# On Solution of Fredholm Integrodifferential Equations Using Composite Chebyshev Finite Difference Method

Authors: Z. Pashazadeh Atabakan; A. Kazemi Nasab; A. Kılıçman
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

A new numerical method is introduced for solving linear
Fredholm integrodifferential equations which is based on a hybrid of
block-pulse functions and Chebyshev polynomials using the well-known
Chebyshev-Gauss-Lobatto collocation points. Composite Chebyshev finite
difference method is indeed an extension of the Chebyshev finite
difference method and can be considered as a nonuniform finite difference
scheme. The main advantage of the proposed method is reducing
the given problem to a set of algebraic equations. Some examples are
given to approve the efficiency and the accuracy of the proposed method.


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## Body

## 1. Introduction

Linear and nonlinear Fredholm integrodifferential equations can be used to model many problems of science and theoretical physics such as engineering, biological models, electrostatics, control theory of industrial mathematics, economics, fluid dynamics, heat and mass transfer, oscillation theory, and queuing theory [1].In recent years, many authors have considered different numerical methods to solve these kinds of problems. In 2012, Dehghan and Salehi employed [2] the meshless moving least square method for solving nonlinear Fredholm integrodifferential equations. A sequential method for the solution of Fredholm integrodifferential equations was presented by Berenguer et al. [3] in 2012. The formulation of the Fredholm integrodifferential equation in terms of an operator and the use of Schauder bases are the main tools of this method.In [4], the operational Adomian-Tau method with Pade approximation was used for solving nonlinear Fredholm integrodifferential equations. This approach is based on two matrices, and Pade approximation was used to improve the accuracy of the method. Chebyshev finite difference method was proposed in [5] in order to solve Fredholm integrodifferential equations. In this scheme the problem is reduced to a set of algebraic equations. In [6], Legendre collocation matrix method was introduced for solving high-order linear Fredholm integrodifferential equations. In this way, the equation and its conditions are converted to matrix equations using collocation points on the interval [-1,1]. Atabakan et al. [7, 8] proposed a modification of homotopy analysis method (HAM) known as spectral homotopy analysis method (SHAM) to solve linear Volterra and Fredholm integrodifferential equations. In this procedure, the Chebyshev pseudospectral method was used to obtain an approximation of solutions to higher-order equation. The semiorthogonal spline method was discussed in [9]. This approach is used to solve Fredholm integral and integrodifferential equations.In this paper, we applied a composite Chebyshev finite difference (ChFD) method for solving Fredholm integrodifferential equations. Fredholm integro differential equation is given by(1)F(x,y(x),y′(x),y′′(x),…,y(n)(x))=f(x)+λ∫0T‍k(x,t)y(t)dt,Gr(y(τ0),…,y(n-1)(τ0),…,y(τn),…,y(n-1)(τn))=0,r=0,…,n-1,
where k(x,t),f(x), and y(x) are analytic functions, λ is a constant value, Gr,r=0,…,n-1, are linear functions and the points τ0,τ1,…,τn lie in [0,T]. It will always be assumed that (1) possesses a unique solution y∈Cn[0,T].The base of the proposed method is a hybrid of block-pulse functions and Chebyshev polynomials using Chebyshev-Gauss-Lobatto points. This method was introduced and applied for solving the optimal control of delay systems with a quadratic performance index in [10, 11].Chebyshev polynomials which are the eigenfunctions of a singular Sturm-Liouville problem have many advantages. They can be considered as a good representation of smooth functions by finite Chebyshev expansions provided that the function is infinitely differentiable. The Chebyshev expansion coefficients converge faster than any finite power of1/m as m goes to infinity for problems with smooth solutions. The numerical differentiation and integration can be performed. Moreover, they have been applied to solve different kinds of boundary value problems [12–14].The paper is organized in the following way. Section2 includes some necessary preliminaries and notations. Chebyshev finite difference method and composite Chebyshev finite difference method for solving Fredholm integrodifferential equations are described in Sections 3 and 4, respectively. Convergence analysis of the proposed method is presented in Section 5. In Section 6 discretization of the method is introduced, and some numerical examples are presented in Section 7. In Section 8, concluding remarks are given.


## 2. Preliminaries and Notations

In this section, we present some notations, definitions, and preliminary facts that will be used further in this work.


