# Solving a Class of Singular Two-Point Boundary Value Problems Using New Modified Decomposition Method

Authors: Randhir Singh; Jitendra Kumar
Journal: ISRN Computational Mathematics
 (2013)

---

## Abstract

We introduce an effective methodology for solving a class of linear as
well as nonlinear singular two-point boundary value problems. This methodology
is based on a modification of Adomian decomposition method (ADM) and a new
two fold integral operator. We use all the boundary conditions to derive an integral
equation before establishing the recursive scheme for the solution components of
solution. Thus, we develop modified recursive scheme without any undetermined
coefficients while computing the successive solution components. This modification
also avoids solving a sequence of nonlinear algebraic or transcendental equations
for the undetermined coefficients. However, most of earlier recursive schemes using
ADM do require computation of undetermined coefficients. The approximate solution is obtained in the form of series with easily calculable components. Numerical
examples are included to demonstrate the accuracy, applicability, and generality
of the present technique. The results reveal that the method is very effective,
straightforward, and simple.


---

## Body

## 1. Introduction

We consider the following class of singular two-point boundary value problems [1–5]:
(1)(xαy′)′=f(x,y),0<x≤1,
subject to the boundary conditions
(2)y(0)=d,ay(1)+by′(1)=c,
where a>0,b,c, and  d are any finite constants and 0≤α<1. We assume that, for (x,y)∈{[0,1]×ℝ}, the function f(x,y) and ∂f/∂y are continuous and ∂f/∂y≥0. In particular, the problem (1) arises very frequently in applied sciences and in physiological studies, for example, in the study of steady-state oxygen diffusion in a spherical cell with Michaelis-Menten uptake kinetics [6] and distribution of heat sources in the human head [7]. In particular, when α=0 and f(x,y)=x-1/2y3/2 (1) is known as Thomas-Fermi equation [8], given the singular equation
(3)y′′=x-1/2y3/2.
Recently, there has been much interest in the study of singular two-point boundary value problems of type (1), (see, e.g., [1–5, 8–15]) and many of the references therein. The main difficulty of problem (1) is that the singularity behavior occurs at x=0. A lot of methods have been applied to tackle this singular boundary value problem. In [9], a standard three-point finite difference scheme was considered with uniform mesh for the solution of problem for α∈(0,1). In [1, 5], the finite difference methods were used to obtain the numerical solutions. In [4], a numerical method based on Green’s function was used to obtain numerical solution of the same problem. A novel approach that combines a modified decomposition method with the cubic B-spline collocation technique is presented in [14] to obtain approximate solution with high accuracy. Recently, in [15], a new modified decomposition method was applied to tackle these problems. These numerical methods have many advantages, but a huge amount of computational work is required to obtain accurate numerical solution especially for nonlinear problems.


