# A Combination of Bernstein and Improved Block-Pulse Functions for Solving a System of Linear Fredholm Integral Equations

Authors: Ji-Huan He; Mahmoud H. Taha; Mohamed A. Ramadan; Galal M. Moatimid
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The present paper illustrates a new numerical technique to solve a system of linear Fredholm integral equations of the second kind. The current work introduces a coupling between hybrid Bernstein functions and improved block-pulse functions (HBI). The current method transforms the linear second kind Fredholm integral equations system into an algebraic system that can be solved by using classical methods. Some numerical examples are introduced to validate the new approach. The results showed that the method is promising, powerful, ultimately accurate, and highly efficient.


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## Body

## 1. Introduction

Integral equations are an integral part of many mathematical formulations for numerous phenomena. Integral equations can be frequently found in many applied fields, such as field theory, mechanics, biology, chemistry, physics, engineering, electrostatics, and economics; for illustration, see references [1–3]. Fredholm equations arise naturally in the theory of signal processing, for example, as the famous spectral concentration problem popularized by David Slepian. The operators involved are the same as linear filters. They also commonly arise in linear forward modeling and inverse problems. In physics, the solution of such integral equations allows for experimental spectra to be related to various underlying distributions, for instance, the mass distribution of polymers in a polymeric melt, or the distribution of relaxation times in the system. In addition, Fredholm integral equations also arise in fluid mechanics problems involving hydrodynamic interactions near finite-sized elastic interfaces. A specific application of the Fredholm equation is the generation of photo-realistic images in computer graphics, in which the Fredholm equation is used to model light transport from the virtual light sources to the image plane. The Fredholm equation is often called the rendering equation in this context.However, most integral equations that arise in our real-life situations are difficult to solve analytically; therefore, a numerical method is required. This paper is going to work with two simultaneous different functions and combine them to create a new efficient technique. A detailed description of these two methods is thoroughly presented as follows.The first one is known as the Bernstein functions. Many researchers have attempted to use Bernstein functions and improved block-pulse functions for solving integral equations. Bernstein functions are an important subclass of complete functions that can be found under many different definitions/names in many fields and branches of mathematics. The most common definition of Bernstein functions is given by probability: “Laplace exponents of infinite divisible nonnegative subdistributions” (or “Laplace exponents of subordinates”). By contrast, in harmonic analysis and Fourier’s analysis, they are generally named negative definite functions. Complete Bernstein functions are employed in the complex analysis under the name of Pick or Nevanlinna functions that can be found in different research papers and books. Concurrently, in the field of operator theory and matrix analysis, the more widely used name is the operator monotone function. Researchers studied the positivity of the solutions of Volterra integral equations and found that different forms of kernels associated with Bernstein functions started to appear, see reference [4]. There can be found a huge number of papers on every subclass of these classes, see reference [5]. On the other hand, only a few texts have brought to light the links between these classes and the existing techniques in other resources; for instance, see reference [6].Bernstein polynomials have a leading role in numerous areas of mathematics. These polynomials are regularly found in analyzing several approximations theory and differential equations; for instance, see references [7–9]. Maleknejad et al. in reference [10] used Bernstein polynomials to effectively solve Volterra integral equations. Ostrovska in reference [11] studied the convergence of Bernstein polynomials. One can find a wide study on Bernstein polynomials and other similar polynomials as well in reference [12]. Mirzaee and Hoseini in reference [13] did the coupling between Bernstein polynomials and block-pulse functions which our work here depends on the way they did that. The two-dimensional Bernstein polynomials and their convergence analysis are also discussed in references [14, 15]. Bernstein polynomials also dealt with fractional integrodifferential equations with ease and it was accurate, see reference [16]. Bernstein’s polynomial is confined to the interval x∈0,1 which will be very important within the style of Bezier curves for computer-aided geometric design (CAGD). They are used for the design of curves, and they are the starting point for several generalizations, particularly for higher dimensions and B-splines [17, 18]. Several characteristics of the Bezier curves and surfaces are inherited from those of the Bernstein polynomials. Because of the growing concerns with Bernstein polynomials, the investigation has emerged in such a way as to depict their properties as far as their coefficients are concerned after they are given inside the Bernstein basis. Farouki and Goodman [19] deduced that the calculation within the Bernstein basis is more stable than computations in other polynomial bases. Moreover, the alteration of basis algorithms is in most cases numerically unstable. Therefore, when the Bernstein basis is used to represent polynomials, it is smarter to make all estimates, utilizing only Bernstein coefficients. Formulae for multiplication, differentiation, integration, and subdivision of polynomials inside the Bernstein structure are given by Farouki and Rajan [20].The class of Bernstein functions is firmly identified with completely monotone functions. The idea of Bernstein functions dates back to the potential theory school of Beurling and Deny [21]. It was then led by Berg et al. [22]. Berg et al. [23] named them as completely monotone functions (or the opposite of completely monotone functions). Therefore, Laplace exponents are still preferred by probabilistic; for instance, see studies by Berge et al. [23] and Bertoin [24].Thenth Bernstein polynomial for a mapping fx over the interval 0,1 is given as(1)Bnf;x=∑j=0nfjnnjxj1−xn−j.A linear transformation of variables can be done to the abovementioned equation; consequently, similar polynomials can be constructed on any finite intervala,b, as an extension to the work done in this paper. It can be thoroughly proven that if fx is continuous on the interval 0,1, it follows that the sequence (Bnf;x) converges uniformly to fx on the interval 0,1 Additionally, derivatives of the Bernstein polynomial converge to derivatives of fx (if there are any). Furthermore, if fx is convex, then each Bernstein polynomial is convex; for instance, see [25].In this paper, the Bernstein polynomial will be coupled with the improved block-pulse functions which was first introduced by Farshid Mirzaee [26] to solve a system of Fredholm integral equations of the second kind. The system of Fredholm integral equations appears in many problems in applied sciences.LetL20,1 be the set of all real-valued functions that are continuous on the interval 0,1. In this paper, the fixed-point theorem is used to discuss the existence and uniqueness of the solution u=u1,u2∈L20,1×L20,1 for the following system of Fredholm integral equations:(2)uix=fix+∑j=1q∫01ki,jx,tujtdt,fori=1,2,where uix,i=1,2, are unknown functions, while fix∈L20,1 and the kernels ki,jx,t∈L20,1,i,j=1,2 are known functions.To crystallize the paper, it is arranged as follows: The existence and uniqueness of the solutions of the system of linear Fredholm integral equations are introduced in Section2. Section 3 describes the coupling of Bernstein polynomial and improved block pulse functions to solve a system of linear Fredholm integral equations. The function approximations are also introduced. Section 4 introduces the solution method. Section 5 investigates the problem's convergence analysis. Section 6 includes numerical results as well as application examples to demonstrate the efficiency and accuracy of the proposed method. Finally, Section 7 provides the main outline of the paper's conclusions.


