Solving a System of Linear Volterra Integral Equations Using the ...

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Sep 30, 2013 - A numerical technique based on reproducing kernel methods for the exact solution of linear Volterra integral equations system of the secondΒ ...
Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 196308, 5 pages http://dx.doi.org/10.1155/2013/196308

Research Article Solving a System of Linear Volterra Integral Equations Using the Modified Reproducing Kernel Method Li-Hong Yang, Hong-Ying Li, and Jing-Ran Wang College of Science, Harbin Engineering University, Heilongjiang 150001, China Correspondence should be addressed to Li-Hong Yang; [email protected] Received 16 May 2013; Accepted 30 September 2013 Academic Editor: Rodrigo Lopez Pouso Copyright Β© 2013 Li-Hong Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A numerical technique based on reproducing kernel methods for the exact solution of linear Volterra integral equations system of the second kind is given. The traditional reproducing kernel method requests that operator a satisfied linear operator equation 𝐴𝑒 = 𝑓, is bounded and its image space is the reproducing kernel space π‘Š21 [π‘Ž, 𝑏]. It limits its application. Now, we modify the reproducing kernel method such that it can be more widely applicable. The n-term approximation solution obtained by the modified method is of high accuracy. The numerical example compared with other methods shows that the modified method is more efficient.

1. Introduction The purpose of this paper is to solve a system of linear Volterra integral equations 𝑏

𝐹 (𝑠) = 𝐺𝑠 + ∫ 𝐾 (𝑠, 𝑑) 𝐹 (𝑑) 𝑑𝑑, π‘Ž

𝑠 ∈ [0, 1] ,

(1)

where 𝑇

𝐹 (𝑠) = [𝑓1 (𝑠) , 𝑓2 (𝑠) , . . . , 𝑓𝑛 (𝑠)] , 𝑇

𝐺 (𝑠) = [𝑔1 (𝑠) , 𝑔2 (𝑠) , . . . , 𝑔𝑛 (𝑠)] , 𝐾 (𝑠, 𝑑) = [π‘˜π‘–,𝑗 ] ,

(2)

𝑖, 𝑗 = 1, 2, . . . , 𝑛.

In (1), the functions 𝐾 and 𝐺 are given, and 𝐹 is the solution to be determined. We assume that (1) has a unique solution. Volterra integral equation arises in many physical applications, for example, potential theory and Dirichlet problems, electrostatics, mathematical problems of radiative equilibrium, the particle transport problems of astrophysics and reactor theory, and radiative heat transfer problems [1–5]. Several valid methods for solving Volterra integral equation have been developed in recent years, including power series method [6], Adomain’s decomposition method [7], homotopy perturbation method [8, 9], block by block method [10], and expansion method [11].

Since the reproducing kernel space π‘Š21 [π‘Ž, 𝑏], which is a special Hilbert space, is constructed in 1986 [12], the reproducing kernel theory has been applied successfully to many linear and nonlinear problems, such as differential equation, population model, and many other equations appearing in physics and engineering [12–21]. The traditional reproducing kernel method is limited, because it requires that the image space of operator 𝐴 in linear operator equation 𝐴𝑒 = 𝑓 is π‘Š21 [π‘Ž, 𝑏] and operator 𝐴 must be bounded. In order to enlarge its application range, the MRKM removes the boundedness of 𝐴 and weakens its image space to 𝐿2 [π‘Ž, 𝑏]. Subsequently, we apply the MRKM to obtain the series expression of the exact solution for (1). The 𝑛-term approximation solution is provided by truncating the series. The final numerical comparisons between our method and other methods show the efficiency of the proposed method. It is worth to mention that the MRKM can be generalized to solve other system of linear equations.

2. Preliminaries 2.1. The Reproducing Kernel Space π‘Š21 [0, 1]. The reproducing kernel space π‘Š21 [0, 1] consists of all absolute continuous realvalued functions, which defined on the closed interval [0, 1], and the first derivative functions belong to 𝐿2 [0, 1].

