Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method

Applied Mathematics and Computation 291 (2016) 137–148

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Numerical investigations for systems of second-order periodic boundary value problems using reproducing kernel method Mohammed Al-Smadi a, Omar Abu Arqub b,∗, Nabil Shawagfeh b,c, Shaher Momani c a

Department of Applied Science, Al-Balqa Applied University, Ajloun 26816, Jordan Department of Mathematics, Al-Balqa Applied University, Salt 19117, Jordan c Department of Mathematics, Faculty of Science, The University of Jordan, Amman 11942, Jordan b

a r t i c l e

i n f o

MSC: 34K28 47B32 34B15 Keywords: System of differential equations Periodic boundary conditions Numerical and analytical solutions Reproducing kernel Hilbert space method

a b s t r a c t The reproducing kernel method is a numerical as well as analytical technique for solving a large variety of ordinary and partial differential equations associated to different kind of boundary conditions, and usually provides the solutions in term of rapidly convergent series in the appropriate Hilbert spaces with components that can be elegantly computed. The aim of the present analysis is to implement a relatively recent computational method, reproducing kernel Hilbert space, for obtaining the solutions for systems of second-order differential equations with periodic boundary conditions. A reproducing kernel space is constructed in which the periodic conditions of the systems are satisfied, whilst, the smooth kernel functions are used throughout the evolution of the method to obtain the required grid points. An efficient construction is given to obtain the approximate solutions for the systems together with an existence proof of the exact solutions is proposed based upon the reproducing kernel theory. Convergence analysis and error behavior of the presented method are also discussed. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed. © 2016 Published by Elsevier Inc.

1. Introduction Boundary value problems (BVPs) are an important branch of modern mathematics that arises naturally in different areas of applied sciences, physics, and engineering. It has wide applications due to the fact that many practical problems in mechanics, astronomy, potential theory, and electrostatics may be converted directly to such problems or to ones that are closely related to BVPs. Generally, it is difficult to obtain closed form solutions for BVPs, especially for nonlinear cases. Factually, in most cases, only approximate solutions or numerical solutions can be expected; therefore, it has attracted much attention and has been studied by many authors. In this regards, there are many iterative methods have been proposed to be one of the suitable and successful classes of numerical techniques for obtaining approximate solutions of numerous types of BVPs (see, for instance, [1–11]).

∗

Corresponding author. E-mail address: [email protected] (O.A. Arqub).

http://dx.doi.org/10.1016/j.amc.2016.06.002 0 096-30 03/© 2016 Published by Elsevier Inc.

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Commonly, systems of ordinary differential equations with periodic boundary conditions constitute a very interesting class of these problems and play an important role in solving real-world problems. A wide variety of natural phenomena are modeled by these systems, which have been used to various applications in biophysics, chemical engineering, and physical sciences. However, systems of periodic BVPs have been investigated systematically in the literature for the development, analysis, and implementation of accurate methods. To mention a few, existence of positive solutions for systems of secondorder, two-point periodic BVPs has been discussed in [12]. Besides, the existence and multiplicity of positive solutions are carried out in [13] for systems of second-order, three-point periodic BVPs. In [14], the authors have described the sufficient conditions for nontrivial periodic solutions for second-order, two-point periodic BVPs. In [15] also, the author has provided the existence and multiplicity of positive solutions to further investigation to second-order, two-point periodic BVPs. In addition, the authors in [16] have studied the solvability and developed an iterative method for systems of first-order, periodic BVPs based on the reproducing kernel Hilbert space method (RKHSM). Recently, the analytical–numerical solutions for systems of second-order, two-point singular periodic BVPs based on the use of the RKHSM is proposed in [17]. We consider in this work the following system of second-order differential equation:

u1 (x ) + p1 (x )u1 (x ) + q1 (x )u1 (x ) = F1 (x, u1 (x ), u2 (x )), u2 (x ) + p2 (x )u2 (x ) + q2 (x )u2 (x ) = F2 (x, u1 (x ), u2 (x )),

(1)

subject to the following periodic boundary conditions:

u1 (0 ) = u1 (1 ), u1 (0 ) = u1 (1 ), u2 (0 ) = u2 (1 ), u2 (0 ) = u2 (1 ),

(2)

where x ∈ [0, 1], us (x ) ∈ W23 [0, 1] are unknown functions to be determined, Fs (x, y, z) are continuous terms in the space W21 [0, 1] as y = y(x ), z = z(x ) ∈ W23 [0, 1], x ∈ [0, 1], −∞ < y, z < ∞, which are depending on the system discussed in which W21 [0, 1], W23 [0, 1] are two Hilbert spaces and s = 1, 2. Through this paper, we assume that Eqs. (1) and (2) have a unique analytical solution on [0, 1]. Investigation about systems of second-order periodic BVPs numerically is rare and sparse. The basic motivation of the present study is the extension of a numerical technique, the RKHSM, to develop an approach for obtaining representation of the solutions for class of systems of periodic BVPs, whereas the conditions for determining solutions can be imposed in the RKHS. The present technique has the following advantages; firstly, it can produce good globally smooth approximate solutions, and with ability to solve many differential equations with complex boundary conditions such as period boundary conditions and integral boundary conditions, which are difficult to solve; secondly, it is accurate, need less effort to achieve the results, and is developed especially for the nonlinear cases; thirdly, it is possible to pick any point in the given domain and as well the approximate solutions and all their derivatives will be applicable; fourthly, the method can be used directly without employing linearization, perturbation, or discretization of the variables, it is not effected by computation round off errors, and it can avoid solving the reduced systems of algebraic equations. Reproducing kernel theory has wide applications in numerical analysis, computational mathematics, probability and statistics, image processing, machine learning, and finance [18–21]. The RKHSM is a useful framework for constructing approximate solutions of great interest to applied sciences. In recent years, based on the reproducing kernel theory, extensive work has been proposed and discussed for the solutions of several integral and differential operators. Anyhow, the reader is kindly requested to go through [16–40] in order to know more details about the RKHSM and the reproducing kernel theory, including their history, their modification for use, their characteristics, and their scientific applications. The rest of the paper is organized as follows. In the next section, two RKHSs needed in this paper are defined, and two reproducing kernel functions are obtained. After that, in Section 3, the solutions and essential theoretical results are presented based upon the reproducing kernel theory. In Section 4, an efficient iterative algorithm for the solutions is described. Finally, numerical examples are discussed to demonstrate the accuracy and the applicability of the presented method as utilized in Section 5. This paper ends with brief conclusions in Section 6. 2. Reproducing kernel Hilbert spaces The reproducing kernel approach builds on a Hilbert space H and requires that all Dirac evaluation functional in H are bounded and continuous. In this section, two useful RKHSs W21 [0, 1] and W23 [0, 1] are constructed. Then, we utilize the reproducing kernel concept to obtain their reproducing kernel functions Gx (y) and Rx (y), respectively, in order to formulate the exact solutions in the space W23 [0, 1], in which every function satisfies the periodic boundary conditions u(0 ) = u(1 ) and u (0 ) = u (1 ), using the proposed method. Before the construction, it is necessary to present some notations, definitions, and preliminary facts upon the reproducing kernel theory that will be used further in the remainder of the paper. Throughout this paper, the symbol C indicates the set 1  2 of complex numbers, while L2 [0, 1] = {u | 0 u2 (x )dx < ∞} and l 2 = {A | ∞ i=1 (Ai ) < ∞}. Definition 1. Let  be a nonempty set. A function R :  ×  → C is a reproducing kernel of the Hilbert space H if and only if the following conditions are met; firstly, for each x ∈ , R( ·, x) ∈ H; secondly, for each x ∈ , and each ϕ ∈ H, ϕ (· ), R(·, x ) = ϕ (x ).

