A Legendre-homotopy method for the solutions of higher order boundary value problems

Maasoomah Sadaf; Ghazala Akram

doi:10.1016/j.jksus.2018.08.002

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32 (

); 537-543

doi:

10.1016/j.jksus.2018.08.002

A Legendre-homotopy method for the solutions of higher order boundary value problems

Maasoomah Sadaf, Ghazala Akram^⁎

Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan

⁎Corresponding author. ghazala.math@pu.edu.pk (Ghazala Akram)

Received: 2018-2-8, Accepted: 2018-8-9, Published: 2018-01-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

In this paper, the Legendre-homotopy analysis method is proposed using orthogonal Legendre polynomials for the approximate solutions of linear and nonlinear higher order boundary value problems. The deformation equations obtained in this case are easily integrable and the calculations involved in the algorithm are much simpler than the standard homotopy analysis method. The method is numerically illustrated by application on linear and nonlinear higher order boundary value problems. The absolute errors in the approximate solution values are calculated and compared with the results available in literature. The approximate solutions are also compared with the exact solutions through graphical illustrations. The numerical and graphical comparisons reveal that the presented method gives highly accurate results.

Keywords

Legendre polynomials

Homotopy analysis method

Higher order boundary value problems

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1 Introduction

Orthogonal polynomials have been of great interest of research due to their application for computation and approximation purposes in different problems of mathematics and physics. These polynomials have many applications to ordinary differential equations, boundary value problems and computational fluid dynamics (Butcher, 1992; Gottlieb and Orszag, 1977; Canuto et al., 1989; Voigt et al., 1984). Doha and Bhrawyb (2008) presented spectral-Galerkin algorithms for solutions of fourth order differential equations using Jacobi polynomials. Yalçinbas et al. (2009) obtained Legendre polynomial solutions of high order Fredholm integro-differential equations using Legendre collocation matrix method. Parand et al. (2010) approximated the solutions to nonlinear Lane-Emden type equations using a collocation method which involved Hermite functions to convert the problem into a system of algebraic equations. Odibat (2011) proposed algorithms for variational iteration method and homotopy analysis method using Legendre polynomials for the solutions of fractional differential equations. Bhrawy and Al-Shomrani (2012) proposed shifted Legendre tau method for the solutions of multi-order fractional differential equations with constant coefficients. Sweilam et al. (2012) used Legendre collocation method for the solutions of Fredholm-Hammerstein integral equations. Liu (2013) used Legendre polynomials to obtain the solutions to Volterra integral equations of second kind. Khader et al. (2014) used Legendre polynomials in an integral collocation approach for solving Riccati, logistic and delay differential equations. Xu and Zhou (2015) approximated the solutions to eighth order initial and boundary value problems using the second kind Chebyshev wavelets. Orthogonal polynomials have also been used to approximate the solutions to Volterra equations using Galerkin method (Mamadu and Njoseh, 2016).

Homotopy analysis method is an effective and reliable mathematical tool to determine the solutions of linear and nonlinear differential equations. It is an analytical approximate solution technique which enables to evaluate the solution to a problem in the form of a convergent series (Liao, 1992). The homotopy analysis is not only an efficient method to solve nonlinear differential equation problems but also allows great freedom to choose the initial approximation and is highly flexible in many respects so that it might overcome restrictions of numerical techniques, perturbation techniques and other non-perturbation methods, such as variational iteration method, homotopy perturbation method, finite element method and collocation method etc (Sadighi and Ganji, 2007; Jalaal and Ganji, 2010; Sheikholeslami et al., 2012, 2014, 2016, 2017; Sheikholeslami and Ganji, 2013; Malvandi and Ganji, 2014; Hosseini et al., 2018; Sheikholeslami and Ganji, 2018). The efficiency and practical usefulness of the homotopy analysis method has caught the attention of many researchers in recent years. It has been successfully implemented to investigate a wide range of problems arising in the study of nonlinear dynamics, micropolar fluids, heat transfer problems and many other areas of science (Ziabakhsh and Domairry, 2009; Sheikholeslami and Ganji, 2017; Sheikholeslami et al., 2018; Shah et al., 2017, 2018; Khan et al., 2018; Muhammad et al., 2018; Dawar et al., 2018; Khan et al., 2017; Tahir et al., 2017; Fiza et al., 2018; Gul, 2018; Alshomrani and Gul, 2017).

In this paper, a modification of homotopy analysis method is introduced for the analytical approximate solutions of higher order boundary value problems using the orthogonal Legendre polynomials. The applications of higher order boundary value problems are reported in various fields of science and engineering. Ninth order boundary value problems arise in astorphysics and aerodynamics, hydrodynamic and hydromagnetic stability (Lyshevski and Dunipace, 1997; Mohyud-Din and Yildirim, 2010; Mohyud-Din and Yildirim, 2010; Chandrasekhar, 1961). Eighth and tenth order boundary value problems also appear in the problems of hydrodynamic and hydromagnetic stability (Chandrasekhar, 1961). The exact solutions of many higher order boundary value problems arising in mathematical models of many real life phenomena cannot be obtained with any existing mathematical technique. Due to this fact, the approximate solutions are often investigated to understand the phenomena modeled, which provide either the numerical or the analytical approximations to the exact solutions. In general, the analytical approximate solutions provide a better account of qualitative behaviour and character of the solutions than the numerical solutions. In this regard, the proposed Legndre-homotopy analysis method can play a significant role to obtain the analytical approximate solutions of various higher order boundary value problems arising in physics and engineering.

2 Preliminaries for Legendre polynomials

Legendre polynomials form a class of functions which are encountered in finding the solutions to many physical problems. For example, Legendre and Associate Legendre polynomials are employed to determine the wave functions of electrons in the orbits of an atom and potential functions in the spherically symmetric geometry,etc. Legendre polynomials also play an important role in the nuclear reactor physics (Anli and Gungor, 2007).

Legendre polynomials are solutions to a very important differential equation, known as Legendre equation, which can be stated, as $(1 - ξ^{2}) \frac{d^{2} v}{d ξ^{2}} - 2 ξ \frac{dv}{d ξ} + j (j + 1) v = 0 .$ This ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace’s equation (and related partial differential equations) in spherical coordinates.