### 2.1. Block-Pulse Functions (BPF)

In order to introduce block-pulse functions, we first suppose the interval[0,T) to be divided into K equidistant subintervals [((k-1)/K)T,(k/K)T), k=1,2,…,K. A set of block-pulse functions B(K)(t) composed of K orthogonal functions with piecewise constant values is defined on the semiopen interval [0,T) as follows:
(2)B(K)(t)=[b1(t),b2(t),…,bk(t),…,bK(t)],
where the kth component is given by
(3)bk(t)={1,(k-1K)T≤t<(kK)T,0,otherwise.
Block-pulse functions have some nice characteristics. They are disjoint and orthogonal; that is,
(4)bk(t)bl(t)={bk(t),k=l,0,k≠l,∫0T‍bk(t)bl(t)dt=∥bk(t)∥2δk,l,
where δk,l is the Kronecker delta function. In addition, the set of block-pulse functionies satisfy Parseval’s identity when K tends to infinity. In other words, for any function v∈£2[0,T),
(5)∥v∥2=∫0T‍v2(t)dt=∑k=1∞‍(∫0T‍v(t)bk(t)dt)2=∑k=1∞‍ck2∥bk(t)∥2,
where
(6)ck=1∥bk(t)∥∫0T‍v(t)bk(t)dt,k=1,2,3,…,
so they are complete. For more information about block-pulse functions, interested reader is referred to [20–30].


### 2.2. Chebyshev Polynomials

Chebyshev polynomials of the first kind of degreem can be defined as follows [12]:
(7)Tm(t)=cosmβ,β=arccost,
which are orthogonal with respect to the weight function w(t)=1/1-t2, where
(8)〈Tm,Tn〉£w2[-1,1]={0,m≠n,π,m=n=0,π2,m=n≥1.
Chebyshev polynomials also satisfy the following recursive formula:
(9)T0(t)=1,T1(t)=t,Tm+1(t)=2tTm(t)-Tm-1(t).
The set of Chebyshev polynomials is a complete orthogonal set in the Hilbert space £w2[-1,1]. The Chebyshev expansion of a function f∈£w2[-1,1] is
(10)f(t)=∑m=0∞‍f^mTm(t),f^m=2πcm∫-11‍f(t)Tm(t)w(t)dx,
where
(11)cm={2,m=0,1,m≥1.
They have also another useful characteristic; see [14]. If
(12)h(t)=12h0T0(t)+∑m=1∞‍hmTm(t),
then
(13)∫-11‍h(t)dt=h0-∑m=2∞‍1+(-1)mm2-1hm.


### 2.3. Hybrid Functions of Block-Pulse and Chebyshev Polynomials

The orthogonal set of hybrid functionsbkm(t),k=1,2,…,K,m=0,1,…M, is defined on the interval [0,T) as
(14)bkm(t)={Tm(2KTt-2k+1),t∈[(k-1K)T,(kK)T),0,otherwise,
where k and m are the order of block-pulse functions and Chebyshev polynomials, respectively. The set of hybrid functions of block-pulse and Chebyshev polynomials is a complete orthogonal set in the Hilbert space £wk2[0,T) because the set of block-pulse functions and the set of Chebyshev polynomials are completely orthogonal. In view of the following formula:
(15)∫0T‍bkl(t)bpq(t)wk(t)dt=πT4Kclδkpδlq,
in which δkp is the Kronecker delta function and wk(t),k=1,2,…,K, are the corresponding weight functions on the kth subinterval [((k-1)/K)T,(k/K)T) and defined as
(16)wk(t)=w(2KTt-2k+1),
we can conclude that the hybrid functions are orthogonal with respect to weight functions wk. The set of hybrid functions is complete, so any function v∈£wk2[0,T) can be written as follows:
(17)v(t)=∑k=1K‍∑m=0∞‍v^kmbkm(t),
in which
(18)v^km=(v,bkm)wk(bkm,bkm)wk=4KπcmT∫((k-1)/K)T(k/K)T‍v(t)bkm(t)wk(t)dt,
where (·,·)wk is the weighted inner product.


## 2.1. Block-Pulse Functions (BPF)

In order to introduce block-pulse functions, we first suppose the interval[0,T) to be divided into K equidistant subintervals [((k-1)/K)T,(k/K)T), k=1,2,…,K. A set of block-pulse functions B(K)(t) composed of K orthogonal functions with piecewise constant values is defined on the semiopen interval [0,T) as follows:
(2)B(K)(t)=[b1(t),b2(t),…,bk(t),…,bK(t)],
where the kth component is given by
(3)bk(t)={1,(k-1K)T≤t<(kK)T,0,otherwise.
Block-pulse functions have some nice characteristics. They are disjoint and orthogonal; that is,
(4)bk(t)bl(t)={bk(t),k=l,0,k≠l,∫0T‍bk(t)bl(t)dt=∥bk(t)∥2δk,l,
where δk,l is the Kronecker delta function. In addition, the set of block-pulse functionies satisfy Parseval’s identity when K tends to infinity. In other words, for any function v∈£2[0,T),
(5)∥v∥2=∫0T‍v2(t)dt=∑k=1∞‍(∫0T‍v(t)bk(t)dt)2=∑k=1∞‍ck2∥bk(t)∥2,
where
(6)ck=1∥bk(t)∥∫0T‍v(t)bk(t)dt,k=1,2,3,…,
so they are complete. For more information about block-pulse functions, interested reader is referred to [20–30].


