Open Access

Direct application of Padé approximant for solving nonlinear differential equations

  • Hector Vazquez-Leal1Email author,
  • Brahim Benhammouda2,
  • Uriel Filobello-Nino1,
  • Arturo Sarmiento-Reyes3,
  • Victor Manuel Jimenez-Fernandez1,
  • Jose Luis Garcia-Gervacio4,
  • Jesus Huerta-Chua5,
  • Luis Javier Morales-Mendoza6 and
  • Mario Gonzalez-Lee6
SpringerPlus20143:563

https://doi.org/10.1186/2193-1801-3-563

Received: 23 June 2014

Accepted: 10 September 2014

Published: 27 September 2014

Abstract

This work presents a direct procedure to apply Padé method to find approximate solutions for nonlinear differential equations. Moreover, we present some cases study showing the strength of the method to generate highly accurate rational approximate solutions compared to other semi-analytical methods. The type of tested nonlinear equations are: a highly nonlinear boundary value problem, a differential-algebraic oscillator problem, and an asymptotic problem. The high accurate handy approximations obtained by the direct application of Padé method shows the high potential if the proposed scheme to approximate a wide variety of problems. What is more, the direct application of the Padé approximant aids to avoid the previous application of an approximative method like Taylor series method, homotopy perturbation method, Adomian Decomposition method, homotopy analysis method, variational iteration method, among others, as tools to obtain a power series solutions to post-treat with the Padé approximant.

AMS Subject Classification

34L30

Keywords

Padé transformNonlinear differential equations

1 Introduction

Solving differential equations is an important issue in sciences because many physical phenomena are modelled using such equations. The Padé method is a well established resummation method from literature. It can increase the domain of convergence of truncate power series (Bararnia et al. 2012; Guerrero et al. 2013; Torabi and Yaghoobi 2011; Vazquez-Leal and Guerrero 2013). It is has been applied to the improve the accuracy of truncated power obtained by power series method (PSM) (Forsyth 1906; Geddes 1979; Ince 1956; Vazquez-Leal and Guerrero 2013), Adomian Decomposition method (ADM) (Wazwaz 2006; Wang et al. 2011), homotopy perturbation method (HPM) (Bararnia et al. 2012; Rashidi and Keimanesh 2010; Torabi and Yaghoobi 2011), homotopy analysis method (HAM) (Guerrero et al. 2013), differential transform method (DTM) (Rashidi and Keimanesh 2010; Rashidi et al. 2010; Rashidi and Pour 2010a, 2010b), among others, during the solution procedure for linear and nonlinear differential equations. Nonetheless, in this work, we propose that the solution of a differential equation can be directly expressed as a rational power series of the independent variable, in other words as a Padé approximant. The proposed procedure will be described by solving several nonlinear problems and comparing results with other semi-analytic methods. The direct application of Padé eradicates the necessity to obtain a power series solution (by some approximative method) to post-treat it with the Padé approximant. Instead, we substitute a Padé approximant of a given order directly to the nonlinear differential equation; it results a residual power series in terms of the independent variable. Next, from the lowest order, we equate each coefficient of such power series to zero, resulting a system of nonlinear algebraic equations (NAEs). Finally, we resolve the NAEs in order to minimize the residual error of the differential equation.

This paper is organized as follows. In Section 2, we introduce the basic concepts of the Padé approximant. Next, the procedure to approximate nonlinear differential equations with Padé is presented in Section 3. In Section 4 some cases study are presented. In Section 5, numerical simulations and a discussion about the results are provided. Finally, a brief conclusion is given in Section 6.

2 Padé approximant

Given an analytical function u(t) with Maclaurin’s expansion
u t = n = 0 u n t n , 0 t T .
(1)
The Padé approximant to u(t) of order [L,M] which we denote by [L/M] u (t) is defined by (Baker 1975)
[ L / M ] u t = p 0 + p 1 t + + p L t L 1 + q 1 t + + q M t M ,
(2)

where we considered q0 = 1, and the numerator and denominator have no common factors.

The numerator and the denominator in (2) 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 t L + M + 1 .
(3)
From (3), we have
u t i = 1 M q i t i - i = 0 L p i t i = O t L + M + 1 .
(4)
From (4), we get the following systems
u L q 1 + + u L - M + 1 q M = - u L + 1 u L + 1 q 1 + + u L - M + 2 q M = - u L + 2 u L + M - 1 q 1 + + u L q M = - u L + M ,
(5)
and
p 0 = u 0 p 1 = u 1 + u 0 q 1 p L = u L + u L - 1 q 1 + + u 0 q L .
(6)

From (5), we calculate first all the coefficients q i ,1 ≤ iM. Then, we determine the coefficient p i ,0 ≤ iL from (6).

Note that for a fixed value of L + M + 1, the error (3) is smallest when the numerator and denominator of (2) have the same degree or when the numerator has degree one higher than the denominator.

3 Padé applied to solve nonlinear differential equations

It can be considered that a nonlinear differential equation can be expressed as
L 1 ( u ) + N ( u ) = 0 , where x Ω ,
(7)
having as boundary condition
B u , u η = 0 , where x Γ ,
(8)

where L1 and N, are a linear and a non-linear operator, respectively; B is a boundary operator, Γ is the boundary of domain Ω, and u/ η denotes differentiation along the normal drawn outwards from Ω.

Now, we assume that the solution for (7) can be written as
u ( x ) = i = 0 L v i ( x - x 0 ) i i = 0 M w i ( x - x 0 ) i ,
(9)

where v0,v1,… and w0,w1,… are unknowns to be determined by the Padé method, L, M are the order of the numerator and denominator, and x0 is an arbitrary constant.