## 2. Existence and Uniqueness

In this section, the existence and uniqueness of solutions for a system of Fredholm integral equations (2) are studied.


### 2.1. Notes

Mmℝ+ denotes the set of all m×m square matrices with positive real elements, Im stands for the identity m×m matrix, and Om refers to the zero m×m matrix. The matrix A∈Mmℝ+ is said to be convergent to zero if An⟶Om as n⟶+∞.Some examples of matrices that converge to zero are as follows:(a)
A=aabb∈M2ℝ+,wherea,b∈ℝ+anda+b<1(b)
A=abab∈M2ℝ+,wherea,b∈ℝ+anda+b<1(c)
A=ab0c∈M2ℝ+,wherea,b,c∈ℝ+andmaxa,c<1Theorem 1.
LetX be a nonempty set endowed with two vector-valued metrics d,ρ∶X×X⟶ℝ+2. Let g∶X⟶X be an operator. The following is assumed:(1)
There exists a matrixP∈Mmℝ+, in such a way that dgu,gv≤Pρu,v,∀u,v∈X(2)
X,d is a complete vector-valued metric space(3)
f∶X,d⟶X,d is continuous(4)
f∶X,ρ⟶X,ρ is a generalized Hardy-Rogers-type contraction, that is, there exist A, B, C∈Mmℝ+, in such a way that the matrices B+C and Λ∶=Im−B−C−1A+B+C are convergent to zero and(3)ρgu,gv≤Aρu,v+Bρu,gu+ρv,gv++Cρu,gv+ρv,gu,forallu,v∈X.Proof.
It can be found in reference [27].
The solutionu1,u2∈L20,12 of the system is sought in equation (2) which can be rewritten in the following form:(4)u1x=f1x+∫01k1x,t,u1t,u2tdt,u2x=f2x+∫01k2x,t,u1t,u2tdt.
It is assumed that there existk11,k12,k21,k22>0 as follows:(5)k1x,t,α,α˜−k1x,t,β,β˜≤k11α−β+k12α˜−β˜k2x,t,α,α˜−k2x,t,β,β˜≤k21α−β+k22α˜−β˜,for all x,t∈0,1,α,α˜,β,β˜∈ℝ.
On a nonempty setX∶=L20,12, the metrics d,ρ∶X×X⟶ℝ+2 are considered, where(6)du,v≔maxt∈0,1u1t−v1tmaxt∈0,1u2t−v2t,(7)ρu,v≔∫01u1t−v1t2dt1/2∫01u2t−v2t2dt1/2for all u=u1,u2,v=v1,v2∈L20,12.
The operatorg:L20,12⟶L20,12 is defined by(8)gxt≔g1xtg2xt≔f1x+∫01k1x,t,u1t,u2tdtf2x+∫01k2x,t,u1t,u2tdt,for all u=u1,u2∈L20,12.
We have(9)gu,gv=∫01g1ut−g1vt2dt1/2∫01g2ut−g2vt2dt1/2.
Then, using the assumptions, one gets(10)≤∫01k11u1t−v1tdt∫01k21u1t−v1tdt+∫01k12u2t−v2tdt∫01k22u2t−v2tdtHolder'sinequality≤∫01k1,12dt1/2∫01u1t−v1t2dt1/2∫01k2,12dt1/2∫01u1t−v1t2dt1/2+∫01k1,22dt1/2∫01u2t−v2t2dt1/2∫01k2,22dt1/2∫01u2t−v2t∫2dt1/2=k1,1ρ˜u1,v1+k1,2ρ˜u2,v2k2,1ρ˜u1,v1+k2,2ρ˜u2,v2,where(11)ρ˜u1,v1≔∫01u1t−v1t2dt1/2,ρ˜u2,v2≔∫01u2t−v2t2dt1/2ρgu,gv≤∫01k1,1ρ˜u1,v1+k1,2ρ˜u2,v22dt1/2∫01k2,1ρ˜u1,v1+k2,2ρ˜u2,v22dt1/2=k1,1k1,2k2,1k2,2ρ˜u1,v1ρ˜u2,v2=Aρu,vA≔k1,1k1,2k2,1k2,2∈M2ℝ+.
The matrixA converges to zero if(12)12k1,1+k2,2±k1,1−k2,22+4k2,1k1,2∈−1,1.
So, under this condition,g becomes an A-contraction with respect to ρ.
In addition, for allx,y∈L20,12, it is to be noted that(13)dx,y≔maxt∈0,1g1xt−g1ytmaxt∈0,1g2xt−g2xt≤Pρx,y,(14)P=k1,1k1,2k2,1k2,2∈M2ℝ+.
By applying Theorem1 with B,C∶=Om, it is found that system (2) has a unique solution in L20,12.


