026 10.1080/00207160412331336026View ArticleGoogle Scholar
  22. Feng X, Mei L, He G: An efficient algorithm for solving Troesch, s problem. Appl Math Comput 2007, 189(1):500-507. 10.1016/j.amc.2006.11.161View ArticleGoogle Scholar
  23. Fereidon A, Rostamiyan Y, Akbarzade M, Ganji DD: Application of He’s homotopy perturbation method to nonlinear shock damper dynamics. Arch Appl Mech 2010, 80(6):641-649. doi:10.1007/s00419-009-0334-x 10.1007/s00419-009-0334-xView ArticleGoogle Scholar
  24. Filobello-Niño U, Vazquez-Leal H, Castañeda-Sheissa R, Yildirim A, Hernandez Martinez L, Pereyra Díaz D, Pérez Sesma A, Hoyos Reyes C: An approximate solution of Blasius equation by using HPM method. Asian J Math Stat 2012, 10. doi:10.3923 /ajms.2012, ISSN 1994–5418Google Scholar
  25. Filobello-Nino U, Vazquez-Leal H, Khan Y, Yildirim A, Pereyra-Diaz D, Perez-Sesma A, Hernandez-Martinez L, Sanchez-Orea J, Castaneda-Sheissa R, Rabago-Bernal F: HPM applied to solve nonlinear circuits: a study case. Appl Math Sci 2012, 6(85–88):4331-4344.Google Scholar
  26. Filobello-Nino U, Vazquez-Leal H, Khan Y, Yildirim A, Jimenez-Fernandez VM, Herrera-May AL, Castaneda-Sheissa R, Cervantes-Perez J: Perturbation method and Laplace–Padé approximation to solve nonlinear problems. Miskolc Math Notes 2013, 14(1):89-101.Google Scholar
  27. Filobello-Niño U, Vazquez-Leal H, Khan Y, Perez-Sesma A, Diaz-Sanchez A, Herrera-May A, Pereyra-Diaz D, Castañeda-Sheissa R, Jimenez-Fernandez VM, Cervantes-Perez J: A handy exact solution for flow due to a stretching boundary with partial slip. Revista Mexicana de Física E 2013, 59(2013):51-55. ISSN 1870–3542Google Scholar
  28. Filobello-Niño U, Vazquez-Leal H, Boubaker K, Khan Y, Perez-Sesma A, Sarmiento Reyes A, Jimenez-Fernandez VM, Diaz-Sanchez A, Herrera-May A, Sanchez-Orea J, Pereyra-Castro K: Perturbation method as a powerful tool to solve highly nonlinear problems: the case of Gelfand,s Equation. Asian J Math and Stat 2013, 2013: 7. doi:10.3923 /ajms.2013, ISSN 1994–5418Google Scholar
  29. Filobello-Nino U, Hector V-L, Brahim B, Luis H-M, Yasir K, Jimenez-Fernandez VM, Herrera-May AL, Roberto C-S, Domitilo P-D, Juan C-P, Jose AA P-S, Hernandez-Machuca SF, Leticia C-H: A handy approximation for a mediated bioelectrocatalysis process, related to Michaelis-Menten equation. Springer Plus 2014, 3: 162. doi:10.1186/2193-1801-3-162View ArticleGoogle Scholar
  30. Filobello-Nino U, Vazquez-Leal H, Khan Y, Perez-Sesma A, Diaz-Sanchez A, Jimenez-Fernandez VM, Herrera-May A, Pereyra-Diaz D, Mendez-Perez JM, Sanchez-Orea J: Laplace transform-homotopy perturbation method as a powerful tool to solve nonlinear problems with boundary conditions defined on finite intervals. Comput Appl Math 2013. ISSN: 0101–8205. doi:10.1007/s40314-013-0073-zGoogle Scholar
  31. Ganji DD, Mirgolbabaei H, Miansari ME, Miansari MO: Application of homotopy perturbation method to solve linear and non-linear systems of ordinary differential equations and differential equation of order three. J Appl Sci 2008, 8: 1256-1261. doi:10.3923/jas.2008.1256.1261View ArticleGoogle Scholar
  32. Hassana HN, El-Tawil MA: An efficient analytic approach for solving two point nonlinear boundary value problems by homotopy analysis method. Math Methods Appl Sci 2011, 34: 977-989. 10.1002/mma.1416View ArticleGoogle Scholar
  33. He JH: A coupling method of a homotopy technique and a perturbation technique for nonlinear problems. Int J Non-Linear Mech 1998, 351: 37-43. doi:10.1016/S0020-7462(98)00085-7Google Scholar
