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 π΄π’ = π, is bounded and its image space is the reproducing kernel space π21 [π, π]. 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 π
πΉ (π ) = πΊπ + β« πΎ (π , π‘) πΉ (π‘) ππ‘, π
π β [0, 1] ,
(1)
where π
πΉ (π ) = [π1 (π ) , π2 (π ) , . . . , ππ (π )] , π
πΊ (π ) = [π1 (π ) , π2 (π ) , . . . , ππ (π )] , πΎ (π , π‘) = [ππ,π ] ,
(2)
π, π = 1, 2, . . . , π.
In (1), the functions πΎ and πΊ are given, and πΉ 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 [1β5]. Several valid methods for solving Volterra integral equation have been developed in recent years, including power series method [6], Adomainβs decomposition method [7], homotopy perturbation method [8, 9], block by block method [10], and expansion method [11].
Since the reproducing kernel space π21 [π, π], 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 [12β21]. The traditional reproducing kernel method is limited, because it requires that the image space of operator π΄ in linear operator equation π΄π’ = π is π21 [π, π] and operator π΄ must be bounded. In order to enlarge its application range, the MRKM removes the boundedness of π΄ and weakens its image space to πΏ2 [π, π]. Subsequently, we apply the MRKM to obtain the series expression of the exact solution for (1). The π-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 π21 [0, 1]. The reproducing kernel space π21 [0, 1] consists of all absolute continuous realvalued functions, which defined on the closed interval [0, 1], and the first derivative functions belong to πΏ2 [0, 1].
2
Abstract and Applied Analysis
3. The Exact Solution of (1)
The inner product and the norm are equipped with 1
(π’, V)π€21 = π’ (0) V (0) + β« π’σΈ (π₯) VσΈ (π₯) ππ₯, 0
βπ’, V β π€21 ,
3.1. Identical Transformation of (1). Consider the ith equation of (1): π
βπ’βπ21 = β(π’, V)π€21 .
π=1 0
(3) π21 [0, 1]
Theorem 1. ducing kernel [22]
is a reproducing kernel space with repro-
1 + π¦, π
π₯ (π¦) = { 1 + π₯,
π¦β€π₯ π¦ > π₯;
(4)
2
(11)
Define operator π΄ ππ : π21 β πΏ2 [0, 1], π = 1, . . . , π, 1
{ {π’ (π ) β β«0 πππ (π , π‘) π’ (π‘) ππ‘, π΄ ππ = { π {β β« πππ (π , π‘) π’ (π‘) ππ‘, { 0
π=π π =ΜΈ π,
(12)
where π’ β π21 . Then, (1) can be turned into
that is, for every π₯ β [0, 1] and π’ β π21 , it follows that (π’ (π¦) , π
π₯ (π¦))π€1 = π’ (π₯) .
π
ππ (π ) β β β« πΎππ (π , π‘) ππ (π‘) ππ‘ = ππ (π ) .
π΄ 11 π1 + π΄ 12 π2 + β
β
β
+ π΄ 1π π1π = π1 (π )
(5)
π΄ 21 π1 + π΄ 22 π2 + β
β
β
+ π΄ 2π π1π = π2 (π )
2.2. The Reproducing Kernel Space π22 [0, 1]. The reproducing kernel space π22 [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 πΏ2 [0, 1]. The inner product and the norm are equipped with 1
.. .
(13)
π΄ π1 π1 + π΄ π2 π2 + β
β
β
+ π΄ ππ π1π = ππ (π ) , where πΉ(π ) = [π1 (π ), π2 (π ), . . . , ππ (π )]π β πΈ.
