Numerical methods for nonlinear fourth-order boundary value problems with applications

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Numerical methods for nonlinearfourth-order boundary value problemswith applicationsMohamed Ali Hajji a & Kamel Al-Khaled ba Department of Mathematical Sciences, United Arab EmiratesUniversity, P.O. Box 17551, Al-Ain, U.A.Eb Department of Mathematics and Statistics, Jordan University ofScience and Technology, Irbid, 22110, JordanVersion of record first published: 31 Dec 2007.

To cite this article: Mohamed Ali Hajji & Kamel Al-Khaled (2008): Numerical methods for nonlinearfourth-order boundary value problems with applications, International Journal of ComputerMathematics, 85:1, 83-104

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International Journal of Computer MathematicsVol. 85, No. 1, January 2008, 83–104

Numerical methods for nonlinear fourth-order boundaryvalue problems with applications

MOHAMED ALI HAJJI† and KAMEL AL-KHALED*‡

†Department of Mathematical Sciences, United Arab Emirates University,P.O. Box 17551 Al-Ain, U.A.E

‡Department of Mathematics and Statistics, Jordan University of Science and Technology,Irbid 22110, Jordan

(Received 06 February 2006; revised version received 06 March 2007; accepted 22 March 2007)

In this paper, we present efficient numerical algorithms for the approximate solution of nonlinearfourth-order boundary value problems. The first algorithm deals with the sinc–Galerkin method (SGM).The sinc basis functions prove to handle well singularities in the problem. The resulting SGM discretesystem is carefully developed. The second method, the Adomian decomposition method (ADM), givesthe solution in the form of a series solution. A modified form of the ADM based on the use of theLaplace transform is also presented. We refer to this method as the Laplace Adomian decompositiontechnique (LADT). The proposed LADT can make the Adomian series solution convergent in theLaplace domain, when the ADM series solution diverges in the space domain. A number of examplesare considered to investigate the reliability and efficiency of each method. Numerical results show thatthe sinc–Galerkin method is more reliable and more accurate.

Keywords: Adomian’s decomposition method; Approximate solutions; Fourth-order BVPs; Laplacetransform; Sinc–Galerkin

AMS Subject Classifications: 34A34; 34K10; 34K28; 65L99; 35C10

1. Introduction

Accurate and fast numerical solution of two-point boundary value ordinary differential equa-tions is necessary in many important scientific and engineering applications, e.g. boundarylayer theory, the study of stellar interiors, control and optimization theory, and flow networksin biology. For example, we consider the problem of bending of a long uniformly rectangularplate supported over the entire surface by an elastic foundation and rigidly supported alongthe edges. The vertical deflection w at every point satisfies the system

Dd4w

dx4= f − kw, w(0) = w′(0) = w(1) = w′(1) = 0 (1.1)

*Corresponding author. Email: [email protected]†Email: [email protected]

International Journal of Computer MathematicsISSN 0020-7160 print/ISSN 1029-0265 online © 2008 Taylor & Francis

http://www.tandf.co.uk/journalsDOI: 10.1080/00207160701363031

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84 M. A. Hajji and K. Al-Khaled

where D is the flexural rigidity of the plate, f is the intensity of the load acting on the plate,and k is the reaction of the foundation. The details are given in [1]. Another mathematicalmodel: suppose v represents an elastic beam of length 1, which is clamped at its left sidex = 0, and resting on a kind of elastic bearing at its right side x = 1, along its length, a loadf is added to cause deformations. Then the differential equation that models this phenomenais given by

v′′′′(x) = f (x, v(x)), 0 < x < 1, v(0) = v′(0) = 0, v′′(1) = 0, v′′′(1) = g(v(1)) (1.2)

where f ∈ C([0, 1] × R) and g ∈ C(R). Corresponding problems modelling vibrating beamson elastic bearings were considered in [2]. The detail of the mechanical interpretation of theabove two models belongs to a general class of boundary value problems of the form

u′′′′(x) + q(x)H(u(x)) = f (x), 0 < x < b (1.3)

subject to the conditions

u(0) = α0, u′(0) = α1, u(b) = β0, u′(b) = β1, (1.4)

where H(u) is a polynomial or an exponential function, and α0, α1, β0, β1 are real constants.For details about the occurrence of two-point boundary value problems, see [3, 4]. The analyticsolution of (1.4) for special choices of q(x), H(u) and f (x) are easily obtained, but forarbitrary choices, the analytic solution cannot be determined. Therefore, numerical methodsfor obtaining an approximate solution are introduced. The literature of numerical analysiscontains many articles on the solution of boundary value problems [5–8]. Theorems which listthe conditions for the existence and uniqueness of solutions of such problems are contained ina comprehensive survey in a book by Agarwal [9], though no numerical methods are containedtherein for solving boundary value problems.

The present work is motivated by the desire to obtain numerical solutions to nonlinear fourth-order boundary value problems with a better accuracy level. In the standard setup of the SGM,the errors are known to be O(exp(−κ

√N) with some κ > 0, where N is the number of nodes

or bases used in the method. The goal of this study can be achieved by implementing both theSGM and ADM. The SGM for ordinary differential equations has many researchers. In [10]the authors applied the SGM to solve the general case for the linear fourth-order boundaryvalue problems. For more details about the applications of SGM see [11, 12]. In recent years,the ADM has been used in obtaining approximate solutions to a wide class of differentialand integral equations. The method provides the solution in a rapidly convergent series withcomponents that are elegantly computed. The main advantage of the method is that it can beused directly without using restrictive assumptions. It is worth mentioning that several authorshave treated many concepts related to Adomain’s method such as the convergence concept andcomparisons with other existing numerical techniques. Wazwaz [13] applied ADM for solvinga special 2m order boundary value problem of the form y(2m)(x) = f (x, y), 0 < x < b.

