Volume 2009, Issue 1 690547

Research Article

Open Access

Solving Nonlinear Boundary Value Problems Using He′s Polynomials and Padé Approximants

Syed Tauseef Mohyud-Din,

Corresponding Author

Syed Tauseef Mohyud-Din

[email protected]

Institute of Aerospace & Avionics Air University Sector E-9 44000 Islamabad, Pakistan , pakrang.com

COMSATS Institute of Information Technology Lahore, Pakistan , ciitlahore.edu.pk

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Ahmet Yildirim,

Ahmet Yildirim

Department of Mathematics Science Faculty Ege University 35100 Bornova İzmir, Turkey , ege.edu.tr

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Syed Tauseef Mohyud-Din,

Corresponding Author

Syed Tauseef Mohyud-Din

[email protected]

Institute of Aerospace & Avionics Air University Sector E-9 44000 Islamabad, Pakistan , pakrang.com

COMSATS Institute of Information Technology Lahore, Pakistan , ciitlahore.edu.pk

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Ahmet Yildirim,

Ahmet Yildirim

Department of Mathematics Science Faculty Ege University 35100 Bornova İzmir, Turkey , ege.edu.tr

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First published: 11 January 2010

https://doi.org/10.1155/2009/690547

Citations: 6

Academic Editor: Ji Huan He

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Abstract

We apply He′s polynomials coupled with the diagonal Padé approximants for solving various singular and nonsingular boundary value problems which arise in engineering and applied sciences. The diagonal Padé approximants prove to be very useful for the understanding of physical behavior of the solution. Numerical results reveal the complete reliability of the proposed combination.

1. Introduction

With the rapid development of nonlinear sciences, many analytical and numerical techniques have been developed by various scientists for solving singular and nonsingular initial and boundary value problems which arise in the mathematical modeling of diversified physical problems related to engineering and applied sciences. The application of these problems involves physics, astrophysics, experimental and mathematical physics, nuclear charge in heavy atoms, thermal behavior of a spherical cloud of gas, thermodynamics, population models, chemical kinetics, and fluid mechanics see [1–68] and the references therein. Several techniques [1–68] including decomposition, variational iteration, finite difference, polynomial spline, differential transform, exp-function and homotopy perturbation have been developed for solving such problems. Most of these methods have their inbuilt deficiencies coupled with the major drawback of huge computational work. He [19–24] developed the homotopy perturbation method (HPM) for solving linear, nonlinear, initial and boundary value problems. The homotopy perturbation method was formulated by merging the standard homotopy with perturbation. Recently, Ghorbani and Saberi-Nadjafi [15, 16] introduced He’s polynomials by splitting the nonlinear term and also proved that He’s polynomials are fully compatible with Adomian’s polynomials but are easier to calculate and are more user friendly. The basic motivation of this paper is to apply He’s polynomials coupled with the diagonal Padé approximants for solving singular and nonsingular boundary value problems. The Padé approximants are applied in order to make the work more concise and for the better understanding of the solution behavior. The use of Padé approximants shows real promise in solving boundary value problems in an infinite domain; see [42, 50, 56–59]. It is well known in the literature that polynomials are used to approximate the truncated power series. It was observed [42, 50, 56–59] that polynomials tend to exhibit oscillations that may give an approximation error bounds. Moreover, polynomials can never blow up in a finite plane and this makes the singularities not apparent. To overcome these difficulties, the obtained series is best manipulated by Padé approximants for numerical approximations. Using the power series, isolated from other concepts, is not always useful because the radius of convergence of the series may not contain the two boundaries. It is now well known that Padé approximants [42, 50, 56–59] have the advantage of manipulating the polynomial approximation into rational functions of polynomials. By this manipulation, we gain more information about the mathematical behavior of the solution. In addition, the power series are not useful for large values of x. It is an established fact that power series in isolation are not useful to handle boundary value problems. This can be attributed to the possibility that the radius of convergence may not be sufficiently large to contain the boundaries of the domain. It is therefore essential to combine the series solution with the Padé approximants to provide an effective tool to handle boundary value problems on an infinite or semi-infinite domain. We apply this powerful combination of series solution and Padé approximants for solving a variety of boundary value problems. Precisely the proposed combination is applied on boundary layer problem, unsteady flow of gas through a porous medium, Thomas-Fermi equation, Flierl-Petviashivili (FP) equation, and Blasius problem. It is worth mentioning that Flierl-Petviashivili equation has singularity behavior at x = 0 which is a difficult element in this type of equations. We transform the FP equation to a first-order initial value problem and He’s polynomials are applied to the reformulated first-order initial value problem which leads the solution in terms of transformed variable. The desired series solution is obtained by implementing the inverse transformation. The fact that the proposed algorithm solves nonlinear problems without using Adomian’s polynomials is a clear advantage of this technique over the decomposition method.