Legendre polynomials can be denoted by $L_{j} (ξ)$ , where j is the degree of the polynomial. These polynomials are defined over the interval $[- 1, 1]$ and satisfy the recurrence relation

(2.1)

L_{j + 1} (ξ) = \frac{2 j + 1}{j + 1} ξ L_{j} (ξ) - \frac{j}{j + 1} L_{j - 1} (ξ), j = 1, 2, 3, \dots,

where

L_{0} (ξ) = 1

and

L_{1} (ξ) = ξ

. The shifted Legendre polynomials over the interval

[0, 1]

are defined as

L_{j}^{*} (ξ) = L_{j} (2 ξ - 1)

and satisfy the recurrence relation

(2.2)

L_{j + 1}^{*} (ξ) = \frac{2 j + 1}{j + 1} (2 ξ - 1) L_{j}^{*} (ξ) - \frac{j}{j + 1} L_{j - 1}^{*} (ξ), j = 1, 2, 3, \dots,

where

L_{0}^{*} (ξ) = 1

and

L_{1}^{*} (ξ) = 2 ξ - 1

. Since Legendre polynomials are orthogonal, therefore any function

g (ξ)

can be expressed in terms of shifted Legendre polynomials, as

(2.3)

g (ξ) = \sum_{k = 0}^{\infty} c_{k} L_{k}^{*} (ξ),

where

c_{k}

’s can be calculated, as

(2.4)

c_{k} = (2 k + 1) \int_{0}^{1} g (ξ) L_{k}^{*} (ξ) d ξ .

3 The Legendre-homotopy analysis method

Homotopy analysis method is found to be easily applicable in many problems but sometimes the higher order deformation equations lead to complicated integrals and tedious calculations. To overcome these difficulties, a modification of homotopy analysis method is proposed in this section using the well known Legendre polynomials.

The nonlinear differential equation is considered, as

(3.5)

N [v (ξ)] + q (ξ) = 0, ξ \in Θ,

where

N

is a nonlinear operator,

ξ

is an independent variable,

v (ξ)

is an unknown function,

Θ

is the interval of domain and

q (ξ)

is a continuous function. In view of the homotopy analysis method, a homotopy

V (ξ, p)

can be defined using an embedding parameter

p \in [0, 1]

(3.6)

(1 - p) L [V (ξ, p) - v_{0} (ξ)] - phH (ξ) (N [V (ξ, p)] + q (ξ)) = 0, ξ \in Θ,

where h is auxiliary parameter,

H (ξ)

is an auxiliary function and

L

is auxiliary linear operator.

An initial approximation $v_{0} (ξ)$ can be obtained, as

(3.7)

v_{0} (ξ) = \sum_{j = 0}^{m} a_{0 j} L_{j}^{*} (ξ),

where

a_{0 j}

’s can be calculated, as

(3.8)

a_{0 j} = (2 j + 1) \int_{0}^{1} V_{0} (ξ) L_{j}^{*} (ξ) d ξ .

Here

V_{0} (ξ)

is the solution of the equation

L [v (ξ)] = 0

, subject to the given boundary conditions and m is some fixed natural number.

The first order deformation equations can be expressed using Legendre polynomials, as

(3.9)

L [v_{1} (ξ)] - hH (ξ) (N (\sum_{j = 0}^{m} a_{0 j} L_{j}^{*} (ξ)) + \sum_{j = 0}^{m} b_{j} L_{j}^{*} (ξ)) = 0 .

For

i = 1, 2, 3, \dots

, the deformation equations can be expressed, as

(3.10)

L [v_{i + 1} (ξ)] - L [v_{i} (ξ)] - hH (ξ) (N (\sum_{j = 0}^{m} a_{ij} L_{j}^{*} (ξ))) = 0 .

Moreover,

(3.11)

a_{ij} = (2 j + 1) \int_{0}^{1} v_{i} (ξ) L_{j}^{*} (ξ) d ξ, j = 0, 1, 2, \dots m, i = 1, 2, 3, \dots

and

(3.12)

b_{j} = (2 j + 1) \int_{0}^{1} q (ξ) L_{j}^{*} (ξ) d ξ, j = 0, 1, 2, \dots m .

Finally, the solution of Eq. (3.5) can be written, as

(3.13)

v (ξ) = \sum_{i = 0}^{\infty} v_{i} (ξ)

and the n-th order analytical approximate solution can be calculated, as

(3.14)

v_{n^{*}} (ξ) = \sum_{i = 0}^{n} v_{i} (ξ) .

4 Convergence of the solution

In this section, convergence of the solution series using the proposed technique is discussed.

Theorem

If the series $v (ξ) = \sum_{i = 0}^{\infty} v_{i} (ξ)$ is convergent, where $v_{i} (ξ)$ is governed by Eqs. (3.9) and (3.10), it must be an exact solution of problem (3.5).

Proof Convergence of the series $\sum_{i = 0}^{\infty} v_{i} (ξ)$ implies

(4.15)

\lim_{i \to \infty} v_{i} (ξ) = 0 .

Consider the series

\sum_{i = 0}^{\infty} hH (ξ) (N (\sum_{j = 0}^{\infty} a_{ij} L_{j}^{*} (ξ)) + (1 - χ_{i + 1}) \sum_{j = 0}^{\infty} b_{j} L_{j}^{*} (ξ))

Using Eqs. (4.15), (3.9) and (3.10) yields

(4.16)

\begin{matrix} \sum_{i = 0}^{\infty} hH (ξ) (N (\sum_{j = 0}^{\infty} a_{ij} L_{j}^{*} (ξ)) + (1 - χ_{i + 1}) \sum_{j = 0}^{\infty} b_{j} L_{j}^{*} (ξ)) \\ = L \sum_{i = 0}^{\infty} (v_{i + 1} (ξ) - χ_{i + 1} v_{i} (ξ)) \\ = L (\lim_{i \to \infty} v_{i + 1} (x)) \\ = 0, \end{matrix}

where

(4.17)

χ_{m} = \{\begin{matrix} 0, m \leq 1, \\ 1, m > 1 \end{matrix}

and the linearity of the operator

L

is used.

Since $h \neq 0, H (x) \neq 0$ , it can be expressed, as

(4.18)

\sum_{i = 0}^{\infty} (N (\sum_{j = 0}^{\infty} a_{ij} L_{j}^{*} (ξ)) + (1 - χ_{i + 1}) \sum_{j = 0}^{\infty} b_{j} L_{j}^{*} (ξ)) = 0 .

Moreover,

(4.19)

\begin{matrix} \sum_{i = 0}^{\infty} (N (\sum_{j = 0}^{\infty} a_{ij} L_{j}^{*} (ξ)) + (1 - χ_{i + 1}) \sum_{j = 0}^{\infty} b_{j} L_{j}^{*} (ξ)) \\ = \sum_{i = 0}^{\infty} (N [v_{i + 1} (ξ)] + (1 - χ_{i + 1}) q (ξ)) \\ = \sum_{i = 0}^{\infty} (N [v_{i + 1} (ξ)] + q (ξ) . \end{matrix}

From Eqs. (4.18) and (4.19), it can be written as

(4.20)

\sum_{i = 0}^{\infty} (N [v_{i + 1} (ξ)] + q (ξ)) = 0,

which shows that the series solution satisfies the differential Eq. (3.5). This completes the proof.

5 Procedure of the Legendre-homotopy method

Step 1 The solution $V_{0} (ξ)$ of the differential equation $L [v (ξ)] = 0$ , subject to the given boundary conditions, is obtained.