## 2.1. Notes

Mmℝ+ denotes the set of all m×m square matrices with positive real elements, Im stands for the identity m×m matrix, and Om refers to the zero m×m matrix. The matrix A∈Mmℝ+ is said to be convergent to zero if An⟶Om as n⟶+∞.Some examples of matrices that converge to zero are as follows:(a)
A=aabb∈M2ℝ+,wherea,b∈ℝ+anda+b<1(b)
A=abab∈M2ℝ+,wherea,b∈ℝ+anda+b<1(c)
A=ab0c∈M2ℝ+,wherea,b,c∈ℝ+andmaxa,c<1Theorem 1.
LetX be a nonempty set endowed with two vector-valued metrics d,ρ∶X×X⟶ℝ+2. Let g∶X⟶X be an operator. The following is assumed:(1)
There exists a matrixP∈Mmℝ+, in such a way that dgu,gv≤Pρu,v,∀u,v∈X(2)
X,d is a complete vector-valued metric space(3)
f∶X,d⟶X,d is continuous(4)
f∶X,ρ⟶X,ρ is a generalized Hardy-Rogers-type contraction, that is, there exist A, B, C∈Mmℝ+, in such a way that the matrices B+C and Λ∶=Im−B−C−1A+B+C are convergent to zero and(3)ρgu,gv≤Aρu,v+Bρu,gu+ρv,gv++Cρu,gv+ρv,gu,forallu,v∈X.Proof.
It can be found in reference [27].
The solutionu1,u2∈L20,12 of the system is sought in equation (2) which can be rewritten in the following form:(4)u1x=f1x+∫01k1x,t,u1t,u2tdt,u2x=f2x+∫01k2x,t,u1t,u2tdt.
It is assumed that there existk11,k12,k21,k22>0 as follows:(5)k1x,t,α,α˜−k1x,t,β,β˜≤k11α−β+k12α˜−β˜k2x,t,α,α˜−k2x,t,β,β˜≤k21α−β+k22α˜−β˜,for all x,t∈0,1,α,α˜,β,β˜∈ℝ.
On a nonempty setX∶=L20,12, the metrics d,ρ∶X×X⟶ℝ+2 are considered, where(6)du,v≔maxt∈0,1u1t−v1tmaxt∈0,1u2t−v2t,(7)ρu,v≔∫01u1t−v1t2dt1/2∫01u2t−v2t2dt1/2for all u=u1,u2,v=v1,v2∈L20,12.
The operatorg:L20,12⟶L20,12 is defined by(8)gxt≔g1xtg2xt≔f1x+∫01k1x,t,u1t,u2tdtf2x+∫01k2x,t,u1t,u2tdt,for all u=u1,u2∈L20,12.
We have(9)gu,gv=∫01g1ut−g1vt2dt1/2∫01g2ut−g2vt2dt1/2.
Then, using the assumptions, one gets(10)≤∫01k11u1t−v1tdt∫01k21u1t−v1tdt+∫01k12u2t−v2tdt∫01k22u2t−v2tdtHolder'sinequality≤∫01k1,12dt1/2∫01u1t−v1t2dt1/2∫01k2,12dt1/2∫01u1t−v1t2dt1/2+∫01k1,22dt1/2∫01u2t−v2t2dt1/2∫01k2,22dt1/2∫01u2t−v2t∫2dt1/2=k1,1ρ˜u1,v1+k1,2ρ˜u2,v2k2,1ρ˜u1,v1+k2,2ρ˜u2,v2,where(11)ρ˜u1,v1≔∫01u1t−v1t2dt1/2,ρ˜u2,v2≔∫01u2t−v2t2dt1/2ρgu,gv≤∫01k1,1ρ˜u1,v1+k1,2ρ˜u2,v22dt1/2∫01k2,1ρ˜u1,v1+k2,2ρ˜u2,v22dt1/2=k1,1k1,2k2,1k2,2ρ˜u1,v1ρ˜u2,v2=Aρu,vA≔k1,1k1,2k2,1k2,2∈M2ℝ+.
The matrixA converges to zero if(12)12k1,1+k2,2±k1,1−k2,22+4k2,1k1,2∈−1,1.
