# An Iterative Scheme for Solving Systems of Nonlinear Fredholm Integrodifferential Equations

Authors: M. I. Berenguer; D. Gámez; A. J. López Linares
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

Using fixed-point techniques and Faber-Schauder systems in adequate Banach spaces, we approximate the
solution of a system of nonlinear Fredholm integrodifferential equations of the second kind.


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## Body

## 1. Introduction

An important area of research interest is the study of systems of nonlinear Fredholm integrodifferential equations. A system of nonlinear Fredholm integrodifferential equations can be written in vectorial form as(1)X′(t)=F(t,X(t))+∫01‍K(t,s,X(s))dsmmmmmmmmmmmmm(t,s∈[0,1]),X(0)=ρ,
where X=[x1,…,xn]T∈C([0,1],Rn) is the solution to be calculated and F=[F1,…,Fn]T∈C([0,1]×Rn,Rn), K=[K1,…,Kn]T∈C([0,1]2×Rn,Rn), and ρ=[ρ1,…,ρn]T∈Rn are known.Observe that, fori=1,…,n, the ith equation of the system (1) adopts the form
(2)xi′(t)=Fi(t,x1(t),…,xn(t))+∫01‍Ki(t,s,x1(s),…,xn(s))ds,
with xi(0)=ρi.The system (1) is linear when for all t,s∈[0,1] and u=[u1,…,un]T∈Rn we have that
(3)K(t,s,u)=[K11(t,s)⋯K1n(t,s)⋮⋱⋮Kn1(t,s)⋯Knn(t,s)][u1⋮un],mmmmmmmmmmwithKij∈C([0,1]2,R).Many problems of physics and engineering lead to the solution of integro or integrodifferential equations or systems of such equations. In most cases, these cannot be solved by direct methods, and this, together with the powerful computer tools available, has led to the development of numerical methods that allow obtaining approximate solutions of these equations or systems of equations. In literature it is easy to find many of them.Danfu and Xufeng [1] utilize the CAS wavelet operational matrix of integration for obtaining numerical solution of linear Fredholm integrodifferential equations. Jafarian and Measoomy Nia [2] offer an architecture of artificial neural networks (NNs) for finding approximate solution of linear Fredholm integral equations system of the second kind. In [3], Maleknejad et al. present a rationalized Haar functions method for solving linear Fredholm integrodifferential systems. In [4] Maleknejad and Tavassoli Kajani use the hybrid Legendre and block-pulse functions on interval [0,1) to solve the systems of linear integrodifferential equations. In [5], a fully discrete version of a piecewise polynomial collocation method is constructed to solve initial or boundary value problems of linear Fredholm integrodifferential equations with weakly singular kernels. In [6], Pour-Mahmoud et al. extend the Tau method for the numerical solution of integrodifferential equations system (IDES). Yalçinbaş et al. [7] present a Legendre collocation matrix method to solve high-order linear Fredholm integrodifferential equations under the mixed conditions in terms of Legendre polynomials. Yusufoğlu in [8] introduce a numerical method for solving initial value problems for a system of integrodifferential equations (the main idea is based on the interpolations of unknown functions at distinct interpolation points). Yüzbaşı et al. [9] present a numerical matrix method based on collocation points for the approximate solution of the systems of high-order linear Fredholm integrodifferential equations with variable coefficients under mixed conditions in terms of the Bessel polynomials. Zarebnia and Ali Abadi [10] use the Sinc-collocation method to solve systems of nonlinear second-order integrodifferential equations. Berenguer et al. used in [11–14] von Neumann series, fixed-point techniques, and Faber-Schauder systems in Banach spaces to solve integro and integrodifferential equations.In the present paper we approximate the solution of (1) and we extend the numerical approximation method given in [14]. This paper is organized as follows. In Section 2 we describe the proposed method and in Section 3 the convergence of the proposed method is investigated. In Section 4 some numerical examples are presented to show the efficiency of the proposed scheme. Finally, in Section 5, we end with some conclusions.