Step 2 $V_{0} (ξ)$ is expressed in terms of Legendre polynomials to get the initial approximate solution, as

(5.21)

v_{0} (ξ) = \sum_{j = 0}^{m} a_{0 j} L_{j}^{*} (ξ),

where

a_{0 j}

’s are calculated using Eq. (3.8).

Step 3 Using Eq. (3.9), $v_{1} (ξ)$ can be calculated, as

(5.22)

v_{1} (ξ) = L^{- 1} hH (ξ) (N (\sum_{j = 0}^{m} a_{0 j} L_{j}^{*} (ξ)) + \sum_{j = 0}^{m} b_{j} L_{j}^{*} (ξ)) .

Step 4 Using Eq. (3.10), $v_{i} (ξ)$ ’s for i = 2,3,4,…, can be calculated, as

(5.23)

v_{i} (ξ) = v_{i - 1} (ξ) + L^{- 1} hH (ξ) (N (\sum_{j = 0}^{m} a_{i - 1 j} L_{j}^{*} (ξ))) = 0,

where

a_{i - 1 j}

’s can be calculated using Eq. (3.11).

Step 5 The n-th order approximate solution is calculated, as

(5.24)

v_{n^{*}} (ξ) = \sum_{i = 0}^{n} v_{i} (ξ) .

When the auxiliary linear operator $L$ is a differential operator, its inverse $L^{- 1}$ gives rise to integrals in Eqs. (5.22) and (5.23). Due to the application of Legendre polynomials, Eqs. (5.22) and (5.23) involve only polynomial functions. Since, polynomials are easier to integrate and use in arithmetic operations, therefore the proposed modification simplifies the calculations to a great extent.

A flow chart of the procedure is shown in Fig. 1. In the next section, the method is applied on different linear and nonlinear higher order boundary value problems. Steps 1–5 are followed to solve the problems, taking $m = 10$ . The calculations are performed using Mathematica 8.0.

Flow chart for the proposed Legendre-homotopy method.

6 Numerical examples

Example 1

The following ninth order linear boundary value problem is considered:

(6.25)

\begin{matrix} v^{(9)} (ξ) - v (ξ) = - 9 e^{ξ}, ξ \in [0, 1] \\ v (0) = 1, v (1) = 0, \\ v^{'} (0) = 0, v^{'} (1) = - e, \\ v^{″} (0) = - 1, v^{″} (1) = - 2 e, \\ v^{‴} (0) = - 2, v^{‴} (1) = - 3 e, \\ v^{(4)} (0) = - 3 . \end{matrix}\}

The exact solution of this problem is

v (ξ) = (1 - ξ) e^{ξ} .

The value of

V_{0} (ξ)

is obtained, as

(6.26)

v_{0} (ξ) = 1 - \frac{ξ^{2}}{2} - \frac{ξ^{3}}{3} - \frac{ξ^{4}}{8} + \frac{1}{24} (- 1012 + 372 e) ξ^{5} + \frac{1}{24} (2642 - 972 e) ξ^{6} + \frac{1}{24} (- 2316 + 852 e) ξ^{7} + \frac{1}{24} (685 - 252 e) ξ^{8} .

The zeroth order approximation is calculated, as

(6.27)

v_{0} (ξ) = \sum_{j = 0}^{10} a_{0 j} L_{j}^{*} (ξ) .

The homogeneous part of the nonlinear operator $N$ is taken as the linear differential operator $L$ . Using the first and second order deformation equations, $v_{1} (ξ)$ and $v_{2} (ξ)$ are obtained, where

(6.28)

\begin{matrix} v_{1} (ξ) = & \frac{1}{379399923302400} (338998227872682409162299 h ξ^{5} \\ - 124710478627904905755180 eh ξ^{5} - 914896108162880458444620 h ξ^{6} \\ + 336571469000861236873836 eh ξ^{6} - 121002746083613568363618 h ξ^{7} \\ + 44514422609488799415984 eh ξ^{7} + 1887818747352851715800526 h ξ^{8} \\ - 694489705809164249335005 eh ξ^{8} + 53363921259166150703070 h ξ^{9} \\ - 19631489531533503984360 eh ξ^{9} - 2702512316820915383304520 h ξ^{10} \\ + 994198720871018535272400 eh ξ^{10} - 53376716177160767428856 h ξ^{11} \\ + 19636196518820603362368 eh ξ^{11} + 2661968899514615252276988 h ξ^{12} \\ - 979283631169195957795008 eh ξ^{12} + 35590387703211667617564 h ξ^{13} \\ - 13092971939332480853280 eh ξ^{13} - 1755339637376751422090400 h ξ^{14} \\ + 645753364864241522176200 eh ξ^{14} - 15254881792516033577016 h ξ^{15} \\ + 5611957388967208766064 eh ξ^{15} + 746082842071026976993876 h ξ^{16} \\ - 274468539008690861139669 eh ξ^{16} + 3814075878247750911005 h ξ^{17} \\ - 1403120102675260687956 eh ξ^{17} - 185314074172190097874740 h ξ^{18} \\ + 68173238047668506052300 eh ξ^{18} - 423817395166793988360 h ξ^{19} \\ + 155913706492696512360 eh ξ^{19} + 20483196329392601606802 h ξ^{20} \\ - 7535346819061888881054 eh ξ^{20}) . \end{matrix}

The value of

v_{2} (ξ)

is not expressed here because it is very bulky and not fit to write. The second order approximation to the exact solution can be determined, as

(6.29)

v (ξ) = v_{0} (ξ) + v_{1} (ξ) + v_{2} (ξ),

where the auxiliary function

H (ξ)

is chosen as

H (ξ) = 1

and the value of h is taken as

h = - 0.01

. The numerical results are summarized in Table 1.

Example 2

For $x \in [0, 1]$ , the ninth order nonlinear boundary value problem is considered:

(6.30)

\begin{matrix} v^{(9)} (ξ) - v (ξ) v^{'} (ξ) = e^{- 2 ξ} (2 + e^{ξ} (ξ - 10) - 3 ξ + ξ^{2}), \\ v (0) = 1, v (1) = 0, \\ v^{'} (0) = - 2, v^{'} (1) = - e^{- 1}, \\ v^{″} (0) = 3, v^{″} (1) = 2 e^{- 1}, \\ v^{‴} (0) = - 4, v^{‴} (1) = - 3 e^{- 1}, \\ v^{(4)} (0) = 5 . \end{matrix}\}

The exact solution of this problem is

v (ξ) = (1 - ξ) e^{- ξ} .