Convergence analyses ofADM have been carried out by Cherruault et al. [14, 15]. However,relatively few papers deal with the comparison of ADM and other methods; we mention forexample a comparison betweenADM and the Taylor series method carried out byWazwaz [16],by using linear and nonlinear problems. The study showed that the decomposition method iseasy to use and produces reliable results with few iterations.Also in [17] the variational iterationmethod is described and used to give approximate solutions for some well-known non-linearproblems. Comparison with Adomian’s decomposition method reveals that the approximatesolutions obtained by the variational iteration method converge to its exact solution faster thanthose of Adomian’s method.

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Numerical methods for nonlinear fourth-order boundary value problems with applications 85

The outline of this paper is as follows. In section 2, we formulate the discrete system byreplacing the differential equation by an explicit system of algebraic equations, which is solvedby Newton’s method. Section 3 is devoted to the implementation of ADM to construct theapproximate solution. In section 4, we address the implementation of the presented algorithmusing the software package Mathematica. Six numerical examples, where the exact solutionis explicitly given, are considered and the results of the numerical algorithms are presented insection 5. Finally, in section 6, we conclude the paper with some remarks.

2. Sinc function properties

The properties of the sinc function are discussed thoroughly in [11, 12]. In this section anoverview of the basic formulation of the sinc function required for our subsequent developmentis presented.

2.1 The sinc function

The sinc function is defined on the whole real line, −∞ < z < ∞, by

sinc(z) ≡

⎧⎪⎨⎪⎩

sin(πz)

πz, z �= 0,

1, z = 0.

(2.1)

For h > 0 and k = 0, ±1, ±2, . . ., the translated sinc function with even spaces nodes aregiven by

S(k, h)(z) ≡

⎧⎪⎨⎪⎩

sin[(π/h)(z − kh)][(π/h)(z − kh)] , z �= kh,

1, z = kh.

(2.2)

The sinc–Galerkin procedure for the boundary value problem (1.3) begins by selecting com-posite sinc functions appropriate to the interval (0, b) as the basis function for the expansionof the approximate solution for u(x). We introduce the map

w = φ(z) = ln

(z

(b − z)

)(2.3)

which is a conformal mapping from DE , the eye-shaped domain in the z-plane, onto the infinitestrip in the w-plane, DS , where

DE ={z = x + iy : |arg

(z

(b − z)

)| < d ≤ π

2

},

DS ={w = u + iv : |v| < d ≤ π

2

}.

These are shown in figure 1.The basis functions are derived from the composite translated sinc functions

S(k, h) ◦ φ(z) = sinc

[(φ(z) − kh)

h

](2.4)

for z ∈ DE . These are shown in figure 2 for real values x.

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Figure 1. Eye-shaped domain and the infinite strip.

S(k, h) ◦ φ(z) in equation (2.4) defines the basis element for equation (1.3) on the interval(0, b). The mesh size h is the mesh size in DE for the uniform grids {kh}, −∞ < k < ∞.The sinc grid points xk ∈ (0, b) in DE will be denoted by xk because they are real. The inverseimages of the equispaced grids are

xk = φ−1(kh) = b ekh

1 + ekh.

For the boundary condition (1.4), the sinc basis functions in (2.4) do not have a derivativewhen z tends to 0 or b. Thus we modify the sinc basis functions as

Sk(z) = S(k, h) ◦ φ(z)

(φ′(z))2= sinc[(φ(z) − kh)/h]

(φ′(z))2. (2.5)

Figure 2. The graph of S(k, h) ◦ φ(x) for k = −2, 0, 2, h = π/12 and b = 1.

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Figure 3. The graph of (S(k, h) ◦ φ(z)/(φ′(z))2) for k = −2, 0, 2, h = π/12 and b = 1.

These are shown in figure 3 for real values x. Note that the derivatives of the modified sincbasis functions are defined as z approaches 0 or b.

2.2 Sinc function interpolation and quadrature

Sinc rules of a special class of functions B(DE) have been developed. A discussion of theproperties of such functions in B(DE) is found in [11, 12].

DEFINITION 2.1 Let φ : DE −→ DS be a conformal mapping of DE to DS with inverse ψ . Let� = {ψ(u) ∈ DE : −∞ < u < ∞} = (0, b). Then B(DE) is the class of functions F whichare analytic in B(DE) and satisfy∫

ψ(t+L)

|F(z)|dz −→ 0, t −→ ±∞,

where L = {iv : |v| < d ≤ π/2}, and on the boundary of DE, denoted by ∂DE satisfy

N (F ) =∫

∂DE

|F(z)| dz < ∞.

Interpolation and quadrature rules for functions in B(DE) are defined in the following theoremswhose proofs are found in [11, 12].

THEOREM 2.1 If φ′, F ∈ B(DE) then, for all z ∈ �,

F(z) =∞∑

j=−∞F(zj )S(j, h) ◦ φ(z) + EF , (2.6)

where

EF = sin(πφ(z)/h)

2πi

∫∂DE

φ′(w) F (w) dw

(φ(w) − φ(z)) sin(πφ(w)/h).

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For the sinc–Galerkin method, the infinite quadrature rule appearing in (2.6) can be evaluateddirectly, but in general it must be truncated to a finite sum. The following theorem indicatesthe conditions under which exponential convergence results.