## 2.2. Chebyshev Polynomials

Chebyshev polynomials of the first kind of degreem can be defined as follows [12]:
(7)Tm(t)=cosmβ,β=arccost,
which are orthogonal with respect to the weight function w(t)=1/1-t2, where
(8)〈Tm,Tn〉£w2[-1,1]={0,m≠n,π,m=n=0,π2,m=n≥1.
Chebyshev polynomials also satisfy the following recursive formula:
(9)T0(t)=1,T1(t)=t,Tm+1(t)=2tTm(t)-Tm-1(t).
The set of Chebyshev polynomials is a complete orthogonal set in the Hilbert space £w2[-1,1]. The Chebyshev expansion of a function f∈£w2[-1,1] is
(10)f(t)=∑m=0∞‍f^mTm(t),f^m=2πcm∫-11‍f(t)Tm(t)w(t)dx,
where
(11)cm={2,m=0,1,m≥1.
They have also another useful characteristic; see [14]. If
(12)h(t)=12h0T0(t)+∑m=1∞‍hmTm(t),
then
(13)∫-11‍h(t)dt=h0-∑m=2∞‍1+(-1)mm2-1hm.


## 2.3. Hybrid Functions of Block-Pulse and Chebyshev Polynomials

The orthogonal set of hybrid functionsbkm(t),k=1,2,…,K,m=0,1,…M, is defined on the interval [0,T) as
(14)bkm(t)={Tm(2KTt-2k+1),t∈[(k-1K)T,(kK)T),0,otherwise,
where k and m are the order of block-pulse functions and Chebyshev polynomials, respectively. The set of hybrid functions of block-pulse and Chebyshev polynomials is a complete orthogonal set in the Hilbert space £wk2[0,T) because the set of block-pulse functions and the set of Chebyshev polynomials are completely orthogonal. In view of the following formula:
(15)∫0T‍bkl(t)bpq(t)wk(t)dt=πT4Kclδkpδlq,
in which δkp is the Kronecker delta function and wk(t),k=1,2,…,K, are the corresponding weight functions on the kth subinterval [((k-1)/K)T,(k/K)T) and defined as
(16)wk(t)=w(2KTt-2k+1),
we can conclude that the hybrid functions are orthogonal with respect to weight functions wk. The set of hybrid functions is complete, so any function v∈£wk2[0,T) can be written as follows:
(17)v(t)=∑k=1K‍∑m=0∞‍v^kmbkm(t),
in which
(18)v^km=(v,bkm)wk(bkm,bkm)wk=4KπcmT∫((k-1)/K)T(k/K)T‍v(t)bkm(t)wk(t)dt,
where (·,·)wk is the weighted inner product.


## 3. Chebyshev Finite Difference Method

We can approximate a functionf(t) in terms of Chebyshev polynomials as follows [31]:
(19)(PM)f(t)=∑′′Mk=0fkTk(t),fk=2M∑′′Mk=0f(tk)Tm(tk),
with double primes meaning that the first and last terms should be halved. Moreover, tk are the extrema of the Mth-order Chebyshev polynomial TM(t) and defined as
(20)tk=cos(kπM),k=0,1,2,…,M.
In view of (7), we have
(21)Tm(tk)=cos(mkπM),
so fm can be rewritten as
(22)fm=2M∑′′Mk=0f(tk)cos(mkπM).
The first three derivatives of the function f(t) at the points tm,m=0,1,…,M, are given in [32, 33] as
(23)f(i)(tm)=∑j=0M‍dm,j(i)f(tj),i=1,2,3,
where
(24)dm,j(1)=4θjM∑n=1M‍∑l=0(n+l)oddn-1‍nθnclTn(tj)Tl(tm)=4θjM∑n=1M‍∑l=0(n+l)oddn-1‍nθnclcos(njπM)cos(lmπM),dm,j(2)=2θjM∑n=2M‍∑l=0(n+l)evenn-2‍n(n2-l2)θnclTn(tj)Tl(tm)=2θjM∑n=2M‍∑l=0(n+l)evenn-2‍n(n2-l2)θnclcos(njπM)×cos(lmπM),(25)dm,j(3)=4θjM∑n=2M‍∑l=1(n+l)evenn-2‍∑r=0(l+r)oddl-1‍nl(n2-l2)θnclcr×Tn(tj)Tr(tm)=4θjM∑n=2M‍∑l=1(n+l)evenn-2‍×∑r=0(l+r)oddl-1‍nl(n2-l2)θnclcrcos(njπM)×cos(lmπM),
with θ0=θM=1/2,θj=1 for j=1,2,…M-1.As can be seen from (23), the first three derivatives of the function f(t) at any point of the Chebyshev-Gauss-Lobatto points is expanded as a linear combination of the values of the function at these points.In view of (13) and (19), we have
(26)∫-11‍f(t)dt≈f0-∑m=2M-1‍1+(-1)mm2-1fm-1+(-1)M2(M2-1)fM.