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The last condition is called “the reproducing property” which means that, the value of the function ϕ at the point x is reproducing by the inner product of ϕ with R( ·, x). Indeed, a Hilbert space which possesses a reproducing kernel is called a RKHS. Definition 2. The reproducing kernel space W23 [0, 1] is defined as W23 [0, 1] = {u | u, u , u are absolutely continuous real-

 valued functions in [0, 1], u, u , u  , u  ∈ L2 [0, 1], and u(0 ) = u(1 ), u (0 ) = u (1 )}. The inner product and the norm are given, respectively, by

u(x ), v(x )W23 = and u W 3 =



2

2 

u ( i ) ( 0 )v ( i ) ( 0 ) +



i=0

1

u (x )v (x )dx,

0

(3)

u, uW 3 , where u, v ∈ W23 [0, 1]. 2

The Hilbert space W23 [0, 1] is called a RKHS if for each fixed y ∈ [0, 1], there exist a function Ry (x ) = R(x, y ) ∈ W23 [0, 1] such that u(x ), Ry (x )W 3 = u(y ) for each u ∈ W23 [0, 1] and x ∈ [0, 1]. 2

Definition 3. The reproducing kernel space W21 [0, 1] is defined as W21 [0, 1] = {u | u is absolutely continuous real-valued function in [0, 1], u, u ∈ L2 [0, 1]}. The inner product and the norm are given, respectively, by

u(x ), v(x )W21 = u(0 )v(0 ) + and u W 1 =



2



1 0

u (x )v (x )dx,

u, uW 1 , where u, v ∈ W21 [0, 1]. 2

Theorem 1. The Hilbert space W23 [0, 1] is a complete reproducing kernel and its reproducing kernel function Ry (x) can be written as



Ry ( x ) =

a1 (y ) + a2 (y )x + a3 (y )x2 + a4 (y )x3 + a5 (y )x4 + a6 (y )x5 , x ≤ y, b1 (y ) + b2 (y )x + b3 (y )x2 + b4 (y )x3 + b5 (y )x4 + b6 (y )x5 , x > y,

(4)

where the unknown coefficients of Ry (x) are given as

a1 ( y ) = 1 , a2 ( y ) = a3 ( y ) = a4 ( y ) = a5 ( y ) = a6 ( y ) =

b1 ( y ) = b2 ( y ) = b3 ( y ) = b4 ( y ) = b5 ( y ) = b6 ( y ) =

1 y(27 − 60y − 20y2 + 85y3 − 32y4 ), 3867 1 y(−240 + 963y − 968y2 + 247y3 − 2y4 ), 15468 1 y(−240 + 963y − 968y2 + 247y3 − 2y4 ), 46404 1 y(−1827 + 1482y + 494y2 − 166y3 + 17y4 ), 9208 1 (3867 − 3840y − 60y2 − 20y3 + 85y4 − 32y5 ), 464040 1 (120 + y5 ), 120 1 y(216 − 480y − 160y2 − 609y3 − 256y4 ), 30936 1 y(−240 + 963y + 321y2 + 247y3 − 2y4 ), 15468 1 y(−240 − 2904y − 968y2 + 247y3 − 2y4 ), 46404 1 y(2040 + 1482y + 494y2 − 166y3 + 17y4 ), 92808 1 y(−3840 − 60y − 20y2 + 85y3 − 32y4 ). 464040

Proof. The proof of the completeness and reproducing property of W23 [0, 1] is similar to the proof in [22]. Let us find out the expression form of Ry (x). Since

v(x ), Ry (x )W23 =

2  i=0

v(i) (0 )Ry(i) (0 ) +



1 0

v(3) (x )Ry(3) (x )dx,

(5)

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through iterative integrations by parts for the sign integral of Eq. (5), we obtain that



1

0

2 

v(3) (x )Ry(3) (x )dx =

i=0

(−1 )i v(i) (x )Ry(5−i) (x ) |yy=1 − =0



1 0

v(x )Ry(6) (x )dx.

Thus, from the definition of the inner product of Eq. (3), one can write

v(x ), Ry (x )W23 =

2 

v(i) (0 )[Ry(i) (0 ) + (−1 )i+1 Ry(5−i) (0 )] +

i=0

2 

(−1 )i v(i) (1 )Ry(5−i) (1 ) −

i=0



1 0

v(x )Ry(6) (x )dx.

Since v(x ), Ry (x ) ∈ W23 [0, 1], it yield that Ry (0 ) = Ry (1 ), Ry (0 ) = Ry (1 ), and v(i ) (a ) = v(i ) (b), i = 0, 1. Hence,

v(x ), Ry (x )W23 =

2 

v(i) (0 )[Ry(i) (0 ) + (−1 )i+1 Ry(5−i) (0 )] +

i=0

(−1 )i v(i) (1 )Ry(5−i) (1 )

i=0



−

2 

1 0

v(x )Ry(6) (x )dx + c1 (v(0 ) − v(1 )) + c2 (v (0 ) − v (1 )).