The value of

V_{0} (ξ)

is obtained, as

(6.31)

\begin{matrix} V_{0} (ξ) = & 1 - 2 ξ + \frac{3 ξ^{2}}{2} - \frac{2 ξ^{3}}{3} + \frac{5 ξ^{4}}{24} + \frac{(660 - 244 e) ξ^{5}}{24 e} \\ + \frac{(- 1788 + 658 e) ξ^{6}}{24 e} + \frac{(1620 - 596 e) ξ^{7}}{24 e} + \frac{(- 492 + 181 e) ξ^{8}}{24 e} . \end{matrix}

The zeroth order approximation is calculated, as

(6.32)

v_{0} (ξ) = \sum_{j = 0}^{10} a_{0 j} L_{j}^{*} (ξ) .

The homogeneous part of the nonlinear operator

N

is taken as the linear differential operator

L

. Moreover, the auxiliary function

H (ξ)

is taken as

H (ξ) = 1

and h is chosen as

h = - 1

. The second order approximation to the exact solution is calculated using the proposed method and the numerical results are summarized in Table 3.

Example 3

The following eighth order linear boundary value problem is considered:

(6.33)

\begin{matrix} v^{(8)} (ξ) - v (ξ) = 8 e^{ξ}, ξ \in [0, 1], \\ v (0) = 1, v^{(4)} (0) = - 3, \\ v^{'} (0) = 0, v^{(5)} (0) = - 4, \\ v^{″} (0) = - 1, v^{'} (1) = - e, \\ v^{‴} (0) = - 2, v^{″} (1) = - 2 e . \end{matrix}\}

The exact solution of the problem is

v (ξ) = (1 - ξ) e^{ξ} .

The value of

V_{0} (ξ)

is obtained, as

V_{0} (ξ) = 1 - \frac{ξ^{2}}{2} - \frac{ξ^{3}}{3} - \frac{ξ^{4}}{8} - \frac{ξ^{5}}{30} + \frac{(4550 - 1680 e) ξ^{6}}{2520} + \frac{(- 2940 + 1080 e) ξ^{7}}{2520},

The zeroth order approximation is calculated, as

(6.34)

v_{0} (ξ) = \sum_{j = 0}^{10} a_{0 j} L_{j}^{*} (ξ) .

The homogeneous part of the nonlinear operator

N

is taken as the linear differential operator

L

and the function

H (ξ)

is chosen as

H (ξ) = 1

. Value of h is chosen as

h = - 0.69

. A second order approximation to the exact solution is calculated using the proposed method and the results are summarized in Table 4.

Example 4

The following tenth order linear boundary value problem is considered:

(6.35)

\begin{matrix} v^{(10)} (ξ)) - ξ v (ξ) + e^{ξ} (89 + 21 ξ + ξ^{2} - ξ^{3}) = 0, ξ \in [- 1, 1], \\ v (- 1) = 0, v (1) = 0, \\ v^{'} (- 1) = \frac{2}{e}, v^{'} (1) = - 2 e, \\ v^{″} (- 1) = \frac{2}{e}, v^{″} (1) = - 6 e, \\ v^{‴} (- 1) = 0, v^{″} (1) = - 12 e, \\ v^{(4)} (- 1) = - \frac{4}{e}, v^{(4)} (1) = - 20 e \end{matrix}\}

The exact solution of this problem is

v (ξ) = (1 - ξ^{2}) e^{ξ} .

The value of

V_{0} (ξ)

is calculated, as

(6.36)

V_{0} (ξ) = \frac{1}{96 e} (91 + 23 e^{2} - 175 ξ + 59 e^{2} ξ - 160 ξ^{2} + 4 e^{2} ξ^{2} + 418 ξ^{3} - 86 e^{2} ξ^{3} + 102 ξ^{4} - 30 e^{2} ξ^{4} - 396 ξ^{5} + 48 e^{2} ξ^{5} - 40 ξ^{6} + 4 e^{2} ξ^{6} + 190 ξ^{7} - 26 e^{2} ξ^{7} + 7 ξ^{8} - e^{2} ξ^{8} - 37 ξ^{9} + 5 e^{2} ξ^{9}) .

The zeroth order approximation is calculated, as

(6.37)

v_{0} (ξ) = \sum_{j = 0}^{10} a_{0 j} L_{j}^{*} (ξ) .

The homogeneous part of the nonlinear operator

N

is taken as the linear operator

N

the auxiliary function

H (ξ)

is taken as

H (ξ) = 1

. Value of h is taken as

h = - 1

. A second order approximation to the exact solution is calculated using the presented method. The numerical results are summarized in Table 6.

Table 1 Absolute errors for Example 1.

$ξ$	Exact value of $v (ξ)$	Approximate value of $v (ξ)$	Absolute error
$0.0$	$1.000000$	$1.000000$	$0.000000$
$0.1$	$0.994654$	$0.994654$	$2.289423 \times 10^{- 10}$
$0.2$	$0.977122$	$0.977122$	$4.623567 \times 10^{- 9}$
$0.3$	$0.944901$	$0.944901$	$2.081388 \times 10^{- 8}$
$0.4$	$0.895095$	$0.895095$	$4.783428 \times 10^{- 8}$
$0.5$	$0.824361$	$0.824361$	$7.122960 \times 10^{- 8}$
$0.6$	$0.728848$	$0.728848$	$7.339381 \times 10^{- 8}$
$0.7$	$0.604126$	$0.604126$	$4.961130 \times 10^{- 8}$
$0.8$	$0.445108$	$0.445108$	$1.629848 \times 10^{- 8}$
$0.9$	$0.245960$	$0.245960$	$1.814813 \times 10^{- 9}$
$1.0$	$0.000000$	$- 6.940873 \times 10^{- 9}$	$6.940873 \times 10^{- 9}$

Figs. 2–5 show the comparison of exact and approximate solutions curves for Examples 1–4 respectively. Solid lines show exact solutions and dashed lines show approximate solutions.

Comparison of exact and approximate solution curves for Example 1.

Comparison of exact and approximate solution curves for Example 2.

Comparison of exact and approximate solution curves for Example 3.

Comparison of exact and approximate solution curves for Example 4.

7 Conclusion

In this paper, the Legendre-homotopy analysis method is proposed using Legendre polynomials to approximate the solutions of linear and nonlinear higher order boundary value problems arising in mathematics and physics. The proposed scheme is a modification of the well-known homotopy analysis method which uses the orthogonal property of the well known Legendre polynomials to simplify the computations involved in each iteration. The resulting higher order deformation equations involve only polynomials which overcomes the difficulty arising in the calculation of integrals. The proposed scheme is effectively applied on different higher order linear and nonlinear boundary value problems. The absolute errors in the approximate solution values are calculated and summarized in Tables 1–7 . Tables 2, 5 and 7 show the comparison of the proposed scheme with different methods available in literature revealing that the proposed method provides better approximations to the exact solutions. Figs. 2–5 show that the approximate solution curves match favorably well with the exact solution curves. The numerical computations and graphical illustrations are performed using Mathematica $8.0$ . The numerical and graphical results depict the efficiency and accuracy of the proposed method.