THEOREM 2.2 Let F ∈ B(DE) and φ be a conformal map with constants α, β, and C so that

∣∣∣∣ F(z)

φ′(z)

∣∣∣∣ ≤ C

{exp(−α|φ(z)|), z ∈ �−,

exp(−β|φ(z)|), z ∈ �+,

where

�− = {z ∈ � : φ(z) = x ∈ (−∞, 0)} =(

0, b

2

),

�+ = {z ∈ � : φ(z) = x ∈ [0, −∞)} =(

b

2, b

)}.

Then the sinc trapezoidal quadrature rule is

∫ b

0F(z)dz = h

N∑j=−M

F(zj )

φ′(zj )+ O(exp(−αMh)) + O(exp(−βNh)) + O

(exp

(−2πd

h

)).

(2.7)

Hence, for the selections N = [|(α/β)M + 1|], h = √2πd/(αM), where [|·|] denotes the

greatest integer, the exponential order of the sinc trapezoidal quadrature rule in (2.7) isO(exp(−√

2πdαM)).

2.3 SGM approach

For clarity of development the method will be presented for the nonlinear fourth-order problem

Lu = u′′′′(x) + q(x)H(u(x)) = f (x), 0 < x < b, (2.8)

subject to the conditions

u(0) = 0, u′(0) = 0, u(b) = 0, u′(b) = 0. (2.9)

An approximate solution is assumed to be of the form

um(x) =N∑

k=−M

ukSk(x), m = M + N + 1, (2.10)

where the basis functions Sk(x) are defined by (2.5) and uk are the expansion coefficients tobe determined. The coefficients uk in (2.10) are determined by orthogonalizing the residual

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Lum − f with respect to the basis functions {Sk}Nk=−M . This yields the discrete system

〈Lum − f, Sk〉 = 0, −M ≤ k ≤ N. (2.11)

Another approach, which we adopt in the present work, is to analyse

〈u′′′′, Sk〉 + 〈qH, Sk〉 − 〈f, Sk〉 = 0, (2.12)

where the general weighted inner product is taken to be

〈g(x), h(x)〉 =∫ b

0g(x)h(x)w(x)dx

where w(x) is a weight function taken to be w(x) = 1/(φ′(x))2. We note that a completediscussion on the choice of the weight function 1/(φ′(x))m can be found in [10, 12].

With the sinc quadrature rule in (2.7), we evaluate the inner products in (2.12). First, weevaluate

〈u′′′′(x), Sk〉 =∫ b

0u′′′′(x)Sk(x)w(x)dx. (2.13)

Integrating by parts the right-hand side of (2.13), we obtain

〈u′′′′(x), Sk〉 = Sk(x)w(x)u′′′(x)

∣∣∣b0−

∫ b

0u′′′(x)

d

dx[Sk(x)w(x)]dx. (2.14)

Since w(0) = w(b) = 0, the first term in the right-hand side of (2.14) is zero. A secondintegration by parts on the second term in the right-hand side of (2.14), noting that w′(0) =w′(b) = 0, gives

〈u′′′′(x), Sk〉 =∫ b

0u′′(x)

d2

dx2 [Sk(x)w(x)]dx. (2.15)

Two more integration by parts to remove the derivatives from the dependent variable u, usingthe conditions in (2.9), leads to

〈u′′′′(x), Sk〉 =∫ b

0u(x)

d4

dx4 [Sk(x)w(x)]dx. (2.16)

Note that (d/dx)[Sk(x)] = (d/dφ)[Sk] φ′(x). Setting

dn

dφn[Sk(x)] = S

(n)k (x), n = 0, 1, 2, 3, 4,

we obtain, by expanding the derivatives under the integral in (2.16) (see, [10, 12, 18]),

〈u′′′′(x), Sk〉 =∫ b

0

(4∑

i=0

u(x)S(i)k (x)g4,i (x)

)dx (2.17)

where g4,i , i = 0, 1, 2, 3, 4 are given by

g4,4(x) = w(φ′)4,

g4,3(x) = 6w(φ′)2φ′′ + 4w′(φ′)3,

g4,2(x) = 3w(φ′′)2 + 4wφ′φ′′′ + 12w′φ′φ′′ + 6w′′(φ′)2,

g4,1(x) = wφ′′′′ + 4w′φ′′′ + 6w′′φ′′ + 4w′′′φ′,

g4,0(x) = w(4).

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Applying the sinc quadrature rule (2.7) to the right-hand side of (2.17), we obtain

〈u′′′′(x), Sk〉 ≈ h

N∑j=−M

4∑i=0

u(xj )

φ′(xj )hiδ

(i)

jk g4,i (xj ), (2.18)

where δ(i)

jk are given by

δ(0)

jk = [S(j, h) ◦ φ(x)]∣∣∣xk

={

1, j = k,

0, j �= k,(2.19)

δ(1)

jk = hd

dφ[S(j, h) ◦ φ(x)]

∣∣∣xk

=

⎧⎪⎨⎪⎩

0, j = k,

(−1)k−j

(k − j), j �= k,

(2.20)

δ(2)

jk = h2 d2

dφ2[S(j, h) ◦ φ(x)]

∣∣∣xk

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

−π2

3, j = k,

−2(−1)k−j

(k − j)2, j �= k,

(2.21)

δ(3)

jk = h3 d3

dφ3[S(j, h) ◦ φ(x)]

∣∣∣xk

=

⎧⎪⎪⎨⎪⎪⎩

0, j = k,

(−1)k−j

(k − j)3[6 − π2(k − j)2], j �= k,

(2.22)

and

δ(4)

jk = h4 d4

dφ4[S(j, h) ◦ φ(x)]

∣∣∣xk

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

π4

5, j = k,

−4(−1)k−j

(k − j)4[6 − π2(k − j)2], j �= k.