There is not a systematic method to choose the optimal Padé order [L/M] for a given problem. However, usually, a finite number of terms are required in order to obtain a highly accurate Padé approximation. The basic process of direct Padé procedure can be described as:
  1. 1.

    The boundary conditions of (7) are substituted in (9) to generate an equation for each boundary condition. It is important to notice, that there is an algebraic equation for each boundary condition, hence, the rest of equations required to generate a NAEs (with the same number of variables and equations) are obtained from the next step.

     
  2. 2.

    u(x) from (9) is substituted into (7), then, we regroup the resulting equation in terms of the x-powers. It is important to notice that the operators L 1 and N will be applied to u(x). After this, the regrouping procedure will include the eradication of the denominator terms emanated from the Padé approximant (9). In this way, the resulting expression is a power series that represents the residual error of the differential equation (7).

     
  3. 3.

    In order to reduce the residual error; from the lowest order, we equate each coefficient of the x-powers in the resulting residual power series to zero to obtain an algebraic equation in terms of the unknown coefficients of (9).

     
  4. 4.

    Aforementioned steps generates a NAEs in terms of the unknowns from (9).

     
  5. 5.

    Finally, we solve the NAEs to obtain v 0,v 1,… and w 0,w 1,….

     

4 Cases study

In this section, we will solve several nonlinear problems of different types to show the validity and power of the direct application of Padé method to solve a broad spectrum of equations.

4.1 A boundary value problem

The Troesch’s equation is a boundary value problem (BVP) derived from research on the confinement of a plasma column by radiation pressure (Weibel 1959) and also from the theory of gas porous electrodes (Gidaspow and Baker 1973; Markin et al. 1966). The problem is expressed as
y = n sinh ( ny ) , y ( 0 ) = 0 , y ( 1 ) = 1 ,
(10)

where prime denotes differentiation with respect to x and n is known as Troesch’s parameter.

In order to facilitate the application of Padé method, we convert the hyperbolic-type nonlinearity from Troesch’s problem into a polynomial type nonlinearity (Chang 2010; Vazquez-Leal et al. 2012c), using the variable transformation
u ( x ) = tanh n 4 y ( x ) .
(11)
After using (11), we obtain the following transformed problem
1 - u 2 u + 2 u ( u ) 2 - n 2 u ( 1 + u 2 ) = 0 ,
(12)

where conditions are obtained by using variable transformation (11).

Then, substituting original boundary conditions y(0) = 0 and y (1) = 1 into (11), results
u ( 0 ) = 0 , and u ( 1 ) = tanh n 4 .
(13)
We suppose that solution for (12) has the following rational expression
u ( x ) = i = 0 L v i ( x - x 0 ) i i = 0 M w i ( x - x 0 ) i ,
(14)

where w0 = 1, x0 = 0, and L = M = 8.

Substituting (14) into (12), rearranging and equating terms having the same x-powers, we obtain
2 v 2 - 2 v 1 w 1 + 2 v 0 w 1 2 - 2 v 0 w 2 + 2 v 0 3 w 2 + 2 v 0 v 1 2 - 2 v 0 2 v 1 w 1 - 2 v 0 2 v 2 - n 2 v 0 3 - n 2 v 0 + 12 v 4 - 4 v 2 w 2 + 12 v 3 w 1 - 12 v 1 w 3 + 4 v 0 w 2 2 - 12 v 0 w 4 - 8 v 1 w 1 w 2 + 8 v 2 v 1 2 - 12 v 4 v 0 2 + 12 v 0 3 w 4 + 4 v 2 2 v 0 + 8 v 0 w 1 2 w 2 + 12 v 1 v 0 2 w 3 - 4 v 0 2 v 2 w 2 - 8 v 2 v 1 v 0 w 1 - 12 v 3 v 0 2 w 1 - v 2 n 2 - 3 n 2 v 0 w 2 - 3 v 2 v 0 2 n 2 - 3 n 2 v 0 w 1 2 - 3 v 1 2 v 0 n 2 - 3 v 1 n 2 w 1 - n 2 v 0 3 w 2 - 3 v 1 v 0 2 n 2 w 1 x + = 0 .
(15)
Next, equating coefficients of x in (15) to zero, we obtain the following system of nonlinear algebraic equations
x 0 : 2 v 2 - 2 v 1 w 1 + 2 v 0 w 1 2 - 2 v 0 w 2 + 2 v 0 3 w 2 + 2 v 0 v 1 2 - 2 v 0 2 v 1 w 1 - 2 v 0 2 v 2 - n 2 v 0 3 - n 2 v 0 = 0 , x 1 : 12 v 4 - 4 v 2 w 2 + 12 v 3 w 1 - 12 v 1 w 3 + 4 v 0 w 2 2 - 12 v 0 w 4 - 8 v 1 w 1 w 2 + 8 v 2 v 1 2 - 12 v 4 v 0 2 + 12 v 0 3 w 4 + 4 v 2 2 v 0 + 8 v 0 w 1 2 w 2 + 12 v 1 v 0 2 w 3 - 4 v 0 2 v 2 w 2 - 8 v 2 v 1 v 0 w 1 - 12 v 3 v 0 2 w 1 - v 2 n 2 - 3 n 2 v 0 w 2 - 3 v 2 v 0 2 n 2 - 3 n 2 v 0 w 1 2 - 3 v 1 2 v 0 n 2 - 3 v 1 n 2 w 1 - n 2 v 0 3 w 2 - 3 v 1 v 0 2 n 2 w 1 = 0 ,
(16)
Now, in order to consider the boundary conditions (13), we substitute them into (14) to obtain
v 0 = 0 , i = 0 8 v i 1 + i = 1 8 w i = tanh n 4 ,
(17)

corresponding to u (0) = 0 and u ( 1 ) = tanh n 4 , respectively.