Table 2 Comparison of maximum absolute error with DTM (Hassan and Erturk, 2009).

DTM (Hassan and Erturk, 2009)	Present method
$3.0 \times 10^{- 7}$	$7.339381 \times 10^{- 8}$

Table 3 Absolute errors for Example 2.

$ξ$	Exact value of $v (ξ)$	Approximate value of $v (ξ)$	Absolute error
$0.0$	$1.000000$	$1.000000$	$0.000000$
$0.1$	$0.814354$	$0.814354$	$3.260858 \times 10^{- 11}$
$0.2$	$0.654985$	$0.654985$	$6.342264 \times 10^{- 10}$
$0.3$	$0.518573$	$0.518573$	$2.721959 \times 10^{- 9}$
$0.4$	$0.402192$	$0.402192$	$5.962918 \times 10^{- 9}$
$0.5$	$0.303265$	$0.303265$	$9.179561 \times 10^{- 9}$
$0.6$	$0.219525$	$0.219525$	$8.321407 \times 10^{- 9}$
$0.7$	$0.148976$	$0.148976$	$8.904865 \times 10^{- 9}$
$0.8$	$0.089866$	$0.089866$	$1.372047 \times 10^{- 8}$
$0.9$	$0.040657$	$0.040657$	$5.576126 \times 10^{- 9}$
$1.0$	$0.000000$	$7.517892 \times 10^{- 8}$	$7.517892 \times 10^{- 8}$

Table 4 Absolute errors for Example 3.

$ξ$	Exact value of $v (ξ)$	Approximate value of $v (ξ)$	Absolute error
$0.0$	$1.000000$	$1.000000$	$0.000000$
$0.1$	$0.994654$	$0.994654$	$1.066014 \times 10^{- 11}$
$0.2$	$0.977122$	$0.977122$	$4.818624 \times 10^{- 10}$
$0.3$	$0.944901$	$0.944901$	$3.701736 \times 10^{- 9}$
$0.4$	$0.895095$	$0.895095$	$1.293098 \times 10^{- 8}$
$0.5$	$0.824361$	$0.824361$	$2.847439 \times 10^{- 8}$
$0.6$	$0.728848$	$0.728848$	$4.114151 \times 10^{- 8}$
$0.7$	$0.604126$	$0.604126$	$2.726661 \times 10^{- 8}$
$0.8$	$0.445108$	$0.445108$	$1.487792 \times 10^{- 8}$
$0.9$	$0.245960$	$0.245960$	$6.976558 \times 10^{- 8}$
$1.0$	$0.000000$	$- 8.648293 \times 10^{- 8}$	$8.648293 \times 10^{- 8}$

Table 5 Comparison of absolute errors with other methods.

$ξ$	Present method	Golbabai and Javidi ( $N = 8$ ) (Golbabai and Javidi, 2007)	Torvattanabun and Koonprasert (Torvattanabun and Koonprasert, 2010)
$0.25$	$1.5137 \times 10^{- 9}$	$2.1630 \times 10^{- 9}$	$3.8922 \times 10^{- 10}$
$0.50$	$2.8474 \times 10^{- 8}$	$1.1571 \times 10^{- 7}$	$1.1571 \times 10^{- 7}$
$0.75$	$2.8074 \times 10^{- 8}$	$1.0479 \times 10^{- 6}$	$1.0479 \times 10^{- 6}$
$1.00$	$8.6423 \times 10^{- 8}$	$4.2188 \times 10^{- 6}$	$4.2188 \times 10^{- 6}$

Table 6 Absolute errors for Example 4.

$ξ$	Exact value of $v (ξ)$	Approximate value of $v (ξ)$	Absolute error
$- 1.0$	$1.00000$	$1.00000$	$8.85958 \times 10^{- 14}$
$- 0.8$	$1.09412$	$1.09412$	$8.65974 \times 10^{- 14}$
$- 0.6$	$1.172550$	$1.172550$	$4.91607 \times 10^{- 13}$
$- 0.4$	$1.228370$	$1.228370$	$5.80425 \times 10^{- 13}$
$- 0.2$	$1.253130$	$1.253130$	$5.32907 \times 10^{- 15}$
0	$1.236540$	$1.236540$	$9.34564 \times 10^{- 12}$
$0.2$	$1.166160$	$1.166160$	$1.18881 \times 10^{- 10}$
$0.4$	$1.027010$	$1.027010$	$4.71345 \times 10^{- 11}$
$0.6$	$0.801195$	$0.801195$	$1.16518 \times 10^{- 9}$
$0.8$	$0.467325$	$0.467325$	$2.18991 \times 10^{- 9}$
$1.0$	$0.000000$	$- 6.33957 \times 10^{- 9}$	$6.33957 \times 10^{- 9}$

Table 7 Comparison of maximum absolute errors with other methods.

Present method	Siddiqi et al. (2009)	Siddiqi and Akram (2007)	Lamnii et al. (2008)
$3.29 \times 10^{- 9}$	$1.97 \times 10^{- 6}$	$3.28 \times 10^{- 6}$	$1.86 \times 10^{- 8}$

