# On the Solution of Quadratic Nonlinear Integral Equation with Different Singular Kernels

Authors: M. Basseem; Ahmad Alalyani
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

All the previous authors discussed the quadratic equation only with continuous kernels by different methods. In this paper, we introduce a mixed nonlinear quadratic integral equation (MQNLIE) with singular kernel in a logarithmic form and Carleman type. An existence and uniqueness of MQNLIE are discussed. A quadrature method is applied to obtain a system of nonlinear integral equation (NLIE), and then the Toeplitz matrix method (TMM) and Nystrom method are used to have a nonlinear algebraic system (NLAS). The Newton–Raphson method is applied to solve the obtained NLAS. Some numerical examples are considered, and its estimated errors are computed, in each method, by using Maple 18 software.


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## Body

## 1. Introduction

Integral equations of various types and kinds play an important role in several mathematical problems modelling. Analytical solutions of integral equations, however, neither exist nor simple to find, so several numerical methods have been developed for finding the solutions of integral equations. The quadratic equation provides an important tool for modelling many numerical phenomena, bio-mathematical problems, and process engineering. In genetics, it represents the reproduction equation, through which events affecting the cells can be predicted. Gripenberg [1] studied the existence and the uniqueness of a bounded continuous solution to the following integral equation of product type:(1)xt=kpt+∫0tAt−sxsdsgt+∫0tBt−sxsds,which arises in the study of the spread of an infectious disease that does not induce permanent immunity.Abdou and Basseem [2] used Chebyshev polynomial in solving mixed integral equation in position and time using spectral relationships. Javidi and Golbabai [3] solved NLFIE by the modified homotopy perturbation method. Alipanah and Esmaeili [4] used radial basis function to find a solution of two-dimensional FIE. Bernstein’s method is used to solve VIE by Maleknejad et al. [5]. The Toeplitz matrix method is used to solve NLIE of Hammerstein by Abdou et al. [6]. Orsi [7] used the product Nystrom method to get the solution of NVIE when its kernel takes a logarithmic form and Carleman function. The degenerate kernel method is discussed in three-dimensional NLIE by Basseem [8, 9]. Guoqiang et al. [10] obtained numerically the solution of two-dimensional NVIE by collocation and iterated collocation methods. Brunner et al. [11] introduced a class of methods to obtain numerically the solution of Abel integral equation. Abdou and Raad [12] used the Adomian decomposition method for solving quadratic NLIE. The radial basis function method with collocation scheme for solving quadratic integral equation of Urysohn’s type is described by Avazzadeh [13]. Assaria et al. used meshless methods for solving NLIE (see [14–17]).In this paper, a new problem in a product type of mixed integral equation with singular kernel is considered. The existence and uniqueness of its solution are discussed.The quadratic method is applied to obtain a NLS of FIE, and then the Toeplitz matrix method or Nystrom method is used to obtain a NLAS which is solved numerically by the Newton–Raphson method.Consider the QNLE(2)γx,t,φx,t−λ∫0tVt,τφx,tdτ∫−11kx−yφy,tdy=fx,t,where Vt,τ is a continuous function of time, belongs to the class C0,T,0,T,t,τ∈0,T,T<1. The singular kernel of position kx−y takes many different forms. The given functionfx,t is in the spaceC0,T×L2−1,1. γx,t,φx,tis a given nonlinear function of the unknown function φx,t. The constant λ has many physical meanings.


## 2. Existence and Uniqueness

In order to guarantee the existence of a unique solution of equation (2), assume(1)
The discontinuous kernel of equation (2) verifies(3)∫−11∫−11k2x−ydxdy1/2=C.(2)
The positive kernel of time is continuous and satisfies(4)max0≤t,τ≤TVt−τ≤M,∀t,τ∈0,T.(3)
The given continuous functionfx,t∈L2−1,1×C0,Tand its norm is defined as(5)fx,t=max0≤t≤T∫0tfx,τdτ∫−11f2x,tdx1/2≤H.(4)
The functionγx,t,φx,tsatisfies(i)
γx,t,φx,t=max0≤t≤T∫0tγx,τ,φx,τdτ∫−11γ2x,t,φx,tdx1/2≤E and φ=D, where D and E are constants.(ii)
For any two functionsφ1andφ2, γx,t,φx,t satisfies Lipchitz condition which is(6)γx,t,φ1x,t−γx,t,φ2x,t≤Nφ1x,t−φ2x,t.