### 1.1. Adomian Decomposition Method (ADM)

In this subsection, we briefly describe standard ADM or MADM for nonlinear second-order equation.Recently, many researchers [3, 8, 12–24] have shown interest in the study of ADM for different scientific models. Adomian [20] asserted that the ADM provides an efficient and computationally suitable method for generating approximate series solution for a large class of differential equations.According to Wazwaz [24], (1) in operator form is given as
(4)Ly=x-αf(x,y),
where L is a linear differential operator defined as
(5)L=x-αddx(xαddx).
The inverse operator L-1 is given as (see [24])
(6)L-1(·)=∫0xx-α∫0xxα(·)dxdx.
Operating the inverse linear operator L-1(·) on both sides of (4) yields
(7)y=y(0)+xy′(0)+L-1[x-αf(x,y)].
Next, we decompose the solution y and the nonlinear function N(y)≡f(x,y) by an infinite series as
(8)y=∑n=0∞‍yn,N(y)=∑n=0∞‍An,
where An are Adomian’s polynomials that can be constructed for various classes of nonlinear functions with the formula given in [17] as
(9)An=1n!dndλn[N(∑k=0∞‍ykλk)]λ=0,n=0,1,2,….
Substituting the series (8) into (7), we obtain
(10)∑n=0∞‍yn=y(0)+xy′(0)+L-1[x-α∑n=0∞‍An].
Upon matching both sides of (10), the standard ADM is given by
(11)y0=y(0)+xy′(0),yk+1=L-1[x-αAk],k≥0
and modified ADM is given as
(12)y0=y(0),y1=xy′(0)+L-1[x-αA0],yk+1=L-1[x-αAk],k≥0.
Having determined the components yn, n≥0 recurrently, the series solution of y follows immediately with the undetermined coefficient y′(0)=η1, is as yet to be determined [12, 21, 22]. For numerical purpose, the n-term truncated series
(13)ϕn(x)=∑m=0n‍ym(x)
may be used to give the approximate solution.The decomposition method has been applied to solve nonlinear boundary value problems for ordinary differential equations by several researchers [3, 12–16, 18, 19, 21–23]. Solving such problems using standard ADM or MADM is always a computationally involved task as it requires the computation of undetermined coefficients in a sequence of nonlinear algebraic equations which increases the computational work (see [3, 14, 15, 21–23]). A major disadvantage of the earlier methods for solving nonlinear BVPs is that we need to solve a sequence of growingly higher-order polynomials or more difficult transcendental equations [3, 21, 23]. For example, consider nonlinear equation
(14)y′′=ey,0≤x≤1,y(0)=0,y(1)=0.
Applying the standard ADM (11) with initial guess y0=η1x, where y′(0)=η1 to above problem (14), we obtain the solution components as
(15)y0=η1x,y1=(-1+eη1x-η1x)η12,y2=(-5+e2η1x-2η1x+eη1x(4-4η1x))4η14,y3=(-22+e3η1x-6η1x+e2η1x(6-6η1x)+3eη1x(5+2η1x(-3+η1x))e3η1x)×(12η16)-1⋮
and n-term approximate solution ϕn(x)=∑m=0n‍ym as
(16)ϕ1(x)=η1x+(-1+eη1x-η1x)η12,ϕ2(x)=η1x+(-1+eη1x-η1x)η12+(-5+e2η1x-2η1x+eη1x(4-4η1x))4η14,ϕ3(x)=η1x+(-1+eη1x-η1x)η12+(-5+e2η1x-2η1x+eη1x(4-4η1x))4η14+(-22+e3η1x-6η1x+e2η1x(6-6η1x)+3eη1x(5+2η1x(-3+η1x))e3η1x)×(12η16)-1⋮
Note that the above calculations become more and more complicated and cannot lead to exact results for the other approximations. To obtain approximate solution, the boundary condition y(1)=0 is imposed into ϕ3, and solving for η1, we can obtain the approximation. However, solving this transcendental equation for η1 requires additional computational work, and η1 may not be uniquely determined (see [3, 15, 21, 23]).In order to avoid solving such nonlinear algebraic or transcendental equations for two-point boundary value problems, in [16], extended ADM was introduced for nonsingular problems with Dirichlet boundary conditions. The method proposed in [13] was based on the new modification of ADM. Recently, a special modified inverse linear integral operators for higher-order boundary value problem was given in [25].In this paper, a new modification of the ADM is proposed to overcome the difficulties occurred in the standard ADM or MADM for solving nonlinear singular boundary value problems (1). To obtain a new modified decomposition method, we operate our proposed twofold integral operator on both sides of (1) and use all the boundary conditions to establish the recursion scheme for the solution components of the solution. We propose a modified recursion scheme which does not require the computation of unknown constants, that is, without solving a sequence of growingly higher-order polynomial or difficult transcendental equations to obtain unknown constant [3, 14, 15, 21, 23]. The main advantage of our proposed method is that it provides a direct recursive scheme for solving the singular boundary value problem.The rest of the paper is organized as follows. In Section2, the description of the proposed recursive scheme based on new definition of differential operator and its inverse operator is given. In Section 3, we illustrate our method with numerical results along with the graphical representation. In Section 4, the conclusion is given.


