# Reproducing Kernel Space Method for the Solution of Linear Fredholm

Authors: Xueqin Lv; Yue Gao
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We present a numerical method to solve the linear Fredholm integro-differential equation in reproducing kernel space. A simple algorithm is given to obtain the approximate solutions of the equation. Through the comparison of approximate and true solution, we can find that the method can effectively solve the linear Fredholm integro-differential equation. At the same time the numerical solution of the equation is stable.


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## Body

## 1. Introduction

In this paper, we consider the following first-order Fredholm type integro-differential equation:(1.1)u′(t)=q(t)u(t)+∫01h(t,s)u(s)ds+f(t),t∈[0,1],
with the initial condition(1.2)u(0)=0.The equation is discussed by Yusufoğlu in [1], where q(t), h(t,s), and f(t) are sufficiently regular given functions. Integro-differential system is an important tool in solving real-world problems. A wide variety of natural phenomena are modelled by Fredholm type integro-differential equations. The ordinary integro-differential system has been applied to many problems in fluid dynamics, engineering, chemical reactions, and so on.In the recent years, there are some methods to solve the Fredholm type integro-differential equations [2–8]. At this point, a new method is presented to solve the integro-differential equations. The method is established in reproducing kernel space; the problem of solving the integro-differential problem with a perturbation can be converted into the simple problem of solving the equation. The representation of all the solutions for Fredholm type integro-differential equation is given if it has solutions. The stability is important and references are there in [9, 10]. There are many discussions about the solutions in reproducing kernel space in [11–15]. In this paper, we discuss the Fredholm type integro-differential equation. In the last section, CAS wavelet approximating methods [5], differential transformation methods [6], HPM [1], and reproducing kernel space method are compared, then we can get some effective data. The numerical experiments show that this kind of method is stable in the reproducing kernel space.


## 2. Two Reproducing Kernel Spaces

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

The reproducing kernel spaceW22[0,1] is defined as follows:(2.1)W22[0,1]={u(t)∣u′(t)isabsolutelycontinuousfunctionin[0,1],u′′(t)∈L2[0,1]}.The inner product and norm inW22[0,1] are defined respectively by(2.2)〈u(t),v(t)〉=∑i=01u(i)(0)v(i)(0)+∫01u′′(t)v′′(t)dt,u,v∈W22[0,1],‖u‖W22[0,1]=〈u(t),u(t)〉1/2,u∈W22[0,1].ThenW22[0,1] is a complete reproducing kernel space. That is, there exists a function Rt(s), for each fixed t∈[0,1],  Rt(s)∈W22[0,1], and for any u(s)∈W22[0,1] and s∈[0,1], satisfying(2.3)〈u(s),Rt(s)〉W22=u(t).
By using Mathematica, Rt(s) is given by(2.4)Rt(s)={-16s[s2-3t(2+s)],t≤s,-16t3+12t(2+t)s,t>s.


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

The construction of reproducing kernel spaceW21[0,1] can be found in [14] and its reproducing kernel function is(2.5)Qs(t)={1+t,t≤s,1+s,t>s.


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

The reproducing kernel spaceW22[0,1] is defined as follows:(2.1)W22[0,1]={u(t)∣u′(t)isabsolutelycontinuousfunctionin[0,1],u′′(t)∈L2[0,1]}.The inner product and norm inW22[0,1] are defined respectively by(2.2)〈u(t),v(t)〉=∑i=01u(i)(0)v(i)(0)+∫01u′′(t)v′′(t)dt,u,v∈W22[0,1],‖u‖W22[0,1]=〈u(t),u(t)〉1/2,u∈W22[0,1].ThenW22[0,1] is a complete reproducing kernel space. That is, there exists a function Rt(s), for each fixed t∈[0,1],  Rt(s)∈W22[0,1], and for any u(s)∈W22[0,1] and s∈[0,1], satisfying(2.3)〈u(s),Rt(s)〉W22=u(t).
By using Mathematica, Rt(s) is given by(2.4)Rt(s)={-16s[s2-3t(2+s)],t≤s,-16t3+12t(2+t)s,t>s.