Then, solving the system composed by (16) and (17), it results
v 0 = 0 , v 1 = . 119880474427 , v 2 = . 380280473821 , v 3 = . 564352544936 , v 4 = 0.706909216018 × 1 0 - 2 , v 5 = 0.0129612620679 , v 6 = - 0.591199285830 × 1 0 - 4 , v 7 = 0.896122363451 × 1 0 - 5 , v 8 = - 6.33691960724 × 1 0 - 7 , w 1 = 3.17216357074 , w 2 = 4.67075066552 , w 3 = - 0.0580095782038 , w 4 = - 0.0645500498446 , w 5 = 0.286395017212 e - 3 , w 6 = 0.449835719687 × 1 0 - 3 , w 7 = - 0.159764848819 × 1 0 - 5 , w 8 = - 0.459066511294 × 1 0 - 5 ,
(18)
and
v 0 = 0 , v 1 = 0.211300671328 , v 2 = 0 , v 3 = 0.0205074952135 , v 4 = 0 , v 5 = - 0.214876109142 × 1 0 - 4 , v 6 = 0 , v 7 = - 0.111221356703 × 1 0 - 4 , v 8 = 0 , w 1 = 0 , w 2 = - 0.0547303880429 , w 3 = 0 , w 4 = 0.109464583871 × 1 0 - 2 , w 5 = 0 , w 6 = - 0.270186358901 × 1 0 - 4 , w 7 = 0 , w 8 = - 3.83815897415 × 1 0 - 7 ,
(19)

for n = 0.5 and n = 1, respectively.

Finally, from (11) and (14), the proposed solution of Troesch’s problem is
y ( x ) = 4 n tanh - 1 i = 0 8 v i x i i = 0 8 w i x i , 0 x 1 ,
(20)

where (18) or (19) are used depending on the value of n.

4.2 Differential-algebraic equation

Consider the index one differential-algebraic equation system (DAEs) (Amat et al. 2012)
y - z = 0 , y ( 0 ) = 2 2 , y 2 + z 2 - 1 = 0 , z ( 0 ) = 2 2 ,
(21)
where prime denotes derivative with respect to t, and the exact solution is
y ( t ) = sin t + π 4 , z ( t ) = cos t + π 4 .
(22)
We suppose that solution for (21) has the following rational form
y ( t ) = i = 0 L 1 v 1 , i t i i = 0 M 1 w 1 , i ( t - t 0 ) i , z ( t ) = i = 0 L 2 v 2 , i t i i = 0 M 2 w 2 , i ( t - t 0 ) i ,
(23)

where w1,0 and w2,0 are considered as 1 to simplify the process of solution, and t0 = 0.

If we consider L1 = M1 = L2 = M2 = 12, and substituting (23) into (21); rearranging and equating terms having the same t-powers, we obtain
v 1 , 1 - v 2 , 0 - v 1 , 0 w 1 , 1 + - v 2 , 1 - 2 v 1 , 0 w 1 , 2 - v 1 , 0 w 1 , 1 w 2 , 1 + v 1 , 1 w 2 , 1 - 2 v 2 , 0 w 1 , 1 + 2 v 1 , 2 t + = 0 , - 1 + v 2 , 0 2 + v 1 , 0 2 + - 2 w 1 , 1 + 2 v 2 , 0 2 w 1 , 1 + 2 v 1 , 0 2 w 2 , 1 + 2 v 2 , 0 v 2 , 1 + 2 v 1 , 0 v 1 , 1 - 2 w 2 , 1 t + = 0 .
(24)
Next, equating coefficients of t in (24) to zero, we obtain the following system of nonlinear algebraic equations
t 0 : v 1 , 1 - v 2 , 0 - v 1 , 0 w 1 , 1 = 0 , t 1 : - v 2 , 1 - 2 v 1 , 0 w 1 , 2 - v 1 , 0 w 1 , 1 w 2 , 1 + v 1 , 1 w 2 , 1 - 2 v 2 , 0 w 1 , 1 + 2 v 1 , 2 = 0 , t 0 : - 1 + v 2 , 0 2 + v 1 , 0 2 = 0 , t 1 : - 2 w 1 , 1 + 2 v 2 , 0 2 w 1 , 1 + 2 v 1 , 0 2 w 2 , 1 + 2 v 2 , 0 v 2 , 1 + 2 v 1 , 0 v 1 , 1 - 2 w 2 , 1 = 0 .
(25)
Now, in order to consider the initial conditions from (21), we substitute them into (23) to obtain
v 1 , 0 = 2 2 , v 2 , 0 = 2 2 ,
(26)

corresponding to y ( 0 ) = 2 2 and z ( 0 ) = 2 2 , respectively.

Solving the NAEs composed by (25) and (26), it results the coefficients shown in Table 1.
Table 1

Coefficients from Padé approximant ( 27) for DAEs ( 21)

i

v 1,i

w 1,i

v 2,i

w 2,i

0

0.7071067812

1

0.7071067812

1

1

0.6944478949

-0.01790236873

-0.710785391

-0.005202339869

2

-0.3552705766

0.015473901

-0.3381563291

0.01657239326

3

-0.1007688245

-0.0002670569686

0.1079008196

-0.0001006409133

4

0.02599872528

0.0001213625776

0.02316102487

0.0001405811897

5

0.003720336719

-1.958219599e-06

-0.004157250847

-9.478457134e-07

6

-0.0006440541915

6.292610474e-07

-0.0005234980593

7.978499480e-07

7

-5.257637767e-05

-9.004870369e-09

6.112387339e-05

-5.582330983e-09

8

6.857858371e-06

2.327029094e-09

4.879283240e-06

3.277470106e-09

9

3.140311387e-07

-2.662291749e-11

-3.785438873e-07

-2.120465882e-11

10

-3.286935291e-08

5.983862645e-12

-1.897258525e-08

9.536760323e-12

11

-6.790217810e-10

-4.175830173e-14

8.457510263e-10

-4.313868912e-14

12

5.933734260e-11

8.658872139e-15

2.337878489e-11

1.599248188e-14

From (23) and Table 1, the proposed solution is
y ( t ) = i = 0 12 v 1 , i t i i = 0 12 w 1 , i t i , z ( t ) = i = 0 12 v 2 , i t i i = 0 12 w 2 , i t i ,
(27)