2

Abstract and Applied Analysis

3. The Exact Solution of (1)

The inner product and the norm are equipped with 1

(𝑒, V)𝑀21 = 𝑒 (0) V (0) + ∫ 𝑒󸀠 (π‘₯) VσΈ€  (π‘₯) 𝑑π‘₯, 0

βˆ€π‘’, V ∈ 𝑀21 ,

3.1. Identical Transformation of (1). Consider the ith equation of (1): 𝑛

β€–π‘’β€–π‘Š21 = √(𝑒, V)𝑀21 .

𝑗=1 0

(3) π‘Š21 [0, 1]

Theorem 1. ducing kernel [22]

is a reproducing kernel space with repro-

1 + 𝑦, 𝑅π‘₯ (𝑦) = { 1 + π‘₯,

𝑦≀π‘₯ 𝑦 > π‘₯;

(4)

2

(11)

Define operator 𝐴 𝑖𝑗 : π‘Š21 β†’ 𝐿2 [0, 1], 𝑗 = 1, . . . , 𝑛, 1

{ {𝑒 (𝑠) βˆ’ ∫0 π‘˜π‘–π‘— (𝑠, 𝑑) 𝑒 (𝑑) 𝑑𝑑, 𝐴 𝑖𝑗 = { 𝑠 {βˆ’ ∫ π‘˜π‘–π‘— (𝑠, 𝑑) 𝑒 (𝑑) 𝑑𝑑, { 0

𝑗=𝑖 𝑗 =ΜΈ 𝑖,

(12)

where 𝑒 ∈ π‘Š21 . Then, (1) can be turned into

that is, for every π‘₯ ∈ [0, 1] and 𝑒 ∈ π‘Š21 , it follows that (𝑒 (𝑦) , 𝑅π‘₯ (𝑦))𝑀1 = 𝑒 (π‘₯) .

𝑠

𝑓𝑖 (𝑠) βˆ’ βˆ‘ ∫ 𝐾𝑖𝑗 (𝑠, 𝑑) 𝑓𝑗 (𝑑) 𝑑𝑑 = 𝑔𝑖 (𝑠) .

𝐴 11 𝑓1 + 𝐴 12 𝑓2 + β‹… β‹… β‹… + 𝐴 1𝑛 𝑓1𝑛 = 𝑔1 (𝑠)

(5)

𝐴 21 𝑓1 + 𝐴 22 𝑓2 + β‹… β‹… β‹… + 𝐴 2𝑛 𝑓1𝑛 = 𝑔2 (𝑠)

2.2. The Reproducing Kernel Space π‘Š22 [0, 1]. The reproducing kernel space π‘Š22 [0, 1] consists of all real-valued functions in

which the first derivative functions are absolute continuous on the closed interval [0, 1] and the second derivative functions belong to 𝐿2 [0, 1]. The inner product and the norm are equipped with 1

.. .

(13)

𝐴 𝑛1 𝑓1 + 𝐴 𝑛2 𝑓2 + β‹… β‹… β‹… + 𝐴 𝑛𝑛 𝑓1𝑛 = 𝑔𝑛 (𝑠) , where 𝐹(𝑠) = [𝑓1 (𝑠), 𝑓2 (𝑠), . . . , 𝑓𝑛 (𝑠)]𝑇 ∈ 𝐸.

(6)

3.2. The Exact solution of (1). Let {π‘₯𝑖 }∞ 𝑖=1 be a dense subset of interval [0, 1], and define 󡄨 󡄨 Ψ𝑖𝑗 (π‘₯) = ( 𝐴 𝑗1 ,𝑦 𝑅π‘₯ (𝑦)󡄨󡄨󡄨󡄨𝑦=π‘₯ , 𝐴 𝑗2 ,𝑦 𝑅π‘₯ (𝑦)󡄨󡄨󡄨󡄨𝑦=π‘₯ , . . . , 𝑖 𝑖 (14) 𝑇 󡄨󡄨 𝐴 𝑗𝑛 ,𝑦 𝑅π‘₯ (𝑦)󡄨󡄨󡄨𝑦=π‘₯ )

Theorem 2. π‘Š22 [0, 1] is a reproducing kernel space with reproducing kernel [22]

for every 𝑗 = 1, 2, . . . , 𝑛, 𝑖 = 1, 2, . . .; the subscript 𝑦 of 𝐴 𝑖𝑗,𝑦 means that the operator 𝐴 𝑖𝑗 acts on the function of 𝑦. It is easy to prove that Ψ𝑖𝑗 ∈ 𝐸.

