Approximate Series Solution of Nonlinear Singular Boundary Value Problems Arising in Physiology

Abstract

We introduce an efficient recursive scheme based on Adomian decomposition method (ADM) for solving nonlinear singular boundary value problems. This approach is based on a modification of the ADM; here we use all the boundary conditions to derive an integral equation before establishing the recursive scheme for the solution components. In fact, we develop the recursive scheme without any undetermined coefficients while computing the solution components. Unlike the classical ADM, the proposed method avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients. The approximate solution is obtained in the form of series with easily calculable components. The uniqueness of the solution is discussed. The convergence and error analysis of the proposed method are also established. The accuracy and reliability of the proposed method are examined by four numerical examples.

1. Introduction

In this paper we propose an efficient recursive scheme based on Adomian decomposition method (ADM) for solving a class of nonlinear singular boundary value problems (SBVPs) that arising in physiology [1–5]: 𝑥^𝛼𝑦^′(𝑥)^′=𝑥^𝛼𝑓(𝑥)𝑦𝑥,𝑦,0<𝑥<1,𝛼≥0,_′(0)=0,𝑎𝑦(1)+𝑏𝑦^′(1)=𝑐. (1) Here 𝑎>0, 𝑏≥0, and 𝑐 are any finite real constants. We assume that for (𝑥,𝑦)∈{(0,1)×ℝ}, the functions 𝑓(𝑥,𝑦) and 𝜕𝑓/𝜕𝑦 are continuous and 𝜕𝑓/𝜕𝑦≥0. Singular boundary value problems frequently arise in the modeling of many problems in biological, physical, and engineering sciences. For example, it arises in the study of steady-state oxygen diffusion in a spherical cell with Michaelis-Menten uptake kinetics [6] with 𝛼=2 and 𝑓(𝑥,𝑦)=𝑁₁𝑦/𝑦+𝐾₁, where 𝑁₁ and 𝐾₁ are positive real constants. In heat conduction model in human head [7, 8] with 𝛼=2, 𝑓(𝑥,𝑦)=−𝑎₀𝑒^−𝑏0𝑦, where 𝑎₀ and 𝑏₀ are positive reals.

There is considerable literature on the numerical treatment of singular boundary value problems [1–11] and many of the references therein. The main difficulty of the problem (1) is that the singularity behavior occurs at 𝑥=0. Various efficient numerical techniques have been used to deal with such SBVPs, such as finite difference method (FDM) [9], cubic spline method (CSM) [1], and B-spline method (BSM) [2]. Although, these techniques are efficient and have many advantages, a huge amount of computational work is needed which combines some root-finding methods to obtain accurate numerical solution especially for nonlinear SBVPs. Recently, some newly developed approximate methods were also applied to deal with such SBVPs, such as variational iteration method (VIM) [5, 11], Adomian decomposition method (ADM) [3], and modified Adomian decomposition method (MADM) [4]. It is well known that solving nonlinear two-point boundary value problems (BVPs) using ADM/MADM is always a computationally involved task. Since it requires computation of unknown constants in a sequence of nonlinear or more difficult transcendental equations. Moreover, in some cases these unknown constants may not be uniquely determined and this may be the major disadvantage of ADM/MADM for solving nonlinear BVPs.

The aim of this work is to introduce an efficient recursive scheme to overcome the difficulties that occur in the ADM or MADM for solving nonlinear SBVPs (1). This approach is based on a modification of the ADM; here we use all the boundary conditions to derive an integral equation before establishing the recursive scheme for the solution components. In fact, we develop the recursive scheme without any undetermined coefficients while computing the solution components. Unlike the ADM, the proposed scheme avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients. The approximations of the solution are obtained in the form of series with easily computable components. The sufficient condition that guarantees the existence of a unique solution of the problem (1) is proved. The convergence analysis and the error estimation are also discussed. Finally, the accuracy and reliability of the proposed method are examined by four numerical examples.

1.1. The Review of ADM

In this subsection, we briefly describe the ADM for solving SBVPs (1). Recently, many researchers [3, 4, 12–23] have applied the ADM to deal with many different scientific models. According to the ADM we rewrite (1) in a operator form as 𝐿𝑦(𝑥)=𝑓𝑥,𝑦(𝑥), (2) where 𝐿=𝑥^−𝛼𝑑/𝑑𝑥(𝑥^𝛼𝑑/𝑑𝑥) is a second-order linear differential operator. In [24] Wazwaz defines the inverse operator 𝐿⁻¹ as 𝐿⁻¹[⋅]=^𝑥₀𝑥^−𝛼𝑥₀𝑥^𝛼[⋅]𝑑𝑥𝑑𝑥. (3)