4.3 Asymptotic problem

The quadratic Riccati equation is a well known, and difficult to solve, asymptotic problem for approximative methods (Abbasbandy 2006, 2007; Tan and Abbasbandy 2008; Tsai and Chen 2010). The problem is expressed as follows
Y - 2 Y + Y 2 - 1 = 0 , Y ( 0 ) = 0 ,
(28)
where prime denotes differentiation with respect to t. The exact solution of (28), was found to be
Y ( t ) = 1 + 2 tanh 2 t + 1 2 log 2 - 1 2 + 1 .
(29)
We suppose that solution for (28) has the following rational form
Y ( t ) = i = 0 L v i ( t - t 0 ) i i = 0 M w i ( t - t 0 ) i ,
(30)

where w0 = 1, t0 = 0, and L = M = 4.

Substituting (30) into (28), rearranging and equating terms having the same t-powers, we obtain the following system of equations
t 0 : - 1 + v 1 - v 0 w 1 - 2 v 0 + v 0 2 = 0 , t 1 : - 4 w 1 + 2 v 0 v 1 + 2 v 2 - 2 v 0 w 2 - 6 v 0 w 1 - 2 v 1 + 2 v 1 w 1 - 2 v 0 w 1 2 + 2 w 1 v 0 2 = 0 ,
(31)
In order to consider the initial condition of Y (0) = 0, we substitute it into (30) to obtain
v 0 w 0 = 0 ,
(32)
Then, using (32) and (31) and solving, results
Y ( t ) = t + 0.19047619 t 3 1 - t + 0.85714286 t 2 - 0.19047619 t 3 + 0.038095238 t 4 .
(33)
Now, we will obtain another Padé approximant from (29), for the expansion point Y (1.7) = 2.28577828560. Therefore, using such point as initial condition, we generate the following extra equation
v 0 w 0 = 2.28577828560 ,
(34)

Next, using (31) and (34) to obtain the coefficients from Padé expression (30), and substituting t by expansion point (t-1.7), results

[!b]
Y ( t ) = - 3.3000449 + 3.2857784 t + 1.9592386 t - 1.7 2 + 0.62586255 t - 1.7 3 + 0.087077272 t - 1.7 4 - 1.1858233 + 1.2857784 t + 0.85714286 t - 1.7 2 + 0.24491017 t - 1.7 3 + 0.038095238 t - 1.7 4 .
(35)
We can see in Figure 1 a comparison between (33) and (35) to exact solution (29). It results that changing the expansion point was useful to increase the domain of convergence of the Padé method for this case study. However, a systematic procedure to choose the optimal expansion point for general problems is a pending task for future research.
Figure 1

Exact solution ( 29) (solid line), Padé approximations ( 33) (diamonds), ( 35) (circles), and a 250 terms power series solution (dash-dot).

5 Numerical simulation and discussion

On one side, semi-analytic methods like: generalized homotopy method (GHM) (Vazquez-Leal 2013), homotopy perturbation method (Araghi and Rezapour 2011; Araghi and Sotoodeh 2012; Bayat et al. 2013, 2014; Biazar and Eslami 2011; Biazar and Ghanbari 2012; Filobello-Nino et al. 2012a, 2012b; He 1999, 2009; Khan et al. 2012a, 2012b; Vazquez-Leal 2012; Vazquez-Leal et al. 2012a, 2012b, 2012d), homotopy analysis method (Hassana and El-Tawil 2011; He 2004; Rashidi et al2012a,2012b; Tan and Abbasbandy2008), variational iteration method (Abbasbandy2007; Chang 2010; Khan et al. 2012c), among others (Khan et al. 2012d), need an initial approximation for the sought solutions and the calculus of one or several adjustment parameters. If the initial approximation is properly chosen, the results can be highly accurate, nonetheless, there is not a general method to choose such initial approximation. This issue motivates the use of adjustment parameters obtained by minimizing the least-squares error with respect to the numerical solution. On the other side, the Padé method obtain its coefficients using a straightforward procedure. Furthermore, at least for low-order approximations, the solution can be easily obtained using the “solve” or “fsolve” commands of MAPLE or equivalent routines from Mathematica or MATLAB.

We presented several cases study to show the successful use of the Padé method to solve directly a wide variety of nonlinear problems. For instance, the Troesch’s BVP problem is a benchmark equation for numerical (Erdogan and Ozis 2011; Lin et al. 2008) and semi-analytical methods (Chang 2010; Deeba 2000; Feng et al. 2007; Hassana and El-Tawil 2011; Khuri 2003; Mirmoradia et al. 2009; Vazquez-Leal et al. 2012c) due to the numerical problems to solve it. Nevertheless, as shown in Table 2, the Padé approximation (20) is exact for n=0.5 compared to the numerical solution reported in (Erdogan and Ozis 2011; Lin et al. 2008). This result is relevant considering the high error values of the solutions reported using other semi-analytical methods: homotopy perturbation method (HPM) (Feng et al. 2007; Mirmoradia et al. 2009; Vazquez-Leal et al. 2012c), Adomian Decomposition method (ADM) (Deeba et al. 2000), homotopy analysis method (HAM) (Hassana and El-Tawil 2011) and Laplace decomposition transform method (LDTM) (Khuri 2003). All of them possess an average absolute relative error (A.A.R.E.) significantly larger that our results. A similar result was found for n=1 as presented in Table 3. Therefore, the direct Padé method can, potentially, be an excellent tool to solve nonlinear BVP problems described over finite intervals. It is important to remark that for boundary conditions over finite intervals, the traditional Padé approximant applied to the power series of the exact solution or to the exact solution, can only guarantee one boundary condition (traditionally at x=0). However, the proposed method build a restriction equation for each non-singular boundary conditions over the finite interval. Such equations are part of the NAEs that is resolved to provide the coefficients of the Padé approximant. Therefore, the resulting modified Padé expression fulfils all the boundary conditions.
Table 2

