# Solving a System of Linear Volterra Integral Equations Using the Modified Reproducing Kernel Method

Authors: Li-Hong Yang; Hong-Ying Li; Jing-Ran Wang
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

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 equationAu=f, is bounded and its image space is the reproducing kernel space W21[a,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.


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## Body

## 1. Introduction

The purpose of this paper is to solve a system of linear Volterra integral equations(1)F(s)=Gs+∫ab‍K(s,t)F(t)dt,s∈[0,1],
where
(2)F(s)=[f1(s),f2(s),…,fn(s)]T,G(s)=[g1(s),g2(s),…,gn(s)]T,K(s,t)=[ki,j],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 spaceW21[a,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 Au=f is W21[a,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[a,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 SpaceW21[0,1]

The reproducing kernel spaceW21[0,1] consists of all absolute continuous real-valued functions, which defined on the closed interval [0,1], and the first derivative functions belong to L2[0,1].The inner product and the norm are equipped with(3)(u,v)w21=u(0)v(0)+∫01‍u′(x)v′(x)dx,∀u,v∈w21,∥u∥W21=(u,v)w21.Theorem 1.
W
2
1
[
0
,
1
]is a reproducing kernel space with reproducing kernel [22]
(4)Rx(y)={1+y,y≤x1+x,y>x;
that is, for every x∈[0,1] and u∈W21, it follows that
(5)(u(y),Rx(y))w21=u(x).


### 2.2. The Reproducing Kernel SpaceW22[0,1]

The reproducing kernel spaceW22[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(6)(u,v)W22=∑k=01‍u(k)(0)v(k)(0)+∫01‍u′′(x)v′′(x)dx,∀u,v∈W22[0,1],∥u∥W22=(u,u)w22.Theorem 2.
W
2
2
[
0
,
1
]is a reproducing kernel space with reproducing kernel [22]
(7)Q(x,y)={1+x×y+x×y22-y36y≤x1+x×y+x2×y2-x36,y>x;
that is, for every x∈[0,1] and u∈W22, it follows that
(8)(u(y),Q(x,y))w22=u(x).The proof of Theorems1 and 2 can be found in [23].


### 2.3.  Hilbert SpaceE

Hilbert spaceE is defined by
(9)E=⨁i=1nW21={(u1,…,un)T∣ui∈w21,i=1,…,n}.
The inner product and the norm are given by
(10)(u,v)E=∑i=1n‍(ui,vi)w21,∥u∥E=(u,u)E.It is easy to prove thatE is a Hilbert space.


## 2.1. The Reproducing Kernel SpaceW21[0,1]

The reproducing kernel spaceW21[0,1] consists of all absolute continuous real-valued functions, which defined on the closed interval [0,1], and the first derivative functions belong to L2[0,1].The inner product and the norm are equipped with(3)(u,v)w21=u(0)v(0)+∫01‍u′(x)v′(x)dx,∀u,v∈w21,∥u∥W21=(u,v)w21.Theorem 1.
W
2
1
[
0
,
1
]is a reproducing kernel space with reproducing kernel [22]
(4)Rx(y)={1+y,y≤x1+x,y>x;
that is, for every x∈[0,1] and u∈W21, it follows that
(5)(u(y),Rx(y))w21=u(x).


## 2.2. The Reproducing Kernel SpaceW22[0,1]

The reproducing kernel spaceW22[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(6)(u,v)W22=∑k=01‍u(k)(0)v(k)(0)+∫01‍u′′(x)v′′(x)dx,∀u,v∈W22[0,1],∥u∥W22=(u,u)w22.Theorem 2.
W
2
2
[
0
,
1
]is a reproducing kernel space with reproducing kernel [22]
(7)Q(x,y)={1+x×y+x×y22-y36y≤x1+x×y+x2×y2-x36,y>x;
that is, for every x∈[0,1] and u∈W22, it follows that
(8)(u(y),Q(x,y))w22=u(x).The proof of Theorems1 and 2 can be found in [23].


## 2.3.  Hilbert SpaceE

Hilbert spaceE is defined by
(9)E=⨁i=1nW21={(u1,…,un)T∣ui∈w21,i=1,…,n}.
The inner product and the norm are given by
(10)(u,v)E=∑i=1n‍(ui,vi)w21,∥u∥E=(u,u)E.It is easy to prove thatE is a Hilbert space.


## 3. The Exact Solution of (1)

### 3.1. Identical Transformation of (1)

Consider theith equation of (1):
(11)fi(s)-∑j=1n‍∫0s‍Kij(s,t)fj(t)dt=gi(s).Define operatorAij:W21→L2[0,1],j=1,…,n,
(12)Aij={u(s)-∫01‍kij(s,t)u(t)dt,j=i-∫0s‍kij(s,t)u(t)dt,j≠i,
where u∈W21. Then, (1) can be turned into
(13)A11f1+A12f2+⋯+A1nf1n=g1(s)A21f1+A22f2+⋯+A2nf1n=g2(s)⋮An1f1+An2f2+⋯+Annf1n=gn(s),
where F(s)=[f1(s),f2(s),…,fn(s)]T∈E.