(6)

On the other aspect as well, if the following relations are hold

⎧ (3 ) ⎪ ⎪Ry (1 ) = 0, ⎪ ⎪ (5 ) ⎪ ⎪Ry (0 ) − Ry (0 ) + c1 = 0, ⎪ ⎪ ⎨R(2) (0 ) − R(3) (0 ) = 0, y y ⎪ Ry(5) (1 ) − c1 = 0, ⎪ ⎪ ⎪ ⎪ ⎪ Ry (0 ) + Ry(4) (0 ) + c2 = 0, ⎪ ⎪ ⎩ (4 ) Ry ( 1 ) + c2 = 0,

(7)

then, Eq. (6) under the assumptions of Eq. (7), implies that v(x ), Ry (x )W 3 = 2

requirement

1 0

v(x )(−Ry(6) (x ))dx. Now, if Ry (x) satisfies the

Ry(6) (x ) = δ (x − y ),

(8)

(6 )

when x = y and Ry (x ) = 0, then, the auxiliary formula of Eq. (8) is given by λ6 = 0, and their auxiliary values are λ = 0 with multiplicity 6 . Therefore, the expression form of the reproducing kernel function Ry (x) is given by

6

Ry ( x ) =

i=1

ai (y )x(i−1) , x ≤ y,

i=1

bi (y )x(i−1) , x > y.

6

Again, for Eq. (8), we have ∂ k Ry (y + 0 ) = ∂ k Ry (y − 0 ), k = 0, 1, 2, 3, 4, and Ry(5 ) (y + 0 ) − Ry(y − 0 ) = −1. Through the last descriptions the unknown coefficients ai (y) and bi (y), i = 1, 2, . . . , 6 of Eq. (4) can be obtained. This completes the proof.  Theorem 2. The Hilbert space W21 [0, 1] is a reproducing kernel and its reproducing kernel function Gy (x) is given by



Gy ( x ) =

1 + x, x ≤ y, 1 + y, x > y.

It is to be noted that, the reproducing kernel functions possess some important properties such as, they are symmetric, unique, and nonnegative for any fixed x, y ∈ [0, 1]. 3. Formulation of approximate solutions In this section, we will show how to solve Eqs. (1) and (2) by using the RKHSM in detail. The formulation and implementation method of the exact and the approximate solutions are given in the RKHS W23 [0, 1]. Meanwhile, we construct an orthogonal function system of the space W23 [0, 1] based on the use of the Gram–Schmidt orthogonalization process. For the remaining sections, the lowercase letter s whenever used means for each s = 1, 2. To do this, as discussed and investigated in [16,17,22–40], we consider the following differential operator Ls :

Ls : W23 [0, 1] → W21 [0, 1], such that

Ls u(x ) = us (x ) + ps (x )us (x ) + qs (x )us (x ). As a result, Eqs. (1) and (2) can be converted into the equivalent form as follows:

Ls u(x ) = Fs (x, u1 (x ), u2 (x )), 0 ≤ x ≤ 1,

(9)

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141

subject to the periodic boundary conditions

us (0 ) = us (1 ), us (0 ) = us (1 ),

(10)

where us (x ) ∈ and Fs (x, y, z ) ∈ for y = y(x ), z = z(x ) ∈ show that Ls is bounded linear operators from W23 [0, 1] into W21 [0, 1]. W23 [0, 1]

W21 [0, 1]

W23 [0, 1],

−∞ < y, z < ∞, and 0 ≤ x ≤ 1. It is easy to {s}

Initially, we construct an orthogonal function system of W23 [0, 1]. To do so, put ϕi (x ) = Gxi (x ) and ψi (x ) = L∗s ϕi (x ), where {xi }∞ is dense on [0, 1] and L∗s is the adjoint operator of Ls . In terms of the properties of the reproducing kernel i=1 {s}

function Gx (y), one obtains us (x ), ψi (x )W 3 = us (x ), L∗s ϕi (x )W 3 = Ls us (x ), ϕi (x )W 1 = Ls us (xi ), i = 1, 2, . . .. 2

2

2

{s}

The orthonormal function systems {ψ i (x )}∞ of W23 [0, 1] can be derived from the Gram–Schmidt orthogonalization i=1 {s}

process of {ψi (x )}∞ as follows: put i=1 i  {s}

ψ¯ i{s} (x ) =

βik ψk{s} (x ),

(11)

k=1

{s}

where βik are orthogonalization coefficients and are given by the following subroutine:

βi{js} = βi{js} =

1

ψ1{s} 1 {s}

dik

βi{js} = − {s}

, for i = j = 1,

, for i = j = 1,

i−1 1  {s} {s} cik βk j , for i > j, {s} dik k= j



such that dik =

ψi{s} 2 −

i−1

It is easy to see that,

ψi{s} (x ) can be written in the of y.

{s} (c{s} )2 , cik = k=1 ik {s} ∗ ψi ( x ) = Ls ϕi ( x ) {s} form of ψi (x ) =

{s}

ψi{s} , ψ k W 3 , and {ψi{s} (x )}∞ are the orthonormal systems in W23 [0, 1]. i=1 2

= L∗s ϕi (x ), Rx (y )W 3 = ϕi (x ), Ls,y Rx (y )W 1 = .Ls,y Rx (y )|y=xi ∈ W23 [0, 1]. Thus, 2

2

.Ls,y Rx (y )|y=xi , where Ls, y indicate that the operators Ls apply to the function {s}

Theorem 3. If {xi }∞ is dense on [0, 1], then {ψi (x )}∞ are complete function systems of the space W23 [0, 1]. i=1 i=1 {s}

{s}

Proof. For each fixed us ∈ W23 [0, 1], let us (x ), ψi (x )W 3 = 0, i = 1, 2, . . .. In other word, one has, us (x ), ψi (x )W 3 = 2

2

us (x ), L∗s ϕi (x )W 3 = Ls us (x ), ϕi (x )W 1 = Ls us (xi ) = 0. Note that {xi }∞ is dense on [0, 1], therefore Ls us (x ) = 0. It follows i=1 2

2

that us (x ) = 0 from the existence of L−1 s . So, the proof of the theorem is complete.



Lemma 1. If us ∈ W23 [0, 1], then there exists positive constants M{s} such that us(i ) C ≤ M{s} us W 3 , i = 0, 1, 2, where us C = 2

max |us (x )|.