## 1.1. Adomian Decomposition Method (ADM)

In this subsection, we briefly describe standard ADM or MADM for nonlinear second-order equation.Recently, many researchers [3, 8, 12–24] have shown interest in the study of ADM for different scientific models. Adomian [20] asserted that the ADM provides an efficient and computationally suitable method for generating approximate series solution for a large class of differential equations.According to Wazwaz [24], (1) in operator form is given as
(4)Ly=x-αf(x,y),
where L is a linear differential operator defined as
(5)L=x-αddx(xαddx).
The inverse operator L-1 is given as (see [24])
(6)L-1(·)=∫0xx-α∫0xxα(·)dxdx.
Operating the inverse linear operator L-1(·) on both sides of (4) yields
(7)y=y(0)+xy′(0)+L-1[x-αf(x,y)].
Next, we decompose the solution y and the nonlinear function N(y)≡f(x,y) by an infinite series as
(8)y=∑n=0∞‍yn,N(y)=∑n=0∞‍An,
where An are Adomian’s polynomials that can be constructed for various classes of nonlinear functions with the formula given in [17] as
(9)An=1n!dndλn[N(∑k=0∞‍ykλk)]λ=0,n=0,1,2,….
Substituting the series (8) into (7), we obtain
(10)∑n=0∞‍yn=y(0)+xy′(0)+L-1[x-α∑n=0∞‍An].
Upon matching both sides of (10), the standard ADM is given by
(11)y0=y(0)+xy′(0),yk+1=L-1[x-αAk],k≥0
and modified ADM is given as
(12)y0=y(0),y1=xy′(0)+L-1[x-αA0],yk+1=L-1[x-αAk],k≥0.
Having determined the components yn, n≥0 recurrently, the series solution of y follows immediately with the undetermined coefficient y′(0)=η1, is as yet to be determined [12, 21, 22]. For numerical purpose, the n-term truncated series
(13)ϕn(x)=∑m=0n‍ym(x)
may be used to give the approximate solution.The decomposition method has been applied to solve nonlinear boundary value problems for ordinary differential equations by several researchers [3, 12–16, 18, 19, 21–23]. Solving such problems using standard ADM or MADM is always a computationally involved task as it requires the computation of undetermined coefficients in a sequence of nonlinear algebraic equations which increases the computational work (see [3, 14, 15, 21–23]). A major disadvantage of the earlier methods for solving nonlinear BVPs is that we need to solve a sequence of growingly higher-order polynomials or more difficult transcendental equations [3, 21, 23]. For example, consider nonlinear equation
(14)y′′=ey,0≤x≤1,y(0)=0,y(1)=0.
Applying the standard ADM (11) with initial guess y0=η1x, where y′(0)=η1 to above problem (14), we obtain the solution components as
(15)y0=η1x,y1=(-1+eη1x-η1x)η12,y2=(-5+e2η1x-2η1x+eη1x(4-4η1x))4η14,y3=(-22+e3η1x-6η1x+e2η1x(6-6η1x)+3eη1x(5+2η1x(-3+η1x))e3η1x)×(12η16)-1⋮
and n-term approximate solution ϕn(x)=∑m=0n‍ym as
(16)ϕ1(x)=η1x+(-1+eη1x-η1x)η12,ϕ2(x)=η1x+(-1+eη1x-η1x)η12+(-5+e2η1x-2η1x+eη1x(4-4η1x))4η14,ϕ3(x)=η1x+(-1+eη1x-η1x)η12+(-5+e2η1x-2η1x+eη1x(4-4η1x))4η14+(-22+e3η1x-6η1x+e2η1x(6-6η1x)+3eη1x(5+2η1x(-3+η1x))e3η1x)×(12η16)-1⋮
Note that the above calculations become more and more complicated and cannot lead to exact results for the other approximations. To obtain approximate solution, the boundary condition y(1)=0 is imposed into ϕ3, and solving for η1, we can obtain the approximation. However, solving this transcendental equation for η1 requires additional computational work, and η1 may not be uniquely determined (see [3, 15, 21, 23]).In order to avoid solving such nonlinear algebraic or transcendental equations for two-point boundary value problems, in [16], extended ADM was introduced for nonsingular problems with Dirichlet boundary conditions. The method proposed in [13] was based on the new modification of ADM. Recently, a special modified inverse linear integral operators for higher-order boundary value problem was given in [25].In this paper, a new modification of the ADM is proposed to overcome the difficulties occurred in the standard ADM or MADM for solving nonlinear singular boundary value problems (1). To obtain a new modified decomposition method, we operate our proposed twofold integral operator on both sides of (1) and use all the boundary conditions to establish the recursion scheme for the solution components of the solution. We propose a modified recursion scheme which does not require the computation of unknown constants, that is, without solving a sequence of growingly higher-order polynomial or difficult transcendental equations to obtain unknown constant [3, 14, 15, 21, 23]. The main advantage of our proposed method is that it provides a direct recursive scheme for solving the singular boundary value problem.The rest of the paper is organized as follows. In Section2, the description of the proposed recursive scheme based on new definition of differential operator and its inverse operator is given. In Section 3, we illustrate our method with numerical results along with the graphical representation. In Section 4, the conclusion is given.