2. Homotopy Perturbation Method and He’s Polynomials

To explain the He’s homotopy perturbation method, we consider a general equation of the type

()

where L is any integral or differential operator. We define a convex homotopy H(u, p) by

()

where F(u) is a functional operator with known solutions v₀, which can be obtained easily. It is clear that, for

()

we have

()

This shows that H(u, p) continuously traces an implicitly defined curve from a starting point H(v₀, 0) to a solution function H(f, 1). The embedding parameter monotonically increases from zero to unit as the trivial problem F(u) = 0, continuously deforms the original problem L(u) = 0. The embedding parameter p ∈ (0, 1] can be considered as an expanding parameter [15, 16, 19–24, 41–50, 60, 63–68]. The homotopy perturbation method uses the homotopy parameter p as an expanding parameter [19–24] to obtain

()

if p→ 1, then (2.5) corresponds to (2.2) and becomes the approximate solution of the form

()

It is well known that series (2.6) is convergent for most of the cases and also the rate of convergence is dependent on L(u); see [19–24]. We assume that (3.2) has a unique solution. The comparisons of like powers of p give solutions of various orders. In sum, according to [15, 16], He’s HPM considers the nonlinear term N(u) as

()

where H_n’s are the so-called He’s polynomials [15, 16], which can be calculated by using the formula

()

of various orders.

3. Padé Approximants

A Padé approximant is the ratio of two polynomials constructed from the coefficients of the Taylor series expansion of a function u(x). The [L/M] Padé approximants to a function y(x) are given by [42, 50, 56–59]

()

where P_L(x) is polynomial of degree at most L and Q_M(x) is a polynomial of degree at most M. The formal power series

()

determine the coefficients of P_L(x) and Q_M(x) by the equation. Since we can clearly multiply the numerator and denominator by a constant and leave [L/M] unchanged, we imposed the normalization condition

()

Finally, we require that P_L(x) and Q_M(x) have noncommon factors. If we write the coefficient of P_L(x) and Q_M(x) as

()

then by (3.6) and (3.7), we may multiply (3.3) by Q_M(x), which linearizes the coefficient equations. We can write out (3.5) in more details as

()

To solve these equations, we start with (3.6), which is a set of linear equations for all the unknown q’s. Once the q’s are known, then (3.7) gives and explicit formula for the unknown p’s, which complete the solution. If (3.6) and (3.7) are nonsingular, then we can solve them directly and obtain (3.8) [42, 50, 56–59], where (3.8) holds, and if the lower index on a sum exceeds the upper, the sum is replaced by zero:

()

To obtain diagonal Padé approximants of different order such as [2/2], [4/4], or [6/6], we can use the symbolic calculus software Maple.

4. Numerical Applications

In this section, we apply He’s polynomials for solving boundary layer problem, unsteady flow of gas through a porous medium, Thomas-Fermi equation, Flierl-Petviashivili equation, and Blasius problem. The powerful Padé approximants are applied for making the work more concise and to get the better understanding of solution behavior.

Example 4.1 (see [51], [59].)Consider the following nonlinear third-order boundary layer problem which appears mostly in the mathematical modeling of physical phenomena in fluid mechanics [51, 59]

()

with boundary conditions

()

By applying the convex homotopy, we have

()

comparing the co-efficient of like powers of p, following approximants are made

()

where f^′′(0) = α < 0 and pⁱs are He’s polynomials. The series solution is given as

()

Example 4.2 (see [51], [57].)Consider the following nonlinear differential equation which governsthe unsteady flow of gas through a porous medium

()

with the following boundary conditions:

()

By applying the convex homotopy method we have

()

By comparing the coefficient of like powers of p, the following approximants are obtained:

()

where A = y^′(0) and pⁱs are He’s polynomials. The series solution is given as

()

The diagonal Padé approximants [51, 57] can be applied to analyze the physical behavior. Based on this, the [2/2] Padé approximants produced the slope A to be

()

and by using [3/3] Padé approximants we find

()

where

()

Using (4.11)–(4.13) gives the values of the initial slope A = y^′(0) listed in Table 3. The formulas (4.11) and (4.12) suggest that the initial slope A = y^′(0) depends mainly on the parameter α, where 0 < α < 1. Table 3 exhibits the initial slopes A = y^′(0) for various values of α. Table 4 exhibits the values of y(x) for α = 0.5 for x = 0.1 to 1.0.