(6)
3.2. The Exact solution of (1). Let {π₯π }β π=1 be a dense subset of interval [0, 1], and define σ΅¨ σ΅¨ Ξ¨ππ (π₯) = ( π΄ π1 ,π¦ π
π₯ (π¦)σ΅¨σ΅¨σ΅¨σ΅¨π¦=π₯ , π΄ π2 ,π¦ π
π₯ (π¦)σ΅¨σ΅¨σ΅¨σ΅¨π¦=π₯ , . . . , π π (14) π σ΅¨σ΅¨ π΄ ππ ,π¦ π
π₯ (π¦)σ΅¨σ΅¨σ΅¨π¦=π₯ )
Theorem 2. π22 [0, 1] is a reproducing kernel space with reproducing kernel [22]
for every π = 1, 2, . . . , π, π = 1, 2, . . .; the subscript π¦ of π΄ ππ,π¦ means that the operator π΄ ππ acts on the function of π¦. It is easy to prove that Ξ¨ππ β πΈ.
(π’, V)π22 = β π’(π) (0) V(π) (0) π=0
1
+ β« π’σΈ σΈ (π₯) VσΈ σΈ (π₯) ππ₯, 0
βπ’, V β π22 [0, 1] ,
βπ’βπ22 = β(π’, π’)π€22 .
π
π₯ Γ π¦ 2 π¦3 { β π¦β€π₯ 1 + π₯ Γ π¦ + { { 2 6 π (π₯, π¦) = { { π₯2 Γ π¦ π₯3 { β , π¦ > π₯; 1+π₯Γπ¦+ { 2 6
(7)
that is, for every π₯ β [0, 1] and π’ β π22 , it follows that (π’ (π¦) , π (π₯, π¦))π€2 = π’ (π₯) . 2
(8)
The proof of Theorems 1 and 2 can be found in [23].
Theorem 3. {Ξ¨π1 , Ξ¨π2 , . . . , Ξ¨ππ }β π=1 is complete in πΈ. Proof. Take π’ = (π’1 , π’2 , . . . , π’π )π β πΈ such that (π’(π₯), Ξ¨ππ (π₯)) = 0 for every π = 1, 2, . . . , π, π = 1, 2, . . .. From this fact, it holds that (π’ (π₯) , Ξ¨ππ (π₯)) π
= ((π’1 , π’2 , . . . , π’π ) ,
2.3. Hilbert Space πΈ. Hilbert space πΈ is defined by π
π
πΈ =β¨ π21 = {(π’1 , . . . , π’π ) | π’π β π€21 , π = 1, . . . , π} .
σ΅¨ ( π΄ π1,π¦ π
π₯ (π¦)σ΅¨σ΅¨σ΅¨σ΅¨π¦=π₯ , π
(9)
π=1
(15) π σ΅¨σ΅¨ σ΅¨ π΄ π2,π¦ π
π₯ (π¦)σ΅¨σ΅¨σ΅¨σ΅¨π¦=π₯ , . . . , π΄ ππ,π¦ π
π₯ (π¦)σ΅¨σ΅¨σ΅¨ ) ) σ΅¨π¦=π₯π π
The inner product and the norm are given by π
π
(π’, V)πΈ = β(π’π , Vπ )π€1 , π=1
2
βπ’βπΈ = β(π’, π’)πΈ . It is easy to prove that πΈ is a Hilbert space.
(10)
σ΅¨σ΅¨ = β π΄ ππ,π¦ (π’π (π₯) , π
π₯ (π¦))π€1 σ΅¨σ΅¨σ΅¨ 2 σ΅¨π¦=π₯π π=1 π
= β π΄ ππ π’π (π₯π ) = 0, π=1
Abstract and Applied Analysis
3 Note that βππ=1 βππ,π β ππ β2π1 = βπΉπ β πΉβ2πΈ β 0. Com2 bining with the expression of π
π₯ (π¦), we have
for every π = 1, 2, . . . , π. The dense {π₯π }β π=1 assumes that π΄ 11 π’1 + π΄ 12 π’2 + β
β
β
+ π΄ 1π π’π = 0 π΄ 21 π’1 + π΄ 22 π’2 + β
β
β
+ π΄ 2π π’π = 0 (16)
.. .
σ΅© σ΅© = σ΅©σ΅©σ΅©ππ,π β ππ σ΅©σ΅©σ΅©π1 βπ
π₯ (π₯)
π΄ π1 π’1 + π΄ π2 π’2 + β
β
β
+ π΄ ππ π’π = 0.