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

The construction of reproducing kernel spaceW21[0,1] can be found in [14] and its reproducing kernel function is(2.5)Qs(t)={1+t,t≤s,1+s,t>s.


## 3. Analysis of the Solution of (1.1)

Letℒ:W22[0,1]→W21[0,1], such that(3.1)Lu(t)=u′(t)-q(t)u(t)-∫01h(t,s)u(s)ds,whereu(t)∈W22[0,1], it is easy to know that ℒ is a linear bounded operator and (1.1) can be converted into the equivalent form(3.2)Lu(t)=f(t),t∈[0,1].In order to obtain the representation of all the solutions of (1.1), let φi(t)=Qti(t), ψi(t)=ℒ*φi(t), where {ti}i=1∞ is dense in [0,1].From the definition of the reproducing kernel, we have(3.3)〈u(t),φi(t)〉W21=u(ti),ψi(t)=(L*φi(s))(t)=〈(L*Qti(s))(a),Rt(a)〉W22=〈Qti(s),(LRt(a))(s)〉W21=(LRt(a))(ti),i=1,2,…,whereℒ* is a conjugate operator of ℒ. Practise Gram-Schmidt orthonormalization for {ψi(t)}i=1∞(3.4)ψ¯i(t)=∑k=1iβikψk(t),
where βik are coefficients of Gram-Schmidt orthonormalization and {ψ¯i(t)}i=1∞ is orthonormal system in W22[0,1].Theorem 3.1.
If (3.2) has solutions, the results are proved by the following formula:
(3.5)u(t)=∑i=1∞∑k=1iβikf(tk)ψ¯i(t).Proof.
The results are proved by the following formula:(3.6)u(t)=∑i=1∞〈u(t),ψ¯i(t)〉W22ψ¯i(t)=∑i=1∞〈u(t),∑k=1iβikψk(t)〉W22ψ¯i(t)=∑i=1∞∑k=1iβik〈u(t),ψk(t)〉W22ψ¯i(t)=∑i=1∞∑k=1iβik〈u(t),L*φk(t)〉W22ψ¯i(t)=∑i=1∞∑k=1iβik〈Lu(t),φk(t)〉W21ψ¯i(t)=∑i=1∞∑k=1iβikLu(tk)ψ¯i(t)=∑i=1∞∑k=1iβikf(tk)ψ¯i(t).Now, the approximate solutionun(t) can be obtained by the n-term truncation of (3.5), that is(3.7)un(t)=∑i=1n∑k=1iβikf(tk)ψ¯i(t).Theorem 3.2.
Assumeu(t) is the solution of (3.2) and rn(t) is the error between the approximate solution un(t) and the exact solution u(t). Then ∥rn(t)∥W22→0.Proof.
In the following we prove the sequencern(t) is monotone decreasing in the sense of ∥·∥W22.
From (3.5) and (3.7), we have(3.8)‖rn‖W22=‖∑i=n+1∞∑k=1iβikf(tk)ψ¯i(t)‖W22=∑i=n+1∞(∑k=1iβikf(tk))2,‖rn-1‖W22=‖∑i=n∞∑k=1iβikf(tk)ψ¯i(t)‖W22=∑i=n∞(∑k=1iβikf(tk))2.
By (3.8), we know that ∥rn∥W22≤∥rn-1∥W22. Then {rn(t)}n=1∞ is monotone decreasing in ∥·∥W22. Since the series ∑i=1∞〈u(t),ψ¯i(t)〉W22ψ¯i(t) is convergent in W22[0,1], we obtain ∥rn(t)∥W22→0.


## 4. The Stability of the Approximate Solution

In order to consider the stability of the approximate solution, we add a perturbationɛ(t) in the right-hand side, then (3.2) becomes(4.1)Lu(t)=f(t)+ɛ(t),t∈[0,1].Now, we discuss the representation of the solution for (4.1).


