# The Combined RKM and ADM for Solving Nonlinear Weakly Singular Volterra Integrodifferential Equations

Authors: Xueqin Lv; Sixing Shi
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The reproducing kernel method (RKM) and the Adomian decomposition method (ADM) are applied to solventh-order nonlinear weakly singular Volterra integrodifferential equations. The numerical solutions of this class of equations have been a difficult topic to analyze. The aim of this paper is to use Taylor’s approximation and then transform the given nth-order nonlinear Volterra integrodifferential equation into an ordinary nonlinear differential equation. Using the RKM and ADM to solve ordinary nonlinear differential equation is an accurate and efficient method. Some examples indicate that this method is an efficient method to solve nth-order nonlinear Volterra integro-differential equations.


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## Body

## 1. Introduction

In this paper, we consider the followingnth-order nonlinear weakly singular Volterra integro-differential equation of the following form [1–4]:
(1.1)∑i=0n‍ai(x)u(i)(x)=f(x)+∫ax‍K(t,x)um(t)dt,a⩽x⩽b,
where m>1, f(x) is a given function, u(x) is the unknown function, and K(t,x) is the kernel of the integro equation. We usually assume that the function u(x) and f(x) are continuous or square integrable on [a,b].Some problems of mathematical physics are described in terms of (1.1) which has been studied by different methods including the spline collocation method [5], piecewise polynomials [6], Haar wavelets [7], the homotopy perturbation method (HPM) [8, 9], the wavelet-Galerkin method [10], Taylor polynomials [11], the Tau method [12], the sinc-collocation method [13], the combined Laplace transform-Adomian decomposition method [14], and the Adomian’s asymptotic decomposition method [15] to determine exact and approximate solutions. But to our knowledge there is still no viable analytic approach for solving weakly singular Volterra integro-differential equations. The present work is motivated by the desire to obtain approximate solution to nth-order nonlinear weakly singular Volterra integro-differential equation, where the integrand is weakly singular in the sense that its integral is continuous at the singular point, that is, its kernel K(t,x)=1/(x-t)α is singular as t→x.Reproducing kernel theory has important application in numerical analysis, differential equation, probability and statistics and so on [16, 17]. And the RKM has been applied successfully to solving linear and nonlinear problems [18–20].The rest of the paper is organized as follows. In the Section2, transforming (1.1) into an differential equation by Taylor’s approximation. In Section 3, the RKM is introduced. Applying RKM and ADM to solving (1.1) is discussed in Section 4. The numerical examples are presented in Section 5. Finally, a brief conclusion is stated in last section.


## 2. Taylor’s Approximation

Consider the followingnth-order nonlinear weakly singular Volterra integro-different equation:
(2.1)∑i=0n‍ai(x)u(i)(x)=f(x)+∫ax‍F(u(t))(x-t)αdt,a⩽x⩽b,0<α<1.
We have by setting
(2.2)y(t)=F(u(t)),(2.3)∑i=0n‍ai(x)u(i)(x)=f(x)+∫ax‍y(t)(x-t)αdt,a⩽x⩽b,0<α<1.
Rewriting (2.3) as
(2.4)∑i=0n‍ai(x)u(i)(x)=f(x)+∫ax‍y(t)-y(x)+y(x)(x-t)αdt,
where the solution under the integral has been replaced by y(t)-y(x)+y(x). Thus,
(2.5)∑i=0n‍ai(x)u(i)(x)=f(x)+y(x)∫ax‍1(x-t)αdt+∫ax‍y(t)-y(x)(x-t)αdt.
So that
(2.6)∑i=0n‍ai(x)u(i)(x)=f(x)+y(x)(x-a)1-α1-α+∫ax‍y(t)-y(x)(x-t)αdt,
or equivalently
(2.7)∑i=0n‍ai(x)u(i)(x)=f(x)+y(x)(x-a)1-α1-α-∫ax‍(x-t)1-αy(t)-y(x)t-xdt.
We use the following Taylor’s approximation of degree n of y(t) about t=x:
(2.8)y(t)≈y(x)+(t-x)y′(x)+(t-x)22!y′′(x)+⋯+(t-x)nn!y(n)(x).
Thus,
(2.9)y(t)-y(x)≈(t-x)y′(x)+(t-x)22!y′′(x)+⋯+(t-x)nn!y(n)(x).
Substituting the approximate relation (2.9) into the right hand side of (2.7) yields
(2.10)∑i=0n‍ai(x)u(i)(x)=f(x)+∑i=0n‍(-1)i(x-a)i+1-αi!(i+1-α)y(i)(x).
Therefore, (2.1) can be approximated by the nth-order nonlinear differential equation (2.10).