(2.23)

We note that |δ(i)

jk | are bounded by (see, [10])

|δ(4)

jk | ≤ π4

5, (2.24)

|δ(3)

jk | ≤ (2π2 − 3)

4, (2.25)

|δ(2)

jk | ≤ π2

3, (2.26)

and

|δ(�)

jk | ≤ 1, � = 0, 1. (2.27)

The bounds (2.24)—(2.27) are used in the next theorem, whose proof resembles the proofof Theorem 3.1 in [10], to show that the error in approximating (2.17) by (2.18) is of orderO(exp(−c/h)) for some positive constants c, h.

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THEOREM 2.3 Let φ be the conformal mapping, and making the selections for h and N asin Theorem 2.2. With the assumptions that ug4,i ∈ B(DE), i = 0, 1, 2, 3, 4, and for someconstant C1 we assume

∣∣∣∣u(x)g4,i (x)

φ′(z)

∣∣∣∣ ≤ C1

{exp(−α|φ(z)|), z ∈ �−,

exp(−β|φ(z)|), z ∈ �+.

Then there exists a constant C2 independent of M such that∣∣∣∣∣∣〈u′′′′(x), Sk〉 − h

N∑j=−M

4∑i=0

u(xj )

φ′(xj )hiδ

(i)

jk g4,i (xj )

∣∣∣∣∣∣ ≤ C2M2 exp(−√

πdαM).

For the other two inner products in (2.12), direct application of (2.7) yields theapproximations

〈qH, Sk〉 ≈ hw(xk)q(xk)H(u(xk))

φ′(xk), (2.28)

and

〈f, Sk〉 ≈ hw(xk)f (xk)

φ′(xk). (2.29)

The following notations are used in writing down the discrete system. For any function g(x)

we define the m × m diagonal matrix, m = M + N + 1,

D(g) = diag[g(x−M), . . . , g(x0), . . . , g(xN)].

We define the m × m matrices I (p), p = 0, 1, 2, 3, 4, by

I (p) = [δ(p)

jk ], j, k = −M, . . . , N,

i.e., I (p) is the m × m matrix whose entries are the components of δ(p)

jk , j, k = −M, . . . , N.

Finally, we denote �u to be the m-vector with entries [u(x−M), u(x−M+1), . . . , u(xN−1), u(xN)],and �1 to be an m-vector each of whose components is 1. Replacing each term of (2.12) withits respective approximation defined in (2.18), (2.28) and (2.29), and replacing u(xj ) by uj

(since u(xj ) = uj ), we obtain the following theorem.

THEOREM 2.4 If the assumed approximate solution of the boundary value problem (2.8)

subject to the conditions in (2.9) is (2.10), then the discrete sinc–Galerkin system for thedetermination of the unknown coefficients �u = {uk}Nk=−M is given by

[4∑

i=0

1

hiI (i)D

(g4,i

φ′

)]�u + D

(wq

φ′

)H(�u) = D

(wf

φ′

)�1. (2.30)

The nonlinear system (2.30) of m = M + N + 1 equations in the m unknown coefficients�u can be solved using the multidimensional version of Newton’s method; for more detailsabout the solution and convergence see [10, 12, 18]. Once the coefficients uk are solved, theapproximate solution um(x) is given by (2.10).

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2.4 Treatment of boundary conditions

In the previous subsection, we showed how to replace the derivatives and integrals by the nec-essary matrices if the boundary conditions are homogeneous (2.9). In particular, we used thehomogeneous boundary conditions are (2.9) in approximating 〈u′′′′, Sk〉 by (2.18). The SGMpresented in this paper is still applicable to equation (1.3) with non-homogeneous bound-ary conditions (1.4). We simply transform the problem to one with homogeneous boundaryconditions. This is accomplished, see [18], by defining

v(x) = u(x) − [α0 + α1x + (3β0 − β1 − 2α1 − 3α0)x2 + (β1 − 2β0 + α1 + 2α0)x

3].(2.31)

The transformation (2.31) will convert (1.3) into another fourth-order differential equation ofthe same type, in terms of v(x), with the new boundary conditions v(0) = v′(0) = v(1) =v′(1) = 0. Therefore, our approximate solution using SGM in (2.10) becomes vm(x) =∑N

k=−M vkSk(x), m = M + N + 1, and the solution u(x) to (1.3) is u(x) = v(x) + [α0 +α1x + (3β0 − β1 − 2α1 − 3α0)x

2 + (β1 − 2β0 + α1 + 2α0)x3].

3. Adomian decomposition

This section presents both theADM, and a LaplaceAdomian decomposition technique (briefly,LADT) for solving the boundary value problem in (1.3). The LADT illustrates how to useboth the Laplace transform integral operator and the ADM to find an approximate solution forour boundary value problem.

3.1 ADM approach

Consider the boundary value problem in (1.3) in operator form

Lu = f (x) − q(x)H(u) (3.1)

where the differential operator L is given by

L = d4

dx4 . (3.2)

The inverse operator L−1 is therefore considered to be a 4-fold integral operator defined by

L−1(·) =∫ x

0

∫ x

0

∫ x

0

∫ x

0(.) dxdxdxdx. (3.3)

Operating with L−1 on (3.1), we obtain

u(x) =3∑

j=0

αj

1

j !xj + L−1[f (x)] − L−1[q(x)H(u)], (3.4)

where α0 = u(0), α1 = u′(0), α2 = u′′(0) and α3 = u′′′(0). The constants α0 and α1 are givenby the boundary conditions (1.4) and the constants α2 and α3 are to be determined later by

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satisfying the boundary conditions at x = b. The ADM expresses the solution u(x) of (1.3)by the decomposition series

u(x) =∞∑

n=0

un(x) (3.5)

and the nonlinear function H(u) by an infinite sum of polynomials

H(u) =∞∑

n=0

An(x), (3.6)

where the components un(x) of the solution u(x) will be determined recurrently, and An

are the so-called Adomian polynomials that can be constructed for various classes of nonlin-earity according to specific algorithms set by Adomian [19, 20]. These polynomials can beconstructed by using the general formula [20]

An = 1

n!dn

dλn

[H

(n∑

i=0

λiui

)]λ=0

, n ≥ 0.