Operating 𝐿⁻¹[⋅] on both sides of (2) and using the condition 𝑦^′(0)=0, we have 𝑦(𝑥)=𝑦(0)+𝐿⁻¹𝑓𝑥,𝑦(𝑥). (4)

The main idea of the ADM depends on decomposing the solution 𝑦(𝑥) and the nonlinear function 𝑓(𝑥,𝑦(𝑥)) by an infinite series as 𝑦(𝑥)=^∞_𝑗=0𝑦_𝑗(𝑥),𝑓𝑥,𝑦(𝑥)=^∞_𝑗=0𝐴_𝑗, (5) where 𝐴_𝑗 are Adomian’s polynomials which can be constructed for various classes of nonlinear functions with the formula given in [13] as 𝐴_𝑛=1𝑑𝑛!𝑑𝜆^𝑛𝑛𝑓𝑥,^∞_𝑘=0𝑦_𝑘𝜆^𝑘_𝜆=0,𝑛=0,1,2,…,. (6) Substituting the series (5) into (4), we obtain ^∞_𝑗=0𝑦_𝑗(𝑥)=𝑦(0)+𝐿^−1∞_𝑗=0𝐴_𝑗. (7) Upon matching both sides of (7), the ADM is given by 𝑦₀𝑥,𝜂=𝜂,𝑦_𝑗𝑥,𝜂=𝐿⁻¹𝐴_𝑗−1,𝑗≥1, (8) where 𝑦(0)=𝜂≠0 is unknown constant to be determined. Having determined the components 𝑦_𝑛(𝑥,𝜂), the series solution of 𝑦(𝑥) follows immediately with the undetermined 𝜂. Hence, the 𝑛-term truncated series solution is given as 𝜙_𝑛𝑥,𝜂=^𝑛_𝑗=0𝑦_𝑗𝑥,𝜂. (9) It should be noted that the approximate solution 𝜙_𝑛(𝑥,𝜂) depends on the unknown constant 𝜂. This constant 𝜂 can be obtained approximately by imposing the boundary condition at 𝑥=1 on 𝜙_𝑛(𝑥,𝜂), which leads to a sequence of transcendental equations 𝑎𝜙_𝑛(1,𝜂)+𝑏𝜙^′_𝑛(1,𝜂)=𝑐𝑛=1,2,3,…. For example, consider 𝑦^′′(𝑥)=−𝑒^𝑦(𝑥),𝑦(0)=𝑦(1)=0. (10) According to the ADM (8), with 𝑦₀(𝑥,𝜂)=𝜂𝑥, we obtain the components as 𝑦₀𝑥,𝜂=𝜂𝑥,𝑦₁=1𝑥,𝜂𝜂²−𝑒^𝜂𝑥𝜂²+𝑥𝜂,𝑦₂=−5𝑥,𝜂4𝜂⁴+𝑒^𝜂𝑥𝜂⁴+𝑒^2𝜂𝑥4𝜂⁴−𝑥2𝜂³−𝑒^𝜂𝑥𝑥𝜂³,⋮ (11) Consequently, the 𝑛-term approximate series solution can be obtained as 𝜙_𝑛𝑥,𝜂=^𝑛_𝑗=0𝑦_𝑗𝑥,𝜂. (12) By imposing the boundary condition at 𝑥=1 on 𝜙_𝑛(𝑥,𝜂), we obtain a sequence of transcendental equations 𝜙_𝑛(1,𝜂)=0, 𝑛=1,2,… as follows 𝜙₁11,𝜂≡𝜂+𝜂²−𝑒^𝜂𝜂²+1𝜂𝜙=0,₂11,𝜂≡𝜂+𝜂²−𝑒^𝜂𝜂²+1𝜂+−54𝜂⁴+𝑒^𝜂𝜂⁴+𝑒^2𝜂4𝜂⁴−12𝜂³−𝑒^𝜂𝜂³⋮=0, (13)

However, solving such transcendental equations for 𝜂 requires additional computational work, and 𝜂 may not be uniquely determined. This may be the major disadvantage of the ADM for solving two-point boundary value problems.

