Applied Mathematics Letters 41 (2015) 1–6

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Applied Mathematics Letters

journal homepage: www.elsevier.com/locate/aml

A novel method for solving a class of second order nonlineardifferential equations with finitely many singularities✩

Boying Wu, Lihua Guo, Dazhi Zhang ∗

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, PR China

a r t i c l e i n f o

Article history:Received 2 August 2014Received in revised form 11 October 2014Accepted 11 October 2014Available online 22 October 2014

Keywords:Nonlinear differential equationReproducing kernel functionNonlinear operatorReproducing kernel space

a b s t r a c t

In this paper, a novel method is presented for solving a class of second order nonlineardifferential equations with finitely many singularities in the reproducing kernel spaceW 3

2 [a, b]. The exact solution u(x) is represented in the form of series. In the mean time,the n term approximate solution un(x) of u(x) is proved to converge to the exact solutionin the sense of norm, the error of the numerical solution is monotonically decreasing withthe increase of n. Numerical examples demonstrate the accuracy of the present method.Numerical results verify that the proposed method is effective and simple for this kind ofproblems.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper, we consider the following second order nonlinear differential equation with finitely many singularities [1]:

u′′(x)+ p(x)u′(x)+ q(x)f (u(x)) = r(x), x ∈ (a, b), (1.1)

subject to the boundary conditions

u(a) = α, u(b) = β, (1.2)

where at least one of the functions p(x), q(x) and r(x) has a singular point, f (u(x)) is a nonlinear term of u(x), a, b, α and βare finite constants.

It is well known that the study of the second order nonlinear differential equation is one of themain classical topics in thetheory of differential equations. The second order nonlinear differential equation with finitely many singularities arises fre-quently in a variety of physics and differential appliedmathematics, such as geophysical fluid dynamics, chemical reactions,optimal control, atmospheric circulation, nuclear physics and gas dynamics. Therefore, such problems have attracted muchattention and have been studied bymany authors in recent years. In [1], Abdelhalim Ebaid discussed the problem (1.1)–(1.2)by the Adomian decompositionmethod. In [2], Zhang andWang discussed the existence and uniqueness of positive solutionsfor the singular nonlinear second-order differential equations. In [3], Cheng and Xin discussed a class of variable coefficientssingular differential equation with super-linearity or sub-linearity assumptions by applications of Green’s function and thefixed point theorem. In [4], Umer Saeed and Mujeeb ur Rehman discussed a class of nonlinear boundary value problems

✩ This work is supported by the National Natural Science Foundation of China (11271100, 11301113, 71303067), Harbin Science and TechnologyInnovative Talents Project of Special Fund (2013RFXYJ044), China Postdoctoral Science Foundation funded project (Grant No. 2013M541400), theHeilongjiang Postdoctoral Fund (Grant No. LBH-Z12102), the Fundamental Research Funds for the Central Universities (Grant No. HIT.HSS.201201).∗ Corresponding author.

E-mail address: zhang-d-z@163.com (D. Zhang).

http://dx.doi.org/10.1016/j.aml.2014.10.0040893-9659/© 2014 Elsevier Ltd. All rights reserved.

2 B. Wu et al. / Applied Mathematics Letters 41 (2015) 1–6

by wavelet-Galerkin and quasilinearization. For these references, one can see [5–9]. The numerical treatment of the secondorder nonlinear differential equation presents some major computational difficulties. Classical numerical methods fail toproduce good approximate solution for these nonlinear problems.

In this paper, we establish the representation of the exact solution u(x) and the approximate solution un(x) of (1.1) in thereproducing kernel space W 3

2 [a, b]. The advantages of our method are as follows. Firstly, we construct the correspondingreproducing kernel space and use the reproducing kernel function to solve (1.1), the boundary conditions for determiningsolution in (1.2) can be imposed in the reproducing kernel spaceW 3

2 [a, b]. Therefore, the reproducing kernel functionwhichsatisfies the boundary conditions can be calculated by homogenizing the boundary conditions. Secondly, the approximatesolution un(x) converges to the exact solution u(x) in the sense of norm. Thirdly, error of the numerical solution ismonotonically decreasing with the increasing of n.

The paper is organized as follows. In Section 2, we show the construction of the nonlinear operator and the reproducingkernel space. In Section 3, a new numerical algorithm is presented. A numerical example is given in Section 4. Section 5 endsthis paper with a brief conclusion.

2. Construction of the nonlinear operator and the reproducing kernel space

2.1. Nonlinear operator

Let Lxx(u) = u′′, Lxx(u) : W 32 [a, b] → W 1

2 [a, b], then (1.1) can be converted into the equivalent form as follows:

Lxx(u) = F(x, u(x), u′(x)), x ∈ (a, b),

where

F(x, u(x), u′(x)) = r(x)− q(x)f (u(x))− p(x)u′(x) ∈ W 12 [a, b].