### 3.2.  The Exact solution of (1)

Let{xi}i=1∞ be a dense subset of interval [0,1], and define
(14)Ψij(x)=(Aj1,yRx(y)|y=xi,Aj2,yRx(y)|y=xi,…,Ajn,yRx(y)|y=xi)T
for every j=1,2,…,n,i=1,2,…; the subscript y of Aij,y means that the operator Aij acts on the function of y. It is easy to prove that Ψij∈E.Theorem 3.
{
Ψ
i
1
,
Ψ
i
2
,
…
,
Ψ
i
n
}
i
=
1
∞ is complete in E.Proof.
Takeu=(u1,u2,…,un)T∈E such that (u(x),Ψij(x))=0 for every j=1,2,…,n,i=1,2,….
From this fact, it holds that(15)(u(x),Ψij(x))=(Ajn,yRx(y)|y=xi)T(Aj1,yRx(y)|y=xi,Aj2,yRx(y)|y=xi,…,Ajn,yRx(y)|y=xi)T(u1,u2,…,un)T,(Aj2,yRx(y)|y=xi,…,Ajn,yRx(y)|y=xiAj1,yRx(y)|y=xi,Aj2,yRx(y)|y=xi,…,Ajn,yRx(y)|y=xi)T)=∑k=1n‍Ajk,y(uk(x),Rx(y))w21|y=xi=∑k=1n‍Ajkuk(xi)=0,
for everyj=1,2,…,n. The dense {xi}i=1∞ assumes that
(16)A11u1+A12u2+⋯+A1nun=0A21u1+A22u2+⋯+A2nun=0⋮An1u1+An2u2+⋯+Annun=0.
Since (16) has a unique solution, it follows that u=u1,u2,…,unT=0. This completes the proof.We arrangeΨ11, Ψ12,…, Ψ1n, Ψ21, Ψ22, …, Ψ2n, …, Ψi1,  Ψi2,…,    Ψin,…, denoted by {ri}i=1∞; that is, r1=Ψ11,r2=Ψ12,…,rn=Ψ1n,rn+1=Ψ21,rn+2=Ψ22,…,rn+n=Ψ2n,…. In a general way, r(i-1)n+j=Ψij,i=1,2,3,…;j=1,2,…,n. The orthogonal basis {r¯i}i=1∞ in E from Gram-Schmidt orthogonalization of {ri}i=1∞ is as follows:
(17)r¯i=∑k=1i‍βikrk,i=1,2,….Theorem 4.
The exact solution of (1) can be expressed by
(18)F(x)=∑i=1∞‍∑k=1i‍βikρkr¯i(x),
where ρk=(F(x),rk)E; if rk=Ψjl, then ρk=gl(xj).Proof.
Assume thatF(x) is the exact solution of (1). F(x) can be expanded to Fourier series in terms of normal orthogonal basis {r¯i(x)}i=1∞ in E:
(19)F(x)=∑i=1∞‍(F,r¯i)Er¯i(x)=∑i=1∞‍∑k=1i‍βik(F,rk)Er¯i(x);
if ρk=(F,rk)E, then
(20)F(x)=∑i=1∞‍∑k=1i‍βijρkr¯i(x).
When rk=Ψjl, it holds that
(21)ρk=(F,Ψjl)=∑k=1n‍Alkuk(xj)=gl(xj).Corollary 5.
The approximate solution of (1) is
(22)Fm(x)=∑i=1m‍∑k=1i‍βikρkr¯i(x)=(f1,m,f2,m,…,fn,m)T,
and fi,m(x) converges uniformly to fi(x) on [0,1] as m→∞ for every i=1,2,…,n.Proof.
Obviously,∥Fm-F∥E2→0 holds as m→∞; that is, Fm(x) is the approximate solution of (1).
Note that∑i=1n‍∥fi,m-fi∥W212=∥Fm-F∥E2→0. Combining with the expression of Rx(y), we have
(23)|fi,m-fi|=|(fi,m(y)-fi(y),Rx(y))W21|≤∥fi,m-fi∥W21·∥Rx(y)∥W21=∥fi,m-fi∥W21Rx(x)≤2∥fi,m-fi∥W21,∀x∈[0,1].
It shows that fi,m converges uniformly to fi on [0,1] as m→∞ for every i=1,2,…,n. So the proof is complete.Remark 6.
Ifkij(s,t)∈C([0,1]×[0,1]) and gi∈W22 in (1), then it is reasonable to regard the unknown functions as the elements of W22.