## 4. Composite Chebyshev Finite Difference Method

In this Section, we present the composite Chebyshev finite difference (ChFD) method. Considertkm,k=1,2,…,K,m=0,1,…,M, as the corresponding Chebyshev-Gauss-Lobatto collocation points at the kth subinterval [(k-1)/K,k/K] such that
(27)tkm=T2K(tm+2k-1).
A function f(t) can be written in terms of hybrid basis functions as follows:
(28)(PM)f(t)=∑k=1K‍∑′′Mm=0fkmbkm(t),
where fkm,n=1,2,…,K,m=0,1,…,M, are the expansion coefficients of the function f(t) at the subinterval [(k-1)/K,k/K] and bk,m(t),k=1,2,…,K,m=0,1,…,M, are defined in (14).In view of (14) and (19), we can obtain the coefficients fkm as
(29)fkm=2M∑′′Mp=0f(tkp)bkm(tkp)=2M∑′′Mp=0f(tkp)cos(mpπM).
Using (23)–(25), the first three derivatives of the function f(t) at the points tkm,k=1,2,…,K,m=0,1,…,M, can be obtained as
(30)f(i)(tkm)=∑j=0M‍dk,m,j(i)f(tkj),i=1,2,3,
where
(31)dk,m,j(1)=8NθjTM∑n=1M‍∑l=0(n+l)oddn-1‍nθnclbkn(tkj)bkl(tkm)=8NθjTM∑n=1M‍∑l=0(n+l)oddn-1‍nθnclcos(njπM)cos(lmπM),dm,j(2)=8K2θjT2M∑n=2M‍∑l=0(n+l)evenn-2‍n(n2-l2)θnclTn(tj)Tl(tm)=8N2θjT2M∑n=2M‍∑l=0(n+l)evenn-2‍n(n2-l2)θnclcos(njπM)×cos(lmπM),dm,j(3)=32K3θjT3M∑n=2M‍∑l=1(n+l)evenn-2‍×∑r=0(l+r)oddl-1‍nl(n2-l2)θnclcrTn(tj)Tr(tm)=32N3θjT3M∑n=2M‍∑l=1(n+l)evenn-2‍×∑r=0(l+r)oddl-1‍nl(n2-l2)θnclcr×cos(njπM)cos(lmπM).In view of (26) and (28), we get
(32)∫0T‍f(t)dt≈T2N∑k=1K‍fk0-∑m=2M-1‍1+(-1)mm2-1fkm-1+(-1)M2(M2-1)fkM.