Table 1. Numerical values for α = f^″(0) for 0 < n < 1 by using diagonal Padé approximants [51, 59].

n	[2/2]	[3/3]	[4/4]	[5/5]	[6/6]
0.2	−0.3872983347	−0.3821533832	−0.3819153845	−0.3819148088	−0.3819121854
1/3	−0.5773502692	−0.5615999244	−0.5614066588	−0.5614481405	−0.561441934
0.4	−0.6451506398	−0.6397000575	−0.6389732578	−0.6389892681	−0.6389734794
0.6	−0.8407967591	−0.8393603021	−0.8396060478	−0.8395875381	−0.8396056769
0.8	−1.007983207	−1.007796981	−1.007646828	−1.007646828	−1.007792100

Table 2. Numerical values for α = f^″(0) for n < 1 by using diagonal Padé approximants [51, 59].

n	α
4	−2.483954032
10	−4.026385103
100	−12.84334315
1000	−40.65538218
5000	−104.8420672

Table 3. [51, 57].

α	B_[2/2] = y^′(0)	B_[3/3] = y^′(0)
0.1	−3.556558821	−1.957208953
0.2	−2.441894334	−1.786475516
0.3	−1.928338405	−1.478270843
0.4	−1.606856838	−1.231801809
0.5	−1.373178096	−1.025529704
0.6	−1.185519607	−0.8400346085
0.7	−1.021411309	−0.6612047893
0.8	−0.8633400217	−0.4776697286
0.9	−0.6844600642	−0.2772628386

Table 4. [51, 57].

x	y^kidder	y_[2/2]	y_[3/3]
0.1	0.8816588283	0.8633060641	0.8979167028
0.2	0.7663076781	0.7301262261	0.7985228199
0.3	0.6565379995	0.6033054140	0.7041129703
0.4	0.5544024032	0.4848898717	0.6165037901
0.5	0.4613650295	0.3761603869	0.5370533796
0.6	0.3783109315	0.2777311628	0.4665625669
0.7	0.3055976546	0.1896843371	0.4062426033
0.8	0.2431325473	0.1117105165	0.3560801699
0.9	0.1904623681	0.04323673236	0.3179966614
1.0	0.1587689826	0.01646750847	0.2900255005

Table 5. Padé approximants and initial slopes y^′(0) [56]

Padé approximants	Initial slope y^′(0)	Error (%)
[2/2]	−1.211413729	23.71
[4/4]	−1.550525919	2.36
[7/7]	−1.586021037	12.9 × 10⁻²
[8/8]	−1.588076820	3.66 × 10⁻⁴
[10/10]	−1.588076779	3.64 × 10⁻⁴

Table 6. Roots of the Padé approximants monopole [42] α, n = 1.

Degree	Roots
[2/2]	−1.5
[4/4]	−2.50746
[6/6]	−2.390278
[8/8]	−2.392214

Table 7. Roots of the Padé approximants monopole [42] α, n = 3.

Degree	Roots
[2/2]	−2.0
[4/4]	−2.0
[6/6]	−2.0
[8/8]	−2.0

Table 8. Roots of the Padé approximants monopole [42] α.

Degree	Roots
[2/2]	0.0
[4/4]	−.2197575908
[6/6]	−1.1918424398
[8/8]	−1.848997181

Example 4.3 (see [56].)Consider the following Thomas-Fermi (T-F) equation [6–13, 17, 31, 33, 34, 54] which arises in the mathematical modeling of various models in physics, astrophysics, solid state physics, nuclear charge in heavy atoms, and applied sciences:

()

with boundary conditions

()

By applying the convex homotopy,

()

Now, we apply a slight modification in the conventional initial value and take y₀(x) = 1, instead of y₀(x) = 1 + Bx, where B = y^′(0). By comparing the coefficient of like powers of p, the following approximants are obtained

()

The series solution is given as

()

Setting x^1/2 = t, the series solution is obtained as

()

The diagonal Padé approximants can be applied [56] in order to study the mathematical behavior of the potential y(x) and to determine the initial slope of the potential y^′(0).

Example 4.4 (see [42].)Consider the generalized variant of the Flierl-Petviashivili equation [37]

()

with boundary conditions

()

Using the transformation u(x) = xy^′(x), the generalized FP equation can be converted to the following first-order initial value problem:

()

with initial conditions

()

By applying the convex homotopy, we have

()

The series solution after four iterations is given by

()

where 𝔄 denote (18n² + 7n + 1)(3nα³ⁿ⁺² + (n + 1)α³ⁿ⁺³), and the inverse transformation will yield

()

where 𝔄 denote (18n² + 7n + 1)(3nα³ⁿ⁺² + (n + 1)α³ⁿ⁺³). Diagonal Padé approximants can be applied [42] to find the roots of the FP monopole α for n ≥ 1. Table 9 shows that the roots of the monopole α converge to −1 as n increases.