Substitution of (3.5) and (3.6) into (3.4) yields

∞∑n=0

un(x) =3∑

j=0

αj

1

j !xj + L−1[f (x)] − L−1

⎡⎣q(x)

∞∑j=0

An(x)

⎤⎦. (3.7)

To determine the components un(x), n ≥ 0, we first identify the zeroth component u0(x) byall terms that arise from the left boundary conditions at x = 0 and the 4-fold integral of thesource term f (x). The remaining components un(x), n ≥ 1 are then determined recursivelyusing the preceding components, in such a way that (3.7) is formally balanced. In other words,the method introduces the recursive relation

u0(x) =3∑

j=0

αj

1

j !xj + L−1[f (x)],

uk+1(x) = −L−1[q(x)Ak(x)], k ≥ 0.

(3.8)

Because uk depends heavily on the zeroth component u0, it is computationally convenient tochoose u0 so as to contain the minimum number of terms. As noted in [13], only part of u0(x)

in (3.8) is assigned to u0(x) and the remaining terms are assigned to y1(x), among other terms.As a result, we formulate a new recursive relation to replace (3.8):

u0(x) = α0,

u1(x) =3∑

j=1

αj

1

j !xj + L−1[f (x)] − L−1[q(x)A0], (3.9)

uk+1(x) = −L−1[q(x)Ak(x)], k ≥ 1.

Relations (3.9) will enable us to determine the components un(x), n ≥ 0, recurrently, and asa result, the series solution of u(x) is readily obtained, where the constants α2, α3 are as of

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yet undetermined. Finally, an approximate solution is obtained by truncating the series (3.5).An N -term approximation is

φN(x, α2, α3) =N−1∑i=0

ui(x). (3.10)

It is important to note that the accuracy of the approximation (3.10) can be dramaticallyimproved by simply taking as many components un(x) as we wish.

The approximate solution (3.10) has yet to satisfy the remaining two boundary conditionsat x = b in equation (1.4). Setting φN(b, α2, α3) = β0 and φ′

N(b, α2, α3) = β1, then solvingthe resulting algebraic system in α2, α3, will determine the two unknown constants α2, α3

and eventually the numerical solution of our nonlinear fourth-order boundary value problem.Finally, the analytic solution u(x) is given by u(x) = limN→∞ φN .

The Adomian decomposition method (ADM) outlined above is easy to implement and doesnot need discretization. However, it has some drawbacks. Its efficiency and accuracy rely onthe convergence and the rate of convergence of the series solution. We found, as the exampleswill show, that the ADM gives a series solution which may have a slow rate of convergenceover wider regions. Furthermore, if the exact solution of the problem is oscillatory, then theADM series solution may be divergent.

To overcome these drawbacks, the ADM needs to be modified in order to work for problemswhere the solutions are of oscillatory nature. In the next subsection a new scheme, the LaplaceAdomian decomposition technique (LADT), is introduced. This technique is based on the useof the Laplace transform together with the Adomian decomposition algorithm.

3.2 The LADT approach

In [21], the author illustrates the use of both the Laplace transform and the Adomian decom-position method to approximate the solution of Bratu’s boundary value problem. With thesame analysis as in [21], we illustrate how the Laplace transform integral operator and theAdomian’s decomposition method (ADM) can be both efficiently used to obtain analytic andnumerical solutions of the nonlinear fourth-order boundary value problems in the interval(0, b) of the form

u(4) + q(x)H(u) = f (x), 0 < x < b, (3.11)

with the boundary conditions

u(0) = α0, u′(0) = α1, u(b) = β0, u′(b) = β1. (3.12)

The technique consists of applying the Laplace transform integral operator (denoted by L) toboth sides of equation (3.11). Hence

L[u(4)] + L[q(x)H(u)] = L[f (x)].Using the formula of the Laplace transform of the derivatives, we obtain

s4L[u] − s3u(0) − s2u′(0) − su′′(0) − u′′′(0) + L[q(x)H(u)] = L[f (x)]. (3.13)

Substituting for the two initial conditions u(0) = α0, u′(0) = α1 in equation (3.13), and setting

A = u′′(0), and B = u′′′(0), it follows that

s4L[u] − α0s3 − α1s

2 − As − B + L[q(x)H(u)] = L[f (x)], (3.14)

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where the constants A, B will be determined later by using the boundary conditions at x = b.Rewrite equation (3.14) as

L[u] = α0

s+ α1

s2+ A

s3+ B

s4+ 1

s4L[f (x)] − 1

s4L[q(x)H(u)]. (3.15)

The Adomian decomposition method assumes the solution u(x) as a decomposition series ofthe form

u(x) =∞∑

n=0

un(x) (3.16)

and the nonlinear function H(u) as

H(u) =∞∑

n=0

An(u0, u1, . . . , un), (3.17)

where the components un(x) of the solution u(x) will be determined recursively, andAn(u0, u1, . . . , un) are the Adomian polynomials which can be constructed for various classesof nonlinearity according to specific algorithms set by Adomian [19, 20] and Wazwaz [16].The first few polynomials are given by

A0 = H(u0),

A1 = u1H′(u0),

A2 = H ′′(u0)u2

1

2! + H ′(u0)u2,

A3 = 1

3!H′′′(u0)u

31 + H ′′(u0)u1u2 + H ′(u0)u3,

A4 = 1

4!H(4)(u0)u

41 + 1

2H(3)(u0)u

21u2 + 1

2H ′′(u0)u

22 + H ′(u0)u4,

...