## 5. Convergence Analysis

A detailed proof of the following results can be found in [11].Lemma 1.
If the hybrid expansion of a continuous functionh(t) converges uniformly, then it converges to the function h(t).Theorem 2.
A functionh(t)∈£wk2[0,T) with bounded second derivative, say |h′′(t)|≤B, can be expanded as an infinite sum of hybrid functions and the series converges uniformly to h(t), that is,
(33)h(t)=∑k=1∞‍∑m=0∞‍h^kmbkm(t).Theorem 3.
Suppose thath(t)∈£wk2[0,T) with bounded second derivative, say |h′′(t)|≤B, and then its hybrid expansion converges uniformly to h(t); that is,
(34)∑k=1K∑′∞m=0‍hkmbkm(t)=h(t),
where the summation symbol with prime denotes a sum with the first term halved.Theorem 4 (accuracy estimation).
Suppose thath(t)∈Lwk2[0,T) with bounded second derivative, say |h′′(t)|≤B, and then one has the following accuracy estimation:
(35)σK,M≤(S+∑k=K+1∞‍∑m=M+1∞‍C2(m2-1)2×πTcm4K)1/2,
where
(36)σK,M=(∫0T‍[h(t)-∑k=1K‍∑′′Mm=0fkmbkm(t)]2wk(t)dt)1/2,C=Rπ312+BT2k2cm,R=max{d2dβ2(h(cos(β)+2k-12KT)cos(mβ)),(h(cos(β)+2k-12KT)cos(mβ))0≤β≤π}S=14h102πT2K+14hKM2πT4K.Proof.
Consider(37)σkM2=14h102∫0T‍b102(t)wk(t)dt+14hKM2∫0T‍bKM2(t)wk(t)dt+∫0T‍[h(t)-∑k=1K‍∑m=0M‍hkmbkm(t)]2wk(t)dt=14h102πT2K+14hKM2πT4K+∫0T‍∑k=K+1∞‍∑m=M+1∞‍hkm2bkm2(t)wk(t)dt=S+∑k=K+1∞‍∑m=M+1∞‍hkm2∫0T‍bkm2(t)wk(t)dt=S+∑k=K+1∞‍∑m=M+1∞‍hkm2×πTcm4K.
With the aid of (15) and the proof of Theorem 3, we will have
(38)σkM2≤S+∑k=K+1∞‍∑m=M+1∞‍C2(m2-1)2×πTcm4K,
where
(39)C=Rπ312+BT2k2cm,R=max{d2dβ2(h(cos(β)+2k-12KT)cos(mβ)),(h(cos(β)+2k-12KT)cos(mβ))0≤β≤π},S=14h102πT2K+14hKM2πT4K.


## 6. Discretization of Problem

In this section, we apply the composite ChFD method to solve Fredholm integrodifferential equations of the form (1). For this purpose, we approximate integral part in (1) using (32). We expand k(x,t)y(t) in terms of hybrid functions:
(40)k(x,t)y(t)≈∑k=1K‍∑′′Mm=0fkmbkm(t),
where
(41)fkm=2M∑′′Mp=0(k(x,tkp)y(tkp))bkm(tkp)=2M∑′′Mp=0(k(x,tkp)y(tkp))cos(mpπM);
with aid of (32), we will have
(42)∫0T‍k(x,t)y(t)dt≈T2K∑k=1K‍fk0-∑m=2M-1‍1+(-1)mm2-1fkm-1+(-1)M2(M2-1)fkM.
In order to obtain the solution y(x) in (1), by applying the composite ChFD method, we first collocate (1) in Chebyshev-Gauss-Lobatto collocation points tkm,k=1,…,K,m=0,1,…,M-n. In addition, substituting (28) and (30) into boundary conditions (1), we get n equations. Moreover, the approximate solution and its first n derivatives should be continuous at the interface of subintervals; that is,
(43)y(i)(tk,0)=y(i)(tk+1,M),k=1,2,…,K-1,i=0,1,…n-1.
Therefore, we have a system of K(M+1) algebraic equations, which can be solved by using Newton's iterative method for the unknowns y(tkm),k=0,1,…,K,m=0,1,…,M. Consequently, the approximate solution y(x) of (1) can be calculated.