## 2. New Modified Decomposition Method

In this section, we propose a new modified ADM based on two-fold integral operator for solving linear as well as nonlinear singular two-point boundary value problems.To overcome the singular behavior atx=0, we again rewrite (1)
(17)(xαy′)′=f(x,y),0<x≤1,
which can be rewritten in operator form as
(18)Ly=f(x,y),
where Ly≡(xαy′)′ is the linear differential operator.Twofold integral operatorL-1(·) regarded as the inverse operator of L(·) is proposed as
(19)L-1(·)≡∫0xx-α(∫x1(·)dx)dx.
To establish a new modified recursive scheme, we operate L-1(·) on the left-hand side of (17) and use the boundary condition y(0)=d, which yields
(20)L-1[(xαy′)′]=∫0xx-α(∫x1(xαy′)′dx)dx,=∫0xx-α(∫x1(xαy′′+αxα-1y′)dx)dx,=∫0xx-α(y′(1)-xαy′(x)-∫x1αxα-1y′dx+∫x1αxα-1y′dx)dx,=c1∫0xx-αdx-y(x)+d,L-1[(x-αy′)′]=c1x1-α1-α-y(x)+d,
where y′(1)=c1.Now, we again operate the inverse operatorL-1(·) on both sides of (17) and use (20), which gives
(21)y(x)=d+c1x1-α1-α-∫0xx-α(∫x1f(x,y)dx)dx.
Imposing the boundary condition ay(1)+by′(1)=c into (21), we get
(22)c1=(c-ad)(1-α)a+b(1-α)+a(1-α)a+b(1-α)×[∫01x-α(∫x1f(x,y)dx)dx].
For simplicity, we set [L-1f(x,y)]x=1=∫01x-α(∫x1f(x,y)dx)dx.Substituting the value ofc1 into (21), we obtain
(23)y(x)=d+(c-ad)x1-αa+b(1-α)+ax1-αa+b(1-α)×[L-1f(x,y)]x=1-[L-1f(x,y)].
Note that the right-hand side of (23) does not involve any undetermined coefficients.Next, we decompose the solutiony(x) by series as
(24)y=∑n=0∞‍yn,
and the nonlinear function f(x,y) is decomposed by series as
(25)f(x,y)=∑n=0∞‍An(y0,y1,…,yn),
where An(y0,y1,…,yn) are Adomian’s polynomials [17].Substituting the series (24) and (25) into (23), we obtain
(26)∑n=0∞‍yn=d+(c-ad)x1-αa+b(1-α)+ax1-αa+b(1-α)×[L-1∑n=0∞‍An]x=1-[L-1∑n=0∞‍An].
Upon matching both sides of (26), the new modified decomposition method is given by
(27)y0=d,y1=(c-ad)x1-αa+b(1-α)+ax1-αa+b(1-α)×[L-1A0]x=1-[L-1A0],yn+1=ax1-αa+b(1-α)[L-1An]x=1-[L-1An],n≥1,oryn+1=ax1-αa+b(1-α)∫01x-α∫x1Andxdx-∫0xx-α∫x1Andxdx,n≥1.
It should be noted that the new modified recursive scheme (27) requires no additional computational work as it does not involve any unknown constant. This slight change plays a major role in minimizing the computational work. This modification also avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients with multiple roots, which is required to complete the calculation of the solution by several earlier modified recursion schemes using the ADM or MADM [3, 14, 22, 23]. The above new modified scheme (27) gives the complete determination of solution components yn of solution y, and hence, the approximate series solution ψn(x) can be obtained by adding the solution components y0,y1,…,yn. However, for numerical purpose, the truncated n-term approximate series solution is given by
(28)ψn(x)=∑m=0n‍ym(x),
and the limiting value of sequence ψn(x) converges to exact solution y. We remark that the convergence of Adomian decomposition method for differential and integral equations has already been established by many authors [26–28].