So, under this condition,g becomes an A-contraction with respect to ρ.
In addition, for allx,y∈L20,12, it is to be noted that(13)dx,y≔maxt∈0,1g1xt−g1ytmaxt∈0,1g2xt−g2xt≤Pρx,y,(14)P=k1,1k1,2k2,1k2,2∈M2ℝ+.
By applying Theorem1 with B,C∶=Om, it is found that system (2) has a unique solution in L20,12.


## 3. Hybrid Bernstein Improved Block-Pulse Functions (HBIs)

Definition 1.
(improved block-pulse functions). The IBPFs were firstly introduced by Farshid Mirzaee [26]. Farshid solved a system of integral equations by using IBPFs. The variable interval block-pulse functions are derived from the regular block-pulse functions, but with a slight change in the interval width. The variable of interval block pulse is a m+1 agreed of functions defined over the interval 0,1 as follows:(15)φ0t=1,t∈0,12m,0,otherwise,φit=1,t∈i−1/2m,i+1/2m,0,otherwise,i=1,2,...,m−1,φmt=1,t∈1−12m,1,0,otherwise,where m is a positive constant that represents the number of subintervals to be decided for the accuracy needed for solving the problem. The IBPFs are disjoint as(16)φitφjt=φit,i=j,0,otherwise,where i,j=0,1,…,m and those functions are orthogonal to each other(17)∫01φitφjtdt=12m,i=j∈0,m,1m,i=j∈1,2,…,m−1,0,otherwise,where t∈[0,1).
The firstm+1 terms of the IBPF can be written in a vector form(18)ϕmt=φ0t,φ1t,…,φmtT,t∈[0,1).
Equation (17) gives(19)ϕmtϕmTt=φ0t0⋯00φ1t⋯0⋮⋮⋱⋮00⋯φmt=diagϕmt.Definition 2.
(HBI, [28]). Bi,jx is a combination of Bernstein polynomials and improved block-pulse functions, where all are orthogonal and complete. Subsequently, the set is a complete orthogonal system. Let Bi,jx where j=0,1,…,M,i=1,2,…,N+1,Bi,jx have two arguments, i and j, representing the order of IBPFs and the degree of Bernstein polynomials, respectively. The function Bx on the interval 0,1 may be defined as follows:(20)Bi,jx=Bj,M2xh,x∈0,h2,0,otherwise,fori=1,j=0,1,…,M,(21)Bi,jx=Bj,Mxh+32−i,x∈i−2h+h2,i−1h+h20,otherwise,fori=2,3,…,N,j=0,1,…,M,(22)Bi,jx=Bj,M2xh−2h+1,x∈1−h2,1,0,otherwise,fori=N+1,j=0,1,…,M.
Therefore, the new basis is given byB1,0,B1,1,…,BN+1,M and the function can be approximated to the base function, where N is an arbitrary positive integer and h=1/N.
The next section deals with the approximating properties of such functions.