Comparison between ( 20), exact solution (Erdogan and Ozis2011; Lin et al.2008), and other reported approximate solutions

x

Exact

This work

HPM

ADM

HPM

HPM

HAM

LDTM

 

(Erdogan and Ozis2011; Lin et al.2008)

(20)

(Vazquez-Leal et al.2012c)

(Deeba et al.2000)

(Feng et al.2007)

(Mirmoradia et al.2009)

(Hassana and El-Tawil2011)

(Khuri2003)

0.1

0.0959443493

0.0959443493

0.0959443155

0.0959383534

0.0959395656

0.095948026

0.0959446190

0.0959443520

0.2

0.1921287477

0.1921287477

0.1921286848

0.1921180592

0.1921193244

0.192135797

0.1921292845

0.1921287539

0.3

0.2887944009

0.2887944009

0.2887943176

0.2887803297

0.2887806940

0.288804238

0.2887952148

0.2887944107

0.4

0.3861848464

0.3861848464

0.3861847539

0.3861687095

0.3861675428

0.386196642

0.3861859313

0.3861848612

0.5

0.4845471647

0.4845471647

0.4845470753

0.4845302901

0.4845274183

0.4845599

0.4845485110

0.4845471832

0.6

0.5841332484

0.5841332484

0.5841331729

0.5841169798

0.5841127822

0.584145785

0.5841348222

0.5841332650

0.7

0.6852011483

0.6852011483

0.6852010943

0.6851868451

0.6851822495

0.685212297

0.6852028604

0.6852011675

0.8

0.7880165227

0.7880165227

0.7880164925

0.7880055691

0.7880018367

0.788025104

0.7880181729

0.7880165463

0.9

0.8928542161

0.8928542161

0.8928542059

0.8928480234

0.8928462193

0.892859085

0.8928553997

0.8928542363

 

Order

[12/12]

2

6

2

2

6

3

 

A.A.R.E.

0

1.83327e(-07)

3.47802e(-05)

3.57932e(-05)

2.44418e(-05)

2.51374e(-06)

3.10957e(-08)

Calculated for n = 0.5.

Table 3

Comparison between ( 20), exact solution (Erdogan and Ozis2011; Lin et al.2008), and other reported approximate solutions

x

Exact

This work

HPM

ADM

HPM

HPM

HAM

LDTM

 

(Erdogan and Ozis2011)

(20)

(Vazquez-Leal et al.2012c)

(Deeba et al.2000)

(Feng et al.2007)

(Mirmoradia et al.2009)

(Hassana and El-Tawil2011)

(Khuri2003)

0.1

0.0846612565

0.0846612565

0.08466075858

0.084248760

0.0843817004

0.084934415

0.0846732692

0.08466308972

0.2

0.1701713582

0.1701713582

0.1701704581

0.169430700

0.1696207644

0.170697546

0.1701954538

0.1701750442

0.3

0.2573939080

0.2573939081

0.2573927827

0.256414500

0.2565929224

0.258133224

0.2574302342

0.2573994845

0.4

0.3472228551

0.3472228551

0.3472217324

0.346085720

0.3462107378

0.348116627

0.3472715981

0.3472303763

0.5

0.4405998351

0.4405998352

0.4405989511

0.439401985

0.4394422743

0.44157274

0.4406610140

0.4406093753

0.6

0.5385343980

0.5385343981

0.5385339413

0.537365700

0.5373300622

0.539498234

0.5386072529

0.5385460046

0.7

0.6421286091

0.6421286092

0.6421286573

0.641083800

0.6410104651

0.642987984

0.7526899495

0.6421421393

0.8

0.7526080939

0.7526080940

0.7526085475

0.751788000

0.7517335467

0.753267551

0.7526899495

0.7526226886

0.9

0.8713625196

0.8713625198

0.8713630450

0.870908700

0.8708835371

0.871733059

0.8714249118

0.8713748860

 

Order

[12/12]

2

6

2

2

6

3

 

A.A.R.E.

1.46588e(-10)

2.54568e(-06)

0.002714577

0.002320107

0.002044737

0.019244326

2.05e(-05)

Calculated for n = 1.

Padé approximation (27) of DAEs problem (21) exhibited highly accurate results for a long period of time as depicted in Figure 2 and Table 4. The differential-algebraic nonlinear problems are of relevance on several fields of science, including microelectronics and chemistry. In addition, there is not any standard analytical method to solve this type of equations, this is what it makes the Padé method in an attractive tool to obtain approximate solutions for DAEs problems. Furthermore, the solution procedure of (21) shows that is possible - potentially - to approximate a wide variety of problems containing several variables.
Figure 2

Exact solution ( 22) (solid circles) of DAEs ( 21) and Padé approximation ( 27) (solid line): a) y ( t ) and b) z ( t ).