## 3. Integral Operator of MQNIE

Eq. (2) can be written in the operator form as(7)Γ=Υφ+F,where(8)Υφ=λ∫0tVt,τφx,tdτ∫−11kx−yφy,tdy.Γ and F are the Nemytskii operator generated by the functions γx,t,φ2x,t and fx,t, respectively.Theorem 1.
The solution of equation (2) exists and is unique under the condition λ<1/MC.
The proof of this theorem can be deduced after the following discussion.Lemma 1.
The operatorY is bounded.Proof.
Since(9)Υφ≤λ∫0tVt,τφx,τdτ∫−11kx−yφy,tdy,applying Cauchy–Schwarz inequality, we have(10)Υφ≤λMCD.Lemma 2.
The operatorΥis continuous.Proof.
We assume two functionsφn and φmsatisfy equation (2), and then we get(11)Υφn−Υφm≤λ∫0tVt,τφnx,t−φmx,tdτ∫−11kx−yφnx,t−φmx,tdy.
Applying Cauchy–Schwarz inequality, we have(12)Υφn−Υφm≤λMCφnx,t−φmx,t,whenever φnx,t−φmx,t⟶0, one can deduce Υφn−Υφm⟶0, which proves the continuity of the operator.
Moreover, under the conditionλ<1/MC, the operator Υ is a contraction mapping, and by fixed point theorem, equation (2) has a unique solution in the space L2−1,1×C0,T.


## 4. Quadratic Numerical Method (See [18])

To obtain a system of NLIE, divide the time interval0,Tas(13)0=t0<t1<t2<…<tl=T.Lett=ti, then equation (2) becomes(14)γx,ti,φx,ti−λ∫0tiVti,τφx,tdτ∫−11kx−yφy,tidy=fx,ti.Applying the quadrature rule, equation (14) reduces to(15)γx,ti,φx,ti−λ∑j=0iωjVi,jφjx∫−11kx−yφiydy=fix,where(16)ωj=h2,j=0,i,h,otherwise.Using the notation(17)φix=φx,ti,Vi,j=Vti,tj,γix,φix=γx,ti,φix,we get(18)γix,φix−λ∫−11kx−yφiydyh2Vi,iφix+∑j=0i−1ωjVi,jφjx=fix,which is the system of NLIE can be solved by two different methods, namely, Toeplitz matrix method and Nystrom method.


### 4.1. Algebraic System of NLIE

Consider(19)Γγφi,x=λUφix⋅Vφix+fix,where(20)Uφix=∫−11kx−yφiydyand(21)Vφix=∑j=0iωjVi,jφjx.In order to guarantee the existence of a unique solution of an algebraic system of NIE, we assume the following conditions:(i)
maxifix≤H∗(ii)
∑j=0imaxjωjVi,j≤M∗(iii)
maxiγx,φix≤E∗Hence, formula (19) has a unique solution under condition λ<1/CM∗.Definition 1.
The estimate local errorR1is determined by the following relation:(22)R1=∫−11kx−yφy,tdy∫0tVt,τφx,τdτ−∫−11kx−yφiydy∑j=0iωjVi,jφjx.


## 4.1. Algebraic System of NLIE

Consider(19)Γγφi,x=λUφix⋅Vφix+fix,where(20)Uφix=∫−11kx−yφiydyand(21)Vφix=∑j=0iωjVi,jφjx.In order to guarantee the existence of a unique solution of an algebraic system of NIE, we assume the following conditions:(i)
maxifix≤H∗(ii)
∑j=0imaxjωjVi,j≤M∗(iii)
maxiγx,φix≤E∗Hence, formula (19) has a unique solution under condition λ<1/CM∗.Definition 1.
The estimate local errorR1is determined by the following relation:(22)R1=∫−11kx−yφy,tdy∫0tVt,τφx,τdτ−∫−11kx−yφiydy∑j=0iωjVi,jφjx.