(𝑒, V)π‘Š22 = βˆ‘ 𝑒(π‘˜) (0) V(π‘˜) (0) π‘˜=0

1

+ ∫ 𝑒󸀠󸀠 (π‘₯) VσΈ€ σΈ€  (π‘₯) 𝑑π‘₯, 0

βˆ€π‘’, V ∈ π‘Š22 [0, 1] ,

β€–π‘’β€–π‘Š22 = √(𝑒, 𝑒)𝑀22 .

𝑖

π‘₯ Γ— 𝑦 2 𝑦3 { βˆ’ 𝑦≀π‘₯ 1 + π‘₯ Γ— 𝑦 + { { 2 6 𝑄 (π‘₯, 𝑦) = { { π‘₯2 Γ— 𝑦 π‘₯3 { βˆ’ , 𝑦 > π‘₯; 1+π‘₯×𝑦+ { 2 6

(7)

that is, for every π‘₯ ∈ [0, 1] and 𝑒 ∈ π‘Š22 , it follows that (𝑒 (𝑦) , 𝑄 (π‘₯, 𝑦))𝑀2 = 𝑒 (π‘₯) . 2

(8)

The proof of Theorems 1 and 2 can be found in [23].

Theorem 3. {Ψ𝑖1 , Ψ𝑖2 , . . . , Ψ𝑖𝑛 }∞ 𝑖=1 is complete in 𝐸. Proof. Take 𝑒 = (𝑒1 , 𝑒2 , . . . , 𝑒𝑛 )𝑇 ∈ 𝐸 such that (𝑒(π‘₯), Ψ𝑖𝑗 (π‘₯)) = 0 for every 𝑗 = 1, 2, . . . , 𝑛, 𝑖 = 1, 2, . . .. From this fact, it holds that (𝑒 (π‘₯) , Ψ𝑖𝑗 (π‘₯)) 𝑇

= ((𝑒1 , 𝑒2 , . . . , 𝑒𝑛 ) ,

2.3. Hilbert Space 𝐸. Hilbert space 𝐸 is defined by 𝑛

𝑇

𝐸 =⨁ π‘Š21 = {(𝑒1 , . . . , 𝑒𝑛 ) | 𝑒𝑖 ∈ 𝑀21 , 𝑖 = 1, . . . , 𝑛} .

󡄨 ( 𝐴 𝑗1,𝑦 𝑅π‘₯ (𝑦)󡄨󡄨󡄨󡄨𝑦=π‘₯ , 𝑖

(9)

𝑖=1

(15) 𝑇 󡄨󡄨 󡄨 𝐴 𝑗2,𝑦 𝑅π‘₯ (𝑦)󡄨󡄨󡄨󡄨𝑦=π‘₯ , . . . , 𝐴 𝑗𝑛,𝑦 𝑅π‘₯ (𝑦)󡄨󡄨󡄨 ) ) 󡄨𝑦=π‘₯𝑖 𝑖

The inner product and the norm are given by 𝑛

𝑛

(𝑒, V)𝐸 = βˆ‘(𝑒𝑖 , V𝑖 )𝑀1 , 𝑖=1

2

‖𝑒‖𝐸 = √(𝑒, 𝑒)𝐸 . It is easy to prove that 𝐸 is a Hilbert space.