### 4.1. Representation of All the Solutions of (4.1)

In order to study the stability of (4.1), let 𝒜 be a projection operator from W22[0,1] to Ψ,(4.2)Ψ={u∣u=∑i=1∞ciψ¯i,for{ci}i=1∞∈l2},
where {ψ¯i}i=1∞ is given in (3.4), and 𝒜 satisfies 𝒜*=𝒜. Moreover(4.3)Aψi(s)=∑k=1∞〈ψi(s),ψ¯k(s)〉ψ¯k(s)=ψi(s).We have(4.4)Au(t)=∑i=1∞〈u(t),ψ¯i(t)〉ψ¯i(t),whereu(t) is a solution of (3.2) in W22[0,1].Define(4.5)uA(t)=Au(t),
then we have(4.6)uA(t)=∑i=1∞〈u(t),ψ¯i(t)〉ψ¯i(t)=∑i=1∞∑k=1iβikf(tk)ψ¯i(t).Theorem 4.1.
If{ti}i=1∞ is dense in [0,1], then the form (4.6) is the solution of (3.2) in Ψ.Proof.
We have(4.7)LuA(ti)=〈LuA(t),φi(t)〉=〈uA(t),L*φi(t)〉=〈Au(t),ψi(t)〉=〈u(t),A*ψi(t)〉=〈u(t),Aψi(t)〉=〈u(t),ψi(t)〉=〈u(t),L*φi(t)〉=〈Lu(t),φi(t)〉=〈f(t),φi(t)〉=f(ti),since {ti}i=1∞ is dense in [0,1], then ℒu𝒜(t)=f(t).
It is easy to know that the solution of (3.2) is unique in Ψ (see [16]).The following lemma holds.Lemma 4.2.
Ψ⊥=N(ℒ), where Ψ⊥={u(t)∣〈u(t),v(t)〉=0,foranyv(t)∈Ψ} and N(ℒ) is a null space of ℒ, that is N(ℒ)={u∣ℒu=0}.Proof.
For anyu(t)∈Ψ⊥,
(4.8)0=〈u(t),ψk(t)〉=〈u(t),L*φk(t)〉=〈Lu(t),φk(t)〉=Lu(tk).Since {ti}i=1∞ is dense in [0,1], then ℒu(t)≡0. We can obtain u(t)∈N(ℒ). Obviously, N(ℒ)⊂Ψ⊥.ThenΨ⊥=N(ℒ).The following theorem is obvious.Theorem 4.3.
Let{ti}i=1∞ be any dense subset of [0,1], if (4.1) has solutions, then its solution ũ(t) can be represented as follows:
(4.9)ũ(t)=∑i=1∞∑k=1iβikf(tk)ψ¯i(t)+τ(t),where τ(t)∈N(ℒ).Assume that{σi}i=1∞ is a basis of N(ℒ). Orthonormalzing {ψ¯1,ψ¯2,…,σ1,σ2,…}, we obtain(4.10)σ¯i=∑k=1∞βikψ¯k(t)+∑j=1iβijσj,i=1,2,….Hence,{ψ¯1,ψ¯2,…,σ¯1,σ¯2,…} is a normal orthogonal basis in W22[0,1].According to Theorem4.3, we can obtain the solution of (4.1). Hence, through the Gram-Schmidt process, we have {ψ¯i}i=1∞∪{σ¯i}i=1∞ are the complete orthonormal system in W22[0,1], and {σ¯i}i=1∞ is a complete orthonormal system in N(ℒ).