0≤x≤1

Proof. For each x, y ∈ [0, 1], we have us(i ) (x ) = us (y ), ∂xi Rx (y )W 3 , i = 0, 1, 2. By the expression form of Rx (y), it follows that 2

∂xi Rx (y ) W 3 ≤ Mi{s} , i = 0, 1, 2. Thus, |us(i) (x )| = |us (x ), ∂xi Rx (x )W 3 | ≤ ∂xi Rx (x ) W 3 us W 3 ≤ Mi{s} us W 3 , i = 0, 1, 2. Hence, 2

us(i) C ≤ max {Mi{s} } us W 3 , i = 0, 1, 2.  i=0,1,2

2

2

2

2

2

The internal structure of the following theorem is as follows: firstly, we will give the representation of the exact solution of Eqs. (9) and (10) in the space W23 [0, 1]. After that, the convergence of approximate solutions us, n (x) to the analytic solutions us (x) will be proved.  ¯ {s} ¯ {s} Theorem 4. For each us in the space W23 [0, 1], the series ∞ i=1 us (x ), ψi (x )ψi (x ) are convergent in the sense of the norm 3 ∞ of W2 [0, 1]. On the other hand, if {xi }i=1 is dense on [0, 1], then the following are hold: (i) the exact solutions of Eqs. (9) and (10) could be represented by

us ( x ) =

∞  i  {s}

βik Fs (xk , u1 (xk ), u2 (xk ))ψ¯ i{s} (x ),

(12)

i=1 k=1

(ii) the approximate solutions of Eqs. (9) and (10)

us,n (x ) =

n  i  {s}

βik Fs (xk , u1 (xk ), u2 (xk ))ψ¯ i{s} (x ),

i=1 k=1

(13)

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(i ) and us,n (x ), i = 0, 1, 2 are converging uniformly to the exact solutions us (x) and all their derivatives as n → ∞, respectively. ∞ Proof. For the first part, let us (x) be solutions of Eqs. (9) and (10) in W23 [0, 1]. Since us ∈ W23 [0, 1], i=1 us (x ), 3 [0, 1] is the Hilbert ψ¯ i{s} (x )ψ¯ i{s} (x ) are the Fourier series expansion about normal orthogonal systems {ψ¯ i{s} (x )}∞ , and W 2 i=1  ¯ {s} ¯ {s} space, then the series ∞ i=1 us (x ), ψi (x )ψi (x ) are convergent in the sense of · W 3 . On the other hand, using Eq. (11), 2

it easy to see that

us ( x ) =

∞  i=1

=

us (x ), ψ¯ i{s} (x )W23 ψ¯ i{s} (x )

∞  i  {s} i=1 k=1

=

∞  i  {s} i=1 k=1

=

βik Ls us (x ), ϕk (x )W21 ψ¯ i{s} (x )

∞  i  {s} i=1 k=1

=

βik us (x ), L∗s ϕk (x )W23 ψ¯ i{s} (x )

∞  i  {s} i=1 k=1

=

βik us (x ), ψk{s} (x )W23 ψ¯ i{s} (x )

βik Fs (x, u1 (x ), u2 (x )), ϕk (x )W21 ψ¯ i{s} (x )

∞  i  {s}

βik Fs (xk , u1 (xk ), u2 (xk ))ψ¯ i{s} (x ).

i=1 k=1

Therefore, the form of Eq. (12) is the exact solutions of Eqs. (9) and (10). For the second part, it easy to see that by Lemma 1, for any x ∈ [0, 1], we have (i ) |us,n (x ) − us(i) (x )| = |us,n (x ) − us (x ), Rx(i) (x )W23 |

≤

∂xi Rx (x ) W23 us,n − us W23 {s}

≤ Mi {s}

where Mi

us,n − us W23 , i = 0, 1, 2,

(i ) are positive constants. Hence, if us,n − us W 3 → 0 as n → ∞, the approximate solutions us, n (x) and us,n ( x ), 2

i = 0, 1, 2 are converge uniformly to the exact solutions us (x) and all their derivatives, respectively. So, the proof of the theorem is complete.  Here, the approximate solutions us, n (x) in Eq. (13) can be obtained directly by taking finitely many terms in the series representation for us (x) of Eq. (12). 4. Construction of iterative technique In this section, an iterative technique for obtaining the solutions of Eqs. (9) and (10) is represented in the RKHS W23 [0, 1] for both linear and nonlinear cases. Indeed, the approximate solutions obtained using the proposed technique with existing periodic boundary conditions are proved to converge to the exact solutions with decreasing absolute error in the same space. The basis of our RKHSM for solving Eqs. (9) and (10) is summarized below for the exact and approximate solutions. Firstly, we shall make use of the following facts about linear and nonlinear case depending on the internal structure of the functions Fs . Case 1: If Eq. (9) is linear, then the exact and the approximate solutions can be obtained directly from Eqs. (12) and (13), respectively. Case 2: If Eq. (9) is nonlinear, then in this case the exact and the approximate solutions can be obtained by using the following iterative algorithm. Algorithm 1. According to Eq. (12), the representation form of the exact solutions of Eqs. (9) and (10) can be denoted by

us ( x ) =

∞  {s} {s} Bi ψ¯ i (x ),

(14)

i=1

{s}

{s}

Bi

i

βik{s} Fs (xk , u1,k−1 (xk ), u2,k−1 (xk )). In fact, B{i s} , i = 1, 2, . . . in Eq. (14) are unknown, we will approximate {s} using known Ai . For the numerical computations, we define initial functions us,0 (x1 ) = 0, put us,0 (x1 ) = us (x1 ), and

where Bi

=

k=1

M. Al-Smadi et al. / Applied Mathematics and Computation 291 (2016) 137–148

143

define the n-term approximations to us (x) by

us,n (x ) =

n  {s} {s} Ai ψ¯ i (x ),

(15)

i=1

{s}

{s} of ψ¯ i (x ), i = 1, 2, . . . , n are given as

where the coefficients Ai {s}

{s}

A1 = β11 Fs (x1 , u1,0 (x1 ), u2,0 (x1 )), {s} {s} us,1 (x ) = A1 ψ¯ 1 (x ), {s}

A2 =

2  {s}

β2k Fs (xk , u1,k−1 (xk ), u2,k−1 (xk )),

k=1 2  {s} {s} Ai ψ¯ i (x ),

us,2 (x ) =

i=1

.. . {s}

An =

n  {s}

βnk Fs (xk , u1,k−1 (xk ), u2,k−1 (xk )),

k=1

us,n (x ) =

n−1  {s} {s} Ai ψ¯ i (x ).

(16)

i=1

In the iterative process of Eq. (15), we can guarantee that the approximations us, n (x) satisfies the periodic boundary conditions of Eq. (10). Now, the approximate solutions uN s,n (x ) can be obtained directly by taking finitely many terms in the series representation of us, n (x) such that

uNs,n (x ) =

N  i  {s}

βik Fs (xk , u1,n−1 (xk ), u2,n−1 (xk ))ψ¯ i{s} (x ).