(10)

󡄨󡄨 = βˆ‘ 𝐴 π‘—π‘˜,𝑦 (π‘’π‘˜ (π‘₯) , 𝑅π‘₯ (𝑦))𝑀1 󡄨󡄨󡄨 2 󡄨𝑦=π‘₯𝑖 π‘˜=1 𝑛

= βˆ‘ 𝐴 π‘—π‘˜ π‘’π‘˜ (π‘₯𝑖 ) = 0, π‘˜=1

Abstract and Applied Analysis

3 Note that βˆ‘π‘›π‘–=1 ‖𝑓𝑖,π‘š βˆ’ 𝑓𝑖 β€–2π‘Š1 = β€–πΉπ‘š βˆ’ 𝐹‖2𝐸 β†’ 0. Com2 bining with the expression of 𝑅π‘₯ (𝑦), we have

for every 𝑗 = 1, 2, . . . , 𝑛. The dense {π‘₯𝑖 }∞ 𝑖=1 assumes that 𝐴 11 𝑒1 + 𝐴 12 𝑒2 + β‹… β‹… β‹… + 𝐴 1𝑛 𝑒𝑛 = 0 𝐴 21 𝑒1 + 𝐴 22 𝑒2 + β‹… β‹… β‹… + 𝐴 2𝑛 𝑒𝑛 = 0 (16)

.. .

σ΅„© σ΅„© = 󡄩󡄩󡄩𝑓𝑖,π‘š βˆ’ 𝑓𝑖 σ΅„©σ΅„©σ΅„©π‘Š1 βˆšπ‘…π‘₯ (π‘₯)

𝐴 𝑛1 𝑒1 + 𝐴 𝑛2 𝑒2 + β‹… β‹… β‹… + 𝐴 𝑛𝑛 𝑒𝑛 = 0.

We arrange Ξ¨11 , Ξ¨12 ,. . ., Ξ¨1𝑛 , Ξ¨21 , Ξ¨22 , . . ., Ξ¨2𝑛 , . . ., Ψ𝑖1 , Ψ𝑖2 , . . ., Ψ𝑖𝑛 ,. . ., denoted by {π‘Ÿπ‘– }∞ 𝑖=1 ; that is, π‘Ÿ1 = Ξ¨11 , π‘Ÿ2 = Ξ¨12 , . . . , π‘Ÿπ‘› = Ξ¨1𝑛 , π‘Ÿπ‘›+1 = Ξ¨21 , π‘Ÿπ‘›+2 = Ξ¨22 , . . . , π‘Ÿπ‘›+𝑛 = Ξ¨2𝑛 , . . .. In a general way, π‘Ÿ(π‘–βˆ’1)𝑛+𝑗 = Ψ𝑖𝑗 , 𝑖 = 1, 2, 3, . . . ; 𝑗 = 1, 2,. . . ,𝑛. The orthogonal basis {π‘Ÿπ‘– }∞ 𝑖=1 in 𝐸 from Gram-Schmidt is as follows: orthogonalization of {π‘Ÿπ‘– }∞ 𝑖=1 𝑖

𝑖 = 1, 2, . . . .

(17)

π‘˜=1

Theorem 4. The exact solution of (1) can be expressed by ∞

𝑖

𝐹 (π‘₯) = βˆ‘ βˆ‘ π›½π‘–π‘˜ πœŒπ‘˜ π‘Ÿπ‘– (π‘₯) ,

(18)

𝑖=1 π‘˜=1

𝑖=1

𝑖=1 π‘˜=1

Taking nodes {π‘₯𝑖 = (𝑖 βˆ’ 1)/(𝑁 βˆ’ 1)}𝑁 𝑖=1 , 𝑓𝑖,𝑁 is the approximate solutions of 𝑓𝑖 , and 𝑒(𝑓𝑖,𝑁) denotes the absolute errors of 𝑓𝑖 , 𝑖 = 1, 2, . . . , 𝑛. According to Remark 6, we solve the following two examples appearing in [11] in π‘Š22 .

𝑠

𝑓1 (𝑠) = 𝑔1 (𝑠) + ∫ (𝑠 βˆ’ 𝑑)3 𝑓1 (𝑑) 𝑑𝑑 + ∫ (𝑠 βˆ’ 𝑑)2 𝑓2 (𝑑) 𝑑𝑑, 0

0

𝑠

𝑓2 (𝑠) = 𝑔2 (𝑠) + ∫ (𝑠 βˆ’ 𝑑)4 𝑓1 (𝑑) 𝑑𝑑 𝑠

(19)

+ ∫ (𝑠 βˆ’ 𝑑)3 𝑓2 (𝑑) 𝑑𝑑, 0

𝑖

𝐹 (π‘₯) = βˆ‘ βˆ‘ 𝛽𝑖𝑗 πœŒπ‘˜ π‘Ÿπ‘– (π‘₯) .