### 4.2. The Stability of the Solution for (3.2) in Reproducing Kernel Space

Let the spaceΨ be complete. ℒΨ be a restricted operator of ℒ in Ψ, we have the converse operator ℒΨ-1:W22[0,1]→Ψ exists and is bounded.Lemma 4.4.
Ifu𝒜(t) is given by (4.6), then u𝒜(t) is the minimal norm solution.Proof.
Letu(t) be a solution of (3.2). We have
(4.11)u(t)=uA(t)+v(t),where u𝒜(t)∈Ψ and v(t)∈Ψ⊥. The following
(4.12)‖u‖2=〈uA+v,uA+v〉=‖uA‖2+〈uA,v〉+〈v,uA〉+‖v‖2=‖uA‖2+‖v‖2≥‖uA‖2holds.
It is pointed thatu𝒜 is the minimal norm solution of (3.2).Theorem 4.5.
If the (4.1) has solutions and let u𝒜(t) be the minimal norm solution, then
(4.13)∥uA(t)-uA,n(t)∥⟶0(n⟶∞),
where u𝒜,n(t) is a truncation of u𝒜(t). Hence, u𝒜(t) is stable in W22[0,1].Proof.
LetℒΨu𝒜,n(t)=fn(t) and f(t)=fn(t)+ɛn(t), where ɛn(t) is a perturbation and ɛn(t)→0(n→∞) in ∥·∥W22.
On the one hand, sinceℒΨ-1ɛn(t)∈Ψ, ℒΨ*φk=ψk, it follows that(4.14)LΨ-1ɛn(t)=∑i=1∞〈LΨ-1ɛn(t),ψ¯i(t)〉ψ¯i(t)=∑i=1∞〈LΨ-1ɛn(t),∑k=1iβikψk(t)〉ψ¯i(t)=∑i=1∞∑k=1iβik〈LΨ-1ɛn(t),ψk(t)〉ψ¯i(t)=∑i=1∞∑k=1iβik〈ɛn(t),LΨ-1*ψk(t)〉ψ¯i(t)=∑i=1∞∑k=1iβik〈ɛn(t),LΨ-1*LΨ*φk(t)〉ψ¯i(t)=∑i=1∞∑k=1iβik〈ɛn(t),φk(t)〉ψ¯i(t)=∑i=1∞∑k=1iβikɛn(tk)ψ¯i(t).
On the other hand,f,fn∈W21[0,1], from the form (4.6), we have(4.15)uA(t)-uA,n(t)=∑i=1∞∑k=1iβik[f(t)-fn(t)]ψ¯i(t)=∑i=1∞∑k=1iβikɛn(tk)ψ¯i(t),where
(4.16)uA,n(t)=∑i=1n∑k=1iβikfn(tk)ψ¯i(t),and then
(4.17)uA(t)-uA,n(t)=LΨ-1ɛn(t).From the continuity of ℒΨ-1 and ɛn(t)→0(n→∞) in ∥·∥W22, we have
(4.18)lim⁡n→∞‖uA(t)-uAn(t)‖W22≤‖LΨ-1‖W22lim⁡n→∞‖ɛn(t)‖W22=0.