References

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Bhrawy A.H., Al-Shomrani M.M., . A shifted Legendre spectral method for fractional-order multi-point boundary value problems. Adv. Difference Equ. 2012
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Butcher J.C., . The role of orthogonal polynomials in numerical ordinary differential equations. J. Comput. Appl. Math.. 1992;43:231-242.
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Canuto C., Hussaini M.Y., Quarteroni A., Zang T.A., . Spectral Methods in Fluid Dynamics. New York: Springer-Verlag; 1989.
Chandrasekhar S., . Hydrodynamic and Hydromagnetic Stability. The International Series of Monographs on Physics. Oxford: Clarendon Press; 1961.
Dawar A., Shah Z., Idrees M., Khan W., Islam S., Gul T., . Impact of thermal radiation and heat source/sink on eyring-powell fluid flow over an unsteady oscillatory porous stretching surface. Math. Comput. Appl.. 2018;23(2):20.
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[Google Scholar]
Fiza M., Islam S., Ullah H., Shah Z., Chohan F., . An asymptotic method with applications to nonlinear coupled partial differential equations. Punjab Univ. J. Math.. 2018;50(1):139-151.
[Google Scholar]
Golbabai A., Javidi M., . Application of homotopy perturbation method for solving eighth-order boundary value problems. Appl. Math. Comput.. 2007;191(2):334-346.
[Google Scholar]
Gottlieb D., Orszag S.A., . Numerical Analysis of Spectral Methods: Theory and Applications. Philadelphia: SIAM; 1977.
Gul T., . Scattering of a thin layer over a nonlinear radially extending surface with magnetohydrodynamic and thermal dissipation. Surf. Rev. Lett.. 2018;1850123
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Hassan I.H.A., Erturk V.S., . Solutions of different types of the linear and non-linear higher order boundary value problems by differential transformation method. Eur. J. Pure Appl. Math.. 2009;2(3):426-447.
[Google Scholar]
Hosseini S.R., Sheikholeslami M., Ghasemian M., Ganji D.D., . Nanofluid heat transfer analysis in a microchannel heat sink (MCHS) under the effect of magnetic field by means of KKL model. Powder Technol.. 2018;324:36-47.
[Google Scholar]
Jalaal M., Ganji, . An analytical study on motion of a sphere rolling down an inclined plane submerged in a Newtonian fluid. Powder Technol.. 2010;198:82-92.
[Google Scholar]
Khader M.M., Mahdy A.M.S., Shehata M.M., . An integral collocation approach based on Legendre polynomials for solving Riccati, logistic and delay differential equations. Appl. Math.. 2014;5:2360-2369.
[Google Scholar]
Khan N.S., Gul T., Islam S., Khan A., Shah Z., . Brownian motion and thermophoresis effects on MHD mixed convective thin film second-grade nanofluid flow with Hall effect and heat transfer past a stretching sheet. J. Nanofluids. 2017;6:1-18.
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Khan H., Haneef M., Shah Z., Islam S., Khan W., Muhammad S., . The combined magneto hydrodynamic and electric field effect on an unsteady Maxwell nanofluid flow over a stretching surface under the influence of variable heat and thermal radiation. Appl. Sci.. 2018;8:160.
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Lamnii A., Mraoui H., Sbibih D., Tijini A., Zidna A., . Spline solution of some linear boundary value problems. Appl. Math. E-Notes. 2008;8:171-178.
[Google Scholar]
Liao S.J., . Proposed homotopy analysis techniques for the solution of nonlinear problems. Shanghai Jiao Tong University; 1992. [Ph.D. dissertation]
Liu Y., . Application of Legendre polynomials in solving volterra integral equations of the second kind. Appl. Math.. 2013;3(5):157-159.
[Google Scholar]
Lyshevski, S.E., Dunipace, K.R., 1997. Identification and tracking control of aircraft from real-time perspectives. In: Proceedings of the 1997 IEEE International Conference on Control Applications Hartford, CT. pp. 499–504.
Malvandi A., Ganji D.D., . Effects of nanoparticle migration on force convection of alumina/water nanofluid in a cooled parallel-plate channel. Adv. Powder Technol.. 2014;25:1369-1375.
[Google Scholar]
Mamadu J.E., Njoseh I.N., . Numerical solutions of Volterra equations using Galerkin method with certain orthogonal polynomials. J. Appl. Math. Phys.. 2016;4:376-382.
[Google Scholar]
Mohyud-Din S.T., Yildirim A., . Solution of tenth and ninth order boundary value problems by homotopy perturbation method. J. Korean Soc. Indus. Appl. Math.. 2010;14(1):17-27.
[Google Scholar]
Mohyud-Din S.T., Yildirim A., . Solutions of tenth and ninth order boundary value problems by modified variational iteration method. Appl. Appl. Math.. 2010;5(1):11-25.
[Google Scholar]
Muhammad S., Ali G., Shah Z., Islam S., Hussain S.A., . The rotating flow of magneto hydrodynamic carbon nanotubes over a stretching sheet with the impact of non-linear thermal radiation and heat generation/absorption. Appl. Sci.. 2018;8:482.
[CrossRef] [Google Scholar]
Odibat Z., . On Legendre polynomial approximation with the VIM or HAM for numerical treatment of nonlinear fractional differential equation. J. Comput. Appl. Math.. 2011;235:2956-2968.
[Google Scholar]
Parand K., Dehghan M., Rezaeia A.R., Ghaderi S.M., . An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method. Comput. Phys. Commun.. 2010;181:1096-1108.
[Google Scholar]
Sadighi A., Ganji D.D., . Solution of the generlaized nonlinear Boussinesq equation using homotopy perturbation and variational iteration methods. Int. J. Nonlinear Sci. Numer. Simul.. 2007;8(3):435-443.
[Google Scholar]
Shah Z., Gul T., Khan M.A., Ali I., Islam S., Husain F., . Effects of Hall current on steady three dimensional non -newtonian nanofluid in a rotating frame with brownian motion and thermophoresis effects. J. Eng. Technol.. 2017;6:280-296.
[Google Scholar]
Shah Z., Islam S., Gul T., Bonyah E., Khan M.A., . The electrical MHD and Hall current impact on micropolar nanofluid flow between rotating parallel plates. Results Phys.. 2018;9:1201-1214.
[Google Scholar]
Sheikholeslami M., Ganji D.D., . Heat transfer of Cu-water nanofluid flow between parallel plates. Powder Technol.. 2013;235:873-879.
[Google Scholar]
Sheikholeslami M., Ganji D.D., . Analytical investigation for Lorentz forces effect on nanofluid Marangoni boundary layer hydrothermal behavior using HAM. Indian J. Phys.. 2017;91(12):1581-1587.
[Google Scholar]
Sheikholeslami M., Ganji D.D., . Influence of electric field on ${Fe}_{3} O_{4}$ -water nanofluid radiative and convective heat transfer in a permeable enclosure. J. Mol. Liq.. 2018;250:404-412.
[Google Scholar]
Sheikholeslami M., Ashorynejada H.R., Ganji D.D., Yildirim A., . Homotopy perturbation method for three-dimensional problem of condensation film on inclined rotating disk. Sci. Iran. B. 2012;19(3):437-442.
[Google Scholar]
Sheikholeslami M., Ellahi R., Ashorynejad H.R., Domairry G., Hayat T., . Effects of heat transfer in flow of nanofluids over a permeable stretching wall in a porous medium. J. Comput. Theor. Nanosci.. 2014;11(2):486-496.
[Google Scholar]
Sheikholeslami M., Soleimani S., Ganji D.D., . Effect of electric field on hydrothermal behavior of nanofluid in a complex geometry. J. Mol. Liq.. 2016;213:153-161.
[Google Scholar]
Sheikholeslami M., Ganji D.D., Moradi R., . Forced convection in existence of Lorentz forces in a porous cavity with hot circular obstacle using nanofluid via Lattice Boltzmann method. J. Mol. Liq.. 2017;246:103-111.
[Google Scholar]
Sheikholeslami M., Jafaryar M., Bateni K., Ganji D.D., . Two phase modeling of nanofluid flow in existence of melting heat transfer by means of HAM. Indian J. Phys.. 2018;92(2):205-214.
[Google Scholar]
Siddiqi S.S., Akram G., . Solutions of tenth-order boundary value problems using eleventh degree spline. Appl. Math. Comput.. 2007;185(1):115-127.
[Google Scholar]
Siddiqi S.S., Akram G., Zaheer S., . Solution of tenth order boundary value problems using variational iteration technique. Eur. J. Sci. Res.. 2009;30(3):326-347.
[Google Scholar]
Sweilam N.H., Khader M.M., Kota W.Y., . On the numerical solution of Hammerstein integral equations using Legendre approximation. Int. J. Appl. Math. Res.. 2012;1(1):65-76.
[Google Scholar]
Tahir F., Gul T., Islam S., Shah Z., Khan A., Khan W., Ali L., Muradullah, . Flow of a nano-liquid film of Maxwell fluid with thermal radiation and magneto hydrodynamic properties on an unstable stretching sheet. J. Nanofluids. 2017;6:1-10.
[Google Scholar]
Torvattanabun M., Koonprasert S., . Variational iteration method for solving eighth-order boundary value problems. Thai J. Math.. 2010;8(4):121-129.
[Google Scholar]
Voigt R.G., Gottlieb D., Hussaini M.Y., . Spectral Methods for Partial Differential Equations. Philadelphia: SIAM; 1984.
Xu, X., Zhou, F., 2015. Numerical solutions for the eighth-order initial and boundary value problems using the second kind Chebyshev wavelets, Adv. Math. Phys. Article ID 964623. 9p.
Yalçinbas S., Sezer M., Sorkun H.H., . Legendre polynomial solutions of high order linear Fredholm integro-differential equations. Appl. Math. Comput.. 2009;210:334-349.
[Google Scholar]
Ziabakhsh Z., Domairry G., . Solution of the laminar viscous flow in a semi-porous channel in the presence of a uniform magnetic field by using the homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul.. 2009;14:1284-1294.
[Google Scholar]