(20)

𝑖=1 π‘˜=1

When π‘Ÿπ‘˜ = Ψ𝑗𝑙 , it holds that

(24)

where 𝑔1 (𝑠) and 𝑔2 (𝑠) are chosen such that the exact solution is 𝑓1 (𝑠) = 1 + 𝑠2 , 𝑓2 (𝑠) = 1 + 𝑠 βˆ’ 𝑠3 . The numerical results obtained by using the present method are compared with [11] in Table 1. Example 8. Consider the following system of linear Volterra integral equations of the second kind [11]:

𝑛

πœŒπ‘˜ = (𝐹, Ψ𝑗𝑙 ) = βˆ‘ 𝐴 π‘™π‘˜ π‘’π‘˜ (π‘₯𝑗 ) = 𝑔𝑙 (π‘₯𝑗 ) .

(21)

π‘˜=1

𝑠

𝑓1 (𝑠) = 𝑔1 (𝑠) + ∫ (sin (𝑠 βˆ’ 𝑑) βˆ’ 1) 𝑓1 (𝑑) 𝑑𝑑 0

Corollary 5. The approximate solution of (1) is 𝑖

4. Numerical Examples

0

if πœŒπ‘˜ = (𝐹, π‘Ÿπ‘˜ )𝐸 , then

π‘š

Remark 6. If π‘˜π‘–π‘— (𝑠, 𝑑) ∈ 𝐢([0, 1] Γ— [0, 1]) and 𝑔𝑖 ∈ π‘Š22 in (1), then it is reasonable to regard the unknown functions as the elements of π‘Š22 .

𝑖

𝐹 (π‘₯) = βˆ‘(𝐹, π‘Ÿπ‘– )𝐸 π‘Ÿπ‘– (π‘₯) = βˆ‘ βˆ‘ π›½π‘–π‘˜ (𝐹, π‘Ÿπ‘˜ )𝐸 π‘Ÿπ‘– (π‘₯) ;

∞

βˆ€π‘₯ ∈ [0, 1] .

It shows that 𝑓𝑖,π‘š converges uniformly to 𝑓𝑖 on [0, 1] as π‘š β†’ ∞ for every 𝑖 = 1, 2, . . . , 𝑛. So the proof is complete.

𝑠

Proof. Assume that 𝐹(π‘₯) is the exact solution of (1). 𝐹(π‘₯) can be expanded to Fourier series in terms of normal orthogonal basis {π‘Ÿπ‘– (π‘₯)}∞ 𝑖=1 in 𝐸: ∞

σ΅„© σ΅„© ≀ √2󡄩󡄩󡄩𝑓𝑖,π‘š βˆ’ 𝑓𝑖 σ΅„©σ΅„©σ΅„©π‘Š1 , 2

Example 7. Consider the following system of Volterra integral equations of the second kind [11]:

where πœŒπ‘˜ = (𝐹(π‘₯), π‘Ÿπ‘˜ )𝐸 ; if π‘Ÿπ‘˜ = Ψ𝑗𝑙 , then πœŒπ‘˜ = 𝑔𝑙 (π‘₯𝑗 ).

∞

(23)

2

Since (16) has a unique solution, it follows that 𝑒 = 𝑒1 , 𝑒2 , . . . , 𝑒𝑛 𝑇 = 0. This completes the proof.