Table 9. Roots [42] of the Padé approximants [8/8] monopole for several values of n.

n	[8/8] roots	n	[8/8] roots
1	−2.392213866	7	−1.000708285
2	−2.0	8	−1.000601615
3	−1.848997181	9	−1.000523005
4	−1.286025892	10	−1.000462636
5	−1.001101141	11	−1.000262137
6	−1.000861533	n → ∞	−1.0

Example 4.5 (see [58], [59].)Consider the two-dimensional nonlinear inhomogeneous initial boundary value problem for the integro-differential equation related to the Blasius problem

()

with boundary conditions

()

where the constant α is positive and defined by

()

By applying the convex homotopy, we have

()

Proceeding as before, the series solution is given as

()

and consequently

()

Now, we apply the diagonal Padé approximants to determine a numerical value for the constant α by using the given condition. Padé approximant of y^′(x) usually converges on the entire real axis [58, 59]. Moreover, y^′(x) is free of singularities on the real axis. Substituting the boundary conditions y^′(−∞) = 0 in each Padé approximant which vanishes if the coefficient of x with the highest power in the numerator vanishes. By solving the resulting polynomials of these coefficients, we obtain the values of α listed in Table 10 [58, 59].

Table 10. Padé approximants and numerical value of α [53].

Padé approximant	α
[2/2]	0.5778502691
[3/3]	0.5163977793
[4/4]	0.5227030798

5. Conclusion

In this paper, we applied a reliable combination of He’s polynomials and the diagonal Padé approximants for obtaining approximate solutions of various singular and nonsingular boundary value problems of diversified physical nature. The proposed algorithm is employed without using linearization, discretization, transformation, or restrictive assumptions. The fact that the suggested technique solves nonlinear problems without using Adomian’s polynomials is a clear advantage of this technique over the decomposition method.

Acknowledgment

The author is highly grateful to Brig (R) Qamar Zaman, the Vice Chancellor of HITEC University, Taxila Cantt, Pakistan for providing excellent research environment and facilities.