## 3. Numerical Illustrations and Discussions

In this section, the new modified ADM (27) is implemented for tackling singular boundary value problems (1). We demonstrate the effectiveness of this method with five examples. All the numerical results obtained by the proposed method are compared with known results. Furthermore, the maximum error functions En(x) and approximate solutions ψn are plotted.Example 1.
Consider the nonlinear singular two-point boundary value problem [10]
(29)(xαy′)′=βxα+β-2ey(βxβey-α-β+1),0<x≤1,y(0)=-ln(4),y(1)=-ln(5),
with exact solution y(x)=ln(1/(4+xβ)).
We apply scheme (27) to (29), where 0≤α<1 and a=1, b=0, c=-ln(5), and d=-ln(4), scheme (27) for (29) can be read as follows
(30)y0=-ln(4),y1=(ln4-ln5)x1-α+x1-α∫01x-α∫x1xα+β-2A0dxdx-∫0xx-α∫x1xα+β-2A0dxdx,yn+1=x1-α∫01x-α∫x1xα+β-2Andxdx-∫0xx-α∫x1xα+β-2Andxdx,n≥1.
The Adomian’s polynomials for nonlinear function f(x,y)=β(βxβe2y-ey(α+β-1)) with y0=-ln(4) are obtained by formula (9) given as
(31)A0=β[0.0625βxβ+0.25(1-α-β)],A1=β[0.125βxβ+0.25(1-α-β)]y1,A2=β[0.125βxβ+0.25(1-α-β)]y2+β[0.25βxβ+0.25(1-α-β)]y122,⋮
For the demonstration purpose, we chose some specific values of α and β.
Forα=0.25, β=1, by using (30) and (31), we obtain the successive solution components yn of solution y(x) as
(32)y0=-1.38629,y1=0.0018564x0.75-0.25x+0.025x2,y2=-0.0016051x0.75-0.000066301x1.75+0.00625x2+0.00004219x2.75-0.0048611x3+0.00024038x4,⋮
Forα=0.5, β=1, we again use (30) and (31) to obtain the successive solution components as
(33)y0=-1.38629,y1=+0.00602312x0.5-0.25x+0.0208333x2,y2=-0.00573744x0.5-0.000501926x1.5+0.0104167x2+0.00015057x2.5-0.00451389x3+0.000186012x4⋮Remark 2.
In a similar manner, we can calculate any solution componentsyn of solution for different values 0≤α<1 and β>0. For simplicity, we have not listed all the solution components yn for different values of α and β, but we have calculated the first ten solution components.Now, we define error functions asEn(x)=|ψn(x)-y(x)| and the maximum absolute error is given as
(34)E(n)=max0<x≤1En(x).
We plot approximate solutions ψn, for n=2,3, and exact solution y in Figures 1 and 2 for different values of α and β. The error functions En(x), for n=3,4,5, are plotted in Figures 3 and 4. In addition, the maximum absolute errors E(n), for n=5,8,10, are listed in Tables 1 and 2.Table 1
Maximum error of Example1, when β=1.
α
E
(
5
)
E
(
8
)
E
(
10
)
0.25
2.24869
×
1
0
-
6
3.14283
×
1
0
-
9
3.90384
×
1
0
-
10
0.5
1.26007
×
1
0
-
6
1.24544
×
1
0
-
9
1.51616
×
1
0
-
11
0.75
7.66693
×
1
0
-
7
4.07955
×
1
0
-
10
9.72399
×
1
0
-
12Table 2
Maximum error of Example1, when β=3.5.
α
E
(
5
)
E
(
8
)
E
(
10
)
0.25
8.41020
×
1
0
-
7
7.75054
×
1
0
-
10
2.30935
×
1
0
-
11
0.5
3.97263
×
1
0
-
7
5.87251
×
1
0
-
9
7.81719
×
1
0
-
11
0.75
3.42314
×
1
0
-
6
2.78721
×
1
0
-
8
9.19921
×
1
0
-
11Figure 1
Comparison of exact solutiony and approximate solutions ψn,n=2,3, of Example 1 when α=0.5,β=1.Figure 2
Comparison of exact solutiony and approximate solutions ψn,n=2,3, of Example 1 when α=0.5,β=3.5.Figure 3
Error functionsEn, n=3,4,5, of Example 1 when α=0.5,β=1.Figure 4
Error functionsEn, n=3,4,5 of Example 1 when α=0.5,β=3.5.Example 3.
Consider the nonlinear singular two-point boundary value problem(35)(x1/2y′)′=316(-4+5x3/2)y5,0<x<1,y(0)=1,y(1)=12=0.7071067812,
with exact solution y(x)=1/(1+x3/2).