  34. He JH: Homotopy perturbation technique. Comput Methods Applied Mech Eng 1999, 178: 257-262. doi:10.1016/S0045-7825(99)00018-3 10.1016/S0045-7825(99)00018-3View ArticleGoogle Scholar
  35. He JH: A coupling method of a homotopy and a perturbation technique for nonlinear problems. Int J Nonlinear Mech 2000, 35(1):37-43. 10.1016/S0020-7462(98)00085-7View ArticleGoogle Scholar
  36. He JH: Homotopy perturbation method for solving boundary value problems. Phys Lett A 2006, 350(1–2):87-88.View ArticleGoogle Scholar
  37. He JH: Some asymptotic methods for strongly nonlinear equations. Int J Modern Phys B 2006, 20(10):1141-1199. doi:10.1142/S0217979206033796 10.1142/S0217979206033796View ArticleGoogle Scholar
  38. He JH: Variational approach for nonlinear oscillators. Chaos Solitons Fractals 2007, 34: 1430-1439. doi:10.1016/j.chaos.2006.10.026 10.1016/j.chaos.2006.10.026View ArticleGoogle Scholar
  39. Holmes MH: Introduction to Perturbation Methods. Springer-Verlag, New York; 1995.View ArticleGoogle Scholar
  40. Ji Huan H: Some asymptotic methods for strongly nonlinear equations. Int J Modern Phys B 2006, 20(10 (2006)):1141-1199.Google Scholar
  41. Kazemnia M, Zahedi SA, Vaezi M, Tolou N: Assessment of modified variational iteration method in BVPs high-order differential equations. J Appl Sci 2008, 8: 4192-4197. doi:10.3923/jas.2008.4192.4197 10.3923/jas.2008.4192.4197View ArticleGoogle Scholar
  42. Khan Y, Qingbiao W: Homotopy perturbation transform method for nonlinear equations using He’s polynomials. Comput Math Appl 2011, 61(8):1963-1967. 10.1016/j.camwa.2010.08.022View ArticleGoogle Scholar
  43. Khan M, Gondal MA, Iqtadar Hussain S, Vanani K: A new study between homotopy analysis method and homotopy perturbation transform method on a semi infinite domain. Math Comput Model 2011, 55: 1143-1150.View ArticleGoogle Scholar
  44. King AC, Billingham J, Otto SR: Differential Equations, Linear, Nonlinear, Ordinary, Partial. 1st edition. Cambridge University Press, New York; 2003.View ArticleGoogle Scholar
  45. Kooch A, Abadyan M: Evaluating the ability of modified Adomian decomposition method to simulate the instability of freestanding carbon nanotube: comparison with conventional decomposition method. J Appl Sci 2011, 11: 3421-3428. doi:10.3923/jas.2011.3421.3428View ArticleGoogle Scholar
  46. Kooch A, Abadyan M: Efficiency of modified Adomian decomposition for simulating the instability of nano-electromechanical switches: comparison with the conventional decomposition method. Trends Appl Sci Res 2012, 7: 57-67. doi:10.3923/tasr.2012.57.67View ArticleGoogle Scholar
  47. Madani M, Fathizadeh M, Khan Y, Yildirim A: On the coupling of the homotopy perturbation method and Laplace transformation. Math Comput Model 2011, 53(9–10):1937-1945.View ArticleGoogle Scholar
  48. Mahmoudi J, Tolou N, Khatami I, Barari A, Ganji DD: Explicit solution of nonlinear ZK-BBM wave equation using Exp-function method. J Appl Sci 2008, 8: 358-363. doi:10.3923/jas.2008.358.363View ArticleGoogle Scholar
  49. Marinca V, Herisanu N: Nonlinear Dynamical Systems in Engineering. 1st edition. Springer-Verlag Berlin, Heidelberg; 2011.View ArticleGoogle Scholar
  50. Mirmoradia SH, Hosseinpoura I, Ghanbarpour S, Barari A: Application of an approximate analytical method to nonlinear Troesch, s problem. Appl Math Sci 2009, 3(32):1579-1585.Google Scholar
  51. Murray R: Spiegel, Teoría y Problemas de Transformadas de Laplace, primera edición. Serie de compendios Schaum, McGraw-Hill, México; 1988.Google Scholar
  52. Murray JD: Mathematical Biology: I. An Introduction. 3rd edition. Springer-Verlag Berlin Heidelberg, USA; 2002. ISBN 0-387-95223-3Google Scholar