2. The Proposed Recursive Scheme

In this section, we propose a new efficient recursive scheme based on the ADM for solving nonlinear SBVPs (1). To overcome the singular behavior at 𝑥=0, we rewrite SBVPs (1) in operator form as ℒ𝑦(𝑥)=𝑥^𝛼𝑓𝑥,𝑦(𝑥),0<𝑥<1 (14) subject to the boundary conditions 𝑦^′(0)=0,𝑎𝑦(1)+𝑏𝑦^′(1)=𝑐, (15) where ℒ=𝑑/𝑑𝑥(𝑥^𝛼𝑑/𝑑𝑥) is a second-order linear singular differential operator. Twofold integral operator ℒ⁻¹[⋅] regarded as the inverse operator of ℒ is proposed as ℒ⁻¹[⋅]=¹_𝑥𝑠^−𝛼𝑠₀[⋅]𝑑𝑥𝑑𝑠. (16) In order to establish the new efficient recursive scheme, we operate ℒ⁻¹[⋅] on left hand side of (14) and impose the boundary condition 𝑦^′(0)=0; we obtain ℒ⁻¹𝑥^𝛼𝑦^′(𝑥)^′=¹_𝑥𝑠^−𝛼𝑠₀𝑥^𝛼𝑦^′(𝑥)^′=𝑑𝑥𝑑𝑠,_1𝑥𝑠^−𝛼𝑠₀𝑥^𝛼𝑦^′′(𝑥)+𝛼𝑥^𝛼−1𝑦^′(𝑥)𝑑𝑥𝑑𝑠,=¹_𝑥𝑠^−𝛼𝑠^𝛼𝑦^′(𝑠)−^𝑠₀𝛼𝑥^𝛼−1𝑦^′(𝑥)𝑑𝑥+^𝑠₀𝛼𝑥^𝛼−1𝑦^′(𝑥)=𝑑𝑥𝑑𝑠,_1𝑥𝑦^′(𝑠)𝑑𝑠=𝑦(1)−𝑦(𝑥). (17) Thus we have ℒ⁻¹𝑥^𝛼𝑦^′(𝑥)^′=𝑐₀−𝑦(𝑥), (18) where 𝑐₀=𝑦(1)≠0 is unknown constant to be determined.

We now operate ℒ⁻¹[⋅] on both sides of (14) and use (18); we get 𝑦(𝑥)=𝑐₀−¹_𝑥𝑠^−𝛼𝑠₀𝑥^𝛼𝑓𝑥,𝑦(𝑥)𝑑𝑥𝑑𝑠. (19) To eliminate unknown constant 𝑐₀, we impose the boundary condition 𝑎𝑦(1)+𝑏𝑦^′(1)=𝑐 on (19) which leads to 𝑐₀=𝑐𝑎−𝑏𝑎¹₀𝑥^𝛼𝑓𝑥,𝑦(𝑥)𝑑𝑥. (20) Substituting the value of 𝑐₀ into (19), we obtain 𝑦(𝑥)=𝑐𝑎−𝑏𝑎¹₀𝑥^𝛼𝑓(𝑥)−𝑥,𝑦𝑑𝑥_1𝑥𝑠^−𝛼𝑠₀𝑥^𝛼𝑓𝑥,𝑦(𝑥)𝑑𝑥𝑑𝑠. (21) Note that (21) does not involve any unknown constants to be determined.

We now decompose the solution 𝑦(𝑥) by the series as 𝑦(𝑥)=^∞_𝑗=0𝑦_𝑗(𝑥),𝑓𝑥,𝑦(𝑥)=^∞_𝑗=0𝐴_𝑗, (22) where 𝐴_𝑛 are the Adomian's polynomials [13]. In 2010, Duan [25, 26] reported several new efficient algorithms for rapid computer generation of the Adomian polynomials. Recently, El-Kalla [27] suggested another programmable formula for Adomian polynomials: 𝐴_𝑛=𝑓𝑥,𝜓_𝑛−^𝑛−1_𝑗=0𝐴_𝑗or𝑓𝑥,𝜓_𝑛=^𝑛_𝑗=0𝐴_𝑗, (23) where 𝜓_𝑛=∑^𝑛_𝑗=0𝑦_𝑗 is partial sum of the series solution ∑^∞_𝑗=0𝑦_𝑗.

Substituting the series (22) into (21), we obtain ^∞_𝑗=0𝑦_𝑗(𝑥)=𝑐𝑎−𝑏𝑎¹₀𝑥^𝛼∞_𝑗=0𝐴_𝑗−𝑑𝑥_1𝑥𝑠^−𝛼𝑠₀𝑥^𝛼∞_𝑗=0𝐴_𝑗𝑑𝑥𝑑𝑠. (24) Comparing both sides of (24), the proposed scheme is given by the following recursive scheme: 𝑦₀(𝑥)=𝑐𝑎,𝑦_𝑗(𝑥)𝑏=−𝑎¹₀𝑥^𝛼𝐴_𝑗−1𝑑𝑥−¹_𝑥𝑠^−𝛼𝑠₀𝑥^𝛼𝐴_𝑗−1𝑑𝑥𝑑𝑠,𝑗≥1. (25)

The recursive scheme (25) provides complete determination of solution components 𝑦_𝑗(𝑥) of the solution 𝑦(𝑥). Hence, the truncated 𝑛-term series solution can be obtained as 𝜓_𝑛(𝑥)=^𝑛_𝑗=0𝑦_𝑗(𝑥). (26)

Unlike ADM or MADM, the proposed recursive scheme (25) does not require any computation of unknown constants.