On applying scheme (27) to (35) where a=1, b=0, c=0.7071067812, and d=1, scheme (27) for (35) becomes as follows:
(36)y0=1,y1=0.2928932x1/2+x1/2∫01x-1/2∫x1g(x)A0dxdx-∫0xx-1/2∫x1g(x)A0dxdx,yn+1=x1/2∫01x-1/2∫x1g(x)Andxdx-∫0xx-1/2∫x1g(x)Andxdx,n≥1,
where g(x)=(3/16)(-4+5x3/2).
As given above, the Adomian polynomials for nonlinear functionf(y)=y5 with y0=1 are given by
(37)A0=1,A1=5y1,A2=10y12+5y2,A3=10y13+20y1y2+5y3,⋮
Using (36) and (37), we have the successive solution components as
(38)y0=1,y1=0.0821068x0.5-0.5x1.5+0.125x3,y2=-0.045527x0.5-0.102633x2+0.25x3+0.0366548x3.5-0.15625x4.5+0.0177557x6,⋮
In order to verify how close the approximate solutions are to the exact solution, we plot approximate solutions ψn,n=1,2,4, and exact solution y in Figure 5 and conclude that the approximate solution ψ4 is very close to exact solution y. In Figure 6, the error functions En(x), for n=5,6,7, are plotted. From Figure 6 we can see that when n increases, the error decreases. In addition, the maximum absolute errors E(n), for n=5,8,10, are listed in Table 3.Table 3
Maximum error of Example3.
n
5
8
10
E
(
n
)
7.385093
×
1
0
-
4
2.3947849
×
1
0
-
5
9.7489281
×
1
0
-
6Figure 5
Comparison of exact solutiony and approximate solutions ψn,n=1,2,4 of Example 3.Figure 6
Error functionsEn, n=5,6,7, of Example 3.Example 4.
Consider the nonlinear singular two-point boundary value problem [15, 23]
(39)y′′+0.5xy′=(0.5ey-e2y),0<x<1y(0)=ln(2),y(1)=0,
with exact solution y(x)=ln[2/(x2+1)].
We apply scheme (27) to (39), where α=0.5 and a=1, b=c=0, and d=ln(2), scheme (27) for (39) becomes
(40)y0=ln(2),y1=-ln(2)x1/2+x1/2∫01x-1/2∫x1x1/2A0dxdx-∫0xx-1/2∫x1x1/2A0dxdx,yn+1=x1/2∫01x-1/2∫x1x1/2Andxdx-∫0xx-1/2∫x1x1/2Andxdx,n≥1.
Proceeding as before, the Adomian’s polynomials for f(y)=(0.5ey-e2y) with y0=ln(2) are given as
(41)A0=-3,A1=-7y1,A2=-7.5y12-7y2,A3=-5.16667y13-15y1y2-7y3,⋮
Making use of (40) and (41), we have successive solution components yn of solution y(x) as
(42)y0=ln(2),y1=0.0211385x0.5-x2+0.285714x4,y2=0.0193055x0.5-0.0126831x2.5+0.214286x4+0.0093949x4.5-0.268398x6+0.0380952x8,⋮
To verify whether the proposed scheme (27) leads to accurate solution, we plot approximations ψn, for n=3,5, and exact y in Figure 7. The figure shows good agreement with exact solution y. In Figure 8 the error functions En(x), for n=3,5,7, are plotted. In addition, numerical results for maximum absolute error E(n), for n=5,8,10, are listed in Table 4.Table 4
Maximum error of Example4.
n
5
8
10
E
(
n
)
6.92128
×
1
0
-
3
8.25260
×
1
0
-
4
2.08089
×
1
0
-
4Figure 7
Comparison of exact solutiony and approximate solutions ψn,n=2,5, of Example 4.Figure 8
Error functionsEn, n=5,6,7, of Example 4.Example 5.
Consider the linear singular two-point boundary value problem [10]
(43)(xαy′)′=βxα+β-2(βxβ+α+β-1)y,0<x≤1,y(0)=1,y(1)=e,
with exact solution y(x)=exβ.
We apply the recursive scheme (27) to (43), where a=1, b=0, c=e, and d=1, scheme (27) for (43) can be read as
(44)y0=1,y1=(e-1)x1-α+x1-α∫01x-α∫x1xα+β-2A0dxdx-∫0xx-α∫x1xα+β-2A0dxdx,yn+1=x1-α∫01x-α∫x1xα+β-2Andxdx-∫0xx-α∫x1xα+β-2Andxdx,n≥1,
where the Adomian’s polynomials are given by An=β(βxβ+α+β-1)yn.
Forα=0.25, β=1: we obtain the successive solution components yn,n>0 using (44)
(45)y0=1,y1=0.318282x0.75+x+0.4x2,y2=-0.39707x0.75+0.0454688x1.75+0.1x2+0.0578694x2.75+0.162963x3+0.0307692x4,⋮