## 4.1. Representation of All the Solutions of (4.1)

In order to study the stability of (4.1), let 𝒜 be a projection operator from W22[0,1] to Ψ,(4.2)Ψ={u∣u=∑i=1∞ciψ¯i,for{ci}i=1∞∈l2},
where {ψ¯i}i=1∞ is given in (3.4), and 𝒜 satisfies 𝒜*=𝒜. Moreover(4.3)Aψi(s)=∑k=1∞〈ψi(s),ψ¯k(s)〉ψ¯k(s)=ψi(s).We have(4.4)Au(t)=∑i=1∞〈u(t),ψ¯i(t)〉ψ¯i(t),whereu(t) is a solution of (3.2) in W22[0,1].Define(4.5)uA(t)=Au(t),
then we have(4.6)uA(t)=∑i=1∞〈u(t),ψ¯i(t)〉ψ¯i(t)=∑i=1∞∑k=1iβikf(tk)ψ¯i(t).Theorem 4.1.
If{ti}i=1∞ is dense in [0,1], then the form (4.6) is the solution of (3.2) in Ψ.Proof.
We have(4.7)LuA(ti)=〈LuA(t),φi(t)〉=〈uA(t),L*φi(t)〉=〈Au(t),ψi(t)〉=〈u(t),A*ψi(t)〉=〈u(t),Aψi(t)〉=〈u(t),ψi(t)〉=〈u(t),L*φi(t)〉=〈Lu(t),φi(t)〉=〈f(t),φi(t)〉=f(ti),since {ti}i=1∞ is dense in [0,1], then ℒu𝒜(t)=f(t).
It is easy to know that the solution of (3.2) is unique in Ψ (see [16]).The following lemma holds.Lemma 4.2.
Ψ⊥=N(ℒ), where Ψ⊥={u(t)∣〈u(t),v(t)〉=0,foranyv(t)∈Ψ} and N(ℒ) is a null space of ℒ, that is N(ℒ)={u∣ℒu=0}.Proof.
For anyu(t)∈Ψ⊥,
(4.8)0=〈u(t),ψk(t)〉=〈u(t),L*φk(t)〉=〈Lu(t),φk(t)〉=Lu(tk).Since {ti}i=1∞ is dense in [0,1], then ℒu(t)≡0. We can obtain u(t)∈N(ℒ). Obviously, N(ℒ)⊂Ψ⊥.ThenΨ⊥=N(ℒ).The following theorem is obvious.Theorem 4.3.
Let{ti}i=1∞ be any dense subset of [0,1], if (4.1) has solutions, then its solution ũ(t) can be represented as follows:
(4.9)ũ(t)=∑i=1∞∑k=1iβikf(tk)ψ¯i(t)+τ(t),where τ(t)∈N(ℒ).Assume that{σi}i=1∞ is a basis of N(ℒ). Orthonormalzing {ψ¯1,ψ¯2,…,σ1,σ2,…}, we obtain(4.10)σ¯i=∑k=1∞βikψ¯k(t)+∑j=1iβijσj,i=1,2,….Hence,{ψ¯1,ψ¯2,…,σ¯1,σ¯2,…} is a normal orthogonal basis in W22[0,1].According to Theorem4.3, we can obtain the solution of (4.1). Hence, through the Gram-Schmidt process, we have {ψ¯i}i=1∞∪{σ¯i}i=1∞ are the complete orthonormal system in W22[0,1], and {σ¯i}i=1∞ is a complete orthonormal system in N(ℒ).