3. Convergence Analysis

In this section, we will give the sufficient condition that guarantees existence of a unique solution of (28) in Theorem 1. Then we discuss the convergence analysis in Theorem 2 and the error analysis in Theorem 3 of the proposed scheme (25) for SBVPs (1). Note that many researcher [28–31] have also established the convergence of the ADM for solving differential as well as integral equations. Let 𝕏=𝐶[0,1] be a Banach space with the norm 𝑦=max_[]𝑥∈0,1𝑦(𝑥),𝑦∈𝕏. (27) Note that (21) can be written in operator form as 𝑦=𝒩𝑦, (28) where 𝒩(𝑦) is given by 𝒩𝑦=𝑐𝑎−𝑏𝑎¹₀𝑥^𝛼𝑓−𝑥,𝑦𝑑𝑥_1𝑥𝑠^−𝛼𝑠₀𝑥^𝛼𝑓𝑥,𝑦𝑑𝑥𝑑𝑠. (29)

Theorem 1. Let 𝕏 be a Banach space with the norm given by (27). Assuming that the nonlinear function 𝑓(𝑥,𝑦) satisfies the Lipschitz condition; that is, |𝑓(𝑥,𝑦)−𝑓(𝑥,𝑦^∗)|≤𝐿|𝑦−𝑦^∗|. Further, let 𝛿 be a constant defined as 𝐿()𝛿∶=𝑎+2𝑏().2𝑎1+𝛼 (30) If 𝛿<1, then (28) has a unique solution in 𝕏.

Proof. For any 𝑦,𝑦^∗∈𝕏, we have 𝒩𝑦−𝒩𝑦^∗=max_[]𝑥∈0,1𝑏𝑎¹₀𝑥^𝛼𝑓𝑥,𝑦−𝑓𝑥,𝑦^∗𝑑𝑥+¹_𝑥𝑠^−𝛼𝑠₀𝑥^𝛼𝑓𝑥,𝑦−𝑓𝑥,𝑦^∗𝑑𝑥𝑑𝑠,≤max_[]𝑥∈0,1𝑓𝑥,𝑦−𝑓𝑥,𝑦^∗×max_𝑥∈[0,1]𝑏𝑎¹₀𝑥^𝛼𝑑𝑥+¹_𝑥𝑠^−𝛼𝑠₀𝑥^𝛼𝑑𝑥𝑑𝑠. (31) Using Lipschitz continuity of 𝑓, we obtain 𝒩𝑦−𝒩𝑦^∗≤𝐿(𝑎+2𝑏)2𝑎(1+𝛼)max_[]𝑥∈0,1𝑦−𝑦^∗=𝛿𝑦−𝑦^∗. (32) Hence, we have 𝒩𝑦−𝒩𝑦^∗≤𝛿𝑦−𝑦^∗. (33) If 𝛿<1, then 𝒩∶𝕏→𝕏 is contraction mapping and hence by the Banach contraction mapping theorem, (28) has a unique solution in 𝕏.

We now prove the convergence of the proposed scheme (25). Let {𝜓_𝑛=∑^𝑛_𝑗=0𝑦_𝑗} be a sequence of partial sums of the series solution ∑^∞_𝑗=0𝑦_𝑗. Using (25) and (26), we have 𝜓_𝑛=^𝑛_𝑗=0𝑦_𝑗=𝑦₀+^𝑛_𝑗=1𝑦_𝑗=𝑐𝑎−^𝑛_𝑗=1𝑏𝑎¹₀𝑥^𝛼𝐴_𝑗−1𝑑𝑥+¹_𝑥𝑠^−𝛼𝑠₀𝑥^𝛼𝐴_𝑗−1=𝑐𝑑𝑥𝑑𝑠𝑎−𝑏𝑎¹₀𝑥^𝛼𝑛_𝑗=1𝐴_𝑗−1−𝑑𝑥_1𝑥𝑠^−𝛼𝑠₀𝑥^𝛼𝑛_𝑗=1𝐴_𝑗−1𝑑𝑥𝑑𝑠. (34) Using (23) in (34), it follows that 𝜓_𝑛=𝑐𝑎−𝑏𝑎¹₀𝑥^𝛼𝑓𝑥,𝜓_𝑛−1−𝑑𝑥_1𝑥𝑠^−𝛼𝑠₀𝑥^𝛼𝑓𝑥,𝜓_𝑛−1𝑑𝑥𝑑𝑠, (35) which is equivalent to the following operator equation: 𝜓_𝑛=𝒩𝜓_𝑛−1,𝑛=1,2,…. (36)

Note that the formulation (36) is used to prove Theorems 2 and 3.