  53. Noorzad R, Tahmasebi Poor A, Omidvar M: Variational iteration method and homotopy-perturbation method for solving Burgers equation in fluid dynamics. J Appl Sci 2008, 8: 369-373. doi:10.3923/jas.2008.369.373View ArticleGoogle Scholar
  54. Patel T, Mehta MN, Pradhan VH: The numerical solution of Burger’s equation arising into the irradiation of tumour tissue in biological diffusing system by homotopy analysis method. Asian J Appl Sci 2012, 5: 60-66. doi:10.3923/ajaps.2012.60.66View ArticleGoogle Scholar
  55. Rashidi MM, Rastegari MT, Asadi M, Bg OA: A study of non-newtonian flow and heat transfer over a non-isothermal wedge using the homotopy analysis method. Chem Eng Commun 2012, 199: 231-256. 10.1080/00986445.2011.586756View ArticleGoogle Scholar
  56. Rashidi M, Pour SM, Hayat T, Obaidat S: Analytic approximate solutions for steady flow over a rotating disk in porous medium with heat transfer by homotopy analysis method. Comput Fluids 2012, 54: 1-9.View ArticleGoogle Scholar
  57. Sharma PR, Methi G: Applications of homotopy perturbation method to partial differential equations. Asian J Math Stat 2011, 4: 140-150. doi:10.3923/ajms.2011.140.150View ArticleGoogle Scholar
  58. Vanani SK, Heidari S, Avaji M: A low-cost numerical algorithm for the solution of nonlinear delay boundary integral equations. J Appl Sci 2011, 11: 3504-3509. doi:10.3923/jas.2011.3504.3509View ArticleGoogle Scholar
  59. Vazquez-Leal H, Khan Y, Fernández-Anaya G, Herrera-May A, Sarmiento-Reyes A, Filobello-Nino U, Jimenez-Fernández VM, Pereyra-Díaz D: A general solution for Troesch’s problem. Math Probl Eng 2012, 208375: 14. doi:10.1155/2012/208375Google Scholar
  60. Vazquez-Leal H, Khan Y, Filobello-Nino U, Sarmiento-Reyes A, Diaz-Sanchez A, Cisneros-Sinencio LF: Fixed-Term Homotopy. J Appl Math 2013, 2013(972704): 11. doi:10.1155/2013/972704Google Scholar
  61. Vazquez-Leal H, Sarmiento-Reyes A, Khan Y, Filobello-Nino U, Diaz-Sanchez A: Rational biparameter homotopy perturbation method and Laplace-Padé coupled version. J Appl Math 2012, 2012(923975):21. doi:10.1155/2012/923975Google Scholar
  62. Vazquez-Leal H, Filobello-Niño U, Castañeda-Sheissa R, Hernandez Martinez L, Sarmiento-Reyes A: Modified HPMs inspired by homotopy continuation methods. Math Probl Eng 2012, 2012(309123):20. doi:10.155/2012/309123Google Scholar
  63. Vazquez-Leal H, Castañeda-Sheissa R, Filobello-Niño U, Sarmiento-Reyes A, Sánchez-Orea J: High accurate simple approximation of normal distribution related integrals. Math Probl Eng 2012, 2012(124029):22. doi:10.1155/2012/124029Google Scholar
  64. Vazquez-Leal H, Hernandez-Martinez L, Khan Y, Jimenez-Fernandez VM, Filobello-Nino U, Diaz-Sanchez A, Herrera-May AL, Castaneda-Sheissa R, Marin-Hernandez A, Rabago-Bernal F, Huerta-Chua J: Multistage HPM applied to path tracking damped oscillations of a model for HIV infection of CD4+ T cells. British J Math Comput Sci 2014, 4(8):1035-1047. 10.9734/BJMCS/2014/7714View ArticleGoogle Scholar
  65. Xu F: A generalized soliton solution of the Konopelchenko-Dubrovsky equation using exp-function method. Zeitschrift für Naturforschung - Section A Journal of Physical Sciences 2007, 62(12):685-688.Google Scholar
  66. Zhang L-N, Xu L: Determination of the limit cycle by He’s parameter expansion for oscillators in a potential. Zeitschrift für Naturforschung - Section A Journal of Physical Sciences 2007, 62(7–8):396-398.Google Scholar
  67. Zill Dennis G: A First Course in Differential Equations with Modeling Applications. 10th edition. Brooks/Cole Cengage Learning, Boston, USA; 2012.Google Scholar

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