## 3.1. Identical Transformation of (1)

Consider theith equation of (1):
(11)fi(s)-∑j=1n‍∫0s‍Kij(s,t)fj(t)dt=gi(s).Define operatorAij:W21→L2[0,1],j=1,…,n,
(12)Aij={u(s)-∫01‍kij(s,t)u(t)dt,j=i-∫0s‍kij(s,t)u(t)dt,j≠i,
where u∈W21. Then, (1) can be turned into
(13)A11f1+A12f2+⋯+A1nf1n=g1(s)A21f1+A22f2+⋯+A2nf1n=g2(s)⋮An1f1+An2f2+⋯+Annf1n=gn(s),
where F(s)=[f1(s),f2(s),…,fn(s)]T∈E.


## 3.2.  The Exact solution of (1)

Let{xi}i=1∞ be a dense subset of interval [0,1], and define
(14)Ψij(x)=(Aj1,yRx(y)|y=xi,Aj2,yRx(y)|y=xi,…,Ajn,yRx(y)|y=xi)T
for every j=1,2,…,n,i=1,2,…; the subscript y of Aij,y means that the operator Aij acts on the function of y. It is easy to prove that Ψij∈E.Theorem 3.
{
Ψ
i
1
,
Ψ
i
2
,
…
,
Ψ
i
n
}
i
=
1
∞ is complete in E.Proof.
Takeu=(u1,u2,…,un)T∈E such that (u(x),Ψij(x))=0 for every j=1,2,…,n,i=1,2,….
From this fact, it holds that(15)(u(x),Ψij(x))=(Ajn,yRx(y)|y=xi)T(Aj1,yRx(y)|y=xi,Aj2,yRx(y)|y=xi,…,Ajn,yRx(y)|y=xi)T(u1,u2,…,un)T,(Aj2,yRx(y)|y=xi,…,Ajn,yRx(y)|y=xiAj1,yRx(y)|y=xi,Aj2,yRx(y)|y=xi,…,Ajn,yRx(y)|y=xi)T)=∑k=1n‍Ajk,y(uk(x),Rx(y))w21|y=xi=∑k=1n‍Ajkuk(xi)=0,
for everyj=1,2,…,n. The dense {xi}i=1∞ assumes that
(16)A11u1+A12u2+⋯+A1nun=0A21u1+A22u2+⋯+A2nun=0⋮An1u1+An2u2+⋯+Annun=0.
Since (16) has a unique solution, it follows that u=u1,u2,…,unT=0. This completes the proof.We arrangeΨ11, Ψ12,…, Ψ1n, Ψ21, Ψ22, …, Ψ2n, …, Ψi1,  Ψi2,…,    Ψin,…, denoted by {ri}i=1∞; that is, r1=Ψ11,r2=Ψ12,…,rn=Ψ1n,rn+1=Ψ21,rn+2=Ψ22,…,rn+n=Ψ2n,…. In a general way, r(i-1)n+j=Ψij,i=1,2,3,…;j=1,2,…,n. The orthogonal basis {r¯i}i=1∞ in E from Gram-Schmidt orthogonalization of {ri}i=1∞ is as follows:
(17)r¯i=∑k=1i‍βikrk,i=1,2,….Theorem 4.
The exact solution of (1) can be expressed by
(18)F(x)=∑i=1∞‍∑k=1i‍βikρkr¯i(x),
where ρk=(F(x),rk)E; if rk=Ψjl, then ρk=gl(xj).Proof.
Assume thatF(x) is the exact solution of (1). F(x) can be expanded to Fourier series in terms of normal orthogonal basis {r¯i(x)}i=1∞ in E:
(19)F(x)=∑i=1∞‍(F,r¯i)Er¯i(x)=∑i=1∞‍∑k=1i‍βik(F,rk)Er¯i(x);
if ρk=(F,rk)E, then
(20)F(x)=∑i=1∞‍∑k=1i‍βijρkr¯i(x).
When rk=Ψjl, it holds that
(21)ρk=(F,Ψjl)=∑k=1n‍Alkuk(xj)=gl(xj).Corollary 5.
The approximate solution of (1) is
(22)Fm(x)=∑i=1m‍∑k=1i‍βikρkr¯i(x)=(f1,m,f2,m,…,fn,m)T,
and fi,m(x) converges uniformly to fi(x) on [0,1] as m→∞ for every i=1,2,…,n.Proof.
Obviously,∥Fm-F∥E2→0 holds as m→∞; that is, Fm(x) is the approximate solution of (1).
Note that∑i=1n‍∥fi,m-fi∥W212=∥Fm-F∥E2→0. Combining with the expression of Rx(y), we have
(23)|fi,m-fi|=|(fi,m(y)-fi(y),Rx(y))W21|≤∥fi,m-fi∥W21·∥Rx(y)∥W21=∥fi,m-fi∥W21Rx(x)≤2∥fi,m-fi∥W21,∀x∈[0,1].
It shows that fi,m converges uniformly to fi on [0,1] as m→∞ for every i=1,2,…,n. So the proof is complete.Remark 6.
Ifkij(s,t)∈C([0,1]×[0,1]) and gi∈W22 in (1), then it is reasonable to regard the unknown functions as the elements of W22.