## 5. Toeplitz Matrix Method (See [6])

We apply the TMM to have a nonlinear algebraic equation. For this, considerh=1/N; therefore,(23)∫−11kx−yφiydy=∑n=−NN−1∫nhnh+hkx−yφiydyand(24)∫nhnh+hkx−yφiydy≈Anxφnh+Bnxφnh+h+R.The functionsAnx and Bnx are arbitrary functions to be determined, and R is the error term. In order to obtain the values of two functions, assume φy=1,y. This yields a set of two equations in terms of two unknown functions. After ignoring the error term, equation (18) becomes(25)γix,φix−hλ2∑n=−NNξnxφinhVi,iφix=λ∑n=−NNξnxφinh∑j=0i−1ωjVi,jφjx+fix.Letx=mh, then using the following notation:(26)φix=φimh=φim,ξnx=ξnm,fix=fim,γix,φix=γimφim,equation (25) becomes(27)γimφim−hλVii2∑n=−NNξnmφinφim=λ∑n=−NNξnmφin∑j=0i−1ωjVi,jφjm+fim,where(28)ξnm=A−nmh,n=−N,Anmh+Bn−1mh,−N<n<N,Bn−1mh,n=N.Equation (27) represents that the NLAS can be solved using the Newton–Raphson method.Definition 2.
The Toeplitz matrix method is said to be convergent of orderr in the interval [−1, 1], if and only if, for sufficient large N, there exists a constant D > 0 independent of N such that(29)φx−φNx≤DN−r.Definition 3.
The estimate local errorR(2) is determined by the following relation:(30)R2=∫−11kx−yφiydy∑j=0iωjVi,jφjx−∑n=−NNξnmφin∑j=0iωjVi,jφjm.


### 5.1. Existence and Uniqueness of NLAS

In order to guarantee the existence of a unique solution of a NLAS, we assume the following conditions:(i)
supn,m∑n=−NNξnm≤C∗(ii)
supi,mfim≤H∗∗(iii)
supi,nγinh,φnh≤E∗∗Definition 4.
The estimate local error R(T) is determined by the following relation:(31)RT=∫−11kx−yφy,tdy∫0tVt,τφx,tdτ−∑n=−NNξnmφin∑j=0iωjVi,jφjm,where RT≤R1+R2.


## 5.1. Existence and Uniqueness of NLAS

In order to guarantee the existence of a unique solution of a NLAS, we assume the following conditions:(i)
supn,m∑n=−NNξnm≤C∗(ii)
supi,mfim≤H∗∗(iii)
supi,nγinh,φnh≤E∗∗Definition 4.
The estimate local error R(T) is determined by the following relation:(31)RT=∫−11kx−yφy,tdy∫0tVt,τφx,tdτ−∑n=−NNξnmφin∑j=0iωjVi,jφjm,where RT≤R1+R2.


## 6. Nystrom Method (See [7])

Here, by using the product integration, we approximate the integral part of equation (18) by a suitable Lagrange interpolation polynomial. For this, let x=xm, and the integral part can be written as(32)∫−11kxm,yφiydy=∑n=0Nℑm,nφiyn=∑n=0N−2/2∫y2ny2n+2kxm,yφiydy,where xm=ym=−1+mh,m=1,2,3,…,Nwith h=2/N and N is even number.Approximate the nonsingular part of the integralφy over each interval y2n,y2n+2 by the Lagrange interpolation polynomial at the points 2n, 2n+1, and 2n+2. Therefore, equation (32) becomes(33)∫−11kxm,yφiydy=∑n=0N−2/2∫y2ny2n+2kxm,yy2n+1−yy2n+2−yy2n+1−y2ny2n+2−y2nφiy2n+y2n−yy2n+2−yy2n−y2n+1y2n−y2n+1φiy2n+1+y2n−yy2n+1−yy2n−y2n+2y2n+1−y2n+2φiy2n+2dy.Comparing equations (32) and (33), we deduce(34)ℑm,0=12h2∫y0y2kxm,yy1−yy2−ydy,ℑm,2n+1=1h2∫y2ny2n+2kxm,yy−y2ny2n+2−ydy,ℑm,2n=12h2∫y2ny2n+2kxm,yy2n+1−yy2n+2−ydy+∫y2n−2y2nkxm,yy−y2n−1y−y2n−2dy,ℑm,N=12h2∫yN−2yNkxm,yy−yN−2y−yN−1dy.Introduce the following notations:(35)αnym=12h2∫y2n−2y2nkxm,yy−y2n−2y−y2n−1dy,βnym=12h2∫y2n−2y2nkxm,yy2n−1−yy2n−ydy,and(36)ζnym=12h2∫y2n−2y2nkxm,yy−y2n−2y2n−ydy.Then,(37)ℑm,0=β1xm,ℑm,2n+1=2ζn+1xm,ℑm,2n=αnxm+βn+1xm,ℑm,N=αN/2xm.By substituting in equation (18), we get(38)γimφim−λ∑n=0Nℑm,nφiynh2Vi,iφim+∑j=0i−1ωjVi,jφjm=fim,where equation (38) represents the NAS in which its existence and uniqueness can be easily shown as in the previous section.Definition 5.
The Nystrom method is said to be convergent of orderr in the interval [−1, 1], if and only if, for sufficient large N, there exists a constant K > 0 independent of N such that(39)φx−φNx≤KN−r.