### 3.1. Function Approximation

In terms of the HBI basis, any functionux can be expressed as follows:(23)ux≅∑i=1N+1∑j=0Mci,j.Bi,jx=CTBx,where Bx is the column vector defined as(24)Bx=B1,0,B1,1,…,BN+1,MT,and C is the row matrix defined as(25)C=c1,0,c1,1,…,cN+1,MT,and we have(26)CT<Bx,Bx>=<ux,Bx>.Or, it can be rewritten as(27)ci,j<Bx,Bx>=<ux,Bx>,where the inner product of the right hand of equation (26) is computed component-wise and the hybrid Bernstein improved block-pulse coefficients ci,j are given by(28)ci,j=ux,Bi,jxBi,jx,Bi,jx,then(29)C=L−1<ux,Bx>,where .,. is the standard inner product and the matrix L is the (N+1M+1×N+1M+1) matrix that is commonly called a dual matrix and defined as(30)L=<Bx,Bx>=∫01Bx.BTxdx=L1000L2000L3⋯0⋯0⋯0⋮⋮⋮000⋱⋮⋯LN+1,and Lii=1,2,…,N+1 is defined as follows:(31)L1i+1,j+1=∫0h/2Bi,M2xhBj,M2xhdx=h2∫01Bi,MxBj,Mxdx=hMiMj22M+12Mi+j,fori,j=0,…,M,Lri+1,j+1=∫i−2h+h/2i−1h+h/2Bi,Mxh+32−iBj,Mxh+32−idx,forr=2,…,N=h∫01Bi,MxBj,Mxdx=hMiMj2M+12Mi+j,fori,j=0,…,M,LN+1i+1,j+1=∫1−h/21Bi,M2xh−2h+1Bj,M2xh−2h+1dx=h2∫01Bi,MxBj,Mxdx=hMiMj22M+12Mi+j,fori,j=0,…,M,and the function kx,t∈L20,1×0,1 can be approximated as follows:(32)kx,t=BTx.K.Bt,where K is an M+1N+1 matrix; it can be obtained as follows:(33)K=L−1Bx,kx,t,BtL−1


## 3.1. Function Approximation

In terms of the HBI basis, any functionux can be expressed as follows:(23)ux≅∑i=1N+1∑j=0Mci,j.Bi,jx=CTBx,where Bx is the column vector defined as(24)Bx=B1,0,B1,1,…,BN+1,MT,and C is the row matrix defined as(25)C=c1,0,c1,1,…,cN+1,MT,and we have(26)CT<Bx,Bx>=<ux,Bx>.Or, it can be rewritten as(27)ci,j<Bx,Bx>=<ux,Bx>,where the inner product of the right hand of equation (26) is computed component-wise and the hybrid Bernstein improved block-pulse coefficients ci,j are given by(28)ci,j=ux,Bi,jxBi,jx,Bi,jx,then(29)C=L−1<ux,Bx>,where .,. is the standard inner product and the matrix L is the (N+1M+1×N+1M+1) matrix that is commonly called a dual matrix and defined as(30)L=<Bx,Bx>=∫01Bx.BTxdx=L1000L2000L3⋯0⋯0⋯0⋮⋮⋮000⋱⋮⋯LN+1,and Lii=1,2,…,N+1 is defined as follows:(31)L1i+1,j+1=∫0h/2Bi,M2xhBj,M2xhdx=h2∫01Bi,MxBj,Mxdx=hMiMj22M+12Mi+j,fori,j=0,…,M,Lri+1,j+1=∫i−2h+h/2i−1h+h/2Bi,Mxh+32−iBj,Mxh+32−idx,forr=2,…,N=h∫01Bi,MxBj,Mxdx=hMiMj2M+12Mi+j,fori,j=0,…,M,LN+1i+1,j+1=∫1−h/21Bi,M2xh−2h+1Bj,M2xh−2h+1dx=h2∫01Bi,MxBj,Mxdx=hMiMj22M+12Mi+j,fori,j=0,…,M,and the function kx,t∈L20,1×0,1 can be approximated as follows:(32)kx,t=BTx.K.Bt,where K is an M+1N+1 matrix; it can be obtained as follows:(33)K=L−1Bx,kx,t,BtL−1