## 3. Analysis of Reproducing Kernel Hilbert Space

Definition 3.1 (reproducing kernel SpaceWn+1[a,b], see [17]).
(3.1)
W
n
+
1
[
a
,
b
]
=
{
u
(
x
)
∣
u
(
n
)
(
x
)
is
an
absolutely
continuous
real
value
function
in
[
a
,
b
]
,
u
(
n
+
1
)
(
x
)
∈
L
2
[
a
,
b
]
,
x
∈
[
a
,
b
]
}
and endowed it with the inner product and norm, respectively,
(3.2)〈u(x),v(x)〉Wn+1=∑i=0n‍u(i)(a)v(i)(a)+∫ab‍u(n+1)(x)v(n+1)(x)dx,∥u∥Wn+1=〈u,u〉1/2.Theorem 3.2.
The spaceWn+1[a,b] is a reproducing kernel space. That is, there exists a function Rx(y), for each fixed x∈[a,b],Rx(y)∈Wn+1[a,b], and for any u(y)∈Wn+1[a,b], satisfying
(3.3)〈u(y),Rx(y)〉Wn+1=u(x),
the reproducing kernel Rx(y) can be denoted by
(3.4)Rx(y)={∑i=12n+2‍ai(x)yi-1,y⩽x,∑i=12n+2‍bi(x)yi-1,y>x,
where ai(x),bi(x)(i=1,2,…,2n+2) are known coefficients.Theorem 3.3.
LetWn+1[a,b] be a reproducing kernel space and um(x),u(x)∈Wn+1[a,b],m=1,2,…. If um(x) converges to u(x) in the sense of ∥·∥Wn+1, then um(k)(x) converges to u(k)(x)(k=0,1,…,n) uniformly.Property 1.
IfWn+1[a,b] is a reproducing kernel space, the reproducing kernel function Rx(y) in Wn+1[a,b] is unique.Definition 3.4 (reproducing kernel SpaceW1[a,b], see [17]).
(3.5)
W
1
[
a
,
b
]
=
{
u
(
x
)
∣
u
(
x
)
is
an
absolutely
continuous
real
value
function
in
[
a
,
b
]
,
u
′
(
x
)
∈
L
2
[
a
,
b
]
,
x
∈
[
a
,
b
]
}
and endowed it with the inner product and norm, respectively,
(3.6)〈u(x),v(x)〉W1=u(a)v(a)+∫ab‍u′(x)v′(x)dx,∥u∥W1=〈u,u〉1/2.There exists a unique reproducing kernel functionQ(x,y)∈W1[a,b], and Q(x,y) can be denoted by
(3.7)Qx(y)={1+y,y⩽x,1+x,y>x.The method of obtainingQx(y), the coefficients of the reproducing kernel Rx(y), and the proof of Theorems 3.2 and 3.3 are given in [17].