Table 4

Relative error (R.E.) of exact solution ( 22) versus Padé approximation ( 27)

t

Exacty(t)

Exactz(t)

R.E.y(t)of (27)

R.E.z(t)of (27)

-10

-0.2086321515

-0.9779941847

0.09330825406

0.09330825406

-9

-0.9356781623

-0.3528546112

0.002548752664

0.002548752664

-8

-0.8024659858

0.5966978646

0.0002522953745

0.0002522953745

-7

0.0685297173

0.9976490755

0.0001597267828

0.0001597267828

-6

0.8765195143

0.4813663272

3.803747675e-07

3.803747675e-07

-5

0.8786413122

-0.4774824024

5.388747443e-09

5.388747443e-09

-4

0.0729443397

-0.9973360132

3.131783529e-10

3.131783529e-10

-3

-0.7998173223

-0.6002434930

2.587566848e-14

2.587566848e-14

-2

-0.9372306267

0.3487101265

9.922514670e-19

9.922514670e-19

-1

-0.2129584152

0.9770612639

1.391674918e-25

1.391674918e-25

0

0.7071067812

0.7071067812

0.0000000000

0.0000000000

1

0.9770612639

-0.2129584152

2.900333665e-26

2.900333665e-26

2

0.3487101265

-0.9372306267

2.436171789e-18

2.436171789e-18

3

-0.6002434930

-0.7998173223

3.003755589e-14

3.003755589e-14

4

-0.9973360132

0.0729443397

1.897834931e-11

1.897834931e-11

5

-0.4774824024

0.8786413122

7.783182416e-09

7.783182416e-09

6

0.4813663272

0.8765195143

5.122538684e-07

5.122538684e-07

7

0.9976490755

0.0685297173

7.591788287e-06

7.591788287e-06

8

0.5966978646

-0.8024659858

0.0002175967642

0.0002175967642

9

-0.3528546112

-0.9356781623

0.003968289586

0.003968289586

10

-0.9779941847

-0.2086321515

0.01052925646

0.01052925646

The accuracy of approximations (33) and (35) for the quadratic Riccati problem (28) is depicted in Figure 1. Moreover, we have suggested a strategy to increase the domain of convergence of the Padé method by changing its expansion point. As depicted in Figure 1, the approximation (35) obtained by expanding at t = 1.7 is far more accurate than (33) obtained by expanding at t = 1.7. It is important to notice that the expansion point was arbitrary choose for this case study; therefore, further work is required to deduce a systematic algorithm to choose optimal expansion points. Furthermore, in (Abbasbandy 2006) was reported a power series solution for the same equation with poor convergence, making necessary to solve the problem by a multi-stage version of HPM method. The advantage of our solution, in this case, is that we do not need to use a complicated segmented method; therefore, this approach generates simpler solutions. In addition, in (Tsai and Chen 2010) was reported the combination of Laplace Adomian Decomposition Method with Padé (LADM-Padé) of order [13/12] to obtain a similar result to our [4/4] order Padé solution. Furthermore, in (Abbasbandy 2007) was reported a power series solutions with short domain of convergence. A HAM solution in terms of exponential expressions was reported in (Tan and Abbasbandy 2008), presenting a high accurate solution with a larger domain than the proposed solution; for this case, we can increase the order of the Padé approximation to obtain a good agreement with HAM solution. Moreover, in order to show the advantage of the proposed method, we calculated 250 terms of the power series solution using the well established series method (using the command dsolve of Maple 16), resulting a poor region of convergence, followed by (33). Finally, as depicted in Figure 1, the best domain of convergence was obtained from the Padé approximant (35) due to the expansion point change.

The direct application of the Padé approximant to obtain rational solutions of nonlinear differential equations circumvent the old requirement of using Taylor series method (Vazquez-Leal et al. 2014), HPM, VIM, HAM, DTM, PSM, ADM and others, as tools to obtain a power series solutions to post-process later by the application Padé approximant. Therefore, this new straightforward methodology reduce the computational effort producing good results.

In general terms, we know from literature (Bararnia et al. 2012; Guerrero et al.2013; Torabi and Yaghoobi 2011; Vazquez-Leal and Guerrero 2013) that larger values for M and L, can lead to better results for Padé approximant, this considering that we count in advance with a suitable power series (large enough) obtained using an extra approximative method as aforementioned. Then, our proposal has a strong advantage because we do not require a power series to post-process with Padé approximant, because the method consist in the direct application of Padé. However, a systematic procedure to obtain the optimal order [L/M] is still a pending issue to study in a future research derived from this paper. Finally, in the present study, we restricted the research to nonsingular initial conditions and Dirichlet finite interval boundary conditions; nonetheless, further work is required to deal with singular initial condition problems, Neumann boundary conditions, infinity boundary conditions, among others.

6 Conclusions

This work presented the direct application of Padé method as a technique with high potential to solve nonlinear differential equations. Also, a comparison between the results of applying the proposed procedure and other semi-analytical was shown. The results showed that Padé is a powerful method to solve different nonlinear equations like the ones for: boundary value problems, differential-algebraic problems, and asymptotic problems. The method provided better results than many of the most used methods like: HPM, ADM, HAM, DTM, VIM, PSM, among others. Finally, further research should be performed to solve other kind of problems as: nonlinear fractional/partial differential equations, Pantograph equations, among others.