## 2. Description of the Proposed Method

We suppose thatF, K satisfy a global Lipschitz condition in its last variable; that is, there exist LF,LK>0 such that for all t,s∈R and u,v∈Rn(4)∥F(t,u)-F(t,v)∥∞≤LF∥u-v∥∞,∥K(t,s,u)-K(t,s,v)∥∞≤LK∥u-v∥∞.If we reformulate the system (1) in terms of an adequate operator, we can derive its unique solvability under a suitable condition. To be more precise, if V:C([0,1],Rn)→C([0,1],Rn) is the operator given for each X as
(5)V(X)(·)∶=ρ+∫0(·)‍F(u,X(u))du+∫0(·)‍∫01‍K(u,s,X(s))dsdu,
then solving (1) is equivalent to finding a fixed point X of the operator V.A direct calculation overV leads to
(6)∥V(Y)-V(Z)∥≤L∥Y-Z∥
for all Y,Z∈C([0,1],Rn), where L:=LF+LK, with LF and LK being the Lipschitz constants of F and K, respectively. Thus, according to the Banach fixed-point theorem (see [15]), (5) has one unique fixed point; equivalently, (1) has one and only one solution X∈C([0,1],Rn) provided that L<1. In addition, for each X~∈C([0,1],Rn),
(7)∥Vm(X~)-X∥≤Lm1-L∥V(X~)-X~∥
and in particular X=lim⁡mVm(X~).Then, givenX~=[x~1,…,x~n]T∈C([0,1],Rn), our next target is to obtain Vm(X~). We consider the functions φ∈C([0,1],Rn) and ξ∈C([0,1]2,Rn) defined by
(8)φ(t)∶=F(t,X~(t))=[φ1(t),…,φn(t)]T,ξ(t,s)∶=K(t,s,X~(s))=[ξ1(t,s),…,ξ1(t,s)]T.Observe thatV(X~)(t)=[(V(X~))1(t),…,(V(X~))n(t)]T, where for all i=1,…,n(9)(V(X~))i(t)=ρi+∫0t‍φi(u)du+∫0t‍∫01‍ξi(u,s)dsdu.Now we will make use of the usual Schauder basis{bi}i≥1 in C([0,1],R) and the usual Schauder basis {Bi}i≥1 for the Banach space C([0,1]2,R) (see [16, 17]), although the numerical method given works equally well by replacing it with any complete biorthogonal system in this space. We denote by {Pi}i≥1 and {Qi}i≥1 the sequences of projections in C([0,1],R) and C([0,1]2,R), respectively (see Section  3 in [11]).Then, for allt∈[0,1] and i=1,…,n,
(10)(V(X~))i(t)=ρi+∑k≥1‍λik∫0t‍bk(u)du+∑k≥1‍δik∫0t‍∫01‍Bk(u,s)dsdu,
where {λik}k≥1 and {δik}k≥1 are the sequences of scalars satisfying φi=∑k≥1‍λikbk and ξi=∑k≥1‍δikBk, where λi1=φi(t1) and δi1=ξi(t1,t1), and for k≥2 is λik=φi(tk)-∑l=1k-1‍bl*(φi)bl(tk) and δik=ξi(tp,tq)-∑l=1k-1‍Bl*(ξi)Bl(tp,tq) with σ(k)=(p,q).In view of (10) we can calculate, at least in a theoretical way, Vm(X~). From a practical point of view, in general these calculations are not possible explicitly, since they are infinite sums. The idea of our numerical method is to truncate them by means of the projections of the Schauder bases {bi}i≥1, {Bi}i≥1 and approximate the solution in this way. Specifically, we consider the sequence {Xr}r≥1 defined as follows. Let X0(t):=X~(t)=[x01(t),…,x0n(t)]T∈C([0,1],Rn), t∈[0,1], and {i1,i2,…}, {j1,j2,…} be subsets of natural numbers and m∈N. Define inductively for r∈{1,…,m} and t,s∈[0,1](11)Xr(t)=[xr1(t),…,xrn(t)]T∶=ρ+[∫0t‍Pir(ψr-1,1(u))du,…,∫0t‍Pir(ψr-1,n(u))du]T+[∫0t‍∫01‍Qjr2(ϕr-1,1(u,s))dsdu,…,∫0t‍∫01‍Qjr2(ϕr-1,n(u,s))dsdu]T,
where
(12)Ψr-1(t)=[ψr-1,1(t),…,ψr-1,n(t)]T∶=F(t,Xr-1(t)),ψr-1,i(t)=Fi(t,Xr-1(t))fori=1,…,n,Φr-1(t,s)=[ϕr-1,1(t,s),…,ϕr-1,n(t,s)]T∶=K(t,s,Xr-1(s)),ϕr-1,i(t,s)=Ki(t,s,Xr-1(s))fori=1,…,n.
Observe that for i=1,…,n(13)xri(t)=ρi+∫0t‍Pir(ψr-1,i(u))du+∫0t‍∫01‍Qjr2(ϕr-1,i(u,s))dsdu.