## 4. Combined ADM and RKM

### 4.1. Representation of the Inverse Operator

Here, we propose a new differential operator, as follows:(4.1)ℒu(x)=∑i=0n‍ai(x)u(i)(x),
letting
(4.2)N(x,u(x))=∑i=0n‍(-1)i(x-a)i+1-αi!(i+1-α)y(i)(x),
where y(x)=F(u(x)), then we convert (2.10) as follows:
(4.3)ℒu(x)=f(x)+N(x,u(x)),a⩽x⩽b.
We now introduce how to determine the inverse operator L-1 of L. Obviously, ℒ:Wn+1[a,b]→W1[a,b] is a bounded linear operator.We choose{xi}i=1∞ as any dense set in [a,b], and let ψx(y)=ℒ*Qx(y), where ℒ* is the conjugate operator of ℒ and Qx(y) is given by (3.7). Furthermore, for simplicity, let ψi(x) denote ψxi(x), namely,
(4.4)ψi(x)=defψxi(x)=ℒ*Qxi(x).Now, several Lemmas are given.Lemma 4.1.
{
ψ
i
(
x
)
}
i
=
1
∞ is the complete system of Wn+1[a,b].Proof.
Foru(x)∈Wn+1[a,b], let 〈u(x),ψi(x)〉=0(i=1,2,…), that is,
(4.5)〈u(x),ℒ*Qxi(x)〉=(ℒu)(xi)=0.
Note that{xi}i=1∞ is the dense set in [a,b], therefore (ℒu)(x)=0. It follows that u(x)=0 from the existence of ℒ-1.Lemma 4.2.
The following formula holds(4.6)ψi(x)=(ℒηRx(η))(xi),
where the subscript η of operator ℒη indicates that the operator ℒ applies to functions of η.Proof.
Consider(4.7)ψi(x)=〈ψi(ξ),Rx(ξ)〉Wn+1[a,b]=〈(ℒ*Qxi)(ξ),Rx(ξ)〉Wn+1[a,b]=〈Qxi(ξ),(ℒηRx(η))(ξ)〉W1[a,b]=(ℒηRx(η))(xi).
This completes the proof.The orthonormal system{ψ¯i(x)}i=1∞ of Wn+1[a,b] can be derived from Gram-Schmidt orthogonalization process of {ψi(x)}i=1∞,
(4.8)ψ¯i(x)=∑k=1i‍βikψk(x),(βii>0,i=1,2,…),
where βik are orthogonal coefficients.Theorem 4.3.
If the inverse operatorL-1, exists and {xi}i=1∞ is dense on [0,1], then the inverse operator L-1 can be determined as
(4.9)u(x)=L-1(f(x)+N(x,u(x)))=∑i=1∞‍Biψ¯i(x),
where Bi=∑k=1iβik(f(xk)+N(xk,u(xk))).Proof.
From (4.9), it holds that
(4.10)u(x)=∑i=1∞‍〈u(x),ψ¯i(x)〉Wn+1ψ¯i(x)=∑i=1∞‍∑k=1i‍βik〈u(x),ℒ*Qxk(x)〉Wn+1ψ¯i(x)=∑i=1∞‍∑k=1i‍βik〈ℒu(x),Qxk(x)〉W1ψ¯i(x)=∑i=1∞‍∑k=1i‍βik〈f(x)+N(x,u(x)),Qxk(x)〉W1ψ¯i(x)=∑i=1∞‍∑k=1i‍βik(f(xk)+N(xk,u(xk)))ψ¯i(x).
The proof of the theorem is complete.From Theorem4.3, obviously, L-1 is determined.


### 4.2. Decomposition Method

By applyingL-1 to both sides of (4.3), we have
(4.11)u(x)=L-1f(x)+L-1N(x,u(x)).
The ADM introduces the solution u(x) and the nonlinear function N(x,u) by infinite series
(4.12)u(x)=∑i=0∞‍ui(x),(4.13)N(x,u(x))=∑i=0∞‍Ai(x),
where An are Adomian polynomials for the nonlinear term N(x,u(x)) and can be found from the following formula:
(4.14)An=1n![dndλnf(∑n=0∞‍λiui)]λ0,n⩾0.
Substituting (4.12) and (4.13) into (4.11) yields
(4.15)∑i=0∞‍ui(x)=L-1f(x)+L-1∑i=0∞‍Ai(x).
According to the ADM, the components ui(x) can be determined as
(4.16)u0(x)=L-1f(x),ui+1(x)=L-1Ai(x),i⩾0,
which gives
(4.17)u0(x)=∑j=0∞‍B0jψ¯j(x),ui+1(x)=∑j=0∞‍B(i+1)jψ¯j(x),i⩾0,
where B0j=∑k=1jβjkf(xk), Bij=∑k=1jβjkAi-1(xk),i⩾1.From (4.17), we can determine the components ui(x), and hence the series solution of u(x) in (4.12) can be immediately obtained.We obtain approximate solution of the following equation:(4.18)Un(x)=∑i=0n‍ui(x).The following use some examples to demonstrate the effectiveness of the algorithm.