(17)

i=1 k=1

Next, we will prove that us,n (x) in the iterative formula of Eq. (14) are converge to the exact solutions us (x) of Eqs. (9) and (10) . The next two lemmas are collected in order to prove Theorem 5. Lemma 2. If us,n−1 − us W 3 → 0, xn → y as n → ∞, and Fs (x, v, w ) is continuous in [0, 1] with respect to x, v, w for x ∈ [0, 2

1] and v, w ∈ (−∞, ∞ ), then Fs (xn , u1,n−1 (xn ), u2,n−1 (xn )) → Fs (y, u1 (y ), u2 (y )) as n → ∞. Proof. Firstly, we will prove that us,n−1 (xn ) → us (y ) in the sense of · W 3 . Since, 2

|us,n−1 (xn ) − us (y )| = |us,n−1 (xn ) − us,n−1 (y ) + us,n−1 (y ) − us (y )| ≤ |us,n−1 (xn ) − us,n−1 (y )| + |us,n−1 (y ) − us (y )|. By reproducing property of Rx (y), we have us,n−1 (xn ) = us,n−1 (x ), Rxn (x ) and us,n−1 (y ) = us,n−1 (x ), Ry (x ). Thus, |us,n−1 (xn ) − us,n−1 (y )| = |us,n−1 (x ), Rxn (x ) − Ry (x )W 3 | ≤ us,n−1 W 3 Rxn − Ry W 3 . From the symmetry of Rx (y), it follows 2

2

2

that Rxn − Ry W 3 → 0 as n → ∞. Hence, |us,n−1 (xn ) − us,n−1 (y )| → 0 as soon as xn → y. On the other hand, by 2

Theorem 4 part (ii), for any y ∈ [0, 1], it holds that |us,n−1 (y ) − us (y )| → 0 as n → ∞. Therefore, us,n−1 (xn ) → us (y ) in the sense of · W 3 as xn → y and n → ∞. Thus, by means of the continuation of Fs it is obtained that 2

Fs (xn , u1,n−1 (xn ), u2,n−1 (xn )) → Fs (y, u1 (y ), u2 (y )) as n → ∞.



Lemma 3. One has Ls us,n (x j ) = Ls us (x j ) = Fs (x j , u1, j−1 (x j ), u2, j−1 (x j )) as j ≤ n. Proof. The proof of Ls us,n (x j ) = Fs (x j , u1, j−1 (x j ), u2, j−1 (x j )) will be obtained by using the mathematical induction as    {s} {s} {s} {s} {s} {s} follows: if j ≤ n, then Ls us,n (x j ) = ni=1 Ai Ls ψ¯ i (x j ) = ni=1 Ai Ls ψ¯ i (x ), ϕ j (x )W 1 = ni=1 Ai ψ¯ i (x ), L∗s, j ϕ (x )W 3 = 2 2 n {s} ¯ {s} {s} ∞ ¯ {s} i=1 Ai ψi (x ), ψ j (x )W 3 . Using the orthogonality of {ψi (x )}i=1 , yields that 2

j  {s}

β jl

j n   {s} {s} Ls us,n (xl ) = Ai ψ¯ i (x ), β {jls} ψl{s} (x )W 3 2

i=1

l=1

=

l=1

n  {s} {s} {s} Ai ψ¯ i (x ), ψ¯ j (x )W 3 2

i=1

{s}

= Aj =

j  {s}

β jl Fs (xl , u1,l−1 (xl ), u2,l−1 (xl )).

l=1

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{s}

{s}

{s}

Now, if j = 1, then Ls us,n (x1 ) = Fs (x1 , u1,0 (x1 ), u2,0 (x1 )). Again, if j = 2, then β21 Ls us,n (x1 ) + β22 Lus,n (x2 ) = β21 Fs (x1 , {s}

u1,0 (x1 ), u2,0 (x1 )) + β22 Fs (x2 , u1,1 (x2 ), u2,1 (x2 )). Thus, Ls us,n (x2 ) = Fs (x2 , u1,1 (x2 ), u2,1 (x2 )). It is easy to see that Ls us,n (x j ) = Fs (x j , u1, j−1 (x j ), u2, j−1 (x j )) by the similar pattern.  On the other aspect as well, from Theorem 4, un (x) converge uniformly to u(x). It follows that, on taking limits in Eq.  {s} ¯ {s} {s} {s} (15), us (x ) = ∞ are an orthogonal projectors from W23 [0, 1] to Span i=1 Ai ψi (x ). Therefore, us,n (x ) = Pn us (x ), where Pn

{ψ1{s} , ψ2{s} , . . . , ψn{s} }. Thus,

Ls us,n (x j ) = Ls us,n (x ), ϕ j (x )W 1 = us,n (x ), L∗s, j ϕ (x )W 3 2

{s}

2

{s}

{s}

= Pn us (x ), ψ j (x )W 3 = us (x ), Pn 2

{s}

ψ j (x )W23

{s}

= us (x ), ψ j (x )W 3 = Ls us (x ), ϕ j (x )W 1 = Ls us (x j ). 2

2

Theorem 5. If us,n W 3 are bounded and

{xi }∞ i=1

is dense on [0, 1], then the n-term approximate solutions us, n (x) in the iterative  {s} ¯ {s} formula of Eq. (15) converge to the exact solutions us (x) of Eqs. (9) and (10) in the space W23 [0, 1] and us (x ) = ∞ i=1 Ai ψi (x ), 2

{s}

where Ai

are given by Eq. (16).

Proof. The proof consists of the following three steps. Firstly, we will prove that the sequence {us,n }∞ n=1 in Eq. (15) are {s}

monotone increasing in the sense of · W 3 . By Theorem 3, {ψ i }∞ are the complete orthonormal systems in W23 [0, 1]. i=1 2    {s} {s} {s} {s} {s} Hence, we have us,n 2 3 = us,n (x ), us,n (x )W 3 =  ni=1 Ai ψ¯ i (x ), ni=1 Ai ψ¯ i (x )W 3 = ni=1 (Ai )2 . Therefore, us,n W 3 W2

2

2

2

are monotone increasing. {s} {s} Secondly, we will prove the convergence of us, n (x). From Eq. (14), we have us,n+1 (x ) = us,n (x ) + An+1 ψ¯ n+1 (x ). From {s} {s} {s} {s} the orthogonality of {ψ¯ i (x )}∞ , it follows that us,n+1 2 3 = us,n 2 3 + (An+1 )2 = us,n−1 2 3 + (An )2 + (An+1 )2 = · · · = i=1 W2 W2 W2  {s} 2 n+1 2 + 2 us,0 W 3 i=1 (Ai ) . Since, the sequence us,n W 3 are monotone increasing. Due to the condition that us,n W23 are 2 2 ∞ {s} 2 {s} bounded, us,n W 3 are convergent as n → ∞. Then, there exists constants c{s} such that i=1 (Ai ) = c . It im2  {s} {s} plies that Ai = ik=1 βik Fs (xk , u1,k−1 (xk ), u2,k−1 (xk )) ∈ l 2 , i = 1, 2, . . .. On the other hand, since (us,m − us,m−1 )⊥(us,m−1 − us,m−2 )⊥ · · · ⊥(us,n+1 − us,n ) it follows for m > n that 2 2 us,m − us,n W 3 = us,m − us,m−1 + us,m−1 − · · · + us,n+1 − us,n W 3 2