## 3. Convergence of the Scheme

This section is devoted to provide a convergence analysis for the numerical scheme{Xr}r≥1. To analyze the convergence we employ the following two results.Theorem 1.
LetF∈C1([0,1]×Rn,Rn) and K∈C1([0,1]2×Rn,Rn) such that F,K and ∂Fi/∂t, ∂Fi/∂uj, ∂Ki/∂t, ∂Ki/∂s, and ∂Ki/∂uj for each i,j=1,…,n satisfy a global Lipschitz condition in the last variables. Then, maintaining the notation above, the sequences {ψr-1,i′}r≥1, {∂ϕr-1,i/∂t}r≥1, and {∂ϕr-1,i/∂s}r≥1, with i=1,…,n, are bounded.Proof.
Let us fixi=1,…,n and write ∂Fi/∂u=[∂Fi/∂u1,…,∂Fi/∂un]T, ∂Ki/∂u=[∂Ki/∂u1,…,∂Ki/∂un]T. Making use of definitions (11), it follows that, for all r≥1, i=1,…,n, and t,s∈[0,1],
(14)ψr-1,i′(t)=∂Fi∂t(t,Xr-1(t))+∂Fi∂u(t,Xr-1(t))·Xr-1′(t),∂ϕr-1,i∂t(t,s)=∂Ki∂t(t,s,Xr-1(s)),∂ϕr-1,i∂s(t,s)=∂Ki∂s(t,s,Xr-1(s))+∂Ki∂u(t,s,Xr-1(s))·Xr-1′(s),
where “·” stands for the usual inner product in Rn.
For allr≥1 and t,s∈[0,1], we have
(15)∥Ψr-1(t)∥∞=∥F(t,Xr-1(t))∥∞≤∥F(t,Xr-1(t))-F(t,0)∥∞+∥F(t,0)∥∞≤LF∥Xr-1(t)∥∞+RF,
with RF:=max⁡t∈[0,1]∥F(t,0)∥∞ and LF being the Lipschitz constant of F and analogous
(16)∥Φr-1(t,s)∥∞=∥K(t,s,Xr-1(s))∥∞≤∥K(t,s,Xr-1(s))-K(t,s,0)∥∞+∥K(t,s,0)∥∞≤LK∥Xr-1(s)∥∞+RK,
with RK:=max⁡(t,s)∈[0,1]2∥K(t,s,0)∥∞ and LK being the Lipschitz constant of K.
Now we will show that the sequence{Xr}r≥1 is bounded.
From the monotonicity of the Schauder bases{bi}i≥1, {Bi}i≥1 and the recursive application of this inequality and the following one,
(17)∥Xr(t)∥∞≤∥ρ∥∞+∫0t‍∥Ψr-1(u)∥∞du+∫0t‍∫01‍∥Φr-1(u,s)∥∞dsdu,
we have
(18)∥Xr(t)∥∞≤∥ρ∥∞+∫0t‍(LF∥Xr-1(u)∥∞+RF)du+∫0t‍∫01‍(LK∥Xr-1(s)∥∞+RK)dsdu≤Γ+L∥Xr-1∥,
with L:=LF+LK, Γ:=∥ρ∥∞+R and R:=RF+RK. Applying it inductively, we arrive at
(19)∥Xr∥≤Γ∑k=1r‍Lk-1+Lr∥X0∥,
for all r≥1, and therefore the sequence {Xr}r≥1 is bounded.
Since the sequence{Xr}r≥1 in (15) and (16) is bounded it follows that {Ψr-1}r≥1 and {Φr-1}r≥1 are uniformly bounded.
Fori=1,…,n, we have
(20)|∂Fi∂t(t,Xr-1(t))|≤|∂Fi∂t(t,Xr-1(t))-∂Fi∂t(t,0)|+|∂Fi∂t(t,0)|≤LFi∥Xr-1∥+Ri,
with Ri:=max⁡t∈[α,α+β]|(∂Fi/∂t)(t,0)| and LFi as the Lipschitz constant of ∂Fi/∂t.
Meanwhile,(21)∥∂Fi∂u(t,Xr-1(t))∥∞≤∥∂Fi∂u(t,Xr-1(t))-∂Fi∂u(t,0)∥∞+∥∂Fi∂u(t,0)∥∞≤LFi*∥Xr-1∥+Ri*,
with Ri*:=max⁡j=1,…,nmax⁡t∈[α,α+β]|(∂Fi/∂uj)(t,0)| and LFi* as the maximum of the Lipschitz constants for each ∂Fi/∂uj, j=1,…,n.