References

1 Abbasbandy S., A new application of He′s variational iteration method for quadratic Riccati differential equation by using Adomian′s polynomials, Journal of Computational and Applied Mathematics. (2007) 207, no. 1, 59–63, MR2332947, https://doi.org/10.1016/j.cam.2006.07.012, ZBL1120.65083.
10.1016/j.cam.2006.07.012
Web of Science® Google Scholar
2 Abbasbandy S., Numerical solution of non-linear Klein-Gordon equations by variational iteration method, International Journal for Numerical Methods in Engineering. (2007) 70, no. 7, 876–881, MR2317382, https://doi.org/10.1002/nme.1924.
10.1002/nme.1924
Web of Science® Google Scholar
3 Abdou M. A. and Soliman A. A., New applications of variational iteration method, Physica D. (2005) 211, no. 1-2, 1–8, MR2200437, https://doi.org/10.1016/j.physd.2005.08.002, ZBL1084.35539.
10.1016/j.physd.2005.08.002
CAS Web of Science® Google Scholar
4 Abdou M. A. and Soliman A. A., Variational iteration method for solving Burger′s and coupled Burger′s equations, Journal of Computational and Applied Mathematics. (2005) 181, no. 2, 245–251, MR2146836, https://doi.org/10.1016/j.cam.2004.11.032.
10.1016/j.cam.2004.11.032
Web of Science® Google Scholar
5 Abassy T. A., El-Tawil M. A., and El Zoheiry H., Solving nonlinear partial differential equations using the modified variational iteration Padé technique, Journal of Computational and Applied Mathematics. (2007) 207, no. 1, 73–91, MR2332949, https://doi.org/10.1016/j.cam.2006.07.024, ZBL1119.65095.
10.1016/j.cam.2006.07.024
Web of Science® Google Scholar
6 Adomian G., Solution of the Thomas-Fermi equation, Applied Mathematics Letters. (1998) 11, no. 3, 131–133, MR1625415, ZBL0947.34501.
10.1016/S0893-9659(98)00046-9
Web of Science® Google Scholar
7 Anderson N., Arthurs A. M., and Robinson P. D., Complementary variational principles for a generalized diffusion equation, Proceedings of the Royal Society A. (1968) 303, 497–502, https://doi.org/10.1098/rspa.1968.0064.
10.1098/rspa.1968.0064
Google Scholar
8 Batiha B., Noorani M. S. M., and Hashim I., Variational iteration method for solving multispecies Lotka-Volterra equations, Computers & Mathematics with Applications. (2007) 54, no. 7-8, 903–909, MR2395627, ZBL1141.65370.
10.1016/j.camwa.2006.12.058
Web of Science® Google Scholar
9 Biazar J. and Ghazvini H., He′s variational iteration method for fourth-order parabolic equations, Computers & Mathematics with Applications. (2007) 54, no. 7-8, 1047–1054, MR2395644.
10.1016/j.camwa.2006.12.049
Web of Science® Google Scholar
10 Bush V. and Caldwell S. H., Thomas-Fermi equation solution by the differential analyzer, Physical Review. (1931) 38, no. 10, 1898–1902, https://doi.org/10.1103/PhysRev.38.1898, ZBL0003.26805.
10.1103/PhysRev.38.1898
Google Scholar
11 Burrows B. L. and Core P. W., A variational-iterative approximate solution of the Thomas-Fermi equation, Quarterly of Applied Mathematics. (1984) 42, no. 1, 73–76, MR736506, ZBL0533.49020.
10.1090/qam/736506
Web of Science® Google Scholar
12 Chan C. Y. and Hon Y. C., A constructive solution for a generalized Thomas-Fermi theory of ionized atoms, Quarterly of Applied Mathematics. (1987) 45, no. 3, 591–599, MR910465, ZBL0639.34021.
10.1090/qam/910465
Web of Science® Google Scholar
13 Cedillo A., A perturbative approach to the Thomas-Fermi equation in terms of the density, Journal of Mathematical Physics. (1993) 34, no. 7, 2713–2717, MR1224172, https://doi.org/10.1063/1.530090, ZBL0777.34057.
10.1063/1.530090
Web of Science® Google Scholar
14 Golbabai A. and Javidi M., A variational iteration method for solving parabolic partial differential equations, Computers & Mathematics with Applications. (2007) 54, no. 7-8, 987–992.
10.1016/j.camwa.2006.12.042
Web of Science® Google Scholar
15 Ghorbani A. and [email protected], Saberi-Nadjafi J., He′s homotopy perturbation method for calculating Adomian polynomials, International Journal of Nonlinear Sciences and Numerical Simulation. (2007) 8, no. 2, 229–232, 2-s2.0-27144469470.
10.1515/IJNSNS.2007.8.2.229
Web of Science® Google Scholar
16 Ghorbani A., [email protected], Beyond Adomian polynomials: He polynomials, Chaos, Solitons and Fractals. (2009) 39, no. 3, 1486–1492, https://doi.org/10.1016/j.chaos.2007.06.034.
10.1016/j.chaos.2007.06.034
Web of Science® Google Scholar
17 He J.-H., Variational approach to the Thomas-Fermi equation, Applied Mathematics and Computation. (2003) 143, no. 2-3, 533–535, MR1981713, https://doi.org/10.1016/S0096-3003(02)00380-6, ZBL1022.65083.
10.1016/S0096-3003(02)00380-6
Web of Science® Google Scholar
18 He J.-H., An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering, International Journal of Modern Physics B. (2008) 22, no. 21, 3487–3578, https://doi.org/10.1142/S0217979208048668, ZBL1149.76607.