We arrange Ξ¨11 , Ξ¨12 ,. . ., Ξ¨1π , Ξ¨21 , Ξ¨22 , . . ., Ξ¨2π , . . ., Ξ¨π1 , Ξ¨π2 , . . ., Ξ¨ππ ,. . ., denoted by {ππ }β π=1 ; that is, π1 = Ξ¨11 , π2 = Ξ¨12 , . . . , ππ = Ξ¨1π , ππ+1 = Ξ¨21 , ππ+2 = Ξ¨22 , . . . , ππ+π = Ξ¨2π , . . .. In a general way, π(πβ1)π+π = Ξ¨ππ , π = 1, 2, 3, . . . ; π = 1, 2,. . . ,π. The orthogonal basis {ππ }β π=1 in πΈ from Gram-Schmidt is as follows: orthogonalization of {ππ }β π=1 π
π = 1, 2, . . . .
(17)
π=1
Theorem 4. The exact solution of (1) can be expressed by β
π
πΉ (π₯) = β β π½ππ ππ ππ (π₯) ,
(18)
π=1 π=1
π=1
π=1 π=1
Taking nodes {π₯π = (π β 1)/(π β 1)}π π=1 , ππ,π is the approximate solutions of ππ , and π(ππ,π) denotes the absolute errors of ππ , π = 1, 2, . . . , π. According to Remark 6, we solve the following two examples appearing in [11] in π22 .
π
π1 (π ) = π1 (π ) + β« (π β π‘)3 π1 (π‘) ππ‘ + β« (π β π‘)2 π2 (π‘) ππ‘, 0
0
π
π2 (π ) = π2 (π ) + β« (π β π‘)4 π1 (π‘) ππ‘ π
(19)
+ β« (π β π‘)3 π2 (π‘) ππ‘, 0
π
πΉ (π₯) = β β π½ππ ππ ππ (π₯) .
(20)
π=1 π=1
When ππ = Ξ¨ππ , it holds that
(24)
where π1 (π ) and π2 (π ) are chosen such that the exact solution is π1 (π ) = 1 + π 2 , π2 (π ) = 1 + π β π 3 . 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]:
π
ππ = (πΉ, Ξ¨ππ ) = β π΄ ππ π’π (π₯π ) = ππ (π₯π ) .
(21)
π=1
π
π1 (π ) = π1 (π ) + β« (sin (π β π‘) β 1) π1 (π‘) ππ‘ 0
Corollary 5. The approximate solution of (1) is π
4. Numerical Examples
0
if ππ = (πΉ, ππ )πΈ , then
π
Remark 6. If πππ (π , π‘) β πΆ([0, 1] Γ [0, 1]) and ππ β π22 in (1), then it is reasonable to regard the unknown functions as the elements of π22 .
π
πΉ (π₯) = β(πΉ, ππ )πΈ ππ (π₯) = β β π½ππ (πΉ, ππ )πΈ ππ (π₯) ;
β
βπ₯ β [0, 1] .
It shows that ππ,π converges uniformly to ππ on [0, 1] as π β β for every π = 1, 2, . . . , π. So the proof is complete.
π
Proof. Assume that πΉ(π₯) is the exact solution of (1). πΉ(π₯) can be expanded to Fourier series in terms of normal orthogonal basis {ππ (π₯)}β π=1 in πΈ: β
σ΅© σ΅© β€ β2σ΅©σ΅©σ΅©ππ,π β ππ σ΅©σ΅©σ΅©π1 , 2
Example 7. Consider the following system of Volterra integral equations of the second kind [11]:
where ππ = (πΉ(π₯), ππ )πΈ ; if ππ = Ξ¨ππ , then ππ = ππ (π₯π ).
β
(23)
2
Since (16) has a unique solution, it follows that π’ = π’1 , π’2 , . . . , π’π π = 0. This completes the proof.