Therefore,{(∂Fi/∂t)(t,Xr-1(t))}r≥1 and {(∂Fi/∂u)(t,Xr-1(t))}r≥1 are bounded.
Next, we will show that the sequence{Xr-1′}r≥1 is bounded.
Givenr≥1, taking into account the definition of Xr, we have for all t∈[0,1] that
(22)Xr′(t)=[Pir(ψr-1,1(t)),…,Pir(ψr-1,n(t))]T+[∫01‍Qjr2(ϕr-1,1(t,s))ds,…,∫01‍Qjr2(ϕr-1,n(t,s))ds]T.
In view of the monotonicity of the Schauder bases{bi}i≥1 and {Bi}i≥1 and (15), (16), and (19), we obtain
(23)∥Xr′∥≤∥Ψr-1∥+∥Φr-1∥≤LF∥Xr-1∥+RF+LK∥Xr-1∥+RK≤∥Xr-1∥(LF+LK)+RF+RK≤Γ∑k=1r‍Lk+Lr∥X0∥+R.
Therefore, the sequence{Xr′}r≥1 is also bounded.
We will prove that the sequences{(∂Ki/∂t)(t,s,Xr-1(s))}r≥1, {(∂Ki/∂s)(t,s,Xr-1(s))}r≥1, and {(∂Ki/∂u)(t,s,Xr-1(s))}r≥1 are bounded.
Fori=1,…,n, we have
(24)|∂Ki∂t(t,s,Xr-1(s))|≤|∂Ki∂t(t,s,Xr-1(s))-∂Ki∂t(t,s,0)|+|∂Ki∂t(t,s,0)|≤M1i∥Xr-1∥+N1i,
with N1i:=max⁡(t,s)∈[0,1]2|(∂Ki/∂t)(t,s,0)| and M1i as the Lipschitz constant of ∂Ki/∂t.
By repeating the previous argument we obtain(25)|∂Ki∂s(t,s,Xr-1(s))|≤M2i∥Xr-1∥+N2i,
with N2i:=max⁡(t,s)∈[0,1]2|(∂Ki/∂s)(t,s,0)| and M2i as the Lipschitz constant of ∂Ki/∂s.
Therefore, the sequences{(∂Ki/∂t)(t,s,Xr-1(s))}r≥1 and {(∂Ki/∂s)(t,s,Xr-1(s))}r≥1 are bounded.
Meanwhile,(26)∥∂Ki∂u(t,s,Xr-1(s))∥∞≤∥∂Ki∂u(t,s,Xr-1(s))-∂Ki∂u(t,s,0)∥∞+∥∂Ki∂u(t,s,0)∥∞≤M3i∥Xr-1∥+N3i,
with N3i:=max⁡j=1,…,nmax⁡(t,s)∈[0,1]2|(∂Ki/∂uj)(t,s,0)| and M3i as the maximum of the Lipschitz constants for each ∂Ki/∂uj, j=1,…,n. Therefore, {(∂Ki/∂u)(t,s,Xr-1(s))} is bounded.
In view of the identities (14), we have that the sequences, {ψr-1,i′}r≥1, {∂ϕr-1,i/∂t}r≥1, and {∂ϕr-1,i/∂s}r≥1, with i=1,…,n, are bounded.For a dense subset{ti}i≥1 of distinct points in [0,1], let Ti be the set {tj,1≤j≤i} ordered in an increasing way for i≥2. Let ΔTi denote the maximum distance between two consecutive points of Ti.Theorem 2.
With the previous notation and the same hypothesis as in Theorem1, for all r≥1, there are ηr,τr>0 and ir,jr≥2 such that
(27)∥[ψr-1,1-Pir(ψr-1,1),…,ψr-1,n-Pir(ψr-1,n)]T∥≤ηrΔTir,∥[ϕr-1,1-Qjr2(ϕr-1,1),…,ϕr-1,n-Qjr2(ϕr-1,n)]T∥≤τrΔTjr.Proof.
The announced estimation follows from the inequalities obtained in Propositions 4 and  5 in [11], respectively, and applying Theorem 1.In the result below we show that the sequence defined in (11) approximates the solution of (1).Theorem 3.