## 4.1. Representation of the Inverse Operator

Here, we propose a new differential operator, as follows:(4.1)ℒu(x)=∑i=0n‍ai(x)u(i)(x),
letting
(4.2)N(x,u(x))=∑i=0n‍(-1)i(x-a)i+1-αi!(i+1-α)y(i)(x),
where y(x)=F(u(x)), then we convert (2.10) as follows:
(4.3)ℒu(x)=f(x)+N(x,u(x)),a⩽x⩽b.
We now introduce how to determine the inverse operator L-1 of L. Obviously, ℒ:Wn+1[a,b]→W1[a,b] is a bounded linear operator.We choose{xi}i=1∞ as any dense set in [a,b], and let ψx(y)=ℒ*Qx(y), where ℒ* is the conjugate operator of ℒ and Qx(y) is given by (3.7). Furthermore, for simplicity, let ψi(x) denote ψxi(x), namely,
(4.4)ψi(x)=defψxi(x)=ℒ*Qxi(x).Now, several Lemmas are given.Lemma 4.1.
{
ψ
i
(
x
)
}
i
=
1
∞ is the complete system of Wn+1[a,b].Proof.
Foru(x)∈Wn+1[a,b], let 〈u(x),ψi(x)〉=0(i=1,2,…), that is,
(4.5)〈u(x),ℒ*Qxi(x)〉=(ℒu)(xi)=0.
Note that{xi}i=1∞ is the dense set in [a,b], therefore (ℒu)(x)=0. It follows that u(x)=0 from the existence of ℒ-1.Lemma 4.2.
The following formula holds(4.6)ψi(x)=(ℒηRx(η))(xi),
where the subscript η of operator ℒη indicates that the operator ℒ applies to functions of η.Proof.
Consider(4.7)ψi(x)=〈ψi(ξ),Rx(ξ)〉Wn+1[a,b]=〈(ℒ*Qxi)(ξ),Rx(ξ)〉Wn+1[a,b]=〈Qxi(ξ),(ℒηRx(η))(ξ)〉W1[a,b]=(ℒηRx(η))(xi).
This completes the proof.The orthonormal system{ψ¯i(x)}i=1∞ of Wn+1[a,b] can be derived from Gram-Schmidt orthogonalization process of {ψi(x)}i=1∞,
(4.8)ψ¯i(x)=∑k=1i‍βikψk(x),(βii>0,i=1,2,…),
where βik are orthogonal coefficients.Theorem 4.3.
If the inverse operatorL-1, exists and {xi}i=1∞ is dense on [0,1], then the inverse operator L-1 can be determined as
(4.9)u(x)=L-1(f(x)+N(x,u(x)))=∑i=1∞‍Biψ¯i(x),
where Bi=∑k=1iβik(f(xk)+N(xk,u(xk))).Proof.
From (4.9), it holds that
(4.10)u(x)=∑i=1∞‍〈u(x),ψ¯i(x)〉Wn+1ψ¯i(x)=∑i=1∞‍∑k=1i‍βik〈u(x),ℒ*Qxk(x)〉Wn+1ψ¯i(x)=∑i=1∞‍∑k=1i‍βik〈ℒu(x),Qxk(x)〉W1ψ¯i(x)=∑i=1∞‍∑k=1i‍βik〈f(x)+N(x,u(x)),Qxk(x)〉W1ψ¯i(x)=∑i=1∞‍∑k=1i‍βik(f(xk)+N(xk,u(xk)))ψ¯i(x).
The proof of the theorem is complete.From Theorem4.3, obviously, L-1 is determined.


## 4.2. Decomposition Method

By applyingL-1 to both sides of (4.3), we have
(4.11)u(x)=L-1f(x)+L-1N(x,u(x)).
The ADM introduces the solution u(x) and the nonlinear function N(x,u) by infinite series
(4.12)u(x)=∑i=0∞‍ui(x),(4.13)N(x,u(x))=∑i=0∞‍Ai(x),
where An are Adomian polynomials for the nonlinear term N(x,u(x)) and can be found from the following formula:
(4.14)An=1n![dndλnf(∑n=0∞‍λiui)]λ0,n⩾0.
Substituting (4.12) and (4.13) into (4.11) yields
(4.15)∑i=0∞‍ui(x)=L-1f(x)+L-1∑i=0∞‍Ai(x).
According to the ADM, the components ui(x) can be determined as
(4.16)u0(x)=L-1f(x),ui+1(x)=L-1Ai(x),i⩾0,
which gives
(4.17)u0(x)=∑j=0∞‍B0jψ¯j(x),ui+1(x)=∑j=0∞‍B(i+1)jψ¯j(x),i⩾0,
where B0j=∑k=1jβjkf(xk), Bij=∑k=1jβjkAi-1(xk),i⩾1.From (4.17), we can determine the components ui(x), and hence the series solution of u(x) in (4.12) can be immediately obtained.We obtain approximate solution of the following equation:(4.18)Un(x)=∑i=0n‍ui(x).The following use some examples to demonstrate the effectiveness of the algorithm.