Declarations

Acknowledgements

We gratefully acknowledge the financial support of 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)
Dirección General de la Unidad de Estudios de Posgrado, Universidad Veracruzana
(5)
Facultad de Ingenieria Civil, Universidad Veracruzana
(6)
Department of Electronics Engineering, Universidad Veracruzana

References

  1. Abbasbandy S: Homotopy perturbation method for quadratic riccati differential equation and comparison with adomian’s decomposition method. Appl Math Comput 2006, 172(1):485-490. 10.1016/j.amc.2005.02.014View ArticleGoogle Scholar
  2. Abbasbandy S: A new application of he’s variational iteration method for quadratic riccati differential equation by using adomian’s polynomials. J Comput Appl Math 2007, 207(1):59-63. 10.1016/j.cam.2006.07.012View ArticleGoogle Scholar
  3. Araghi MF, Rezapour B: Application of homotopy perturbation method to solve multidimensional schrodinger’s equations. Int J Math Archive (IJMA) ISSN 2229-5046 2011, 2(11):1-6.Google Scholar
  4. Araghi MF, Sotoodeh M: An enhanced modified homotopy perturbation method for solving nonlinear volterra and fredholm integro-differential equation. World Appl Sci J 2012, 20(12):1646-1655.Google Scholar
  5. Amat S, Legaz MJ, Pedregal P: On a newton-type method for differential-algebraic equations. J Appl Math 2012, 2012: 1-15.View ArticleGoogle Scholar
  6. Baker GA: Essentials of Padé approximants. Academic, New York; 1975.Google Scholar
  7. Bararnia H, Ghasemi E, Soleimani S, Ghotbi AR, Ganji DD: Solution of the falkner-skan wedge flow by hpm-pade method́. Adv Eng Softw 2012, 43(1):44-52. 10.1016/j.advengsoft.2011.08.005View ArticleGoogle Scholar
  8. Bayat M, Pakar I, Emadi A: Vibration of electrostatically actuated microbeam by means of homotopy perturbation method. Struct Eng Mech 2013, 48(6):823-831. 10.12989/sem.2013.48.6.823View ArticleGoogle Scholar
  9. 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
  10. Biazar J, Eslami M: A new homotopy perturbation method for solving systems of partial differential equations. Comput Math Appl 2011, 62(1):225-234. 10.1016/j.camwa.2011.04.070View ArticleGoogle Scholar
  11. Biazar J, Ghanbari B: The homotopy perturbation method for solving neutral functional-differential equations with proportional delays. J King Saud Univ - Sci 2012, 24(1):33-37. 10.1016/j.jksus.2010.07.026View ArticleGoogle Scholar
  12. Chang SH: A variational iteration method for solving troesch’s problem. J Comput Appl Math 2010, 234(10):3043-3047. 10.1016/j.cam.2010.04.018View ArticleGoogle Scholar
  13. Deeba E, Khuri SA, Xie S: An algorithm for solving boundary value problems. J Comput Phys 2000, 159: 125-138. 10.1006/jcph.2000.6452View ArticleGoogle Scholar
  14. 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
  15. Forsyth AR: Theory of differential equations. University Press: Cambridge, Cambridge; 1906.Google Scholar
  16. 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
  17. Filobello-Nino U, Vazquez-Leal H, Khan Y, Castaneda-Sheissa R, Yildirim A, Hernandez-Martinez L, Sanchez-Orea J, Castaneda-Sheissa R, Bernal FR: Hpm applied to solve nonlinear circuits: a study case. Appl Math Sci 2012a, 6(85-88):4331-4344.Google Scholar
  18. Filobello-Nino U, Vazquez-Leal H, Castaneda-Sheissa R, Yildirim A, Hernandez-Martinez L, Pereyra-Diaz D, Perez-Sesma A, Hoyos-Reyes C: An approximate solution of blasius equation by using hpm method. Asian J Math Stat 2012b, 5: 50-59. 10.3923/ajms.2012.50.59View ArticleGoogle Scholar
  19. Gidaspow D, Baker B: A model for discharge of storage batteries. J Electrochem Soc 1973, 120: 1005-1010. 10.1149/1.2403617View ArticleGoogle Scholar
  20. Geddes K: Convergence behaviour of the newton iteration for first order differential equations. Proc EUROSAM 1979, 72: 189-199.Google Scholar
  21. Guerrero F, Santonja F, Villanueva R: Solving a model for the evolution of smoking habit in spain with homotopy analysis method. Nonlinear Anal: Real World Appl 2013, 14(1):549-558. 10.1016/j.nonrwa.2012.07.015View ArticleGoogle Scholar
  22. 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
  23. He J-H: Homotopy perturbation technique. Comput Meth Appl Mech Eng 1999, 178(3-4):257-262. 10.1016/S0045-7825(99)00018-3View ArticleGoogle Scholar
  24. He J-H: Comparison of homotopy perturbation method and homotopy analysis method. Appl Math Comput 2004, 156(2):527-539. 10.1016/j.amc.2003.08.008View ArticleGoogle Scholar
  25. He J-H: An elementary introduction to the homotopy perturbation method. Comput Math Appl 2009, 57(3):410-412. 10.1016/j.camwa.2008.06.003View ArticleGoogle Scholar
  26. Ince EL: Ordinary differential equations. Dover Publications: New York 1956, 189-199.Google Scholar
  27. Khuri S: A numerical algorithm for solving troesch’s problem. Int J Comput Math 2003, 80(4):493-498. 10.1080/0020716022000009228View ArticleGoogle Scholar
  28. Khan Y, Vazquez-Leal H, Hernandez-Martinez L: Removal of noise oscillation term appearing in the nonlinear equation solution. J Appl Math 2012a, 2012: 1-9. doi:10.1155/2012/387365.View ArticleGoogle Scholar
  29. Khan Y, Vazquez-Leal H, Wu Q: An efficient iterated method for mathematical biology model. Neural Comput Appl 2012b, 23(3-4):1-6.Google Scholar