Theorem 2. Let 𝒩(𝑦) be the nonlinear operator defined by (28) as contractive; that is, 𝒩𝑦−𝒩𝑦^∗≤𝛿𝑦−𝑦^∗,∀𝑦,𝑦^∗𝑦∈𝕏,𝑤𝑖𝑡ℎ𝛿<1,₁<∞. (37) Then the sequence {𝜓_𝑛} of partial sums defined by (26) converges to the exact solution 𝑦.

Proof. Using the relation (36) and the estimate (33), we have 𝜓_𝑚+1−𝜓_𝑚=𝒩𝜓_𝑚−𝒩𝜓_𝑚−1≤𝛿𝜓_𝑚−𝜓_𝑚−1. (38) Thus we have 𝜓_𝑚+1−𝜓_𝑚≤𝛿𝜓_𝑚−𝜓_𝑚−1≤𝛿²𝜓_𝑚−1−𝜓_𝑚−2≤⋯≤𝛿^𝑚𝜓₁−𝜓₀. (39) We now show that the sequence {𝜓_𝑛} is convergent. Now for all 𝑛, 𝑚∈ℕ, with 𝑛≥𝑚, we consider 𝜓_𝑛−𝜓_𝑚=𝜓_𝑛−𝜓_𝑛−1+𝜓_𝑛−1−𝜓_𝑛−2+⋯+𝜓_𝑚+1−𝜓_𝑚≤𝜓_𝑛−𝜓_𝑛−1+𝜓_𝑛−1−𝜓_𝑛−2+⋯+𝜓_𝑚+1−𝜓_𝑚≤𝛿^𝑛−1+𝛿^𝑛−2+⋯+𝛿^𝑚𝜓₁−𝜓₀=𝛿^𝑚1+𝛿+𝛿²+⋯+𝛿^{𝑛−𝑚−1}𝜓₁−𝜓₀=𝛿^𝑚1−𝛿^𝑛−𝑚𝑦1−𝛿¹. (40) Since 0<𝛿<1, (1−𝛿^𝑛−𝑚)<1, and ‖𝑦₁‖<∞, it follows 𝜓_𝑛−𝜓_𝑚≤𝛿^𝑚𝑦1−𝛿¹. (41) Taking limit as 𝑚→∞, we obtain 𝜓_𝑛−𝜓_𝑚⟶0. (42) Hence {𝜓_𝑛} is cauchy sequence in 𝕏. Hence there exits 𝜓 in 𝕏 such that lim_𝑛→∞𝜓_𝑛=𝜓. Note that 𝜓 is the exact solution of (28) as 𝑦=^∞_𝑗=0𝑦_𝑗=lim_𝑛→∞𝜓_𝑛=𝜓. (43) This completes the proof.

Theorem 3. Let 𝑦(𝑥) be the exact solution of the operator equation (28). Let 𝜓_𝑚(𝑥) be the sequence of partial sums of series solution defined by (26). Then there holds max_[]𝑥∈0,1𝑦(𝑥)−^𝑚_𝑗=0𝑦_𝑗(𝑥)≤𝑀𝛿^𝑚+1𝐿(1−𝛿), (44) where 𝑀=max_𝑥∈[0,1]|𝑓(𝑥,𝑦₀)|.

Proof. For any 𝑛≥𝑚 and using the estimate (41), we have 𝜓_𝑛−𝜓_𝑚≤𝛿^𝑚𝑦1−𝛿¹. (45) Since lim_𝑛→∞𝜓_𝑛=𝑦, fixing 𝑚 and letting 𝑛→∞ in above estimate, we obtain 𝑦−𝜓_𝑚≤𝛿^𝑚1−𝛿max_[]𝑥∈0,1𝑦₁(𝑥). (46) Since, we have 𝑦₁(𝑥)𝑏=−𝑎¹₀𝑥^𝛼𝐴₀𝑑𝑥−¹_𝑥𝑠^−𝛼𝑠₀𝑥^𝛼𝐴₀𝐴𝑑𝑥𝑑𝑠,₀=𝑓𝑥,𝑦₀, (47) so max_[]𝑥∈0,1𝑦₁(𝑥)=max_[]𝑥∈0,1𝑏𝑎¹₀𝑥^𝛼𝐴₀𝑑𝑥+¹_𝑥𝑠^−𝛼𝑠₀𝑥^𝛼𝐴₀≤()𝑑𝑥𝑑𝑠𝑎+2𝑏2𝑎(1+𝛼)max_[]𝑥∈0,1𝑓𝑥,𝑢₀. (48) Combining the estimates (46) and (48) and using 𝛿/𝐿=(𝑎+2𝑏)/(2𝑎(1+𝛼)), we obtain max_𝑥∈[0,1]𝑦(𝑥)−^𝑚_𝑗=0𝑦_𝑗(𝑥)≤𝑀𝛿^𝑚+1𝐿(1−𝛿). (49) This completes the proof.