## 4. Numerical Examples

Taking nodes{xi=(i-1)/(N-1)}i=1N, 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 W22.Example 7.
Consider the following system of Volterra integral equations of the second kind [11]:
(24)f1(s)=g1(s)+∫0s‍(s-t)3f1(t)dt+∫0s‍(s-t)2f2(t)dt,f2(s)=g2(s)+∫0s‍(s-t)4f1(t)dt+∫0s‍(s-t)3f2(t)dt,
where g1(s) and g2(s) are chosen such that the exact solution is f1(s)=1+s2,f2(s)=1+s-s3. The numerical results obtained by using the present method are compared with [11] in Table 1.Table 1
Absolute errors for Example7.
Nodesxi
Errorse(f1) [11]
Errorse(f1,100)
Errorse(f2) [11]
Errorse(f2,100) [11]
0.0
0
1.58309
E
-
10
0
3.98245
E
-
10
0.1
2.63472
E
-
7
3.92220
E
-
12
2.11685
E
-
8
3.94493
E
-
10
0.2
1.62592
E
-
5
3.21563
E
-
10
2.61132
E
-
6
4.72710
E
-
10
0.3
1.74905
E
-
4
5.95890
E
-
10
4.18979
E
-
5
6.19366
E
-
10
0.4
8.93799
E
-
4
5.11051
E
-
10
2.86285
E
-
4
6.95422
E
-
10
0.5
3.00491
E
-
3
2.46104
E
-
10
1.19940
E
-
3
4.19959
E
-
10
0.6
7.47528
E
-
3
1.98685
E
-
9
3.56141
E
-
3
5.90035
E
-
10
0.7
1.40733
E
-
2
5.01512
E
-
9
7.74239
E
-
3
2.83080
E
-
9
0.8
1.78384
E
-
2
9.62848
E
-
9
1.09171
E
-
2
6.94058
E
-
9
0.9
4.97756
E
-
3
1.61180
E
-
8
2.27326
E
-
3
1.36984
E
-
8
1.0
3.84378
E
-
2
2.49043
E
-
8
3.32111
E
-
2
2.42565
E
-
8Example 8.
Consider the following system of linear Volterra integral equations of the second kind [11]:
(25)f1(s)=g1(s)+∫0s‍(sin(s-t)-1)f1(t)dt+∫0s‍(1-tcoss)f2(t)dt,f2(s)=g2(s)+∫0s‍(f1(t))dt+∫0s‍(s-t)f2(t)dt,
where g1(s) and g2(s) are chosen such that the exact solution is f1(s)=coss,f2(s)=sins. The numerical results obtained by using the present method are compared with [11] in Table 2.Table 2
Absolute errors for Example8.
Nodesxi
Errorse(f1) [11]
Errorse(f1,100)
Errorse(f2) [11]
Errorse(f2,100) [11]
0.0
0
6.93348
E
-
11
0
3.60316
E
-
11
0.1
1.37735
E
-
4
4.53518
E
-
09
1.52721
E
-
4
2.75123
E
-
08
0.2
9.27188
E
-
4
8.84879
E
-
09
1.14715
E
-
3
3.10611
E
-
08
0.3
2.67117
E
-
3
1.28253
E
-
08
3.71248
E
-
3
3.53307
E
-
08
0.4
5.45507
E
-
3
1.65442
E
-
08
8.57201
E
-
3
4.03402
E
-
08
0.5
9.22670
E
-
3
2.00881
E
-
08
1.64412
E
-
2
4.61209
E
-
08
0.6
1.38644
E
-
2
2.35657
E
-
09
2.78243
E
-
2
5.27214
E
-
08
0.7
1.92960
E
-
2
2.71160
E
-
08
4.25337
E
-
2
6.02041
E
-
08
0.8
2.56349
E
-
2
3.09302
E
-
08
5.91212
E
-
2
6.86601
E
-
08
0.9
3.31574
E
-
2
3.52645
E
-
08
7.48883
E
-
2
7.82029
E
-
08
1.0
4.19808
E
-
2
3.67322
E
-
08
8.70896
E
-
2
1.02387
E
-
07


## 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.



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*Source: 196308-2013-11-21.xml*