Introduction

Preliminaries for Legendre polynomials

The Legendre-homotopy analysis method

Convergence of the solution

Procedure of the Legendre-homotopy method

Numerical examples

Conclusion

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[1] Alshomrani A.S., Gul T., . The convective study of the ${Al}_{2} O_{3}$ - $H_{2} O$ and Cu- $H_{2} O$ nano-liquid films sprayed over a stretching cylinder with viscous dissipation. Eur. Phys. J. Plus. 2017;132:495.
[Google Scholar]

[2] Anli F., Gungor S., . Some useful properties of Legendre polynomials and its applications to neutron transport equation in slab geometry. Appl. Math. Model.. 2007;31:727-733.
[Google Scholar]

[3] Bhrawy A.H., Al-Shomrani M.M., . A shifted Legendre spectral method for fractional-order multi-point boundary value problems. Adv. Difference Equ. 2012
[CrossRef] [Google Scholar]

[4] Butcher J.C., . The role of orthogonal polynomials in numerical ordinary differential equations. J. Comput. Appl. Math.. 1992;43:231-242.
[Google Scholar]

[5] Canuto C., Hussaini M.Y., Quarteroni A., Zang T.A., . Spectral Methods in Fluid Dynamics. New York: Springer-Verlag; 1989.

[6] Chandrasekhar S., . Hydrodynamic and Hydromagnetic Stability. The International Series of Monographs on Physics. Oxford: Clarendon Press; 1961.

[7] Dawar A., Shah Z., Idrees M., Khan W., Islam S., Gul T., . Impact of thermal radiation and heat source/sink on eyring-powell fluid flow over an unsteady oscillatory porous stretching surface. Math. Comput. Appl.. 2018;23(2):20.
[CrossRef] [Google Scholar]

[8] Doha E.H., Bhrawyb A.H., . Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials. Applied Numerical Mathematics. 2008;58:1224-1244.
[Google Scholar]

[9] Fiza M., Islam S., Ullah H., Shah Z., Chohan F., . An asymptotic method with applications to nonlinear coupled partial differential equations. Punjab Univ. J. Math.. 2018;50(1):139-151.
[Google Scholar]

[10] Golbabai A., Javidi M., . Application of homotopy perturbation method for solving eighth-order boundary value problems. Appl. Math. Comput.. 2007;191(2):334-346.
[Google Scholar]

[11] Gottlieb D., Orszag S.A., . Numerical Analysis of Spectral Methods: Theory and Applications. Philadelphia: SIAM; 1977.

[12] Gul T., . Scattering of a thin layer over a nonlinear radially extending surface with magnetohydrodynamic and thermal dissipation. Surf. Rev. Lett.. 2018;1850123
[CrossRef] [Google Scholar]

[13] Hassan I.H.A., Erturk V.S., . Solutions of different types of the linear and non-linear higher order boundary value problems by differential transformation method. Eur. J. Pure Appl. Math.. 2009;2(3):426-447.
[Google Scholar]

[14] Hosseini S.R., Sheikholeslami M., Ghasemian M., Ganji D.D., . Nanofluid heat transfer analysis in a microchannel heat sink (MCHS) under the effect of magnetic field by means of KKL model. Powder Technol.. 2018;324:36-47.
[Google Scholar]

[15] Jalaal M., Ganji, . An analytical study on motion of a sphere rolling down an inclined plane submerged in a Newtonian fluid. Powder Technol.. 2010;198:82-92.
[Google Scholar]

[16] Khader M.M., Mahdy A.M.S., Shehata M.M., . An integral collocation approach based on Legendre polynomials for solving Riccati, logistic and delay differential equations. Appl. Math.. 2014;5:2360-2369.
[Google Scholar]

[17] Khan N.S., Gul T., Islam S., Khan A., Shah Z., . Brownian motion and thermophoresis effects on MHD mixed convective thin film second-grade nanofluid flow with Hall effect and heat transfer past a stretching sheet. J. Nanofluids. 2017;6:1-18.
[Google Scholar]

[18] Khan H., Haneef M., Shah Z., Islam S., Khan W., Muhammad S., . The combined magneto hydrodynamic and electric field effect on an unsteady Maxwell nanofluid flow over a stretching surface under the influence of variable heat and thermal radiation. Appl. Sci.. 2018;8:160.
[CrossRef] [Google Scholar]

[19] Lamnii A., Mraoui H., Sbibih D., Tijini A., Zidna A., . Spline solution of some linear boundary value problems. Appl. Math. E-Notes. 2008;8:171-178.
[Google Scholar]

[20] Liao S.J., . Proposed homotopy analysis techniques for the solution of nonlinear problems. Shanghai Jiao Tong University; 1992. [Ph.D. dissertation]

[21] Liu Y., . Application of Legendre polynomials in solving volterra integral equations of the second kind. Appl. Math.. 2013;3(5):157-159.
[Google Scholar]

[22] Lyshevski, S.E., Dunipace, K.R., 1997. Identification and tracking control of aircraft from real-time perspectives. In: Proceedings of the 1997 IEEE International Conference on Control Applications Hartford, CT. pp. 499–504.