Forα=0.5, β=1, we have the successive solution components yn,n>0, using (44):
(46)y0=1,y1=0.384948x0.5+x+0.333333x2,y2=-0.551338x0.5+0.128316x1.5+0.166667x2+0.0769897x2.5+0.155556x3+0.0238095x4,⋮
We plot approximate solutions ψn, for n=1,4, and exact solution y in Figure 9. The maximum absolute errors E(n), for n=5,8,10, are displayed in Table 5. In addition, the error functions En(x), for n=5,6,7, are plotted in Figure 10.Table 5
Maximum error of Example5, when β=1.
α
E
(
5
)
E
(
8
)
E
(
10
)
0.25
1.58242
×
1
0
-
4
1.24094
×
1
0
-
6
4.52334
×
1
0
-
8
0.5
8.72985
×
1
0
-
4
7.42702
×
1
0
-
5
8.03711
×
1
0
-
6Figure 9
Comparison of exact solutiony and approximate solutions ψn,n=1,4, of Example 5 when α=0.5,β=1.Figure 10
Error functionsEn, n=5,6,7, of Example 5 when α=0.5,β=1.Example 6.
Consider the nonlinear singular two-point boundary value problem known as Thomas-Fermi equation [8]:
(47)y′′=x-1/2y3/2,0<x<1,y(0)=1,y(1)=0.
On applying scheme (27) to (47), where a=1, b=c=0, and d=1, scheme (27) for (47) becomes as follows:
(48)y0=1,y1=-x+x∫01∫x1x-1/2A0dxdx-∫0x∫x1x-1/2A0dxdx,yn+1=x∫01∫x1x-1/2Andxdx-∫0x∫x1x-1/2Andxdx,n≥1.
Proceeding as before, the Adomian polynomials for f(y)=y3/2 with y0=ln(2) are given by
(49)A0=1,A1=1.5y1,A2=0.375y12+1.5y2,A3=-0.0625y13+0.75y1y2+1.5y3,⋮
Using (48) and (49), we have solution components yn of solution y(x) as follows:
(50)y0=1,y1=-2.33333x+1.33333x1.5,y2=0.6x-0.933333x2.5+0.333333x3,y3=-0.236296x+0.24x2.5+0.233333x3.5-0.311111x4+0.0740741x4.5,⋮
Since the exact solution y of this problem is not known, we instead investigate the error remainder function, which is a measure of how well the approximate solutions satisfy the original equation,
(51)ERn(x)=ψn′′(x)-x-1/2ψn3/2(x),
where the n-term approximate solution ψn is used in place of y in order to check the convergence of ψn to y, since ψn→y as n→∞. We plot the error remainder function ERn(x), for n=8,9,10, in Figure 12. From the figure we see that ERn(x) is converging to 0 as expected. In addition, we have also plotted the approximate solutions ψn, n=1,3,5,7, in Figure 11. It is clear from the figure that the numerical solution converges to a certain function as the number n increases.Figure 11
The approximate solutionsψn,n=1,3,5,7, of Example 6.Figure 12
Error remainder functionERn(x) for n=8,9,10, of Example 6.In order to verify our numerical results, we also solve the same problem using modified ADM given by [12, 21, 22]. We apply scheme (12) to (47), scheme (12) for (47) can be read as follows:
(52)y0=1,y1=Bx+∫0x∫0xx-1/2A0dxdx,yn+1=∫0x∫0xx-1/2Andxdx,n≥1,
where B=y′(0).Using (52) and (49), we have solution components yn of solution y(x) as follows:
(53)y0=1,y1=Bx+1.33333x1.5,y2=0.4Bx2.5+0.333333x3,y3=0.0428571B2x3.5+0.133333Bx4+0.0740741x4.5,⋮
and the approximate solutions ϕn=y0+y1+y2+⋯+yn given by
(54)ϕ1(x)=1+Bx+1.33333x1.5,ϕ2(x)=1+Bx+1.33333x1.5+0.4Bx2.5+0.333333x3,ϕ3(x)=1+Bx+1.33333x1.5+0.4Bx2.5+0.333333x3+0.0428571B2x3.5+0.133333Bx4+0.0740741x4.5,⋮
Next, we match the boundary value at x=1, that is, ϕn(1)=y(1)=0, n=1,2,…,10, and solving for undetermined coefficient B. The real roots of the matching equations in B, for ϕn(1)=0, are listed in Table 6.Table 6
The roots of matching equationsϕn(1)=y(1) in B.
n
Real roots forB
1
-
2.333333333333
2
-
1.904761904762
3
-
1.886959992813
4
-