4. Numerical Examples

In this section, we demonstrate the accuracy and reliability of the proposed scheme (25) by implementing it to four SBVPs, arising various physical models. All the numerical results obtained by the proposed (25) are compared with the results obtained by various numerical methods.

Example 1. Consider the following nonlinear SBVPs: 𝑥^𝛼𝑦^′(𝑥)^′=𝑥^𝛼𝑁₁𝑦(𝑥)𝑦(𝑥)+𝐾₁𝑦,0<𝑥<1,_′(0)=0,5𝑦(1)+𝑦^′(1)=5, (50) where 𝑁₁ and 𝐾₁ are positive constants involving the reaction rate and Michaelis constants. Thus we take 𝑁₁=0.76129 and 𝐾₁=0.03119 as used in [1, 3, 5].

According to the proposed method (25) with 𝑎=5, 𝑏=1, and 𝑐=5. Consequently, we have the following recursive scheme: 𝑦₀(𝑥)𝑦=1,_𝑗(𝑥)1=−5¹₀𝑥^𝛼𝐴_𝑗−1𝑑𝑥−¹_𝑥𝑠^−𝛼𝑠₀𝑥^𝛼𝐴_𝑗−1𝑑𝑥𝑑𝑠,𝑗≥1. (51) Using the formula (6), Adomian’s polynomials for 𝑓(𝑦(𝑥))=0.76129𝑦(𝑥)/(𝑦(𝑥)+0.03119) with 𝑦₀=1 are obtained as 𝐴₀𝐴=0.738264,₁=0.02233𝑦₁(𝑥),𝐴₂=−0.0216546𝑦²₁(𝑥)+0.02233𝑦₂(𝑥),𝐴₃=0.0209996𝑦³₁(𝑥)−0.0433091𝑦₁(𝑥)𝑦₂(𝑥)+0.02233𝑦₃(𝑥)⋮ (52)

Case (𝛼=1). Using (51) and (52), the components 𝑦_𝑗(𝑥) of the solution 𝑦(𝑥) are obtained as 𝑦₀(𝑥)𝑦=1,₁(𝑥)=−0.25839224+0.18456588𝑥²,𝑦₂(𝑥)=0.00155580−0.00144247𝑥²+0.00025758𝑥⁴,𝑦₃(𝑥)=0.00030986−0.0003527𝑥²+0.0001270𝑥⁴−0.0000203𝑥⁶⋮ (53) The 6-term approximate series solution 𝜓₆(𝑥)=∑⁶_𝑗=0𝑦_𝑗(𝑥) is given by 𝜓₆(𝑥)=0.743551+0.182662𝑥²+0.0004499𝑥⁴−0.0000453𝑥⁶+5.181586×10⁻⁶𝑥⁸−5.061711×10⁻⁷𝑥¹⁰+2.801167×10⁻⁸𝑥¹². (54)

Case (𝛼=2). In a similar manner, we obtain the components 𝑦_𝑗(𝑥) of solution 𝑦(𝑥) as 𝑦₀(𝑥)𝑦=1,₁(𝑥)=−0.17226149+0.12304392𝑥²,𝑦₂(𝑥)=0.0006502−0.0006410𝑥²+0.0001373𝑥⁴,𝑦₃(𝑥)=0.0000822−0.0001046𝑥²+0.000045𝑥⁴−7.732815×10⁻⁶𝑥⁶⋮ (55) Hence, we obtain the 6-term approximate series solution 𝜓₆(𝑥)=∑⁶_𝑗=0𝑦_𝑗 as 𝜓₆(𝑥)=0.828483+0.122278𝑥²+0.000196𝑥⁴−0.000013𝑥⁶+1.013102×10⁻⁶𝑥⁸−7.367823×10⁻⁸𝑥¹⁰+3.366698×10⁻⁹𝑥¹². (56)

Case (𝛼=3). The components 𝑦_𝑗(𝑥) of the solution 𝑦(𝑥) are obtained as 𝑦₀(𝑥)𝑦=1,₁(𝑥)=−0.1291961+0.0922829𝑥²,𝑦₂(𝑥)=0.000350−0.000360𝑥²+0.0000858𝑥⁴,𝑦₃(𝑥)=0.0000321−0.0000442𝑥²+0.000021𝑥⁴−3.802000×10⁻⁶𝑥⁶⋮ (57) The series solution is given as follows: 𝜓₆(𝑥)=0.87119+0.0918721𝑥²+0.0001116𝑥⁴−5.600223×10⁻⁶𝑥⁶+3.279844×10⁻⁷𝑥⁸−1.8874803×10⁻⁸𝑥¹⁰+7.358761×10⁻¹⁰𝑥¹². (58) Comparison of the numerical results obtained by the proposed method (25), the VIM used in [5], and the CSM used in [1] are shown in Tables 1, 2, and 3. These tables show that the numerical results obtained by the proposed scheme (25) are comparable with those in [1, 5]. In addition, we have also presented numerical results for 𝛼=0,4,5 in Tables 1 and 3.