In order to put (1.2) into the reproducing kernel space W 32 [a, b], we must homogenize these conditions. Through

transformation of the function, (1.1)–(1.2) can be converted into the equivalent form as follows:Lxx(u) = F(x, u(x), u′(x)), x ∈ (a, b),u(a) = u(b) = 0, (2.1)

where u(x) ∈ W 32 [a, b], F(x, u(x), u′(x)) ∈ W 1

2 [a, b], W 32 [a, b] and W 1

2 [a, b] are introduced in the following section.

2.2. Reproducing kernel space

In this section, two reproducing kernel spaces are introduced for solving (2.1). The reproducing kernel spaceW 32 [a, b] is

defined as follows:

W 32 [a, b] = {u(x) | u′′(x) is an absolutely continuous function, u′′′(x) ∈ L2[a, b], x ∈ [a, b], u(a) = u(b) = 0}.

The inner product and norm inW 32 [a, b] are defined respectively by

⟨u(x), v(x)⟩W32 [a,b] =

2i=0

ui(a)vi(a)+

b

au′′′(x)v′′′(x)dx, ∀ u(x), v(x) ∈ W 3

2 [a, b],

∥u(x)∥W32 [a,b] =

⟨u(x), u(x)⟩W3

2 [a,b], ∀ u(x) ∈ W 32 [a, b].

The reproducing kernel spaceW 12 [a, b] is defined as follows:

W 12 [a, b] = {u(x) | u(x) is an absolutely continuous function, u′(x) ∈ L2[a, b], x ∈ [a, b]}.

The inner product and norm in W 12 [a, b] are defined respectively by

⟨u(x), v(x)⟩W12 [a,b] = u(a)v(a)+

b

au′(x)v′(x)dx, ∀ u(x), v(x) ∈ W 1

2 [a, b],

∥u(x)∥W12 [a,b] =

⟨u(x), u(x)⟩W1

2 [a,b], ∀ u(x) ∈ W 12 [a, b].

Theorem 1. For any u(y) ∈ W 32 [a, b] and each fixed x ∈ [a, b], there exists a reproducing kernel function K(x, y) ∈ W 3

2 [a, b],such that

⟨u(y), K(x, y)⟩W32 [a,b] = u(x), (2.2)

B. Wu et al. / Applied Mathematics Letters 41 (2015) 1–6 3

(2.2) is called the reproducibility of reproducing kernel function K(x, y), where the reproducing kernel function K(x, y)can be denoted by

K(x, y) =

6

i=1

ai(x)yi−1, y ≤ x,

6i=1

bi(x)yi−1, y > x.

The proof of Theorem 1 and the coefficients of the reproducing kernel function K(x, y) are given in [10,11].

3. The solution of (2.1)

In this section, we will solve (2.1) and give the representation of the exact solution u(x) of (2.1) in the reproducing kernelspaceW 3

2 [a, b].For any u(x) ∈ W 3

2 [a, b] and each fixed point x ∈ [a, b], let

ϕi(x) = K(xi, y), ψi(x) = L∗

xxϕi(x),

where {xi}∞i=1 is a dense point set on [a, b], L∗xx is the adjoint operator of Lxx and K(x, y) is the reproducing kernel function of

W 32 [a, b]. In view of the properties of (2.2) and the inner product, one obtains

⟨u(y), ψi(x)⟩W32 [a,b] = ⟨u(y), L∗

xxϕi(x)⟩W32 [a,b]

= ⟨Lxxu(y), ϕi(x)⟩W12 [a,b]

= ⟨Lxxu(y), K(xi, y)⟩W12 [a,b]

= Lxxu(xi), i = 1, 2, . . . .

The orthonormal system {ψ i(x)}∞

i=1 of W 32 [a, b] can be derived from the Gram–Schmidt orthogonalization process of

{ψi(x)}∞i=1. That is

ψ i(x) =

ik=1

βikψk(x), (3.1)

where βik are the orthogonalization coefficients.

Theorem 2. If {xi}∞i=1 is a dense point set on [a, b] and u(x) is the exact solution of (2.1) in the reproducing kernel spaceW 32 [a, b],

then u(x) satisfies the form

u(x) =

∞i=1

ik=1

β ikF(xk, u(xk), u′(xk))ψ i(x). (3.2)

Proof. Due to u(x) ∈ W 32 [a, b] and {ψ i(x)}

∞

i=1 is a normal complete orthogonal system, u(x) can be expanded in terms ofseries with the normal orthogonal basis, that is

u(x) =

∞i=1

⟨u(x), ψ i(x)⟩W32 [a,b]ψ i(x).