1.900137008160
5
-
1.904680051375
6
-
1.905652841857
7
-
1.905958152343
8
-
1.906114487541
9
-
1.906205619766
10
-
1.906260891987The numerical results of approximate solutionsψn, for n=4,6,8,10, obtained by proposed method and approximate solutions ϕn, for n=4,6,8,10, obtained by modified ADM are listed in Tables 7 and 8. From Tables 7 and 8, we conclude that our numerical results show good agreement with those obtained by modified ADM.Table 7
Comparison of numerical solutions betweenψn and ϕn, for n=4,6.
x
ψ
4
ϕ
4
ψ
6
ϕ
6
0
1
1
1
1
0.1
0.8522423635
0.8501067213
0.8500236323
0.8495484088
0.2
0.7323982960
0.7285222308
0.7282578059
0.7273830360
0.3
0.6262858379
0.6212711270
0.6206821720
0.6195269347
0.4
0.5285276286
0.5230884400
0.5220226768
0.5207318595
0.5
0.4360215033
0.4308844855
0.4292273964
0.4279528751
0.6
0.3467535427
0.3425502678
0.3402853521
0.3391664153
0.7
0.2593472508
0.2565019303
0.2537849766
0.2529290646
0.8
0.1728401288
0.1714556547
0.1687032793
0.1681679638
0.9
0.0865524972
0.0862975946
0.0842902775
0.0840670588
1
0
0
0
0Table 8
Comparison of numerical solutions betweenψn and ϕn, for n=8,10.
x
ψ
8
ϕ
8
ψ
10
ϕ
10
0
1
1
1
1
0.1
0.8495969286
0.8495016788
0.8495046727
0.8494868590
0.2
0.7274614883
0.7272876062
0.7272890033
0.7272573380
0.3
0.6196064297
0.6193802967
0.6193729507
0.6193337312
0.4
0.5207781835
0.5205320369
0.5205073481
0.5204682806
0.5
0.4279329340
0.4277006140
0.4276499517
0.4276191499
0.6
0.3390569614
0.3388691705
0.3387861666
0.3387709561
0.7
0.2527284963
0.2526072061
0.2524919010
0.2524970231
0.8
0.1679120917
0.1678634079
0.1677297778
0.1677541091
0.9
0.0838498011
0.0838543745
0.0837432823
0.0837736317
1
0
0
0
0Remark 7.
The major advantage of the proposed scheme is that it does not require additional computational work for unknown constant, whereas most of previous recursive schemes using ADM or MADM do require the computational work for unknown constant (see for more details [3, 14, 15, 22, 23]).


## 4. Conclusion

In this paper, we have illustrated that proposed recursive scheme (27) can be used to solve a class of linear as well as nonlinear singular two-point boundary value problems. The accuracy of the numerical results indicates that the method is well suited for the solution of such type of problems. The main advantage of this approach is that it provides a direct scheme to obtain approximate solutions, and we have also shown graphically that these approximate solutions are almost identical to the analytic solution. Another major advantage of the proposed method is that it does not require the computation of undetermined coefficients. However, most of the previous recursive schemes using ADM or MDAM do require the computation of undetermined coefficients which increases computational work (see for more details [3, 14, 15, 22, 23]). The proposed method provides a reliable technique which requires less work compared to the traditional techniques such as finite difference method, cubic spline method, and standard ADM or MADM. The numerical results of the examples are presented, and only a few terms are required to obtain accurate solutions. By comparing the results with other existing methods, it has been proved that the proposed method is a powerful method for solving singular two-point boundary value problems.



---

*Source: 262863-2013-02-21.xml*