2

2 2 = us,m − us,m−1 W 3 + · · · + us,n+1 − us,n W 3 . 2

Furthermore,

2 us,m − us,m−1 W 3 2

2

{s} = (Am )2 . Consequently, as n, m → ∞, we have

2 = us,m − us,n W 3 2

m

{s} 2 ) → 0. Con-

i=n+1 (Ai

sidering the completeness of W23 [0, 1], there exists us (x ) ∈ W23 [0, 1] such that us, n (x) → us (x) as n → ∞ in the sense of · W 3 . 2

Thirdly, we will prove that us (x) are the solutions of Eqs. (9) and (10). Since {xi }∞ is dense on [0, 1] , for any i=1 x ∈ [0, 1], there exists subsequence {xn j }∞ , such that x → x as j → ∞ . From Lemma 3 , It is clear that Ls us (xn j ) = n j j=1 Fs (xn j , u1,n j −1 (xk ), u2,n j −1 (xk )). Hence, let j → ∞, by Lemma 2 and the continuity of Fs , we have Ls us (x ) = Fs (x, u1 (x ), u2 (x )). {s} That is, us (x) satisfies Eq. (9). Also, since ψ¯ i (x ) ∈ W23 [0, 1], clearly, us (x) satisfy the periodic boundary conditions of Eq. (10).  {s} ¯ {s} {s} In other words, us (x) are the solution of Eqs. (9) and (10), where us (x ) = ∞ are given by Eq. (16). The i=1 Ai ψi (x ) and Ai proof is complete. 

It obvious that, if we let us (x) denote the exact solutions of Eqs. (9) and (10), us, n (x) denote the approximate solu{s}

tions obtained by the RKHSM as given by Eq. (15), and rn (x ) are the differences between us, n (x) and us (x), where x ∈  ∞ ∞ {s} {s} ¯ {s} {s} 2 {s} 2 {s} 2 {s} 2 [0, 1], then rn 2 3 = us − us,n 2 3 = ∞ i=n+1 Ai ψi (x ) 3 = i=n+1 (Ai ) and rn−1 3 = i=n (Ai ) or rn W 3 ≤ W2

W2

W2

W2

} rn{s−1 W 3 . Consequently, this show that the differences rn{s} (x ) are monotone decreasing in the sense of · W 3 . 2

2

2

5. Numerical results To demonstrate the simplicity and effectiveness of the proposed method, approximate solutions for a class of system of BVPs is constructed using the RKHSM for different values of n and N. The method is applied in a direct way without using linearization, transformation, or restrictive assumptions. The approximate solutions us, n (x) are calculated for linear case by Eq. (13), whilst, the approximate solutions uN s,n (x ) are calculated for nonlinear case by Eq. (17). Results obtained by the RKHSM are compared with the exact solutions of each example and are found to be in good agreement with each other. These results reveal that the present method is an accurate and reliable analytical numerical method for solving such

M. Al-Smadi et al. / Applied Mathematics and Computation 291 (2016) 137–148

145

Table 1 Numerical results of u1 (x) for Example 1. x

u1 (x)

u51 1,2 ( x )

|u51 1,2 ( x ) − u1 ( x )|

−1 |u51 1,2 ( x ) − u1 ( x ) u1 ( x )|

0.16 0.32 0.48 0.64 0.80 0.96

0.01806340 0.04734980 0.06230020 0.05308420 0.02560 0 0 0 0.00147456

0.018066378900530353 0.047352305694442064 0.062300481694450930 0.053082067390799940 0.025596843619117605 0.001473370698343928

3.01890 × 10−6 2.54569 × 10−6 3.21694 × 10−7 2.09261 × 10−6 3.15638 × 10−6 1.18930 × 10−6

1.67128 × 10−5 5.37636 × 10−5 5.16362 × 10−6 3.94206 × 10−5 1.23296 × 10−5 8.06547 × 10−5

Table 2 Numerical results of u2 (x) for Example 1. x

u2 (x)

u51 2,2 ( x )

|u51 2,2 ( x ) − u2 ( x )|

−1 |u51 2,2 ( x ) − u2 ( x ) u2 ( x )|

0.16 0.32 0.48 0.64 0.80 0.96

0.990982 0.976418 0.969011 0.973575 0.987227 0.999263

0.9909811288608564 0.9764176987846284 0.9690111768830832 0.9735756768455183 0.9872281563381624 0.9992631287612721

7.78163 × 10−7 6.90555 × 10−7 1.28896 × 10−7 5.50740 × 10−7 8.72963 × 10−7 3.18169 × 10−7

7.85245 × 10−7 7.07233 × 10−7 1.33018 × 10−7 5.65689 × 10−7 8.84257 × 10−7 3.18403 × 10−7

problems. In the process of computation, all the symbolic and numerical computations performed by using Mathematica 7.0 software package. Let us consider the following periodic differential systems of different types: Example 1. Consider the following nonlinear second-order differential system:

u1 (x ) + 2u1 (x ) +

u2 ( x ) = f 1 ( x ), 1 + (u2 (x ))2

u2 (x ) + u2 (x ) + ex u1 (x ) + x(u2 (x ))2 = f2 (x ), subject to the periodic boundary conditions

u1 (0 ) = u1 (1 ), u1 (0 ) = u1 (1 ), u2 (0 ) = u2 (1 ), u2 (0 ) = u2 (1 ), where f1 (x) and f2 (x) are chosen such that the exact solutions are u1 (x ) = x2 (x − 1 )2 and u2 (x ) = cos(x(1 − x )). Example 2. Consider the following nonlinear second-order differential system:

u1 (x ) + 2u1 (x ) + sinh(u1 (x ) − (2x3 − 3x2 )) − (u2 (x ))2 = f1 (x ), u2 (x ) +

x2

x3 u (x ) + e−u2 (x ) + (u1 (x ))3 = f2 (x ), ( 1 − x )2 + 1 2

subject to the periodic boundary conditions

u1 (0 ) = u1 (1 ), u1 (0 ) = u1 (1 ), u2 (0 ) = u2 (1 ), u2 (0 ) = u2 (1 ), where f1 (x) and f2 (x) are chosen such that the exact solutions are u1 (x ) = 2x3 − 3x2 + x and u2 (x ) = ln(x2 (x − 1 )2 + 1 ). Example 3. Consider the following linear second-order differential system:

u1 (x ) − 3x3 u1 (x ) + u1 (x ) − xu2 (x ) = f1 (x ),

u2 (x ) − cosh(x )u2 (x ) + u1 (x ) + u2 (x ) = f2 (x ), subject to the periodic boundary conditions

u1 (0 ) = u1 (1 ), u1 (0 ) = u1 (1 ), u2 (0 ) = u2 (1 ), u2 (0 ) = u2 (1 ), 2 2 where f1 (x) and f2 (x) are chosen such that the exact solutions are u1 (x ) = ex (x−1 ) and u2 (x ) = x4 − 2x3 + x2 .