ππ = β π½ππ ππ ,
σ΅¨σ΅¨ σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ππ,π β ππ σ΅¨σ΅¨σ΅¨ = σ΅¨σ΅¨σ΅¨(ππ,π (π¦) β ππ (π¦) , π
π₯ (π¦))π1 σ΅¨σ΅¨σ΅¨ 2 σ΅¨ σ΅¨ σ΅© σ΅©σ΅© σ΅©σ΅© σ΅©σ΅© σ΅© β€ σ΅©σ΅©ππ,π β ππ σ΅©σ΅©π1 β
σ΅©σ΅©π
π₯ (π¦)σ΅©σ΅©π1 2 2
π
+ β« (1 β π‘ cos π ) π2 (π‘) ππ‘, π
πΉπ (π₯) = β β π½ππ ππ ππ (π₯) = (π1,π , π2,π , . . . , ππ,π ) ,
0
(22)
π=1 π=1
and ππ,π (π₯) converges uniformly to ππ (π₯) on [0, 1] as π β β for every π = 1, 2, . . . , π. Proof. Obviously, βπΉπ β πΉβ2πΈ β 0 holds as π β β; that is, πΉπ (π₯) is the approximate solution of (1).
π
π
0
0
(25)
π2 (π ) = π2 (π ) + β« (π1 (π‘)) ππ‘ + β« (π β π‘) π2 (π‘) ππ‘, where π1 (π ) and π2 (π ) are chosen such that the exact solution is π1 (π ) = cos π , π2 (π ) = sin π . 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₯π 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Errors π(π1 ) [11] 0 2.63472πΈ β 7 1.62592πΈ β 5 1.74905πΈ β 4 8.93799πΈ β 4 3.00491πΈ β 3 7.47528πΈ β 3 1.40733πΈ β 2 1.78384πΈ β 2 4.97756πΈ β 3 3.84378πΈ β 2
Errors π(π1,100 ) 1.58309πΈ β 10 3.92220πΈ β 12 3.21563πΈ β 10 5.95890πΈ β 10 5.11051πΈ β 10 2.46104πΈ β 10 1.98685πΈ β 9 5.01512πΈ β 9 9.62848πΈ β 9 1.61180πΈ β 8 2.49043πΈ β 8
Errors π(π2 ) [11] 0 2.11685πΈ β 8 2.61132πΈ β 6 4.18979πΈ β 5 2.86285πΈ β 4 1.19940πΈ β 3 3.56141πΈ β 3 7.74239πΈ β 3 1.09171πΈ β 2 2.27326πΈ β 3 3.32111πΈ β 2
Errors π(π2,100 ) [11] 3.98245πΈ β 10 3.94493πΈ β 10 4.72710πΈ β 10 6.19366πΈ β 10 6.95422πΈ β 10 4.19959πΈ β 10 5.90035πΈ β 10 2.83080πΈ β 9 6.94058πΈ β 9 1.36984πΈ β 8 2.42565πΈ β 8
Table 2: Absolute errors for Example 8. Nodes π₯π 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Errors π(π1 ) [11] 0 1.37735πΈ β 4 9.27188πΈ β 4 2.67117πΈ β 3 5.45507πΈ β 3 9.22670πΈ β 3 1.38644πΈ β 2 1.92960πΈ β 2 2.56349πΈ β 2 3.31574πΈ β 2 4.19808πΈ β 2
Errors π(π1,100 ) 6.93348πΈ β 11 4.53518πΈ β 09 8.84879πΈ β 09 1.28253πΈ β 08 1.65442πΈ β 08 2.00881πΈ β 08 2.35657πΈ β 09 2.71160πΈ β 08 3.09302πΈ β 08 3.52645πΈ β 08 3.67322πΈ β 08
Errors π(π2 ) [11] 0 1.52721πΈ β 4 1.14715πΈ β 3 3.71248πΈ β 3 8.57201πΈ β 3 1.64412πΈ β 2 2.78243πΈ β 2 4.25337πΈ β 2 5.91212πΈ β 2 7.48883πΈ β 2 8.70896πΈ β 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.
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