Substitution of equations (3.16) and (3.17) into equation (3.15) yields

L[ ∞∑

n=0

un

]= α0

s+ α1

s2+ A

s3+ B

s4+ 1

s4L[f (x)] − 1

s4L

[q(x)

∞∑n=0

An

]. (3.18)

The decomposition method [16] identifies the zeroth component u0(x) by all terms that arisefrom the boundary conditions at x = 0 and from the Laplace transform of the source term.Based on this identification, matching both sides of equation (3.18) yields the followingiterative algorithm:

L[u0(x)] = α0

s+ α1

s2+ A

s3+ B

s4+ 1

s4L[f (x)], (3.19)

L[u1(x)] = − 1

s4L[q(x)A0], (3.20)

L[u2(x)] = − 1

s4L[q(x)A1]. (3.21)

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In general,

L[uk+1(x)] = − 1

s4L[q(x)Ak], k ≥ 0. (3.22)

The solution components un(x) are determined from their Laplace transforms. That is, we firstfind L[un(x)] then we take the inverse Laplace transform. Thus, applying the inverse Laplacetransform to equation (3.19) we obtain the zeroth component

u0(x) = α0 + α1x + A

2!x2 + 1

3!Bx3 + L−1

[1

s4L[f (x)]

]. (3.23)

Next, substituting u0(x) and A0 into equation (3.20), we obtain

L[u1(x)] = − 1

s4L[q(x)H(u0)]. (3.24)

Applying the inverse Laplace transform to equation (3.24) we obtain u1(x). Substituting u0, u1

and A1 into equation (3.21), and then applying the inverse Laplace transform, we find u2(x).Continuing in this way we can find all the solution components. As mentioned in [22], wenote that the existence and uniqueness of the solution is guaranteed by a result in Agarwal’sbook [9]. It is important to note that, since the solution u(x) is of a series form, we can find asmany solution components un(x) as we wish to get a good approximate solution. An N -termapproximate solution is

φN(x) =N−1∑n=0

un(x; A; B) = u0(x; A; B) + u1(x; A; B)

+ u2(x; A; B) + · · · + uN−1(x; A; B).

To determine the constants A and B, we require that φN(x) satisfies the boundary conditionsat x = b (see equation (3.12)). Setting φN(b) = β0 and φ′

N(b) = β1 results in an algebraicsystem for A and B. Solving the system, we obtain the required constants A and B whichcomplete the numerical solution of our nonlinear forth-order boundary value problem.

To give a clear overview of the content of the present work, in section 5, illustrative exampleshave been selected to demonstrate the efficiency of the methods presented.

4. Implementation issues

The proposed algorithms in the previous sections were implemented using the symbolic soft-ware package Mathematica. In particular, the LADT approach was successfully implementedsymbolically in Mathematica using the commands LaplaceTransform[f[t], t, s] and Inverse-LaplaceTransform[F[s], s, t] which, respectively, find the Laplace and the inverse Laplacetransforms. The SGM was also implemented in Mathematica numerically. We made use ofthe command Solve[{eq1, eq2},{A, B}] in Mathematica to solve for the still undeterminedconstants A and B, by the imposed boundary conditions at x = b, i.e., φN(b; A, B) = β0 andφ′

N(b; A, B) = β1, where φN is the approximate solution.

5. Numerical applications

In this section we provide numerical examples which illustrate various features of the methodspresented in this paper. The six examples reported in this section were selected from a large col-lection of problems to which both SGM and ADM were applied. These examples demonstrate

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the ease of implementation and assembly of the discrete system in SGM and the iterativetechnique in using the ADM techniques. Also, they show that the LADT is always better thanor equivalent to the ADM. The numerical results show that the SGM is better than both theADM and the LADT. One interesting example (Example 5.2) shows that the ADM divergeswhereas the LADT converges.

In all of the examples, unless otherwise indicated, the parameters used in the SGM methodare d = π/2, α = β = 1/2, and N = 64. We note that there are conservative choices for d, α

and β which work reasonably well in most cases [10]. The errors are reported on the set ofsinc grid-points xk , where

xk = bekh

1 + ekh, k = −M, . . . , N.

Example 5.1 [22] Consider the nonlinear boundary value problem

y(4)(x) = −6e−4y(x), 0 < x < 1, (5.1)


y(0) = 1, y ′(0) = 1

e, y(1) = ln(e + 1), y ′(1) = 1

(e + 1). (5.2)

The exact solution for this problem is y(x) = ln(e + x). In this example the nonlinearoperator H(y) = 6e−4y , q(x) = 1 and the source term f (x) = 0. For this example, both theADM and the LADT produce the same solution components yk . Thus, we only write downthe details of the LADT. The first four Adomian polynomials are calculated to be

A0 = 6e−4y0 , A1 = −24y1e−4y0 ,

A2 = 48e−4y0y21 − 24e−4y0y2, A3 = −64e−4y0y3

1 + 96e−4y0y1y2 − 24e−4y0y3.