i−1 Using RKHSM, taking xi = N−1 , i = 1, 2, . . . , N in uN s,n (xi ) of Eq. (17), with the reproducing kernel functions Rx (y) and Gx (y) on [0, 1]; the numerical results and tabulate data are presented and discussed quantitatively at some selected grid points on [0, 1] to illustrate the numerical solutions for the dependent variables u1 (x) and u2 (x), respectively, for the first two examples as shown in Tables 1–4. Anyhow, it is clear from the tables that, the numerical solutions are in close agreement

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M. Al-Smadi et al. / Applied Mathematics and Computation 291 (2016) 137–148

Table 3 Numerical results of u1 (x) for Example 2. x 0.16 0.32 0.48 0.64 0.80 0.96

u1 (x) 0.091392 0.078336 0.009984 −0.064512 −0.0960 0 0 −0.035328

u51 1,3 ( x ) 0.091391245274867360 0.078335363576389440 0.009983919576387133 −0.06451147684770 0 09 −0.09599921090477949 −0.03532770267458599

|u51 1,3 ( x ) − u1 ( x )| 7.54725 × 10−7 6.36424 × 10−7 8.04236 × 10−8 5.23152 × 10−7 7.89095 × 10−7 2.97325 × 10−7

−1 |u51 1,3 ( x ) − u1 ( x ) u1 ( x )|

8.25811 × 10−6 8.12428 × 10−6 8.05525 × 10−6 8.10938 × 10−6 8.21974 × 10−6 8.41614 × 10−6

Table 4 Numerical results of u2 (x) for Example 2. x

u2 (x)

u51 2,3 ( x )

|u51 2,3 ( x ) − u2 ( x )|

−1 |u51 2,3 ( x ) − u2 ( x ) u2 ( x )|

0.16 0.32 0.48 0.64 0.80 0.96

0.01790220 0.04626290 0.06043650 0.05172320 0.02527780 0.00147347

0.0179025515762960700 0.0462636036494551840 0.0604370904198937300 0.0517228429937984700 0.0252767369727549600 0.0014731357326304949

3.95699 × 10−7 6.68330 × 10−7 5.70999 × 10−7 3.10991 × 10−7 1.07021 × 10−6 3.38171 × 10−7

2.21034 × 10−5 1.44463 × 10−5 9.44792 × 10−6 6.01260 × 10−6 4.23380 × 10−5 2.29506 × 10−4

Fig. 1. The numerical values of the absolute error functions for the solutions of Example 3: blue: |u1,51 (x ) − u1 (x )| and red: |u2,51 (x ) − u2 (x )|. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 2. The numerical values of the absolute error functions for the first derivatives of Example 3: blue: |u1,51 (x ) − u1 (x )| and red: |u2,51 (x ) − u2 (x )|. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

with the exact solutions for the two examples, while the accuracy is in advanced by using only few tens of the RKHS iterations. Indeed, we can conclude that higher accuracy can be achieved by computing further RKHS iterations. Next, numerical results for Example 3 are presented and investigated geometrically for the two dependent variables ( p) ( p) u1 (x ) and u2 (x ), p = 0, 1, 2 at various x in [0, 1]. As we mentioned earlier, it is possible to pick any point in [0, 1] and as well the numerical solutions and all their numerical derivatives up to order two will be applicable. Anyhow, the numerical ( p) values of the absolute errors for approximating us (x ), p = 0, 1, 2, where xi = ni−1 −1 , i = 1, 2, . . . , n in us, n (xi ) of Eq. (13) have been plotted in Figs. 1–3, respectively. It is observed that the increase in the number of node results in a reduction in the absolute errors and correspondingly an improvement in the accuracy of the obtained solutions. This goes in agreement with the known fact, the error is decreasing, where more accurate solutions are achieved using an increase in the number of

M. Al-Smadi et al. / Applied Mathematics and Computation 291 (2016) 137–148

147

Fig. 3. The numerical values of the absolute error functions for the second derivatives of Example 3: blue: |u1,51 (x ) − u1 (x )| and red: |u2,51 (x ) − u2 (x )|. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