With the same hypothesis as in Theorem1, suppose that V:C([0,1],Rn)→C([0,1],Rn) is the integral operator (5), X~∈C([0,1],Rn), and that {Xr}r≥1 is the sequence defined by (11). Let us also assume that m∈N, ir,jr≥2, and {ε1,…,εm} is a set of positive numbers such that for all r∈{1,…,m} we have
(28)ΔTir≤εr2ηr,ΔTjr≤εr2τr.
Then
(29)∥V(Xr-1)-Xr∥≤εr.
Moreover, if X is the exact solution of the integral equation (1), then the error ∥X-Xm∥ is given by
(30)∥X-Xm∥≤Lm1-L∥VX~-X~∥+∑r=1m‍Lm-rεr.Proof.
Form≥1, from (7), we have
(31)∥X-VmX~∥≤Lm1-L∥VX~-X~∥.
First we deal with proving (29). For all r∈{1,…,m} and t∈[0,1], Theorem 2 gives
(32)∥VXr-1(t)-Xr(t)∥∞=∥ρ+[∫0t‍ψr-1,1(u)du,…,∫0t‍ψr-1,n(u)du]T+[∫0t‍∫01‍ϕr-1,1(u,s)dsdu,…,∫0t‍∫01‍ϕr-1,n(u,s)dsdu]T-(…,∫0t‍∫01‍Qjr2(ϕr-1,n(u,s))dsdu]Tρ+[∫0t‍Pir(ψr-1,1(u))du,…,∫0t‍Pir(ψr-1,n(u))du]T+[∫0t‍∫01‍Qjr2(ϕr-1,1(u,s))dsdu,…,∫0t‍∫01‍Qjr2(ϕr-1,n(u,s))dsdu]T)∥∞≤∥∫0t‍(ψr-1,n(u))]T[∫0t‍(ψr-1,1(u)-Pir(ψr-1,1(u)))du,…,∫0t‍(ψr-1,n(u)-Pir(ψr-1,n(u)))du]T∥∞+∥[∫0t‍∫01‍(ϕr-1,1(u,s)-Qjr2(ϕr-1,1(u,s)))dsdu,…,∫0t‍∫01‍(Qjr2(ϕr-1,n(u,s))ϕr-1,n(u,s)-Qjr2(ϕr-1,n(u,s)))dsdu∫0t‍∫01‍(ϕr-1,1(u,s)-Qjr2(ϕr-1,1(u,s)))dsdu,]T∥∞≤∫0t∥…,ψr-1,n(u)-Pir(ψr-1,n(u))]T[ψr-1,1(u)-Pir(ψr-1,1(u)),…,ψr-1,n(u)-Pir(ψr-1,n(u))]T∥∞du+∫0t‍∫01∥,ϕr-1,n(u,s)-Qjr2(ϕr-1,n(u,s))]T[ϕr-1,1(u,s)-Qjr2(ϕr-1,1(u,s)),…,‍ϕr-1,n(u,s)-Qjr2(ϕr-1,n(u,s))]T∥∞dsdu≤ηrΔTir+τrΔTjr≤εr.
And, in turn, applying (29) and recursively (6), we obtain
(33)∥Vm(X~)-Xm∥=∥Vm(X0)-Xm∥≤∑r=1m‍∥Vm-r+1(Xr-1)-Vm-r(Xr)∥=∑r=1m‍∥Vm-rV(Xr-1)-Vm-r(Xr)∥≤∑r=1m‍Lm-r∥V(Xr-1)-Xr∥≤∑r=1m‍Lm-rεr.
Finally, using the triangle inequality,(34)∥X-Xm∥≤∥X-VmX~∥+∥Vm(X~)-Xm∥,
the proof is complete in view of (31) and (33).Observe that under the hypotheses of Theorem3, by inequality (30), we have
(35)∥X-Xm∥≤Lm1-L∥VX~-X~∥+1-Lm1-Lmax⁡r≥1{εr}.
Therefore, given ε>0, there exists m≥1 such that ∥X-Xm∥<ε for sufficiently small εr, since the points of the partition can be chosen in such a way that ΔTir and ΔTjr become so close to zero as we desire and the first sum on the right hand side approach zero when m increases.Remark 4.
If we consider an interval[a,b], then L=(b-a)LF+(b-a)2Lk and the bound obtained in Theorem 3 for ∥VXr-1(t)-Xr(t)∥∞ is given by
(36)∥VXr-1(t)-Xr(t)∥∞≤ηrΔTir(b-a)+τrΔTjr(b-a)2≤εr,
when
(37)ΔTir≤εr2ηr(b-a),ΔTjr≤εr2τr(b-a)2.