The other polynomials can be easily generated as needed.In order to avoid evaluating the Laplace transform of some difficult terms, in matching both

sides of equation (3.18), we employ the modified decomposition method [22] and introducethe recursive relation

y0(x) = α0,

y1(x) = x

e+ A

2x2 + B

6x3 + L−1

[1

s4L(f (x))

]− L−1

[1

s4L(H(y0))

],

yk+1(x) = −L−1

[1

s4L(Ak)

], k ≥ 1

where L−1 denotes the inverse Laplace transform. The first few components of the solutiony(x) are calculated to be

y0(x) = 1,

y1(x) = 1

ex + A

2x2 + B

6x3 − 1

4e4x4,

y2(x) = 24 x5

e4

(0.00306566 + 0.00138889 A x + 0.000198413 B x2 − 2.72554 10−6 x3

),

and similarly for yk(x), k ≥ 3. A four-term approximate solution is φ4 = ∑3i=0 yi(x; A; B).

This has yet to satisfy the remaining two boundary conditions (5.2). Imposing the boundary

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Table 1. Numerical values for Example 5.1.

xi LADT/ADM

0.0001 1.675E−12 6.668E−130.2 4.993E−06 3.819E−100.4 1.322E−05 9.202E−100.6 1.566E−05 9.481E−100.8 8.247E−06 4.169E−100.9999 3.803E−12 9.243E−13

conditions y(1) = ln(e + 1), y ′(1) = 1/(e + 1), on the four-term approximation φ4 gives anonlinear algebraic system for the unknowns A and B. Upon solving this system we getA = −0.13567 and B = 0.10085, and eventually the numerical solution is obtained.

The SGM outline in section 2 has been applied to this example. The absolute error betweenthe approximate solution for both the LADT/ADM and the SGM methods and the exact solu-tion are tabulated in table 1. The errors are reported in the form a.aaa − Eγ which representa.aaa × 10−γ . The results indicate that the SGM performs better than the ADM/LADT andclearly show exponential convergence.

In the next example, where the solution is oscillatory, we show that the ADM produces adivergent series solution whereas the proposed LADT produces a convergent series solutionin the Laplace domain.

Example 5.2 [1] We consider the problem of bending a rectangular clamped beam of lengthπ resting on an elastic foundation. The vertical deflection w of the beam satisfies the system

d4w

dx4 + 64w = sin(2x), w(0) = w′(0) = w(π) = w′(π) = 0. (5.3)

The exact solution of (5.3) is

w(x) = −(−1 + e2 x) (−e2 π + e2 x) sin(2x)

80 e2 x (1 + e2 π ). (5.4)

Following the ADM procedure outlined in the previous section, we find that the general formof the solution components wk(x), where the solution w(x) = ∑∞

k=0 wk(x), is

wk(x) =3∑

j=0

αjxj (−64x4)k

(j + 4k)! + (−64)k

24k+4sin(2x)

from which the formal series solution is

w(x) =∞∑

k=0

wk(x) =∞∑

k=0

⎡⎣ 3∑

j=0

αjxj (−64x4)k

(j + 4k)! + (−64)k

24k+4sin(2x)

⎤⎦

=3∑

j=0

αjxj

( ∞∑k=0

(−64x4)k

(j + 4k)!

)+ 1

24sin(2x)

∞∑k=0

(−64

24

)k

.

The second infinite sum in the last equation is a divergent geometric series, since |(−64/24)| =4 > 1. This shows that the regular ADM diverges and hence is not suitable for this type ofproblem. However, the LADT overcomes this shortcoming of the ADM.

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Following the LADT outlined in the previous section, we find

L[w0] = A

s3+ B

s4+ 2

s4(s2 + 4)(5.5)

L[wk] =(−64

s4

)k

L[w0], k ≥ 1. (5.6)

Then the Laplace transform of the solution w(x) is

L[w(x)] =∞∑

k=0

L[wk] = L[w0]∞∑

k=0

(−64

s4

)k

.

The infinite sum on the right hand side of the above equation is convergent for |(−64/s4)| < 1.The exact solution w(x) is given by its Laplace transform by

L[w(x)] = B

64 + s4+ A s

64 + s4+ 2

(4 + s2)(64 + s4).

For this example, it can be easily verified that the exact solution w(x) in (5.4) can be recoveredby taking the inverse Laplace transform. By the above analysis, we see that the coupling of theLaplace transform with the Adomian decomposition made it possible to obtain a convergentseries expansion in the Laplace domain.

An N -term approximate solution is obtained by calculating N solution components, w0(x),w1(x), . . . , wN−1(x), as described above. LetφN(x; A; B) = ∑N−1

k=0 wk(x)be the approximatesolution. The constants A and B are determined by imposing the boundary conditions atx = π . Using Mathematica A and B were found to be A = 0.0996272 and B = −0.3. Theapproximate solution φ7(x) reads

φ7(x) = 1638.38 x + 0.0498136 x2 − 1092.3 x3 + 218.467 x5 − 0.00885575 x6

− 20.8025 x7 + 1.15556 x9 + 0.000112454 x10 − 0.042051 x11 + 0.00107876 x13

− 2.99578 × 10−7 x14 − 0.0000204876 x15 + 3.00699 × 10−7 x17

+ 2.6107 × 10−10 x18 − 3.55832 × 10−9 x19 + 3.41515 × 10−11 x21

− 9.51724 × 10−14 x22 − 2.57512 × 10−13 x23 + 1.66137 × 10−15 x25

+ 1.69761 × 10−17 x26 − 1.13598 × 10−17 x27 + 6.21772 × 10−20 x29

− 1.65188 × 10−21 x30 − 1.06971 × 10−22 x31 − 819.19 sin(2 x).