nodes. On the other hand, the cost to be paid while going in this direction is the rapid increase in the number of iterations required for convergence. 6. Conclusions The RKHSM is a powerful method for solving various linear and nonlinear problems. In this paper, we introduce the reproducing kernel approach to enlarge its application range. It is analyzed that the proposed method is well suited for use in systems of differential equations and resides in its simplicity in dealing with periodic boundary conditions. However, the RKHSM does not require discretization of the variables, it provides the best solution in a less number of iterations and reduces the computational work. The numerical results we obtained justify these advantages of this methodology. The approximate solutions obtained by this method and their derivative are both uniformly convergent. Numerical experiments are carried out to illustrate that the present method is an accurate and reliable analytical technique for treating systems of periodic BVPs. It is worth to be pointed out that the RKHSM is still suitable and can be employed for solving other strongly linear and nonlinear systems of differential equations. Acknowledgment The authors express their thanks to unknown referees for the careful reading and helpful comments. References [1] U.M. Ascher, R.M. Mattheij, R.D. Russell, Numerical solution of boundary value problems for ordinary differential equations, in: Classics in Applied Mathematics, 13, Siam, 1995. [2] R.B. Agarwal, in: Boundary Value Problems for High Ordinary Differential Equations, World Scientific, Singapore, 1986. [3] V. Seda, J.J. Nieto, M. Gera, Periodic boundary value problem for nonlinear higher ordinary differential equations, Appl. Math. Comput. 48 (1992) 71–82. [4] F.M. Atici, G.S. Guseinov, On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions, J. Comput. Appl. Math. 132 (2001) 341–356. [5] Y. Lin, J. Lin, A numerical algorithm for solving a class of linear nonlocal boundary value problems, Appl. Math. Lett. 23 (2010) 997–1002. [6] S. Bellew, E. O’Riordan, A parameter robust numerical method for a system of two singularly perturbed convection-diffusion equations, Appl. Numer. Math. 51 (2004) 171–186. [7] S. Matthews, E. O’Riordan, G.I. Shishkin, A numerical method for a system of singularly perturbed reaction-diffusion equations, J. Comput. Appl. Math. 145 (2002) 151–166. [8] C. Xenophontos, L. Oberbroeckling, A numerical study on the finite element solution of singularly perturbed systems of reaction-diffusion problems, Appl. Math. Comput. 187 (2007) 1351–1367. ˚ [9] I. Rachunkov , O. Kochb, G. Pulvererb, E. Weinmüller, On a singular boundary value problem arising in the theory of shallow membrane caps, J. Math. Anal. Appl. 332 (2007) 523–541. [10] O.A. Arqub, Z. Abo-Hammour, Numerical solution of systems of second-order boundary value problems using continuous genetic algorithm, Inf. Sci. 279 (2014) 396–415. [11] O.A. Arqub, A. El-Ajou, Z.A. Zhour, S. Momani, Multiple solutions of nonlinear boundary value problems of fractional order: a new analytic iterative technique, Entropy 16 (2014) 471–493. [12] W. Liu, L. Liu, Y. Wu, Positive solutions of a singular boundary value problem for systems of second-order differential equations, Appl. Math. Comput. 208 (2009) 511–519. [13] Y. Zhou, Y. Xu, Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations, J. Math. Anal. Appl. 320 (2006) 578–590. [14] B. Liu, L. Liu, Y. Wuc, Existence of nontrivial periodic solutions for a nonlinear second order periodic boundary value problem, Nonlinear Anal. Theory Methods Appl. 72 (2010) 3337–3345. [15] Q. Yao, Positive solutions of nonlinear second-order periodic boundary value problems, Appl. Math. Lett. 20 (2007) 583–590. [16] M. Al-Smadi, O.A. Arqub, . El-Ajuo, A numerical method for solving systems of first-order periodic boundary value problems, J. Appl. Math. 2014 (2014) 10, doi:10.1155/2014/135465. Article ID 135465. [17] O.A. Arqub, Reproducing kernel algorithm for the analytical-numerical solutions of nonlinear systems of singular periodic boundary value problems, Math. Probl. Eng. 2015 (2015) 13, doi:10.1155/2015/518406. Article ID 518406.

148 [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

M. Al-Smadi et al. / Applied Mathematics and Computation 291 (2016) 137–148 M. Cui, Y. Lin, in: Nonlinear Numerical Analysis in the Reproducing Kernel Space, Nova Science, New York, NY, USA, 2009. A. Berlinet, C.T. Agnan, Reproducing Kernel Hilbert Space in Probability and Statistics, Kluwer Academic Publishers, Boston, Mass, USA, 2004. A. Daniel, Reproducing Kernel Spaces and Applications, Springer, Basel, Switzerland, 2003. H.L. Weinert, Reproducing Kernel Hilbert Spaces: Applications in Statistical Signal Processing, Hutchinson Ross, 1982. L.H. Yang, Y. Lin, Reproducing kernel methods for solving linear initial-boundary-value problems, Electron. J. Differ. Equ. 2008 (2008) 1–11. O.A. Arqub, M. Al-Smadi, Numerical algorithm for solving two-point, second-order periodic boundary value problems for mixed integro-differential equations, Appl. Math. Comput. 243 (2014) 911–922. S. Momani, O.A. Arqub, T. Hayat, H. Al-Sulami, A computational method for solving periodic boundary value problems for integro-differential equations of Fredholm-Voltera type, Appl. Math. Comput. 240 (2014) 229–239. M. Al-Smadi, O.A. Arqub, S. Momani, A computational method for two-point boundary value problems of fourth-order mixed integro-differential equations, Math. Probl. Eng. 2013 (2012) 10, doi:10.1155/2013/832074. Article ID 832074. O.A. Arqub, M. Al-Smadi, N. Shawagfeh, Solving Fredholm integro-differential equations using reproducing kernel Hilbert space method, Appl. Math. Comput. 219 (2013) 8938–8948. O.A. Arqub, M. Al-Smadi, S. Momani, Application of reproducing kernel method for solving nonlinear Fredholm-Volterra integro-differential equations, Abstr. Appl. Anal. 2012 (2012) 1–16, doi:10.1155/2012/839836. Article ID 839836. O.A. Arqub, An iterative method for solving fourth-order boundary value problems of mixed type integro-differential equations, J. Comput. Anal. Appl. 8 (2015) 857–874. N. Shawagfeh, O.A. Arqub, S. Momani, Analytical solution of nonlinear second-order periodic boundary value problem using reproducing kernel method, J. Comput. Anal. Appl. 16 (2014) 750–762. O.A. Arqub, M. AL-Smadi, S. Momani, T. Hayat, Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method, Soft Comput. (2015) 1–20, doi:10.10 07/s0 050 0- 015- 1707- 4. F.Z. Geng, S.P. Qian, S. Li, A numerical method for singularly perturbed turning point problems with an interior layer, J. Comput. Appl. Math. 255 (2014) 97–105. F.Z. Geng, S.P. Qian, Solving singularly perturbed multipantograph delay equations based on the reproducing kernel method, Abstr. Appl. Anal. 2014 (2014) 6, doi:10.1155/2014/794716. Article ID 794716. F.Z. Geng, S.P. Qian, Modified reproducing kernel method for singularly perturbed boundary value problems with a delay, Appl. Math. Model. 39 (2015) 5592–5597. F.Z. Geng, S.P. Qian, M.G. Cui, Improved reproducing kernel method for singularly perturbed differential-difference equations with boundary layer behavior, Appl. Math. Comput. 252 (2015) 58–63. F.Z. Geng, S.P. Qian, Reproducing kernel method for singularly perturbed turning point problems having twin boundary layers, Appl. Math. Lett. 26 (2013) 998–1004. Y. Wang, M. Du, F. Tan, Z. Li, T. Nie, Using reproducing kernel for solving a class of fractional partial differential equation with non-classical conditions, Appl. Math. Comput. 219 (2013) 5918–5925. H. Du, G. Zhao, C. Zhao, Reproducing kernel method for solving Fredholm integro-differential equations with weakly singularity, J. Comput. Appl. Math. 255 (2014) 122–132. W. Jiang, Y. Lin, Representation of exact solution for the time-fractional telegraph equation in the reproducing kernel space, Commun. Nonlinear Sci. Numer. Simul. 16 (9) (2011) 3639–3645. W. Jiang, Z. Chen, Solving a system of linear Volterra integral equations using the new reproducing kernel method, Appl. Math. Comput. 219 (2013) 10225–10230. W. Jiang, Z. Chen, A collocation method based on reproducing kernel for a modified anomalous subdiffusion equation, Numer. Methods Partial Differ. Equ. 30 (2014) 289–300.