## 4. Numerical Examples

We now turn our attention to the application of the method presented in this paper for the numerical solution of six test problems. In order to construct the Schauder basis, we consider the subset{ti}i≥1 defined by t1=0, t2=1 and for n∈N∪{0}, ti+1=(2k+1)/2n+1, if i=2n+k+1, where 0≤k<2n are integers. To define the sequence {Xr}r≥1, we take X0(t)=ρ and ir=jr=j (for all r≥1). We include, for different values of j, the absolute errors committed in some representative points of [0,1] when we approximate the exact solution X(t) by the iteration Xr(t), where r is shown in each table. The algorithms associated with the numerical methods were performed using Mathematica 7. In Examples 1, 2, and 3, X(t)=x(t) and Xr(t)=xr(t). In the other examples, X(t)=[x1(t),x2(t)]T and Xr(t)=[xr1(t),xr2(t)]T.Example 1.
Consider the Fredholm integrodifferential equation appearing in [1]:
(38)x′(t)=-2πsin(2πt)-12sin(4πt)+∫01‍sin(4πt+2πs)y(s)ds,x(0)=1,
whose exact solution is x(t)=cos⁡(2πt). Numerical results obtained for this problem when we apply the method described in this paper and the results obtained in [1] are given in Table 1.Table 1
Absolute errors for Example1.
t
j
=
33
Method in [1]
|
x
4
(
t
)
-
x
(
t
)
|
withk=4,M=1
0
.
1
4.93
×
1
0
-
4
2.40
×
1
0
-
3
0.2
8.90
×
1
0
-
4
5.07
×
1
0
-
3
0.3
4.00
×
1
0
-
3
6.25
×
1
0
-
3
0.4
5.70
×
1
0
-
3
3.87
×
1
0
-
3
0.5
6.40
×
1
0
-
3
1.74
×
1
0
-
2
0.6
3.72
×
1
0
-
3
1.58
×
1
0
-
2
0.7
7.78
×
1
0
-
4
8.41
×
1
0
-
3
0.8
8.96
×
1
0
-
4
9.65
×
1
0
-
3
0.9
9.40
×
1
0
-
4
9.49
×
1
0
-
3Example 2.
Consider the Fredholm integrodifferential equation:(39)x′(t)=f(t)+1125∫05‍stx(s)ds,x(0)=1125,
where f(t) is chosen so that the exact solution is given by x(t)=e-5t/250. The numerical results are given in Table 2.Table 2
Absolute errors for Example2.
t
j
=
33
t
j
=
33
|
x
3
(
t
)
-
x
(
t
)
|
|
x
3
(
t
)
-
x
(
t
)
|
0
.
125
1.91
×
1
0
-
3
2.625
5.93
×
1
0
-
3
0.250
2.41
×
1
0
-
3
2.750
6.11
×
1
0
-
3
0.375
3.51
×
1
0
-
3
2.825
6.29
×
1
0
-
3
0.500
3.83
×
1
0
-
3
3.00
6.49
×
1
0
-
3
0.625
4.02
×
1
0
-
3
3.125
6.69
×
1
0
-
3
0.750
4.15
×
1
0
-
3
3.250
6.91
×
1
0
-
3
0.825
4.25
×
1
0
-
3
3.375
7.13
×
1
0
-
3
1.00
4.34
×
1
0
-
3
3.500
7.35
×
1
0
-
3
1.125
4.43
×
1
0
-
3
3.625
7.59
×
1
0
-
3
1.250
4.50
×
1
0
-
3
3.750
7.84
×
1
0
-
3
1.375
4.60
×
1
0
-
3
3.875
8.09
×
1
0
-
3
1.500
4.69
×
1
0
-
3
4
8.35
×
1
0
-
3
1.625
4.81
×
1
0
-
3
4.125
8.62
×
1
0
-
3
1.750
4.91
×
1
0
-
3
4.250
8.90
×
1
0
-
3
1.875
5.50
×
1
0
-
3
4.375
9.19
×
1
0
-
3
2
5.51
×
1
0
-
3
4.500
9.48
×
1
0
-
3
2.125
5.53
×
1
0
-
3
4.625
9.78
×
1
0
-
3
2.250
5.54
×
1
0
-
3
4.750
1.01
×
1
0
-
2
2.375
5.56
×
1
0
-
3
4.875
1.04
×
1
0