## 4. Outline of the Solution of a System of Linear Fredholm Integral Equations

This section presents the algorithm of the method for solving a system of linear Fredholm equations as given in (2).Step 1.
The functionsuix maybe approximated as(34)uix≅UiTBx,where Ui is an unknown N+1M+1−vector and B are given by (23).Step 2.
The functionsfix and ki,jx,t are also expanded into the HBI function as follows:(35)ki,jx,t≅BTx.Ki,j.Bt,(36)fix≅FiTBxwhere Ki,j is a known N+1M+1×N+1M+1-matrix and Fi is a known N+1M+1−vector.Step 3.
Substituting approximation (34)–(36) into equation (2). For i=1,2,…,q, one gets(37)UiTBx=FiTBx+∑j=1q∫01BTx.Ki,j.Bt.UjTBtdt,UiTBx=FiTBx+∑j=1qBTx.Ki,j∫01Bt.BTtdtUj,UiTBx=FiTBx+∑j=1qBTx.Ki,j.D.Uj,Ui=Fi+∑j=1qKi,j.D.Uj,fori=1,2,…,q.
So, system (2) can be written in a matrix form as follows:(38)U=F+KD^U,where Ux and Fx are qM+1N+1 vectors in the following form:(39)Ux=U1x,U2x,…,Uqx,andFx=F1x,F2x,…,Fqx.
Also,KD^ is qM+1N+1×qM+1N+1 matrix as follows:(40)KD^=K1,1.DK1,2.DK2,1.DK2,2.D⋯K1,q.D⋯K2,q.D⋮⋮Kq,1.DKq,2.D⋱⋮⋯Kq,q.D,where Ki,ji,j=1,2,…,q and D is M+1N+1×M+1N+1 matrix. Therefore, the equation can be written as follows:(41)I−KD^U=F,where I is the qM+1N+1×qM+1N+1 identity matrix. After solving this linear system, the solution can be approximated as(42)uix=UiTBx,i=1,2,…,q.
To validate our approach, some numerical examples will be introduced in the next section.


## 5. Convergence Analysis

We assume the following conditions onki,jx,t and Guix=Uitm,m≥1, where ki,jx,t are known functions and uix are unknown functions, in such a way that(a)
S=sup0≤x,t≤1ki,jx,t<∞,(b)
Guix is continuous on 0,1 and satisfies Lipschitz condition(43)Guix−Gzix≤Luix−zix.The error of term of the solution obtained by the HBI method is denoted byeB=uix−u˜ix, where uix and u˜ix are the exact and approximate solution of (2), respectively. u˜ix is given as(44)u˜ix=∑r=0N+1∑j=0Mci,jBr,jxTheorem 2.
The solution to Fredholm integral equations system (2) using the HBI method converges if0<SL<1.Proof.
We have(45)eBx=uix−u˜ix=maxx∈0,1uix−u˜ix≤maxx∈0,1∫01ki,jx,tGuit−Gu˜itdt.
This implies that1−SLeBx<1.
Hence, by choosing0<1−SL<1 when N+1M⟶+∞, it implies that eB⟶0.