Table 1 

Numerical results for 𝛼=0,1 of Example 1.

Table 2 

Numerical results for 𝛼=2 of Example 1.

Table 3 

Numerical results for 𝛼=3,4,5 of Example 1.

Example 2. Consider the following nonlinear SBVPs [9]: 𝑥²𝑦^′(𝑥)^′=𝑥²𝑦⁵(𝑥)𝑦,0<𝑥<1,_′(0)(1)=√=0,𝑦32. (59) The analytical solution is 𝑦(𝑥)=3/(3+𝑥²).

According to the proposed method (25) with 𝛼=2, 𝑎=1, 𝑏=0, and 𝑐=√3/2. Consequently, we have the following recursive scheme as 𝑦₀(𝑥)=√32,𝑦_𝑗(𝑥)0=−1¹₀𝑥²𝐴_𝑗−1𝑑𝑥−¹_𝑥𝑠^−2𝑠₀𝑥²𝐴_𝑗−1𝑑𝑥𝑑𝑠,𝑗≥1. (60) Similar to the previous example, Adomian’s polynomials for 𝑓(𝑦(𝑥))=𝑦⁵(𝑥) are given as 𝐴₀=𝑦⁵₀(𝑥),𝐴₁=5𝑦⁴₀𝑦₁(𝑥),𝐴₂=10𝑦³₀(𝑥)𝑦²₁(𝑥)+5𝑦⁴₀(𝑥)𝑦₂(𝑥),𝐴₃=10𝑦²₀(𝑥)𝑦³₁(𝑥)+20𝑦³₀(𝑥)𝑦₁(𝑥)𝑦₂(𝑥)+5𝑦⁴₀(𝑥)𝑦₃(𝑥)⋮ (61) Using (60) and (61), we obtain the components 𝑦_𝑗(𝑥) of the solution 𝑦(𝑥) as 𝑦₀(𝑥)=0.8660254,𝑦₁(𝑥)=0.0811899−0.0811899𝑥²,𝑦₂(𝑥)=0.0266404−0.0380578𝑥²+0.0114173𝑥⁴,𝑦₃(𝑥)=0.0117741−0.0196235𝑥²+0.00963337𝑥⁴−0.00178396𝑥⁶⋮ (62) Hence, we obtain approximate series solution as 𝜓₆(𝑥)=0.996812−0.159895𝑥²+0.0354173𝑥⁴−0.00743284𝑥⁶+0.00126204𝑥⁸−0.000146626𝑥¹⁰+8.48888×10⁻⁶𝑥¹². (63) For quantitative comparison, we now define the maximum absolute error by 𝐸_𝑛=max_0<𝑥≤1𝜓_𝑛(𝑥)−𝑦(𝑥). (64) Here 𝜓_𝑛(𝑥) is the 𝑛-term approximate series solution obtained by the proposed method (25) and 𝑦(𝑥) is the analytical solution. Table 4 shows a comparison between the numerical results obtained by the proposed method (25) and the FDM used in [9]. The numerical results obtained by the proposed method (25) show good agreement with those in [9].

Table 4 

Comparison of the maximum absolute error of Example 2.

Example 3. Consider the following nonlinear SBVPs [3]: 𝑥𝑦^′(𝑥)^′=−𝑥𝑒^𝑦(𝑥)𝑦,0<𝑥<1,_′(0)=0,𝑦(1)=0. (65) The analytical solution is 𝑦(𝑥)=2ln((𝐶+1)/(𝐶𝑥²+1)), where 𝐶=3−2√2.