The SGM was applied to this example as well and the errors between the exact solution andthe numerical solutions obtained by the LADT and the SGM are displayed in figures 4 and 5.These results show that the performances of both methods are relatively the same.

The LADT relies heavily on finding the Laplace transform the source function f (x) andthat of the subsequent Adomian polynomials. This is not always easily accomplished if notimpossible. In these situations, we find that the SGM to be an excellent efficient alternative asdemonstrated in the next two examples.

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Figure 4. The graph of the absolute error for the SGM applied to Example 5.2.

Figure 5. The graph of the absolute error for the LADT applied to Example 5.2.

Example 5.3 [18] Consider the nonlinear boundary value problem

y(4)(x) = 6e−4y(x) − 12

(1 + x)4, 0 < x < 1, (5.7)


y(0) = 0, y ′(0) = 1, y(1) = ln 2, y ′(1) = 1/2. (5.8)

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Note that this example is similar to Example 5.1, except that (5.7) contains a source term. Thiskind of boundary value problems occur in plate deflection theory [23]. The exact solution forproblem (5.7) subject to (5.8) is y(x) = ln(1 + x). The nonlinear operator is H(y) = −6e−4y ,the source term is f (x) = −(12/(1 + x)4). For this example, we found that the best of thethree methods is the SGM. The ADM algorithm was not successful in obtaining a goodapproximate solution. The difficulty was in determining the unknown constants A and B, ifmore than four solution components were taken. If only four components were taken, theapproximate solution was found to be poor. The LADT was not suitable as well. The reason isthat the Laplace transform of f (x) involves the incomplete Gamma function. This makes theiterative Adomian algorithm difficult in computing the solution components yk . On the otherhand, the SGM performs well in obtaining an accurate approximate solution to (5.7) subjectto (5.8). The absolute errors between the exact and numerical solution at selected grid pointsxi are displayed in table 2.

Example 5.4 In this example we consider the model equation

y(4)(x) + 1

x4y(x) = f (x), 0 < x < 1 (5.9)

with homogeneous boundary conditions

y(0) = y ′(0) = y(1) = y ′(1) = 0, (5.10)

The function f (x) = (24 + 21 log(x) − 3 log(x)2 + (log(x)3/16)/x(3/2)) is consistent withthe singular exact solution y(x) = x5/2 ln(x)3. This solution has both an algebraic and log-arithmic singularity at x = 0. The absolute errors using SGM at selected xi are reported intable 3, where excellent results are shown. The accuracy of the approximate solution is alsoreflected in figure 6, in which the solid curve represents the exact solution, while the dottedcurve is the approximate solution. We can observe the perfect match of these two solutionseven for small N = 8.


xi SGM

0.0001 4.944E−120.2 2.438E−110.4 3.943E−120.6 3.274E−120.8 4.789E−110.9999 1.333E−11

Table 3. Numerical results for Example 4.4.

xi SGM

0.0001 5.553E−100.2 7.816E−100.4 6.785E−110.6 7.106E−110.8 6.905E−100.9999 5.276E−10

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Figure 6. The analytic solution (solid line) and the approximate solution (dotted line) for Example 4.4 using SGMfor N = 8.

Example 5.5 [2] In this example we consider the model equation (1.2),

u(4)(x) = u2 − x10 + 4x9 − 4x8 − 4x7 + 8x6 − 4x4 + 120x − 48, 0 < x < 1, (5.11)

with linear boundary conditions involving third-order derivatives

u(0) = u′(0) = u′′(1) = 0, u′′′(1) = g(u(1)), g(u) = 12u. (5.12)

The exact solution is u(x) = x5 − 2x4 + 2x2. This kind of problem appears naturally in thestudy of deformations of elastic beams on elastic bearings. For this example, we use theADM to seek an approximate solution. First we obtain an N -term approximate solution ofthe Adomian series (3.10) satisfying the first two boundary conditions u(0) = u′(0) = 0.The constants α2, α3, (see (3.10)), are determined by applying the remaining two boundaryconditions u′′(1) = 0, u′′′(1) = g(u(1)). Table 4 exhibits the absolute error of a three-termapproximate solution.

Example 5.6 [24] In this last example we consider the model equation

y(4)(x) + xy(x) = −(8 + 7x + x3)ex, 0 < x < 1, (5.13)


xi ADM

0.0001 1.873E−100.2 6.381E−040.4 2.108E−030.6 3.771E−030.8 5.139E−030.9999 6.119E−03

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xi SGM

0.0001 3.226E−100.2 9.836E−100.4 9.300E−110.6 6.321E−110.8 2.564E−100.9999 2.007E−10


y(0) = y(1) = 0, y ′(0) = 1, y ′(1) = −e, (5.14)

and y(x) = x(1 − x)ex as its exact solution. The SGM is applied to this example to obtainan approximate solution. The absolute errors are tabulated in table 5. This example has beennumerically solved in [24] using the spline method. Our results show that the proposed SGMmethod is better than the spline method used in [24].

6. Conclusion

In this work, we have applied the ADM, the proposed LADT and the SGM to a number offourth-order boundary value problems. This study shows that for some examples for whichthe ADM diverges, the proposed LADT may produce a convergent series solution and givesgood results. However, the LADT strongly depends on whether the Laplace and the inverseLaplace transforms are easily obtainable. On the other hand, the sinc–Galerkin method worksvery well. Another benefit of the SGM is that the scheme presented handles singularities withease. The examples considered demonstrate the good accuracy of the SGM. These features ofthe SGM suggest that the SGM can be considered to be a better alternative over other methodssuch as the ADM.

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Numerical methods for nonlinear fourth-order boundary value problems with applications

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