## 6. Numerical Examples

In this part, some mathematical models will be introduced to show the exactness of our methodology. Most of these models are taken from different references. In this way, one can reevaluate the obtained mathematical outcomes by the proposed technique for obtaining the exact solution. All results are computed by using a program implemented in MATLAB R2018b. The absolute errorsei are defined as(46)ei=max∀x∈[0,1)u˜ix−uix,i=1,2,where u˜ix and uix are the approximate and the exact solutions of the integral equation system, respectively.Example 1.
Consider the system of two linear Fredholm integral equations that was given in references [29, 30].(47)u1x=116x+1115−∫t=01x+tu1tdt−∫t=01x+2t2u2tdt,u2x=54x2+14x−∫t=01xt2u1tdt−∫t=01x2tu2tdt,with exact solution (u1x,u2x)=x,x2.
HBI method is applied to solve Example 1. The numerical results are tabulated, as can be seen in Table 1 for M=2,N=3. Table 2 shows the comparison of the absolute error among the approximate solutions obtained by the proposed method, the block-pulse function method which works with equally spaced points intervals while taking nonequal spaced points in the improved block-pulse method with different intervals [30], and the iterative triangular function method for m=32 [31] at different values of x.Table 1
The numerical results for Example1 using our presented method when M=2,N=3.
xExact solutionPresented methodAbsolute erroru1xu2xu˜1xu˜2xmax∀x∈[0,1)u˜1x−u1xmax∀x∈[0,1)u˜2x−u1x000000.0000000e + 000.0000000e + 000.10.10.010.10.019.0949470e − 142.2737368e − 150.20.20.040.20.041.7735147e − 131.8758328e − 140.30.30.090.30.092.0008883e − 133.8653525e − 140.40.40.160.40.161.4097168e − 133.2400749e − 140.50.50.250.50.250.0000000e + 000.0000000e + 000.60.60.360.60.362.0008883e − 132.0008883e − 130.70.70.490.70.497.2759576e − 144.3655746e − 130.80.80.640.80.643.8198777e − 137.0940587e − 130.90.90.810.90.817.2759576e − 145.8207661e − 13Table 2
Absolute error of the approximate solutions obtained by the proposed method, block-pulse function method [30], and triangular function method [31] at different values of x.
xBPF method [30]TFs method [31]Presented methode1e2e1e2e1e201.421E − 024.331E − 027.06640825E − 051.43660814E − 020.0000000e + 000.0000000e + 000.11.980E − 032.900E − 046.05982093E − 051.31923746E − 029.0949470e − 142.2737368e − 150.23.450E − 031.720E − 035.05322133E − 051.17262107E − 021.7735147e − 131.8758328e − 140.38.540E − 031.340E − 034.04661140E − 051.00042018E − 022.0008883e − 133.8653525e − 140.41.210E − 028.120E − 033.03999306E − 058.06296018E − 031.4097168e − 133.2400749e − 140.51.386E − 021.411E − 022.03336826E − 055.93909818E − 030.0000000e + 000.0000000e + 000.66.410E − 032.405E − 021.02673894E − 053.66922804E − 032.0008883e − 132.0008883e − 130.79.140E − 039.000E − 052.01070365E − 071.28996204E − 037.2759576e − 144.3655746e − 130.81.134E − 021.180E − 029.86525506E − 061.16208753E − 033.8198777e − 137.0940587e − 130.91.512E − 024.490E − 031.99315675E − 053.65030840E − 037.2759576e − 145.8207661e − 13Example 2.
Consider the system of linear Fredholm integral equations (Tables3 and 4):(48)u1x=34+∫t=01xu1tdt+∫t=01x−tu2tdt,u2x=−112−x+3x2+∫t=01x−tu1tdt+∫t=01tu2tdt,with exact solution (u1x,u2x)=2x,3x2.Table 3
The numerical results for Example2 using our presented method when M=2,N=3.
xExact solutionPresented methodu1xu2xu˜1xu˜2x000000.10.20.030.20.030.20.40.120.40.120.30.60.270.60.270.40.80.480.80.480.510.7510.750.61.21.081.21.080.71.41.471.41.470.81.61.921.61.920.91.82.431.82.43Table 4
Absolute error of the approximate solutions obtained by the proposed method and triangular function method [31] at different values of x.
xPresented methodTFs method [31]u˜1xu˜2xe1xe2x00.0000000e + 000.0000000e + 007.19349008e − 035.99302621e − 020.11.8189894e − 132.7284841e − 145.47004118e − 036.00726375e − 020.23.5470293e − 136.1390892e − 143.74659228e − 036.02150130e − 020.34.0017767e − 132.7284841e − 142.02314338e − 036.03573884e − 020.42.8194336e − 136.8212103e − 152.99694479e − 046.04997638e − 020.50.0000000e + 000.0000000e + 001.42375442e − 036.06421393e − 020.64.0017767e − 132.1827873e − 133.14720332e − 036.07845147e − 020.71.4551915e − 138.7311491e − 134.87065222e − 036.09268902e − 020.87.6397555e − 131.9645086e − 126.59410112e − 036.10692656e − 020.91.4551915e − 138.7311491e − 138.3175501e − 036.1211641e − 02Example 3.
Consider the following system of two linear Fredholm integral equations,(49)u1x=x2−13x−712+∫t=01x−tu1tdt+∫t=01tu2tdt,u2x=34−56x+∫t=01x+tu1tdt+∫t=01xu2tdt,with exact solution (u1x,u2x)=x2,1+x.
HBI method is applied to solve Example 3. The numerical results are tabulated, as seen in Table 5 for M=2,N=3. Table 6 shows the comparison of the absolute error between the approximate solutions obtained by the proposed method and the iterative triangular function method for m=32 [31] at different values of x.Table 5