10.1142/S0217979208048668
Web of Science® Google Scholar
19 He J.-H., Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B. (2006) 20, no. 10, 1141–1199, MR2220565, ZBL1102.34039.
10.1142/S0217979206033796
Web of Science® Google Scholar
20 He J.-H., Homotopy perturbation method for solving boundary value problems, Physics Letters A. (2006) 350, no. 1-2, 87–88, MR2199322, https://doi.org/10.1016/j.physleta.2005.10.005.
10.1016/j.physleta.2005.10.005
CAS Web of Science® Google Scholar
21 He J.-H., Comparison of homotopy perturbation method and homotopy analysis method, Applied Mathematics and Computation. (2004) 156, no. 2, 527–539, MR2087530, https://doi.org/10.1016/j.amc.2003.08.008, ZBL1062.65074.
10.1016/j.amc.2003.08.008
Web of Science® Google Scholar
22 He J.-H., Homotopy perturbation method for bifurcation of nonlinear problems, International Journal of Nonlinear Sciences and Numerical Simulation. (2005) 6, no. 2, 207–208, 2-s2.0-4243080805.
10.1515/IJNSNS.2005.6.2.207
Web of Science® Google Scholar
23 He J.-H., The homotopy perturbation method nonlinear oscillators with discontinuities, Applied Mathematics and Computation. (2004) 151, no. 1, 287–292, MR2037967, https://doi.org/10.1016/S0096-3003(03)00341-2, ZBL1039.65052.
10.1016/S0096-3003(03)00341-2
Web of Science® Google Scholar
24 He J.-H., A coupling method of a homotopy technique and a perturbation technique for non-linear problems, International Journal of Non-Linear Mechanics. (2000) 35, no. 1, 37–43, MR1723761, https://doi.org/10.1016/S0020-7462(98)00085-7.
10.1016/S0020-7462(98)00085-7
Web of Science® Google Scholar
25 He J.-H., Variational iteration method—a kind of non-linear analytical technique: some examples, International Journal of Non-Linear Mechanics. (1999) 34, no. 4, 699–708, 2-s2.0-0002228705, https://doi.org/10.1016/S0020-7462(98)00048-1.
10.1016/S0020-7462(98)00048-1
Web of Science® Google Scholar
26 He J.-H., Variational iteration method for autonomous ordinary differential systems, Applied Mathematics and Computation. (2000) 114, no. 2-3, 115–123, MR1778658, https://doi.org/10.1016/S0096-3003(99)00104-6, ZBL1027.34009.
10.1016/S0096-3003(99)00104-6
Web of Science® Google Scholar
27 He J.-H. and Wu X.-H., Construction of solitary solution and compacton-like solution by variational iteration method, Chaos, Solitons and Fractals. (2006) 29, no. 1, 108–113, MR2213340, https://doi.org/10.1016/j.chaos.2005.10.100, ZBL1147.35338.
10.1016/j.chaos.2005.10.100
Web of Science® Google Scholar
28 He J.-H., Variational iteration method—some recent results and new interpretations, Journal of Computational and Applied Mathematics. (2007) 207, no. 1, 3–17, MR2332941, https://doi.org/10.1016/j.cam.2006.07.009, ZBL1119.65049.
10.1016/j.cam.2006.07.009
Web of Science® Google Scholar
29 He J.-H. and Wu X.-H., Variational iteration method: new development and applications, Computers & Mathematics with Applications. (2007) 54, no. 7-8, 881–894, MR2395625, ZBL1141.65372.
10.1016/j.camwa.2006.12.083
Web of Science® Google Scholar
30 He J.-H., The variational iteration method for eighth-order initial-boundary value problems, Physica Scripta. (2007) 76, no. 6, 680–682, MR2393451, ZBL1134.34307.
10.1088/0031-8949/76/6/016
CAS Web of Science® Google Scholar
31 Hon Y. C., A decomposition method for the Thomas-Fermi equation, Southeast Asian Bulletin of Mathematics. (1996) 20, no. 3, 55–58, MR1430609, ZBL0858.34017.
Google Scholar
32 Inokuti M., Sekine H., and Mura T., S. Nemat-Naseer, General use of the Lagrange multiplier in nonlinear mathematical physics, Variational Method in the Mechanics of Solids, 1978, Pergamon Press, New York, NY, USA, 156–162.
Google Scholar
33 Kobayashi S., Matsukuma T., Nagai S., and Umeda K., Some coefficients of the TFD function, Journal of the Physical Society of Japan. (1955) 10, 759–765.
10.1143/JPSJ.10.759
Web of Science® Google Scholar
34 Laurenzi B. J., An analytic solution to the Thomas-Fermi equation, Journal of Mathematical Physics. (1990) 31, no. 10, 2535–2537, MR1072970, https://doi.org/10.1063/1.528998, ZBL0743.34021.
10.1063/1.528998
Web of Science® Google Scholar
35 Lu J., Variational iteration method for solving two-point boundary value problems, Journal of Computational and Applied Mathematics. (2007) 207, no. 1, 92–95, MR2332950, https://doi.org/10.1016/j.cam.2006.07.014, ZBL1119.65068.
10.1016/j.cam.2006.07.014
Web of Science® Google Scholar
36 Ma W. X. and Zhou D. T., Explicit exact solution of a generalized KdV equation, Acta Mathematica Scientia. (1997) 17, 168–174.
Google Scholar
37 Ma W.-X. and You Y., Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions, Transactions of the American Mathematical Society. (2005) 357, no. 5, 1753–1778, MR2115075, ZBL1062.37077.
10.1090/S0002-9947-04-03726-2