[23] Malvandi A., Ganji D.D., . Effects of nanoparticle migration on force convection of alumina/water nanofluid in a cooled parallel-plate channel. Adv. Powder Technol.. 2014;25:1369-1375.
[Google Scholar]

[24] Mamadu J.E., Njoseh I.N., . Numerical solutions of Volterra equations using Galerkin method with certain orthogonal polynomials. J. Appl. Math. Phys.. 2016;4:376-382.
[Google Scholar]

[25] Mohyud-Din S.T., Yildirim A., . Solution of tenth and ninth order boundary value problems by homotopy perturbation method. J. Korean Soc. Indus. Appl. Math.. 2010;14(1):17-27.
[Google Scholar]

[26] Mohyud-Din S.T., Yildirim A., . Solutions of tenth and ninth order boundary value problems by modified variational iteration method. Appl. Appl. Math.. 2010;5(1):11-25.
[Google Scholar]

[27] Muhammad S., Ali G., Shah Z., Islam S., Hussain S.A., . The rotating flow of magneto hydrodynamic carbon nanotubes over a stretching sheet with the impact of non-linear thermal radiation and heat generation/absorption. Appl. Sci.. 2018;8:482.
[CrossRef] [Google Scholar]

[28] Odibat Z., . On Legendre polynomial approximation with the VIM or HAM for numerical treatment of nonlinear fractional differential equation. J. Comput. Appl. Math.. 2011;235:2956-2968.
[Google Scholar]

[29] Parand K., Dehghan M., Rezaeia A.R., Ghaderi S.M., . An approximation algorithm for the solution of the nonlinear Lane-Emden type equations arising in astrophysics using Hermite functions collocation method. Comput. Phys. Commun.. 2010;181:1096-1108.
[Google Scholar]

[30] Sadighi A., Ganji D.D., . Solution of the generlaized nonlinear Boussinesq equation using homotopy perturbation and variational iteration methods. Int. J. Nonlinear Sci. Numer. Simul.. 2007;8(3):435-443.
[Google Scholar]

[31] Shah Z., Gul T., Khan M.A., Ali I., Islam S., Husain F., . Effects of Hall current on steady three dimensional non -newtonian nanofluid in a rotating frame with brownian motion and thermophoresis effects. J. Eng. Technol.. 2017;6:280-296.
[Google Scholar]

[32] Shah Z., Islam S., Gul T., Bonyah E., Khan M.A., . The electrical MHD and Hall current impact on micropolar nanofluid flow between rotating parallel plates. Results Phys.. 2018;9:1201-1214.
[Google Scholar]

[33] Sheikholeslami M., Ganji D.D., . Heat transfer of Cu-water nanofluid flow between parallel plates. Powder Technol.. 2013;235:873-879.
[Google Scholar]

[34] Sheikholeslami M., Ganji D.D., . Analytical investigation for Lorentz forces effect on nanofluid Marangoni boundary layer hydrothermal behavior using HAM. Indian J. Phys.. 2017;91(12):1581-1587.
[Google Scholar]

[35] Sheikholeslami M., Ganji D.D., . Influence of electric field on ${Fe}_{3} O_{4}$ -water nanofluid radiative and convective heat transfer in a permeable enclosure. J. Mol. Liq.. 2018;250:404-412.
[Google Scholar]

[36] Sheikholeslami M., Ashorynejada H.R., Ganji D.D., Yildirim A., . Homotopy perturbation method for three-dimensional problem of condensation film on inclined rotating disk. Sci. Iran. B. 2012;19(3):437-442.
[Google Scholar]

[37] Sheikholeslami M., Ellahi R., Ashorynejad H.R., Domairry G., Hayat T., . Effects of heat transfer in flow of nanofluids over a permeable stretching wall in a porous medium. J. Comput. Theor. Nanosci.. 2014;11(2):486-496.
[Google Scholar]

[38] Sheikholeslami M., Soleimani S., Ganji D.D., . Effect of electric field on hydrothermal behavior of nanofluid in a complex geometry. J. Mol. Liq.. 2016;213:153-161.
[Google Scholar]

[39] Sheikholeslami M., Ganji D.D., Moradi R., . Forced convection in existence of Lorentz forces in a porous cavity with hot circular obstacle using nanofluid via Lattice Boltzmann method. J. Mol. Liq.. 2017;246:103-111.
[Google Scholar]

[40] Sheikholeslami M., Jafaryar M., Bateni K., Ganji D.D., . Two phase modeling of nanofluid flow in existence of melting heat transfer by means of HAM. Indian J. Phys.. 2018;92(2):205-214.
[Google Scholar]

[41] Siddiqi S.S., Akram G., . Solutions of tenth-order boundary value problems using eleventh degree spline. Appl. Math. Comput.. 2007;185(1):115-127.
[Google Scholar]

[42] Siddiqi S.S., Akram G., Zaheer S., . Solution of tenth order boundary value problems using variational iteration technique. Eur. J. Sci. Res.. 2009;30(3):326-347.
[Google Scholar]

[43] Sweilam N.H., Khader M.M., Kota W.Y., . On the numerical solution of Hammerstein integral equations using Legendre approximation. Int. J. Appl. Math. Res.. 2012;1(1):65-76.
[Google Scholar]

[44] Tahir F., Gul T., Islam S., Shah Z., Khan A., Khan W., Ali L., Muradullah, . Flow of a nano-liquid film of Maxwell fluid with thermal radiation and magneto hydrodynamic properties on an unstable stretching sheet. J. Nanofluids. 2017;6:1-10.
[Google Scholar]

[45] Torvattanabun M., Koonprasert S., . Variational iteration method for solving eighth-order boundary value problems. Thai J. Math.. 2010;8(4):121-129.
[Google Scholar]

[46] Voigt R.G., Gottlieb D., Hussaini M.Y., . Spectral Methods for Partial Differential Equations. Philadelphia: SIAM; 1984.

[47] Xu, X., Zhou, F., 2015. Numerical solutions for the eighth-order initial and boundary value problems using the second kind Chebyshev wavelets, Adv. Math. Phys. Article ID 964623. 9p.

[48] Yalçinbas S., Sezer M., Sorkun H.H., . Legendre polynomial solutions of high order linear Fredholm integro-differential equations. Appl. Math. Comput.. 2009;210:334-349.
[Google Scholar]

[49] Ziabakhsh Z., Domairry G., . Solution of the laminar viscous flow in a semi-porous channel in the presence of a uniform magnetic field by using the homotopy analysis method. Commun. Nonlinear Sci. Numer. Simul.. 2009;14:1284-1294.
[Google Scholar]

A Legendre-homotopy method for the solutions of higher order boundary value problems

Abstract

Keywords

Legendre polynomials

Homotopy analysis method

Higher order boundary value problems

1 Introduction

2 Preliminaries for Legendre polynomials

3 The Legendre-homotopy analysis method

4 Convergence of the solution

5 Procedure of the Legendre-homotopy method

6 Numerical examples

7 Conclusion

References

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