  30. Khan Y, Vazquez-Leal H, Hernandez-Martinez L, Faraz N: Variational iteration algorithm-ii for solving linear and non-linear odes. Int J Phys Sci 2012c, 7(25):3099-4002.View ArticleGoogle Scholar
  31. Khan Y, Vazquez-Leal H, Faraz N: An auxiliary parameter method using adomian polynomials and laplace transformation for nonlinear differential equations. Appl Math Model 2012d, 37(5):2702-2708.View ArticleGoogle Scholar
  32. Lin Y, Enszer JA, Stadtherr MA: Enclosing all solutions of two-point boundary value problems for odes. Comput Chem Eng 2008, 32(8):1714-1725. 10.1016/j.compchemeng.2007.08.013View ArticleGoogle Scholar
  33. Markin V, Chernenko A, Chizmadehev Y, Chirkov Y: Aspects of the theory of gas porous electrodes. In Fuel Cells: Their Electrochemical Kinetics. Edited by: Bagotskii VS, Vasilev YB. New York: Consultants Bureau; 1966:22-33.Google Scholar
  34. Mirmoradia S, 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
  35. Rashidi MM, Keimanesh M: Using differential transform method and padé approximant for solving mhd flow in a laminar liquid film from a horizontal stretching surface. Math Problems Eng 2010, 2010(Article ID 491319): 14.Google Scholar
  36. Rashidi MM, Laraqi N, Sadri SM: A novel analytical solution of mixed convection about an inclined flat plate embedded in a porous medium using the dtm-padé. Int J Thermal Sci 2010, 49(12):2405-2412. 10.1016/j.ijthermalsci.2010.07.005View ArticleGoogle Scholar
  37. Rashidi MM, Pour SAM: A novel analytical solution of steady flow over a rotating disk in porous medium with heat transfer by dtm-padé. Afr J Math Comput Sci Res 2010a, 3(6):93-100.Google Scholar
  38. Rashidi MM, Pour SAM: Explicit solution of axisymmetric stagnation flow towards a shrinking sheet by dtm-padé. Appl Math Sci 2010b, 4(53):2617-2632.Google Scholar
  39. Rashidi MM, Rastegari MT, Asadi M, Bég OA: A study of non-newtonian flow and heat transfer over a non-isothermal wedge using the homotopy analysis method. Chem Eng Commun 2012a, 199(2):231-256. 10.1080/00986445.2011.586756View ArticleGoogle Scholar
  40. Rashidi MM, Pour SAM, 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 2012b, 54(0):1-9.View ArticleGoogle Scholar
  41. Tan Y, Abbasbandy S: Homotopy analysis method for quadratic riccati differential equation. Commun Nonlinear Sci Numer Simul 2008, 13(3):539-546. 10.1016/j.cnsns.2006.06.006View ArticleGoogle Scholar
  42. Tsai P-Y, Chen C-K: An approximate analytic solution of the nonlinear riccati differential equation. J Franklin Inst 2010, 347(10):1850-1862. 10.1016/j.jfranklin.2010.10.005View ArticleGoogle Scholar
  43. Torabi M, Yaghoobi H: Novel solution for acceleration motion of a vertically falling spherical particle by hpm-pade approximant́. Adv Powder Technol 2011, 22(5):674-677. 10.1016/j.apt.2011.02.013View ArticleGoogle Scholar
  44. Vazquez-Leal H: Rational homotopy perturbation method. J Appl Math 2012, 1-14. doi:10.1155/2012/490342.Google Scholar
  45. Vazquez-Leal H, Castaneda-Sheissa R, Filobello-Nino U, Sarmiento-Reyes A, Sanchez-Orea J: High accurate simple approximation of normal distribution related integrals. Math Probl Eng 2012a, 2012: 22.Google Scholar
  46. Vazquez-Leal H, Filobello-Nino U, Castaneda-Sheissa R, Hernandez-Martinez L, Sarmiento-Reyes A: Modified hpms inspired by homotopy continuation methods. Math Probl Eng 2012b, 2012: 19.Google Scholar
  47. Vazquez-Leal H, Khan Y, Fernandez-Anaya G, Herrera-May A, Sarmiento-Reyes A, Filobello-Nino U, Jimenez-Fernandez VM, Pereyra-Diaz D: A general solution for troesch’s problem. Math Probl Eng 2012c, 2012: 1-14. doi:10.1155/2012/208375Google Scholar
  48. 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 2012d, 2012: 1-21. doi:10.1155/2012/923975.Google Scholar
  49. Vazquez-Leal H, Guerrero F: Application of series method with padé and laplace-padé resummation methods to solve a model for the evolution of smoking habit in spain. Comput Appl Math 2013, 33(1):1-12.Google Scholar
  50. Vazquez-Leal H: Generalized homotopy method for solving nonlinear differential equations. Comput Appl Math 2013, 33(1):1-14.Google Scholar
  51. Vazquez-Leal H, Benhammouda B, Filobello-Nino U, Sarmiento-Reyes A, Jimenez-Fernandez V, Marin-Hernandez A, Herrera-May A, Diaz-Sanchez A, Huerta-Chua J: Modified taylor series method for solving nonlinear differential equations with mixed boundary conditions defined on finite intervals. SpringerPlus 2014, 3(1):160. 10.1186/2193-1801-3-160View ArticleGoogle Scholar
  52. Weibel ES: On the confinement of a plasma by magnetostatic fields. Phys Fluid 1959, 2(1):5. 10.1063/1.1724393View ArticleGoogle Scholar
  53. Wazwaz A-M: Paé approximants and adomian decomposition method for solving the flierl-petviashivili equation and its variants. Appl Math Comput 2006, 182(2):1812-1818. 10.1016/j.amc.2006.06.018View ArticleGoogle Scholar
  54. Wang Z, Zou L, Zong Z: Adomian decomposition and padé approximate for solving differential-difference equation. Appl Math Comput 2011, 218(4):1371-1378. 10.1016/j.amc.2011.06.019View ArticleGoogle Scholar

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This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.