-
2
2.500
5.57
×
1
0
-
3
5
1.07
×
1
0
-
2Example 3.
Consider the Fredholm integrodifferential equation:(40)x′(t)=-e-t+110(-2+e-2+cos⁡(2))+cos⁡(t)+∫02‍x(s)10ds,x(0)=1,
whose exact solution is x(t)=e-t+sin(t). The numerical results are given in Table 3.Table 3
Absolute errors for Example3.
t
j
=
33
t
j
=
33
|
x
5
(
t
)
-
x
(
t
)
|
|
x
5
(
t
)
-
x
(
t
)
|
0
.
125
1.00
×
1
0
-
4
1.125
7.05
×
1
0
-
4
0.250
1.95
×
1
0
-
4
1.250
7.54
×
1
0
-
4
0.375
2.85
×
1
0
-
4
1.375
7.97
×
1
0
-
4
0.500
3.69
×
1
0
-
4
1.500
8.34
×
1
0
-
4
0.625
4.48
×
1
0
-
4
1.625
8.64
×
1
0
-
4
0.750
5.21
×
1
0
-
4
1.750
8.88
×
1
0
-
4
0.825
5.89
×
1
0
-
4
1.875
9.06
×
1
0
-
4
1.00
6.50
×
1
0
-
4
2
9.19
×
1
0
-
4Example 4.
Consider now the following system of Fredholm integrodifferential equations with the exact solutionsx1(t)=cos⁡(t) and x2(t)=t:
(41)x1′(t)=-112-15t2sin(1)x2(s)-sin(t)+∫01‍(t35x1(s)+s23x2(s))ds,x2′(t)=2021-15tx2(t)sin(1)+∫01‍(t25x1(s)+s7x2(s))ds,x1(0)=1,x2(0)=0.
The numerical results are given in Table 4.Table 4
Absolute errors for Example4.
t
j
=
9
X
5
=
[
x
51
,
x
52
]
T
j
=
17
X
5
=
[
x
51
,
x
52
]
T
j
=
33
X
5
=
[
x
51
,
x
52
]
T
|
x
51
(
t
)
-
x
1
(
t
)
|
|
x
52
(
t
)
-
x
2
(
t
)
|
|
x
51
(
t
)
-
x
1
(
t
)
|
|
x
52
(
t
)
-
x
2
(
t
)
|
|
x
51
(
t
)
-
x
1
(
t
)
|
|
x
52
(
t
)
-
x
2
(
t
)
|
0
.
125
1.7
E
-
4
4.8
E
-
5
4.4
E
-
5
1.2
E
-
5
1.1
E
-
5
3.2
E
-
6
0.250
3.7
E
-
4
9.7
E
-
5
9.3
E
-
5
2.4
E
-
5
2.3
E
-
5
6.3
E
-
6
0.375
5.9
E
-
4
1.4
E
-
4
1.4
E
-
4
3.6
E
-
5
3.7
E
-
5
9.5
E
-
6
0.5
8.2
E
-
4
1.9
E
-
4
2.0
E
-
4
4.8
E
-
5
5.2
E
-
5
1.2
E
-
5
0.625
1.0
E
-
3
2.3
E
-
4
2.7
E
-
4
5.9
E
-
5
6.8
E
-
5
1.5
E
-
5
0.750
1.3
E
-
3
2.7
E
-
4
3.3
E
-
4
7.0
E
-
5
8.5
E
-
5
1.8
E
-
5
0.875
1.6
E
-
3
3.1
E
-
4
4.0
E
-
4
8.0
E
-
5
1.0
E
-
4
2.0
E
-
5
1
1.9
E
-
3
3.5
E
-
4
4.7
E
-
4
8.9
E
-
5
1.2
E
-
4
2.3
E
-
5Example 5.
Consider now the following system of Fredholm integrodifferential equations with the exact solutionsx1(t)=t2 and x2(t)=t:
(42)x1′(t)=2t-15x1(t)(-cos⁡(1)+sin(t))+∫01‍t2s5sin(x2(s))ds,x2′(t)=1-148x2(t)(π-log⁡(4))+∫01‍ts6arctg(x1(s))ds,x1(0)=0,x2(0)=0.
The numerical results are given in Table 5.Table 5
Absolute errors for Example5.
t
j
=
9
X
4
=
[
x
41
,
x
42
]
T
j
=
17
X
4
=
[
x
41
,
x
42
]
T
j
=
33
X
4
=
[
x
41
,
x
42
]
T
|
x
41
(
t
)
-
x
1
(
t
)
|
|
x
42
(
t
)
-
x
2
(
t
)
|
|
x
41
(
t
)
-
x
1
(
t
)
|
|
x
42
(
t
)
-
x
2
(
t
)
|
|
x
41
(
t
)
-
x
1
(
t
)
|
|
x
42
(
t
)
-
x
2
(
t
)
|
0.125
3.4
E
-
7
3.1
E
-
6
6.1
E
-
8
8.1
E
-
7
1.1
E
-
8
2.3
E
-
7
0.250
2.1
E
-
6
1.2
E
-
5
4.4
E
-
7
3.2
E
-
6
7.3
E
-
8
9.4
E
-
7
0.375
6.5
E
-
6
2.7
E
-
5
1.4
E
-
6
7.2
E
-
6
2.2
E
-
7
2.1
E
-
6
0.5
1.5
E
-
5
4.9
E
-
5
3.3
E
-
6
1.3
E
-
5
4.8
E
-
7