Web of Science® Google Scholar
38 Ma W. X. and You Y., Rational solutions of the Toda lattice equation in Casoratian form, Chaos, Solitons and Fractals. (2004) 22, no. 2, 395–406, MR2060875, https://doi.org/10.1016/j.chaos.2004.02.011, ZBL1090.37053.
10.1016/j.chaos.2004.02.011
Web of Science® Google Scholar
39 Ma W.-X., Wu H., and He J., Partial differential equations possessing Frobenius integrable decompositions, Physics Letters A. (2007) 364, no. 1, 29–32, MR2288009, https://doi.org/10.1016/j.physleta.2006.11.048.
10.1016/j.physleta.2006.11.048
CAS Web of Science® Google Scholar
40 Momani S. and Abuasad S., Application of He′s variational iteration method to Helmholtz equation, Chaos, Solitons and Fractals. (2006) 27, no. 5, 1119–1123, MR2164845, https://doi.org/10.1016/j.chaos.2005.04.113, ZBL1086.65113.
10.1016/j.chaos.2005.04.113
Web of Science® Google Scholar
41 Mohyud-Din S. T., Noor M. A., and Noor K. I., Travelling wave solutions of seventh-order generalized KdV equations using He′s polynomials, International Journal of Nonlinear Sciences and Numerical Simulation. (2009) 10, no. 2, 227–233.
10.1515/IJNSNS.2009.10.2.227
Web of Science® Google Scholar
42 Mohyud-Din S. T. and Noor M. A., Homotopy perturbation method and Padé approximants for solving Flierl-Petviashivili equation, Applications and Applied Mathematics. (2008) 3, no. 2, 224–234, MR2469968.
Google Scholar
43 Noor M. A. and [email protected], Mohyud-Din S. T., Homotopy perturbation method for solving nonlinear higher-order boundary value problems, International Journal of Nonlinear Sciences and Numerical Simulation. (2008) 9, no. 4, 395–408, 2-s2.0-34748871953.
10.1515/IJNSNS.2008.9.4.395
Web of Science® Google Scholar
44 Noor M. A. and [email protected], Mohyud-Din S. T., Variational iteration method for solving higher-order nonlinear boundary value problems using He′s polynomials, International Journal of Nonlinear Sciences and Numerical Simulation. (2008) 9, no. 2, 141–157, 2-s2.0-0034276401.
10.1515/IJNSNS.2008.9.2.141
Web of Science® Google Scholar
45 Noor M. A. and Mohyud-Din S. T., Variational iteration technique for solving higher order boundary value problems, Applied Mathematics and Computation. (2007) 189, no. 2, 1929–1942, MR2332146, https://doi.org/10.1016/j.amc.2006.12.071, ZBL1122.65374.
10.1016/j.amc.2006.12.071
Web of Science® Google Scholar
46 Noor M. A. and Mohyud-Din S. T., An efficient method for fourth-order boundary value problems, Computers & Mathematics with Applications. (2007) 54, no. 7-8, 1101–1111, MR2398132, ZBL1141.65375.
10.1016/j.camwa.2006.12.057
Web of Science® Google Scholar
47 Noor M. A. and Mohyud-Din S. T., Modified variational iteration method for heat and wave-like equations, Acta Applicandae Mathematicae. (2008) 104, no. 3, 257–269, MR2453425, ZBL1162.65397.
10.1007/s10440-008-9255-x
Web of Science® Google Scholar
48 Noor M. A. and Mohyud-Din S. T., Variational homotopy perturbation method for solving higher dimensional initial boundary value problems, Mathematical Problems in Engineering. (2008) 2008, 11, 696734, MR2417233, ZBL1155.65082.
10.1155/2008/696734
Web of Science® Google Scholar
49 Noor M. A., [email protected], Noor K. I., and [email protected], Mohyud-Din S. T., Modified variational iteration technique for solving singular fourth-order parabolic partial differential equations, Nonlinear Analysis Series A: Theory, Methods & Applications. (2009) 71, e630–e640, https://doi.org/10.1016/j.na.2008.11.011.
10.1016/j.na.2008.11.011
Web of Science® Google Scholar
50 Noor M. A. and [email protected], Mohyud-Din S. T., Variational iteration method for unsteady flow of gas through a porous medium using He′s polynomials and Pade approximants, Computers and Mathematics with Applications. (2009) 58, 2182–2189, https://doi.org/10.1016/j.camwa.2009.03.016.
10.1016/j.camwa.2009.03.016
Web of Science® Google Scholar
51 Noor M. A. and Mohyud-Din S. T., Solution of singular and nonsingular initial and boundary value problems by modified variational iteration method, Mathematical Problems in Engineering. (2008) 2008, 23, 917407, MR2430646, ZBL1155.65083.
10.1155/2008/917407
Web of Science® Google Scholar
52 Sweilam N. H., Harmonic wave generation in non linear thermoelasticity by variational iteration method and Adomian′s method, Journal of Computational and Applied Mathematics. (2007) 207, no. 1, 64–72, MR2332948, https://doi.org/10.1016/j.cam.2006.07.013.
10.1016/j.cam.2006.07.013
Web of Science® Google Scholar
53 Sweilam N. H., Fourth order integro-differential equations using variational iteration method, Computers & Mathematics with Applications. (2007) 54, no. 7-8, 1086–1091, MR2398130, ZBL1141.65399.
10.1016/j.camwa.2006.12.055
Web of Science® Google Scholar