π‘Ÿπ‘– = βˆ‘ π›½π‘–π‘˜ π‘Ÿπ‘˜ ,

󡄨󡄨 󡄨 󡄨󡄨 󡄨󡄨 󡄨󡄨𝑓𝑖,π‘š βˆ’ 𝑓𝑖 󡄨󡄨󡄨 = 󡄨󡄨󡄨(𝑓𝑖,π‘š (𝑦) βˆ’ 𝑓𝑖 (𝑦) , 𝑅π‘₯ (𝑦))π‘Š1 󡄨󡄨󡄨 2 󡄨 󡄨 σ΅„© σ΅„©σ΅„© σ΅„©σ΅„© σ΅„©σ΅„© σ΅„© ≀ 󡄩󡄩𝑓𝑖,π‘š βˆ’ 𝑓𝑖 σ΅„©σ΅„©π‘Š1 β‹… 󡄩󡄩𝑅π‘₯ (𝑦)σ΅„©σ΅„©π‘Š1 2 2

𝑠

+ ∫ (1 βˆ’ 𝑑 cos 𝑠) 𝑓2 (𝑑) 𝑑𝑑, 𝑇

πΉπ‘š (π‘₯) = βˆ‘ βˆ‘ π›½π‘–π‘˜ πœŒπ‘˜ π‘Ÿπ‘– (π‘₯) = (𝑓1,π‘š , 𝑓2,π‘š , . . . , 𝑓𝑛,π‘š ) ,

0

(22)

𝑖=1 π‘˜=1

and 𝑓𝑖,π‘š (π‘₯) converges uniformly to 𝑓𝑖 (π‘₯) on [0, 1] as π‘š β†’ ∞ for every 𝑖 = 1, 2, . . . , 𝑛. Proof. Obviously, β€–πΉπ‘š βˆ’ 𝐹‖2𝐸 β†’ 0 holds as π‘š β†’ ∞; that is, πΉπ‘š (π‘₯) is the approximate solution of (1).

𝑠

𝑠

0

0

(25)

𝑓2 (𝑠) = 𝑔2 (𝑠) + ∫ (𝑓1 (𝑑)) 𝑑𝑑 + ∫ (𝑠 βˆ’ 𝑑) 𝑓2 (𝑑) 𝑑𝑑, where 𝑔1 (𝑠) and 𝑔2 (𝑠) are chosen such that the exact solution is 𝑓1 (𝑠) = cos 𝑠, 𝑓2 (𝑠) = sin 𝑠. The numerical results obtained by using the present method are compared with [11] in Table 2.

4

Abstract and Applied Analysis Table 1: Absolute errors for Example 7.

Nodes π‘₯𝑖 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Errors 𝑒(𝑓1 ) [11] 0 2.63472𝐸 βˆ’ 7 1.62592𝐸 βˆ’ 5 1.74905𝐸 βˆ’ 4 8.93799𝐸 βˆ’ 4 3.00491𝐸 βˆ’ 3 7.47528𝐸 βˆ’ 3 1.40733𝐸 βˆ’ 2 1.78384𝐸 βˆ’ 2 4.97756𝐸 βˆ’ 3 3.84378𝐸 βˆ’ 2

Errors 𝑒(𝑓1,100 ) 1.58309𝐸 βˆ’ 10 3.92220𝐸 βˆ’ 12 3.21563𝐸 βˆ’ 10 5.95890𝐸 βˆ’ 10 5.11051𝐸 βˆ’ 10 2.46104𝐸 βˆ’ 10 1.98685𝐸 βˆ’ 9 5.01512𝐸 βˆ’ 9 9.62848𝐸 βˆ’ 9 1.61180𝐸 βˆ’ 8 2.49043𝐸 βˆ’ 8

Errors 𝑒(𝑓2 ) [11] 0 2.11685𝐸 βˆ’ 8 2.61132𝐸 βˆ’ 6 4.18979𝐸 βˆ’ 5 2.86285𝐸 βˆ’ 4 1.19940𝐸 βˆ’ 3 3.56141𝐸 βˆ’ 3 7.74239𝐸 βˆ’ 3 1.09171𝐸 βˆ’ 2 2.27326𝐸 βˆ’ 3 3.32111𝐸 βˆ’ 2

Errors 𝑒(𝑓2,100 ) [11] 3.98245𝐸 βˆ’ 10 3.94493𝐸 βˆ’ 10 4.72710𝐸 βˆ’ 10 6.19366𝐸 βˆ’ 10 6.95422𝐸 βˆ’ 10 4.19959𝐸 βˆ’ 10 5.90035𝐸 βˆ’ 10 2.83080𝐸 βˆ’ 9 6.94058𝐸 βˆ’ 9 1.36984𝐸 βˆ’ 8 2.42565𝐸 βˆ’ 8