## 7. Numerical Examples

In this section, we apply the technique described in Section6 to some illustrative examples of higher-order linear Fredholm integrodifferential equations.Example 1.
Consider the second-order Fredholm integrodifferential equation [5, 15]
(44)y′′(x)+4xy′(x)=-8x4(x2+1)3-2∫01‍t2+1(x2+1)2y(t)dt,0≤x,t≤1
subject to the boundary conditions
(45)y(0)=1,y(1)=1,
with the exact solution y(x)=1/(x2+1).We solve the problem withM=10, and K=8. A comparison between absolute errors in solutions obtained by composite Chebyshev finite difference method, wavelet collocation method, wavelet Galerkin and Chebyshev finite difference method is tabulated in Table 1. As can been seen in Table 1, our results are much more accurate than those K obtained by other methods specially wavelet collocation method.Table 1
A comparison of absolute errors between Wc, WG, Cfd, and present method.
x
Wavelet collocation [15]
Wavelet Galerkin [15]
Chebyshev finite difference [5]
Present method
0.125
9.3
×
1
0
-
4
7.9
×
1
0
-
7
2.1
×
1
0
-
9
1.16
×
1
0
-
15
0.250
1.6
×
1
0
-
3
1.3
×
1
0
-
6
2.0
×
1
0
-
8
2.28
×
1
0
-
15
0.375
2.0
×
1
0
-
3
1.6
×
1
0
-
6
1.8
×
1
0
-
7
1.27
×
1
0
-
15
0.500
1.9
×
1
0
-
3
1.6
×
1
0
-
6
1.9
×
1
0
-
8
3.15
×
1
0
-
16
0.625
1.6
×
1
0
-
3
1.5
×
1
0
-
6
1.9
×
1
0
-
7
2.79
×
1
0
-
17
0.750
1.1
×
1
0
-
3
1.1
×
1
0
-
6
4.9
×
1
0
-
8
1.63
×
1
0
-
16
0.875
5.5
×
1
0
-
4
6.5
×
1
0
-
7
4.2
×
1
0
-
8
1.52
×
1
0
-
15The maximum errors for approximate solutionyKM(x) can be defined as
(46)EKM=∥yKM-yexact(x)∥∞=max{|yKM(x)-yexact(x)|,0≤x≤1},
where the computed result with K is shown by yKM and yexact(x) is the exact solution. For different values of K, the errors of EKM are presented in Table 2.Table 2
The maximum errors ofEKM for different values of K and M.
K
4
10
8
M
8
8
10
E
K
M
6.66
×
1
0
-
10
3.01
×
1
0
-
13
2.28
×
1
0
-
15Example 2.
Consider the second-order Fredholm integrodifferential equation [5, 15]
(47)x2y′′(x)+50xy′(x)-35y(x)=1-ex+1x+1+(x2+50x-35)ex+∫01‍exty(t)dt,0≤x,t≤1,
subject to the boundary condition
(48)y(0)=1,y(1)=e.
The exact solution of this equation is y(x)=ex.The problem is solved withM=7, and K=8. A comparison between absolute errors in solutions by composite Chebyshev finite difference method, wavelet collocation method, wavelet Galerkin and Chebyshev finite difference method is tabulated in Table 3. It is clear from Table 3 that our method is reliable and applicable to handle Fredholm integrodifferential equations. For different values of K, the errors of EKM are shown in Table 4.Table 3
A comparison of absolute errors between Wc, WG, Cfd, and present method.
x
Wavelet collocation [15]
Wavelet Galerkin [15]
Chebyshev  finite difference [5]
Present method
0.125
2.6
×
1
0
-
2
2.7
×
1
0
-
4
1.8
×
1
0
-
10
1.45
×
1
0
-
15
0.250
1.5
×
1
0
-
2
3.0
×
1
0
-
5
4.4
×
1
0
-
10
1.78
×
1
0
-
15
0.375
9.3
×
1
0
-
3
2.6
×
1
0
-
4
1.4
×
1
0
-
9
1.71
×
1
0
-
15
0.500
5.1
×
1
0
-
3
4.3
×
1
0
-
4
2.4
×
1
0
-
10
5.00
×
1
0
-
15
0.625
2.5
×
1
0
-
3
5.6
×
1
0
-
4
1.7
×
1
0
-
9
1.22
×
1
0
-
15
0.750
1.0
×
1
0
-
3
6.5
×
1
0
-
4
7.7
×
1
0
-
10
8.66
×
1
0
-
16
0.875
2.3
×
1
0
-
4
7.2
×
1
0
-
4
1.3
×
1
0
-
9
4.52
×
1
0
-
16Table 4
The maximum errors ofEKM for different values of K and M.
K
8
10
8
M
6
6
7
E
K
M
6.01
×
1
0
-
13
8.72
×
1
0
-
14
1.22
×
1
0
-
15Example 3.
Consider the second-order Fredholm integrodifferential equation [5, 16]
(49)exy′′(x)+cos(x)y′(x)+sin(x)y(x)+∫-11‍e(x+1)ty(t)dt=(cos(x)+sin(x)+ex)ex-2sinh(x+2)x+2,-1≤x,t≤1,
subject to
(50)y(-1)+y(1)=e+1e,y(-1)+y′(-1)+y(1)=e,