## 5. Numerical Examples

To illustrate the applicability and effectiveness of our method, we consider the following examples. Symbolic and numerical computations performed by using Mathematica 5.0.Example 5.1.
Consider the following first-order nonlinear weakly singular Volterra integro-differential equation:(5.1)u′(x)+p(x)u(x)=f(x)+∫ax‍F(u(t))(x-t)αdt,a⩽x⩽b,0<α<1
with α=1/2, a=0, u(0)=0 and u(x)=x(x-1).LetF(u(x))=u2, p(x)=(16/315)x5/2(21+4x(4x-9))+1, f(x)=x2+x-1.On[0,1] select 100 points and get the approximate solution U2(x)=∑n=02un(x), the results are shown in Figure 1.Figure 1
The approximate valuesU2(x) in comparison with its exact values x(x-1).Example 5.2.
Consider the following second-order nonlinear weakly singular Volterra integro-differential equation:(5.2)u′′(x)+p(x)u′(x)+q(x)u(x)=f(x)+∫ax‍F(u(t))(x-t)αdt,a⩽x⩽b,0<α<1
with α=1/3, a=0, u(0)=u(1)=0 and u(x)=x(x-1).LetF(u(x))=u3, p(x)=x2+1, q(x)=(243/52360)x11/3(-119+9x(9x2-30x+34)), f(x)=2x3+ x2+2x+3.On[0,1] select 100 points and get the approximate solution U1(x)=∑n=01un(x), the results are shown in Table 1.Table 1
Comparison of the errors ofu(x) and U1(x).
Node
Exact solution
Approximate solution
Relative error
1/100
−0.00980296
−0.00980291
-
5.55234
E
-
06
11/100
−0.0970493
−0.0970705
-
2.18212
E
-
04
21/100
−0.16469
−0.164731
-
2.50108
E
-
04
31/100
−0.212724
−0.212782
-
2.70663
E
-
04
41/100
−0.241153
−0.241219
-
2.72492
E
-
04
51/100
−0.249975
−0.250037
-
2.44481
E
-
04
61/100
−0.239192
−0.239236
-
1.84461
E
-
04
71/100
−0.208803
−0.208825
-
1.04903
E
-
04
81/100
−0.158808
−0.158813
-
2.96234
E
-
05
91/100
−0.0892069
−0.0892053
-
1.80668
E
-
05Example 5.3.
Consider the following third-order nonlinear weakly singular Volterra integro-differential equation:(5.3)u′′′(x)+p(x)u′′(x)+q(x)u′(x)+r(x)u(x)=f(x)+∫ax‍F(u(t))(x-t)αdt,a⩽x⩽b,0<α<1
with α=1/2, a=0, u(0)=u(1)=0, u′(0)=0 and u(x)=x2(x-1).LetF(u(x))=u2, p(x)=x, q(x)=x-1, r(x)=(256/45045)x9/2(143+20x(6x-13)), f(x)=3x3+ x2+6.On[0,1] select 100 points and get the approximate solution U2(x)=∑n=02un(x), the results are shown in Figure 2.Figure 2
The approximate valuesU2(x) in comparison with its exact values x2(x-1).


## 6. Conclusion

In this paper, we have reduced the solution of nonlinear weakly singular Volterra integro-differential equations to the solution of ordinary nonlinear differential equations by removing the singularity using an appropriate Taylor’s approximation. Then we have demonstrated the solution of these ordinary nonlinear differential equations by RKM and ADM. The ADM is an accurate and efficient method to solve nonlinear weakly singular Volterra integro-differential equations.



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*Source: 258067-2012-12-03.xml*