## 7. Numerical Examples

### 7.1. Example 1

Consider the equation(40)γφ,x,t−λ∫0tVt,τφx,τdτ⋅∫−11kx−yφy,tdydt=fx,t,where fx,t is given by putting φx,t=x2t2 as an exact value with γ=φ2,Vt,τ=t−τ2, and (41)kx−y=lny−x,x−yν,0<ν<1.(1)
The following table is selected among a large amount of data to compare between the exact solution and its numerical solution in the case of logarithmic kernel for both of the previous methods in some points in the regionx∈0,1 and for different values of timeT=0.009,0.02,0.8.(2)
In the following table, we compare between the Toeplitz matrix method and Nystrom method for differentν and fixed time T=0.3 in Carleman kernel form.


### 7.2. Example 2

In the next example, the Nystrom method and Toeplitz matrix method are used with fixed timeT=0.4 and the position interval is divided with N=2,4,8,16,32 points. The rate of errors is evaluated using the formula(42)Rate=log2Error2NErrorN.The negative sign means that by increasingN, the error decreases (see Table 3).(43)eφx,t−λ∫0tt−τ2φx,τdτ⋅∫−11lnx−yφy,tdydt=fx,t,where fx,tis given by setting ϕx,t=xt as an exact value.Table 1
Comparison between the Toeplitz matrix method and Nystrom method in different time when kernel takes logarithmic form.
Toeplitz matrix methodNystrom methodTxφErrorφError0.009–0.870.00006132.630 × 10−150.00006134.950 × 10−15–0.370.00001112.157 × 10−150.00001118.298 × 10−150.130.00000151.637 × 10−70.00000151.642 × 10−70.630.00003211.219 × 10−140.00003214.992 × 10−150.90.00006565.469 × 10−150.00006561.139 × 10−140.02–0.870.00030281.179 × 10−130.00030284.435 × 10−13–0.370.00005485.055 × 10−140.00005482.952 × 10−130.130.00000689.062 × 10−120.00000689.582 × 10−120.630.00015881.417 × 10−130.00015883.315 × 10−130.90.00032406.330 × 10−150.00032403.040 × 10−130.8–0.870.48443071.468 × 10−50.48446254.647 × 10−5–0.370.08762327.192 × 10−60.08764492.889 × 10−50.130.01081144.647 × 10−60.01083211.606 × 10−50.630.25400511.094 × 10−50.25405413.807 × 10−50.90.51839475.367 × 10−60.51843013.009 × 10−5


## 7.1. Example 1

Consider the equation(40)γφ,x,t−λ∫0tVt,τφx,τdτ⋅∫−11kx−yφy,tdydt=fx,t,where fx,t is given by putting φx,t=x2t2 as an exact value with γ=φ2,Vt,τ=t−τ2, and (41)kx−y=lny−x,x−yν,0<ν<1.(1)
The following table is selected among a large amount of data to compare between the exact solution and its numerical solution in the case of logarithmic kernel for both of the previous methods in some points in the regionx∈0,1 and for different values of timeT=0.009,0.02,0.8.(2)
In the following table, we compare between the Toeplitz matrix method and Nystrom method for differentν and fixed time T=0.3 in Carleman kernel form.