## 4.2. The Stability of the Solution for (3.2) in Reproducing Kernel Space

Let the spaceΨ be complete. ℒΨ be a restricted operator of ℒ in Ψ, we have the converse operator ℒΨ-1:W22[0,1]→Ψ exists and is bounded.Lemma 4.4.
Ifu𝒜(t) is given by (4.6), then u𝒜(t) is the minimal norm solution.Proof.
Letu(t) be a solution of (3.2). We have
(4.11)u(t)=uA(t)+v(t),where u𝒜(t)∈Ψ and v(t)∈Ψ⊥. The following
(4.12)‖u‖2=〈uA+v,uA+v〉=‖uA‖2+〈uA,v〉+〈v,uA〉+‖v‖2=‖uA‖2+‖v‖2≥‖uA‖2holds.
It is pointed thatu𝒜 is the minimal norm solution of (3.2).Theorem 4.5.
If the (4.1) has solutions and let u𝒜(t) be the minimal norm solution, then
(4.13)∥uA(t)-uA,n(t)∥⟶0(n⟶∞),
where u𝒜,n(t) is a truncation of u𝒜(t). Hence, u𝒜(t) is stable in W22[0,1].Proof.
LetℒΨu𝒜,n(t)=fn(t) and f(t)=fn(t)+ɛn(t), where ɛn(t) is a perturbation and ɛn(t)→0(n→∞) in ∥·∥W22.
On the one hand, sinceℒΨ-1ɛn(t)∈Ψ, ℒΨ*φk=ψk, it follows that(4.14)LΨ-1ɛn(t)=∑i=1∞〈LΨ-1ɛn(t),ψ¯i(t)〉ψ¯i(t)=∑i=1∞〈LΨ-1ɛn(t),∑k=1iβikψk(t)〉ψ¯i(t)=∑i=1∞∑k=1iβik〈LΨ-1ɛn(t),ψk(t)〉ψ¯i(t)=∑i=1∞∑k=1iβik〈ɛn(t),LΨ-1*ψk(t)〉ψ¯i(t)=∑i=1∞∑k=1iβik〈ɛn(t),LΨ-1*LΨ*φk(t)〉ψ¯i(t)=∑i=1∞∑k=1iβik〈ɛn(t),φk(t)〉ψ¯i(t)=∑i=1∞∑k=1iβikɛn(tk)ψ¯i(t).
On the other hand,f,fn∈W21[0,1], from the form (4.6), we have(4.15)uA(t)-uA,n(t)=∑i=1∞∑k=1iβik[f(t)-fn(t)]ψ¯i(t)=∑i=1∞∑k=1iβikɛn(tk)ψ¯i(t),where
(4.16)uA,n(t)=∑i=1n∑k=1iβikfn(tk)ψ¯i(t),and then
(4.17)uA(t)-uA,n(t)=LΨ-1ɛn(t).From the continuity of ℒΨ-1 and ɛn(t)→0(n→∞) in ∥·∥W22, we have
(4.18)lim⁡n→∞‖uA(t)-uAn(t)‖W22≤‖LΨ-1‖W22lim⁡n→∞‖ɛn(t)‖W22=0.


## 5. Numerical Experiments

To illustrate the effectiveness of the above method, we give an example as follows:Example 5.1.
Consider the following first-order Fredholm type integro-differential equation (see [1]):
(5.1)u′(t)=(t+1)et-t+∫01tu(s)ds,t∈[0,1],u(0)=0.The analytic solution of this equation is u(t)=tet.
From Tables1, 2, and 3, we can see that the absolute error, relative error and the root-mean-square are small.Table 1
Comparison of absolute errors of approximate solutions, for example, obtained by reproducing kernel space methods, CAS wavelet approximating [5], differential transformation [6] methods and HPM [1].
NodeMethod in [5]Method in [6]Method in [1]Reproducing kernel methods0.11.34917637e-031.00118319e-022.304814815e-044.77378e-050.21.15960044e-032.78651355e-029.259259259e-044.80048e-050.35.67152531e-035.08730892e-022.083333333e-034.97945e-050.45.93405645e-027.55356316e-023.703703704e-035.31631e-050.51.32330751e-029.71888592e-025.787037037e-035.81740e-050.64.39287720e-021.09551714e-018.333333333e-036.48985e-050.71.41201624e-021.04133232e-011.134259259e-027.34170e-050.81.34514117e-026.94512700e-021.481481481e-028.38198e-050.91.32045209e-021.00034260e-021.875000000e-029.62085e-05Table 2
Relative error of approximate solutions withɛ=0.0001.
NodeApproximate solutionu100(t)True solutionu(t)Relative error0.10.1105750.1105175.25129e-40.20.2443500.2442812.83292e-40.30.4050400.4049582.03706e-40.40.5968280.5967301.64163e-40.50.8244760.8243611.40317e-40.61.0934101.0932701.24119e-40.71.4097901.4096301.12167e-40.81.7806201.7804301.02793e-40.92.2138502.2136409.50945e-512.7185202.7182808.85463e-5Table 3
The root-mean-square, for example, withɛ=0.0001.
∑i=1100[ũ(0.01i)-ũn(0.01i)]21001.38171e-04∑i=1100[ũ′(0.01i)-ũn′(0.01i)]21001.67175e-04


## 6. Conclusions

From the previous numerical results, we can see that the error is quite small and the numerical solution is stable when the right-hand side with a small perturbation. It illustrates that the method is given in the paper is valid.



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