In view of the completeness ofW 32 [a, b], u(x) is uniformly convergent in the sense of ∥ · ∥W3

2 [a,b]. Note that u(x) ∈ W 32 [a, b],

u(x) is an absolutely continuous function, in view of (2.2) and (3.1), one obtains

u(x) =

∞i=1

⟨u(x), ψ i(x)⟩W32 [a,b]ψ i(x)

=

∞i=1

u(x),

ik=1

βikψk(x)

W3

2 [a,b]

ψ i(x)

=

∞i=1

ik=1

β ik⟨u(x), ψk(x)⟩W32 [a,b]ψ i(x)

=

∞i=1

ik=1

β ik⟨u(x), L∗

xxϕk(x)⟩W32 [a,b]ψ i(x)

4 B. Wu et al. / Applied Mathematics Letters 41 (2015) 1–6

=

∞i=1

ik=1

β ik⟨Lxxu(x), ϕk(x)⟩W12 [a,b]ψ i(x)

=

∞i=1

ik=1

β ik⟨Lxxu(x), K(xk, y)⟩W12 [a,b]ψ i(x)

=

∞i=1

ik=1

β ikLxxu(xk)ψ i(x)

=

∞i=1

ik=1

β ikF(xk, u(xk), u′(xk))ψ i(x).

The proof is complete. �

Next, in order to discuss the error of the exact solution u(x) and the n-term approximate solution un(x) of (2.1) in thereproducing kernel spaceW 3

2 [a, b], we will construct an iterative sequence. Letu0(x) ∈ W 3

2 [a, b],

un(x) =

ni=1

Ciψ i(x), n = 1, 2, . . . ,(3.3)

where

C1 = β11F(x1, u0(x1), u′

0(x1)),

C2 =

2k=1

β2kF(xk, uk−1(xk), u′

k−1(xk)),

· · · · · ·

Cn =

nk=1

βnkF(xk, uk−1(xk), u′

k−1(xk)).

Since {ψ i(x)}∞

i=1 is a normal complete orthogonal system inW 32 [a, b], hence, we have

∥εn(x)∥ = ∥u(x)− un−1(x)∥

=

∞i=1

Ciψ i(x)−

n−1i=1

Ciψ i(x)

=

∞i=n

Ciψ i(x)

=

∞i=n

Ciψ i(x),∞i=n

Ciψ i(x)

=

∞i=n

C2i .

In a similar way, one obtains

∥εn+1(x)∥ =

∞i=n+1

C2i .

Clearly, ∥εn(x)∥ ≥ ∥εn+1(x)∥.Summing up the above parts, we have the following results.

Theorem 3. The error of the numerical solution of (2.1) is monotonically decreasing with the increasing of n, that is εn(x) → 0,as n → ∞.

B. Wu et al. / Applied Mathematics Letters 41 (2015) 1–6 5

Table 1The comparison of M.A. error in solutions for example.

x u(x) |u(x)− Padé[5/5](x)| Methodin [12]

|u(x)− φ10(x)| Methodin [1]

n Our method

10−1 0.6831968497 4.432E−04 1.734E−17 2 3.19856(−6)10−2 0.6930471856 1.368E−04 0.000E−00 4 6.89292(−7)10−3 0.6931461806 1.300E−05 1.602E−17 8 1.25454(−7)10−4 0.6931471706 1.219E−06 1.077E−17 16 8.32115(−8)10−5 0.6931471805 2.659E−06 8.279E−18 32 6.36363(−8)10−6 0.6931471806 2.803E−06 2.212E−17 64 4.20512(−9)

Fig. 1. Comparison of u(x) and u128(x).

Fig. 2. R.M.S of exact solution um(x) and approximate solution umn (x), (m = 0, 1, 2).

4. Numerical example

Consider the following second order nonlinear differential equation with finitely many singularities [1,12]:u′′(x)+ 0.5x−1u′(x) = eu(x)(0.5 − eu(x)), x ∈ (0, 1),u(0) = ln 2, u(1) = 0, (4.1)

where p(x) = 0.5x−1, q(x) = −1, f (u(x)) = eu(x)(0.5 − eu(x)), r(x) = 0 and the exact solution is u(x) = ln( 21+x2

).In order to put boundary conditions of (4.1) into the reproducing kernel space W 3

2 [a, b], they must be homogenized bytransformation of function

u(x) = ln(1 + x2)− x ln 2,

that is u(0) = u(1) = 0. So we can solve (4.1) with the aid of our proposed method. We choose xk =k−1n−1 (k = 1, 2,

. . . , n, n = 21, 22, 23, 24, 25, 26) and obtain the n-term approximate solution un(x), the numerical results are as givenin Table 1. We compare the u128(x)with the exact solution in Fig. 1. The Root-mean-square (R.M.S) for m-order derivatives(m = 0, 1, 2) of u128(x) and u(x) are displayed in Fig. 2.

5. Conclusions

In this paper, we construct the corresponding reproducing kernel space and use the reproducing kernel function to solvea class of second order nonlinear differential equations with finitely many singularities in the reproducing kernel spaceW 3

2 [a, b]. The nonlinear differential equation is converted into the equivalent nonlinear operator equation by homogenizingthe boundary conditions. The exact solution u(x) is represented in the form of series. In the mean time, the n termapproximate solution un(x) of u(x) is proved to converge to the exact solution in the sense of norm and error of the numericalsolution is monotonically decreasing with the increasing of n. At last, Numerical example demonstrates the accuracy of thepresent method, numerical results show that the method is effective and simple for this kind of problem.

6 B. Wu et al. / Applied Mathematics Letters 41 (2015) 1–6

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