## 7.2. Example 2

In the next example, the Nystrom method and Toeplitz matrix method are used with fixed timeT=0.4 and the position interval is divided with N=2,4,8,16,32 points. The rate of errors is evaluated using the formula(42)Rate=log2Error2NErrorN.The negative sign means that by increasingN, the error decreases (see Table 3).(43)eφx,t−λ∫0tt−τ2φx,τdτ⋅∫−11lnx−yφy,tdydt=fx,t,where fx,tis given by setting ϕx,t=xt as an exact value.Table 1
Comparison between the Toeplitz matrix method and Nystrom method in different time when kernel takes logarithmic form.
Toeplitz matrix methodNystrom methodTxφErrorφError0.009–0.870.00006132.630 × 10−150.00006134.950 × 10−15–0.370.00001112.157 × 10−150.00001118.298 × 10−150.130.00000151.637 × 10−70.00000151.642 × 10−70.630.00003211.219 × 10−140.00003214.992 × 10−150.90.00006565.469 × 10−150.00006561.139 × 10−140.02–0.870.00030281.179 × 10−130.00030284.435 × 10−13–0.370.00005485.055 × 10−140.00005482.952 × 10−130.130.00000689.062 × 10−120.00000689.582 × 10−120.630.00015881.417 × 10−130.00015883.315 × 10−130.90.00032406.330 × 10−150.00032403.040 × 10−130.8–0.870.48443071.468 × 10−50.48446254.647 × 10−5–0.370.08762327.192 × 10−60.08764492.889 × 10−50.130.01081144.647 × 10−60.01083211.606 × 10−50.630.25400511.094 × 10−50.25405413.807 × 10−50.90.51839475.367 × 10−60.51843013.009 × 10−5


## 8. General Conclusion

From the above tables and our numerical results, we can deduce the following:(1)
The estimated error increases by time, where its mean errors by using Toeplitz and Nystrom methods, whenT = 0.02, are 1.876 × 10−12 and 2.191 × 10−12, respectively, while, its mean errors when T = 0.8 are 8.565 × 10−6 and 3.192 × 10−5, respectively.(2)
The Toeplitz matrix method is comparatively better than the Nystrom method for different kernels (see Tables1 and 2).(3)
By increasingN, the error is extremely stable in both methods, but in the Toeplitz matrix method, the error almost decreases by increasing in N, where the convergence rate with +ve sign means the increasing of errors, while its −ve sign means the errors decreasing (see Table 3).(4)
In Carleman kernel form, the estimated error decreases in small values ofν, where its mean errors take 1.816 × 10−8 when ν = 0.07, while it takes 5.925 × 10−8 when ν = 0.47 by using the Toeplitz matrix method (see Table 2).Table 2
Errors in the Toeplitz matrix method and Nystrom method in different values ofν in Carleman kernel form.
Toeplitz matrix methodNystrom methodνxErrorError0.07–0.872.665 × 10−83.198 × 10−8–0.372.218 × 10−82.205 × 10−80.131.439 × 10−81.734 × 10−80.631.204 × 10−81.563 × 10−80.91.556 × 10−82.184 × 10−80.17–0.874.191 × 10−85.629 × 10−8–0.372.924 × 10−82.716 × 10−80.139.517 × 10−91.726 × 10−80.639.449 × 10−101.125 × 10−80.91.209 × 10−83.042 × 10−80.47–0.871.352 × 10−72.110 × 10−7–0.376.733 × 10−84.547 × 10−80.131.876 × 10−81.748 × 10−80.636.492 × 10−83.365 × 10−100.91.004 × 10−81.268 × 10−7Table 3
Convergence rate in both methods with fixed timeT=0.4.
Toeplitz matrix methodNystrom methodNMean errorRateMean errorRate26.641 × 10−6—7.144 × 10−6—46.298 × 10−6−0.07650671.079 × 10−50.595699986.482 × 10−60.04154531.110 × 10−50.0408648165.691 × 10−6−0.18775681.102 × 10−5−0.0104355325.556 × 10−6−0.03463561.096 × 10−5−0.0078764



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