3.9
E
-
6
0.625
2.9
E
-
5
7.7
E
-
5
6.3
E
-
6
2.1
E
-
5
8.3
E
-
7
6.2
E
-
6
0.750
4.9
E
-
5
1.1
E
-
4
1.1
E
-
5
2.9
E
-
5
1.2
E
-
6
9.2
E
-
6
0.875
7.7
E
-
5
1.5
E
-
4
1.6
E
-
5
4.1
E
-
5
1.5
E
-
6
1.2
E
-
5
1
1.1
E
-
4
1.9
E
-
4
2.4
E
-
5
5.3
E
-
5
1.6
E
-
6
1.7
E
-
5Example 6.
Consider now the following system of Fredholm integrodifferential equations with the exact solutionsx1(t)=sin(t) and x2(t)=cos⁡(t):
(43)x1′(t)=x2(t)+110(-1+cos⁡(2)-sin(2))+110∫02‍(x1(s)+x2(s))ds,x2′(t)=-x1(t)+115(1-cos⁡(2)-sin(2))+115∫02‍(x2(s)-x1(s))ds,x1(0)=0,x2(0)=1.
The numerical results are given in Table 6.Table 6
Absolute errors for Example6.
t
j
=
17
X
3
=
[
x
31
,
x
32
]
T
j
=
33
X
3
=
[
x
31
,
x
32
]
T
j
=
65
X
3
=
[
x
31
,
x
32
]
T
|
x
31
(
t
)
-
x
1
(
t
)
|
|
x
32
(
t
)
-
x
2
(
t
)
|
|
x
31
(
t
)
-
x
1
(
t
)
|
|
x
32
(
t
)
-
x
2
(
t
)
|
|
x
31
(
t
)
-
x
1
(
t
)
|
|
x
32
(
t
)
-
x
2
(
t
)
|
0.125
1.1
E
-
3
4.8
E
-
5
8.5
E
-
4
8.3
E
-
6
9.3
E
-
5
9.8
E
-
7
0.250
2.2
E
-
3
1.1
E
-
4
9.2
E
-
4
2.1
E
-
5
1.8
E
-
4
1.1
E
-
6
0.375
3.4
E
-
3
2.0
E
-
4
2.9
E
-
3
4.0
E
-
5
3.3
E
-
4
1.5
E
-
6
0.5
4.5
E
-
3
3.1
E
-
4
4.1
E
-
3
6.3
E
-
5
5.4
E
-
4
1.8
E
-
6
0.625
5.7
E
-
3
4.3
E
-
4
5.6
E
-
3
9.0
E
-
5
7.8
E
-
4
4.3
E
-
6
0.750
6.8
E
-
3
5.7
E
-
4
6.7
E
-
3
1.2
E
-
4
9.2
E
-
4
8.5
E
-
6
0.875
8.0
E
-
3
7.3
E
-
4
7.9
E
-
3
1.5
E
-
4
3.5
E
-
3
1.3
E
-
5
1
9.2
E
-
3
9.0
E
-
4
8.8
E
-
3
1.9
E
-
4
5.3
E
-
3
1.9
E
-
5
1.125
1.0
E
-
2
1.0
E
-
3
9.7
E
-
3
2.3
E
-
4
7.9
E
-
3
2.6
E
-
5
1.250
1.1
E
-
2
1.2
E
-
3
1.0
E
-
2
2.8
E
-
4
8.2
E
-
3
3.3
E
-
5
1.375
1.2
E
-
2
1.4
E
-
3
1.1
E
-
2
3.2
E
-
4
9.0
E
-
3
4.1
E
-
5
1.5
1.4
E
-
2
1.6
E
-
3
1.3
E
-
2
3.7
E
-
4
9.9
E
-
3
4.9
E
-
5
1.625
1.5
E
-
2
1.8
E
-
3
1.5
E
-
2
4.1
E
-
4
1.0
E
-
2
5.7
E
-
5
1.750
1.6
E
-
2
2.0
E
-
3
1.6
E
-
2
4.6
E
-
4
1.2
E
-
2
6.4
E
-
5
1.875
1.8
E
-
2
2.2
E
-
3
1.7
E
-
2
5.1
E
-
4
1.5
E
-
2
7.2
E
-
5
2
1.9
E
-
2
2.4
E
-
3
1.8
E
-
2
5.5
E
-
4
1.7
E
-
2
7.9
E
-
5


## 5. Conclusion

In this paper we have successfully approximated the solution of systems of nonlinear Fredholm integrodifferential equations. To this end, we have used the Banach fixed-point theorem and the Schauder basis. Moreover, the convergence of the proposed scheme is analyzed and some illustrative examples were included to demonstrate the validity and applicability of the method. The approximating functionsxr and [xr1,xr2]T are the sum of integrals of piecewise univariate and bivariate polynomials of degree 2 and the calculation of the coefficients of such polynomials just requires linear combinations of several evaluations of the basic functions at sufficient number of points.



---

*Source: 401541-2014-07-06.xml*