with the exact solution y(x)=ex.In order to apply the presented method for solving the given problem, we should transform using an appropriate change of variables as(51)x=2ζ-1,ζ∈[0,1].
In this example, we set M=9, and K=10. In Table 5, absolute errors in solutions obtained by composite Chebyshev finite difference method are compared with Tau method and Chebyshev finite difference method. According to Table 5 using the proposed method, we can obtain approximate solution which is almost same as exact solution. For different values of K the errors of EKM are shown in Table 6.Table 5
A comparison of absolute errors between Tm, Cfm, and present method.
x
Exact solution
Tau method [16]
Chebyshev finite difference [5]
Present method
−1.0
0.367879441
1.52
×
1
0
-
6
1.19
×
1
0
-
8
1.32
×
1
0
-
16
−0.8
0.449328964
1.74
×
1
0
-
6
1.33
×
1
0
-
8
1.36
×
1
0
-
16
−0.6
0.548811636
1.95
×
1
0
-
6
1.29
×
1
0
-
8
1.38
×
1
0
-
16
−0.4
0.670320046
2.02
×
1
0
-
6
1.43
×
1
0
-
8
1.40
×
1
0
-
16
0.2
0.818730753
1.97
×
1
0
-
6
1.27
×
1
0
-
8
1.20
×
1
0
-
16
0.0
1.000000000
1.83
×
1
0
-
6
1.02
×
1
0
-
8
9.99
×
1
0
-
16
0.2
1.221402758
1.63
×
1
0
-
6
1.04
×
1
0
-
8
7.91
×
1
0
-
17
0.4
1.491824698
1.36
×
1
0
-
6
8.68
×
1
0
-
9
7.31
×
1
0
-
17
0.6
1.822118800
1.04
×
1
0
-
6
2.92
×
1
0
-
9
3.08
×
1
0
-
17
0.8
2.225540928
5.56
×
1
0
-
7
1.65
×
1
0
-
9
3.69
×
1
0
-
17
1.0
2.718281828
1.52
×
1
0
-
6
1.19
×
1
0
-
8
1.32
×
1
0
-
16Table 6
The maximum errors ofEKM for different values of K and M.
K
4
10
8
M
8
8
10
E
K
M
4.07
×
1
0
-
12
3.93
×
1
0
-
15
2.79
×
1
0
-
17Example 4.
Consider the first-order Fredholm integrodifferential equation [3, 17–19]
(52)y′(x)=(x+1)ex-x+∫01‍xy(t)dt,0≤x,t≤1,
subject to
(53)y(0)=0.M=9, and K=4 are considered to solve Example 4. In Table 7, absolute errors in solutions obtained by composite Chebyshev finite difference method are compared with CAS wavelet method, differential transfer method, Improved homotopy perturbation method, and a sequential method. It is illustrated in Table 7 that the results obtained using current method are very closed to exact solution. The graph of absolute errors for M=9, and K=4 is shown in Figure 1.Table 7
A comparison of absolute errors between DTM, IHPM, Sa, and present method.
x
CAS  wavelet method [17]
DT method [18]
Improved  homotopy perturbation [19]
Sequential approach [3]
Present method
0.1
1.34
×
1
0
-
3
1.00
×
1
0
-
2
0.23
×
1
0
-
5
1.01
×
1
0
-
7
1.25
×
1
0
-
17
0.2
1.15
×
1
0
-
3
2.78
×
1
0
-
2
0.92
×
1
0
-
5
4.82
×
1
0
-
7
4.27
×
1
0
-
17
0.3
5.67
×
1
0
-
3
5.08
×
1
0
-
2
0.20
×
1
0
-
4
1.017
×
1
0
-
6
1.46
×
1
0
-
16
0.4
5.93
×
1
0
-
2
7.08
×
1
0
-
2
0.37
×
1
0
-
4
1.61
×
1
0
-
6
1.53
×
1
0
-
16
0.5
1.32
×
1
0
-
2
9.71
×
1
0
-
2
0.57
×
1
0
-
4
2.30
×
1
0
-
6
1.44
×
1
0
-
16
0.6
4.39
×
1
0
-
2
1.09
×
1
0
-
1
0.83
×
1
0
-
4
3.09
×
1
0
-
6
1.68
×
1
0
-
16
0.7
1.41
×
1
0
-
2
1.04
×
1
0
-
1
0.11
×
1
0
-
3
3.97
×
1
0
-
6
1.74
×
1
0
-
16
0.8
1.34
×
1
0
-
2
6.94
×
1
0
-
2
0.14
×
1
0
-
3
4.90
×
1
0
-
6
5.40
×
1
0
-
17
0.9
1.32
×
1
0
-
2
1.00
×
1
0
-
2
0.18
×
1
0
-
3
6.13
×
1
0
-
6
1.72
×
1
0
-
17Figure 1
The graph of absolute errors for Example4 for K=4, and M=9.Example 5.
Consider the first-order Fredholm integrodifferential equation [6, 34, 35]
(54)y′′(x)+xy′(x)-xy=ex-2sin(x)+∫-11‍sin(x)e-ty(t)dt,-1≤x,t≤1,
subject to
(55)y(0)=1,y′(0)=1.
This example is solved for M=8 and K=4. In order to apply the presented method for solving the given problem, we should transform using an appropriate change of variables as
(56)x=2ζ-1,ζ∈[0,1].