54 Sabirov R. K., Solution of the Thomas-Fermi-Dirac of the statistical model of an atom at small distances from the nucleus, Optical Spectra. (1993) 75, no. 1, 1–2.
Google Scholar
55 Tatari M. and Dehghan M., On the convergence of He′s variational iteration method, Journal of Computational and Applied Mathematics. (2007) 207, no. 1, 121–128, MR2332953, https://doi.org/10.1016/j.cam.2006.07.017, ZBL1120.65112.
10.1016/j.cam.2006.07.017
Web of Science® Google Scholar
56 Wazwaz A.-M., The modified decomposition method and Padé approximants for solving the Thomas-Fermi equation, Applied Mathematics and Computation. (1999) 105, no. 1, 11–19, MR1706059, https://doi.org/10.1016/S0096-3003(98)10090-5, ZBL0956.65064.
10.1016/S0096-3003(98)10090-5
Web of Science® Google Scholar
57 Wazwaz A.-M., The modified decomposition method applied to unsteady flow of gas through a porous medium, Applied Mathematics and Computation. (2001) 118, no. 2-3, 123–132, MR1812950, https://doi.org/10.1016/S0096-3003(99)00209-X, ZBL1024.76056.
10.1016/S0096-3003(99)00209-X
Web of Science® Google Scholar
58 Wazwaz A.-M., A reliable algorithm for solving boundary value problems for higher-order integro-differentiable equations, Applied Mathematics and Computation. (2001) 118, no. 2-3, 327–342, MR1812962, https://doi.org/10.1016/S0096-3003(99)00225-8, ZBL1023.65150.
10.1016/S0096-3003(99)00225-8
Web of Science® Google Scholar
59 Wazwaz A.-M., A study on a boundary-layer equation arising in an incompressible fluid, Applied Mathematics and Computation. (1997) 87, no. 2-3, 199–204, MR1468298, https://doi.org/10.1016/S0096-3003(96)00281-0, ZBL0904.76067.
10.1016/S0096-3003(96)00281-0
Web of Science® Google Scholar
60 Xu L., He′s homotopy perturbation method for a boundary layer equation in unbounded domain, Computers & Mathematics with Applications. (2007) 54, no. 7-8, 1067–1070, MR2398127.
10.1016/j.camwa.2006.12.052
Web of Science® Google Scholar
61 Xu L., Variational iteration method for solving integral equations, Computers & Mathematics with Applications. (2007) 54, no. 7-8, 1071–1078, MR2398128, ZBL1141.65400.
10.1016/j.camwa.2006.12.053
Web of Science® Google Scholar
62 Zhou X.-W., Wen Y.-X., and He J.-H., Exp-function method to solve the nonlinear dispersive K(m, n) equations, International Journal of Nonlinear Sciences and Numerical Simulation. (2008) 9, no. 3, 301–306, 2-s2.0-40549134152.
10.1515/IJNSNS.2008.9.3.301
Web of Science® Google Scholar
63 Mohyud-Din S. T. and Noor M. A., Homotopy perturbation method for solving fourth-order boundary value problems, Mathematical Problems in Engineering. (2007) 2007, 15, 98602, MR2278494, ZBL1144.65311.
10.1155/2007/98602
Web of Science® Google Scholar
64 Mohyud-Din S. T. and Noor M. A., Homotopy perturbation method for solving partial differential equations, Zeitschrift für Naturforschung A. (2009) 64, 157–170.
10.1515/zna-2009-3-402
CAS Google Scholar
65 Mohyud-Din S. T., Noor M. A., and [email protected], Noor K. I., [email protected], Parameter-expansion techniques for strongly nonlinear oscillators, International Journal of Nonlinear Sciences and Numerical Simulation. (2009) 10, no. 5, 581–583, 2-s2.0-33846784840.
Web of Science® Google Scholar
66 Mohyud-Din S. T., Noor M. A., and [email protected], Noor K. I., [email protected], Traveling wave solutions of seventh-order generalized kdv equations by variational iteration method using adomian′s polynomials, International Journal of Modern Physics B. (2009) 23, no. 15, 3265–3277, https://doi.org/10.1142/S0217979209052467, ZBL1168.35427.
10.1142/S0217979209052467
Web of Science® Google Scholar
67 Mohyud-Din S. T., Noor M. A., and [email protected], Noor K. I., [email protected], Some relatively new techniques for nonlinear problems, Mathematical Problems in Engineering. (2009) 2009, 25, 234849, 2-s2.0-34250696383, https://doi.org/10.1155/2009/234849.
10.1155/2009/234849
Web of Science® Google Scholar
68 Mohyud-Din S. T., Noor M. A., and Noor K. I., Variational iteration method for Flieral-Petviashivili equations using He′s polynomials and Padé approximants, Journal of Applied Sciences. (2009) 6, no. 8, 1139–1146.
Google Scholar

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Solving Nonlinear Boundary Value Problems Using He′s Polynomials and Padé Approximants

Abstract

1. Introduction

2. Homotopy Perturbation Method and He’s Polynomials

3. Padé Approximants

4. Numerical Applications

5. Conclusion

Acknowledgment

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Solving Nonlinear Boundary Value Problems Using He′s Polynomials and Padé Approximants

Abstract

1. Introduction

2. Homotopy Perturbation Method and He’s Polynomials

3. Padé Approximants

4. Numerical Applications

5. Conclusion

Acknowledgment

References

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