Table 2: Absolute errors for Example 8. Nodes π‘₯𝑖 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Errors 𝑒(𝑓1 ) [11] 0 1.37735𝐸 βˆ’ 4 9.27188𝐸 βˆ’ 4 2.67117𝐸 βˆ’ 3 5.45507𝐸 βˆ’ 3 9.22670𝐸 βˆ’ 3 1.38644𝐸 βˆ’ 2 1.92960𝐸 βˆ’ 2 2.56349𝐸 βˆ’ 2 3.31574𝐸 βˆ’ 2 4.19808𝐸 βˆ’ 2

Errors 𝑒(𝑓1,100 ) 6.93348𝐸 βˆ’ 11 4.53518𝐸 βˆ’ 09 8.84879𝐸 βˆ’ 09 1.28253𝐸 βˆ’ 08 1.65442𝐸 βˆ’ 08 2.00881𝐸 βˆ’ 08 2.35657𝐸 βˆ’ 09 2.71160𝐸 βˆ’ 08 3.09302𝐸 βˆ’ 08 3.52645𝐸 βˆ’ 08 3.67322𝐸 βˆ’ 08

Errors 𝑒(𝑓2 ) [11] 0 1.52721𝐸 βˆ’ 4 1.14715𝐸 βˆ’ 3 3.71248𝐸 βˆ’ 3 8.57201𝐸 βˆ’ 3 1.64412𝐸 βˆ’ 2 2.78243𝐸 βˆ’ 2 4.25337𝐸 βˆ’ 2 5.91212𝐸 βˆ’ 2 7.48883𝐸 βˆ’ 2 8.70896𝐸 βˆ’ 2

5. Conclusion In this paper, we modify the traditional reproducing kernel method to enlarge its application range. The new method named MRKM is applied successfully to solve a system of linear Volterra integral equations. The numerical results show that our method is effective. It is worth to be pointed out that the MRKM is still suitable for solving other systems of linear equations.

[4]

[5]

[6]

Acknowledgments The research was supported by the Fundamental Research Funds for the Central Universities.

References [1] Y. Ren, B. Zhang, and H. Qiao, β€œA simple Taylor-series expansion method for a class of second kind integral equations,” Journal of Computational and Applied Mathematics, vol. 110, no. 1, pp. 15–24, 1999. [2] K. Maleknejad and Y. Mahmoudi, β€œNumerical solution of linear Fredholm integral equation by using hybrid Taylor and blockpulse functions,” Applied Mathematics and Computation, vol. 149, no. 3, pp. 799–806, 2004. [3] S. YalcΒΈΔ±nbasΒΈ and M. Sezer, β€œThe approximate solution of highorder linear Volterra-Fredholm integro-differential equations

[7]

[8]

[9]

[10]

Errors 𝑒(𝑓2,100 ) [11] 3.60316𝐸 βˆ’ 11 2.75123𝐸 βˆ’ 08 3.10611𝐸 βˆ’ 08 3.53307𝐸 βˆ’ 08 4.03402𝐸 βˆ’ 08 4.61209𝐸 βˆ’ 08 5.27214𝐸 βˆ’ 08 6.02041𝐸 βˆ’ 08 6.86601𝐸 βˆ’ 08 7.82029𝐸 βˆ’ 08 1.02387𝐸 βˆ’ 07