In Table 8, absolute errors in solutions obtained by composite Chebyshev finite difference method are compared with Legendre polynomial method. As can be shown in Table 8, the introduced method is more efficient than Legendre polynomial method, and the numerical results are in good agreement with exact solutions up to 13 decimal places. The graph of absolute errors for K=4, and M=8 is shown in Figure 2.Table 8
A comparison of absolute errors between Lps and ChFd.
x
Legendre  polynomial solutions [6]
Present method
−1.0
1.00
×
1
0
-
8
0
−0.8
1.00
×
1
0
-
8
2.98
×
1
0
-
13
−0.6
0.00
6.56
×
1
0
-
13
−0.4
1.00
×
1
0
-
8
9.80
×
1
0
-
13
−0.2
0.00
1.13
×
1
0
-
13
0.0
0.00
1.18
×
1
0
-
12
0.2
1.00
×
1
0
-
8
9.18
×
1
0
-
12
0.4
0.00
8.34
×
1
0
-
13
0.6
2.00
×
1
0
-
8
7.80
×
1
0
-
13
0.8
4.60
×
1
0
-
7
4.75
×
1
0
-
13
1.0
5.25
×
1
0
-
6
0Figure 2
The graph of absolute errors for Example5 for K=4, and M=8.Example 6.
Consider the third-order linear Fredholm integrodifferential equation [6]
(57)y′′′(x)-y′(x)=2x(cos1-sin1)-2cosx+∫-11‍xty(t)dt,
subject to
(58)y(0)=0,y′(0)=1,y′′(0)-2y′(0)=-2.
The exact solution for this problem is y(x)=sinx. We solve the problem with m=9, and n=10. In Table 9, absolute errors in solutions obtained by composite Chebyshev finite difference method are compared with Legendre polynomial solutions. For different values of K the errors of EKM are shown in Table 10. The graph of absolute errors for K=10, and M=9 is shown in Figure 3.Table 9
A comparison of absolute errors between Lps and present method.
x
Exact solution
Legendre polynomial solutions [6]
Present method
−1.0
−0.8414709848
4.39
×
1
0
-
9
7.00
×
1
0
-
20
−0.8
−0.7173560909
4.69
×
1
0
-
9
1.50
×
1
0
-
15
−0.6
−0.5646424734
1.19
×
1
0
-
9
4.91
×
1
0
-
15
−0.4
−0.3894183423
2.30
×
1
0
-
9
8.38
×
1
0
-
15
−0.2
−0.1986693308
9.50
×
1
0
-
11
1.05
×
1
0
-
14
0.0
0.0
1.99
×
1
0
-
17
1.07
×
1
0
-
14
0.2
0.1986693308
1.04
×
1
0
-
10
8.79
×
1
0
-
15
0.4
0.3894183423
1.06
×
1
0
-
8
5.34
×
1
0
-
15
0.6
0.5646424734
5.00
×
1
0
-
8
1.48
×
1
0
-
15
0.8
0.7173560909
1.35
×
1
0
-
6
1.11
×
1
0
-
15
1.0
0.8414709848
4.65
×
1
0
-
7
7.00
×
1
0
-
20Table 10
The maximum errors ofEKM for different values of K and M.
K
4
5
9
10
M
5
5
8
9
E
K
M
4.13
×
1
0
-
6
3.32
×
1
0
-
6
3.77
×
1
0
-
14
1.07
×
1
0
-
14Figure 3
The graph of absolute errors for Example6 for K=10, and M=9.


## 8. Conclusion

In this paper, we presented the composite Chebyshev finite difference method for solving Fredholm integrodifferential equations. The composite ChFD method is indeed an extension of the ChFD scheme withK=1. This method is based on a hybrid of block-pulse functions and Chebyshev polynomials using Chebyshev-Gauss-Lobatto collocation points.The useful properties of Chebyshev polynomials and block-pulse functions make it a computationally efficient method to approximate the solution of Fredholm integrodifferential equations. We converted the given problem to a system of algebraic equations using collocation points.The main advantage of the present method is the ability to represent smooth and especially piecewise smooth functions properly. It was also shown that the accuracy can be improved either by increasing the number of subintervals or by increasing the number of collocation points in subintervals. Several examples have been provided to demonstrate the powerfulness of the proposed method. A comparison was made among the present method, some other well-known approaches, and exact solution which confirms that the introduced method is more accurate and efficient.



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*Source: 694043-2013-06-11.xml*