According to the proposed method (25) with 𝛼=1, 𝑎=1, 𝑏=0, and 𝑐=0, we have the following recursive scheme: 𝑦₀(𝑥)𝑦=0,_𝑗(𝑥)=01¹₀𝑥𝐴_𝑗−1𝑑𝑥+¹_𝑥𝑠^−1𝑠₀𝑥𝐴_𝑗−1𝑑𝑥𝑑𝑠,𝑗≥1. (66) Proceeding as before, Adomian's polynomials for 𝑓(𝑦(𝑥))=𝑒^𝑦(𝑥) with 𝑦₀(𝑥)=0 are given as 𝐴₀=1,𝐴₁=𝑦₁(𝑥),𝐴₂=𝑦²₁(𝑥)2+𝑦₂(𝑥),𝐴₃=𝑦³₁(𝑥)6+𝑦₁(𝑥)𝑦₂(𝑥)+𝑦₃(𝑥)⋮ (67) Making use of (66) and (67), we can calculate the components 𝑦_𝑗(𝑥) of solution 𝑦(𝑥) as 𝑦₀(𝑥)=0,𝑦₁(𝑥)=0.25−0.25𝑥²,𝑦₂(𝑥)=0.046875−0.0625𝑥²+0.015625𝑥⁴,𝑦₃(𝑥)=0.0130208−0.0195313𝑥²+0.0078125𝑥⁴−0.00130208𝑥⁶⋮ (68) Hence we obtain approximate series solution as 𝜓₆(𝑥)=0.316294−0.342438𝑥²+0.0289497𝑥⁴−0.00310771𝑥⁶+0.000328064𝑥⁸−0.000027465𝑥¹⁰+1.2715657×10⁻⁶𝑥¹². (69)

Table 5 shows a comparison of the numerical results obtained by the proposed method (25), the FDM used in [9], and the BSM used in [2]. Once again, the approximate series solutions using proposed method (25) are comparable with the results reported earlier in the literature [2, 9].

Table 5 

Comparison of the maximum absolute error of Example 3.

Example 4. Consider the following nonlinear SBVPs [8]: 𝑥²𝑦^′(𝑥)^′=−𝑥²𝑒^{−𝑦(𝑥)}𝑦,0<𝑥<1,_′(0)=0,2𝑦(1)+𝑦^′(1)=0. (70)

According to the proposed method (25) with 𝛼=2, 𝑎=2, 𝑏=1, and 𝑐=0, we have the following scheme: 𝑦₀(𝑥)𝑦=0,_𝑗(𝑥)=12¹₀𝑥²𝐴_𝑗−1𝑑𝑥+¹_𝑥𝑠^−2𝑠₀𝑥²𝐴_𝑗−1𝑑𝑥𝑑𝑠,𝑗≥1. (71) Adomian’s polynomials, for 𝑓(𝑦(𝑥))=𝑒^{−𝑦(𝑥)} with 𝑦₀(𝑥)=0 are given as 𝐴₀𝐴=1,₁=−𝑦₁(𝑥),𝐴₂=𝑦²₁(𝑥)2−𝑦₂(𝑥),𝐴₃𝑦=−³¹(𝑥)6+𝑦₁(𝑥)𝑦₂(𝑥)−𝑦₃(𝑥)⋮ (72) Making use of (71) and (72), we obtain the component 𝑦_𝑛(𝑥) as follows: 𝑦₀(𝑥)=0,𝑦₁(𝑥)=0.333333−0.166667𝑥²,𝑦₂(𝑥)=−0.0861111+0.0555556𝑥²−0.00833333𝑥⁴,𝑦₃(𝑥)=0.032672−0.0236111𝑥²+0.00555556𝑥⁴−0.000529101𝑥⁶⋮ (73) Hence, the 6-term approximate series solution is given by 𝜓₆(𝑥)=0.270736−0.127835𝑥²−0.00461112𝑥⁴−0.00030756𝑥⁶+3.17554×10⁻⁶𝑥⁸−3.22937×10⁻⁶𝑥¹⁰+2.18422×10⁻⁷𝑥¹²−1.74996×10⁻⁸𝑥¹⁴. (74)

Table 6 presents a comparison of the approximate solutions obtained by the proposed method (25), the FDM used in [32], and tangent chord technique used in [8]. We again obtain the numerical results using the proposed scheme (25) which are comparable with those in [8, 32].

Table 6 

Numerical results of Example 4.

5. Conclusions

(i)In this work, the simplicity, efficiency, and reliability of the proposed method (25) have been tested by solving four nonlinear SBVPs that arise in physiology.(ii)The accuracy of the numerical results indicates that the proposed method is well suited for the solution of SBVPs. It has also been shown that only 6 terms are sufficient to obtain comparable solution with those in [2, 9].(iii)Unlike ADM or MDAM, the proposed method (25) does not require any computation of unknown constants. In fact, it provides a direct scheme to obtain approximations to the solution.(iv)Unlike the finite difference and the cubic spline methods, the proposed method does not require any linearization and discretization of the variables.(v)Uniqueness of the solution of (28) has been discussed in Theorem 1. The convergence analysis and the error estimation of the proposed scheme (25) have also been established in Theorems 2 and 3.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the anonymous referees for their useful comments and suggestions that led to improvement of the presentation and content of this paper. The authors thankfully acknowledge the financial assistance provided by the Council of Scientific and Industrial Research (CSIR), New Delhi, India.