The numerical results for Example3 using our presented method when M=2,N=3.
xExact solutionPresented methodu1xu2xu˜1xu˜2x001010.10.011.10.011.10.20.041.20.041.20.30.091.30.091.30.40.161.40.161.40.50.251.50.251.50.60.361.60.361.60.70.491.70.491.70.80.641.80.641.80.90.811.90.811.9Table 6
Absolute error of the approximate solutions obtained by the proposed method and triangular function method [31].
xPresented methodTFs method in [30]e1xe2xe1xe2x0001.81717963e − 038.57684144e − 020.12.273736754e − 151.455191523e − 131.81867423e − 038.56561331e − 020.21.875832822e − 147.639755495e − 131.82505093e − 038.55438518e − 020.33.865352483e − 141.455191523e − 131.83630971e − 038.54315705e − 020.43.240074875e − 144.001776688e − 131.85245058e − 038.53192892e − 020.5001.87347354e − 038.52070079e − 020.62.000888344e − 134.001776688e − 131.89937858e − 038.50947266e − 020.74.365574569e − 131.455191523e − 131.93016571e − 038.49824453e − 020.87.094058674e − 137.639755495e − 131.96583493e − 038.48701640e − 020.95.820766091e − 131.455191523e − 132.00638624e − 038.47578827e − 02Example 4.
Consider the system of two linear Fredholm integral equations [31, 32].(50)u1x=x18+1736+∫t=01x+t3u1t+u2tdt,u2x=x2−1912x+1+∫t=01xtu1t+u2tdt,with exact solution (u1x,u2x)=x+1,x2+1.
Table7 shows the comparison among the results yielded by our presented method, iterative triangular function method [31], and the decomposition method proposed with m=32 [32] as far as Example 4 is concerned. Furthermore, in Table 8, the comparison of absolute error, among the approximate solutions obtained by the proposed method, iterative triangular function method for m=32 [31], and the decomposition method proposed in [32], at different values of x is shown.Table 7
The numerical results for Example4 using our presented method when M=2,N=3.
xExact solutionMethod [32]Method [31]Presented methodResults ofu˜1x010.9884981.00007710.11.11.0866321.1000971.10.21.21.1847661.2001171.20.31.31.2828991.3001361.30.41.41.3810331.4001561.40.51.51.4791671.5001751.50.61.61.5773011.6001951.60.71.71.6754351.7002141.70.81.81.7735691.8002341.80.91.91.98717021.9002531.9Results ofu˜2x0111.07036310.11.011.0065491.0677081.010.21.041.0330991.0860821.040.31.091.0796481.1254861.090.41.161.1461981.1859181.160.51.251.2327471.2673791.250.61.361.3392961.3698691.360.71.491.4658461.4933871.490.81.641.6126951.6379351.640.91.811.7789451.8035111.81Table 8
Absolute error of the approximate solutions obtained by the proposed method, iterative triangular function method [31], and the decomposition method proposed with m=32 [32].
xPresented methodTFs method [31]Method [32]e1xe2xe1xe2xe1xe2x0007.74578E − 057.03625E − 021.15020E − 020.00000E + 000.11.4551915e − 131.4551915e − 139.70027E − 055.77080E − 021.33680E − 023.45100E − 030.27.6397555e − 135.4569682e − 131.16548E − 044.60824E − 021.52340E − 026.90100E − 030.31.4551915e − 137.2759576e − 131.36093E − 043.54856E − 021.71010E − 021.03520E − 020.44.0017767e − 135.4569682e − 131.55637E − 042.59177E − 021.89670E − 021.38020E − 020.5001.75182E − 041.73787E − 022.08330E − 021.72530E − 020.64.0017767e − 135.8207661e − 131.94727E − 049.86860E − 032.26990E − 022.07040E − 020.71.4551915e − 138.7311491e − 132.14272E − 043.38735E − 032.45650E − 022.41540E − 020.87.6397555e − 138.7311491e − 132.33817E − 042.06503E − 032.64310E − 022.73050E − 020.91.4551915e − 132.9103830e − 132.53362E − 046.48854E − 038.71702E − 023.10550E − 02


### 6.1. Work Extension

We intend to extend our work to apply our method to solve weakly singular Fredholm integral equation [33]. Also, fractional Fredholm integral equations will be investigated in future work [34–36] as well as two-dimensional Fredholm integral equations [37].


## 6.1. Work Extension

We intend to extend our work to apply our method to solve weakly singular Fredholm integral equation [33]. Also, fractional Fredholm integral equations will be investigated in future work [34–36] as well as two-dimensional Fredholm integral equations [37].


## 7. Conclusion

The hybrid Bernstein and improved block-pulse functions method proposed by Ramadan and Osheba [28] is used in this paper to achieve a numerical solution for a system of linear Fredholm integral equations of the second kind. Illustrative examples to test the accuracy and the applicability of the proposed HBI are given. The obtained numerical results indicate that the accuracy of the proposed hybrid method is more satisfactory than the usage of triangular functions, iterative triangular functions, block-pulse function, and the decomposition method. Error obtained by the present method is reduced at lower iterations, that is why the new hybrid method is highly recommended. The software package used to obtain those results is the MATLAB R2018b program.



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