in terms of Taylor polynomials,” Applied Mathematics and Computation, vol. 112, no. 2-3, pp. 291–308, 2000. S. YalcΒΈinbasΒΈ, β€œTaylor polynomial solutions of nonlinear Volterra-Fredholm integral equations,” Applied Mathematics and Computation, vol. 127, no. 2-3, pp. 195–206, 2002. E. Deeba, S. A. Khuri, and S. Xie, β€œAn algorithm for solving a nonlinear integro-differential equation,” Applied Mathematics and Computation, vol. 115, no. 2-3, pp. 123–131, 2000. A. Tahmasbi and O. S. Fard, β€œNumerical solution of linear Volterra integral equations system of the second kind,” Applied Mathematics and Computation, vol. 201, no. 1-2, pp. 547–552, 2008. E. Babolian, J. Biazar, and A. R. Vahidi, β€œOn the decomposition method for system of linear equations and system of linear Volterra integral equations,” Applied Mathematics and Computation, vol. 147, no. 1, pp. 19–27, 2004. J. Biazar and H. Ghazvini, β€œHe’s homotopy perturbation method for solving systems of Volterra integral equations of the second kind,” Chaos, Solitons & Fractals, vol. 39, no. 2, pp. 770–777, 2009. E. Yusufo˘glu (Agadjanov), β€œA homotopy perturbation algorithm to solve a system of Fredholm-Volterra type integral equations,” Mathematical and Computer Modelling, vol. 47, no. 11-12, pp. 1099–1107, 2008. R. Katani and S. Shahmorad, β€œBlock by block method for the systems of nonlinear Volterra integral equations,” Applied Mathematical Modelling, vol. 34, no. 2, pp. 400–406, 2010.

Abstract and Applied Analysis [11] M. Rabbani, K. Maleknejad, and N. Aghazadeh, β€œNumerical computational solution of the Volterra integral equations system of the second kind by using an expansion method,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 1143–1146, 2007. [12] M. G. Cui and Z. X. Deng, β€œOn the best operator of interpolation in π‘Š21 [π‘Ž, 𝑏],” Mathematica Numerica Sinica, vol. 2, pp. 209–216, 1986. [13] M. Cui and F. Geng, β€œA computational method for solving onedimensional variable-coefficient Burgers equation,” Applied Mathematics and Computation, vol. 188, no. 2, pp. 1389–1401, 2007. [14] M. Cui and F. Geng, β€œA computational method for solving thirdorder singularly perturbed boundary-value problems,” Applied Mathematics and Computation, vol. 198, no. 2, pp. 896–903, 2008. [15] M. Cui and Z. Chen, β€œThe exact solution of nonlinear agestructured population model,” Nonlinear Analysis. Real World Applications, vol. 8, no. 4, pp. 1096–1112, 2007. [16] F. Z. Geng and M. G. Cui, β€œSolving singular nonlinear two-point boundary value problems in the reproducing kernel space,” Journal of the Korean Mathematical Society, vol. 45, no. 3, pp. 631–644, 2008. [17] M. G. Cui and Y. B. Yan, β€œThe representation of the solution of a kind operator equation 𝐴𝑒 = 𝑓,” Journal of Chinese Universities, vol. 3, pp. 82–86, 1995. [18] O. Abu Arqub, M. Al-Smadi, and S. Momani, β€œApplication of reproducing kernel method for solving nonlinear FredholmVolterra integrodifferential equations,” Abstract and Applied Analysis, vol. 2012, Article ID 839836, 16 pages, 2012. [19] O. A. Arqub, M. Al-Smadi, and N. Shawagfeh, β€œSolving Fredholm integro-differential equations using reproducing kernel Hilbert space method,” Applied Mathematics and Computation, vol. 219, no. 17, pp. 8938–8948, 2013. [20] M. Al-Smadi, O. Abu Arqub, and S. Momani, β€œA computational method for two-point boundary value problems of fourth-order mixed integrodifferential equations,” Mathematical Problems in Engineering, vol. 2013, Article ID 832074, 10 pages, 2013. [21] M. Al-Smadi, O. Abu Arqub, and N. Shawagfeh, β€œApproximate solution of BVPs for 4th-order IDEs by using RKHS method,” Applied Mathematical Sciences, vol. 6, no. 49–52, pp. 2453–2464, 2012. [22] Z. Chen and Y. Lin, β€œThe exact solution of a linear integral equation with weakly singular kernel,” Journal of Mathematical Analysis and Applications, vol. 344, no. 2, pp. 726–734, 2008. [23] M. Cui and Y. Lin, Nonlinear Numerical Analysis in the Reproducing Kernel Space, Nova Science Publishers, New York, NY, USA, 2009.

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