# Solution of Nonlinear Volterra Integral Equations with Weakly Singular Kernel by Using the HOBW Method

Authors: Mohamed R. Ali; Mohamed M. Mousa; Wen-Xiu Ma
Journal: Advances in Mathematical Physics
 (2019)

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## Abstract

We present a new numerical technique to discover a new solution of Singular Nonlinear Volterra Integral Equations (SNVIE). The considered technique utilizes the Hybrid Orthonormal Bernstein and Block-Pulse functions wavelet method (HOBW) to solve the weakly SNVIE including Abel’s equations. We acquire the HOBW implementation matrix of the integration to derive the procedure of solving these kind integral equations. The explained technique is delineated with two numerical cases to demonstrate the benefit of the technique used by us. At last, the exchange uncovers the way that the strategy utilized here is basic in usage.


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## Body

## 1. Introduction

In the current literature, there are many different applications of SNVIE in various areas, such as mathematical physics, electrochemistry, scattering theory, heat conduction, semiconductors, population dynamics, and fluid flow [1, 2]. Numerical strategies for the SNVIE are spline collocation methods [3], Newton–Cotes methods [4], extrapolation algorithm [5], and Hermite-collocation method [6]. The most popular methods for talking about the such equations are introduced, such as homotopy asymptotic method [7], Nyström interpolant method [8], Mesh method [9], Tau method [10], Laplace transform [11], orthonormal Bernstein, and block-pulse functions [12–17].Wavelet theory is a moderately new and considered as a rising territory in the field of applied science and engineering. Wavelets allow the accurate representation of a lot of functions. The wavelet technique is a new numerical technique utilized for dissolving the fractional equations. SNVIE has numerous applications in different zones, for example, semiconductors’ mathematical chemistry, chemical reactions, physics, scattering theory, electrochemistry, seismology, metallurgy, fluid flow, and population dynamics [2, 18–20].In 1823, Niels Henrik Abel derived the equation(1)fx+∫0xutx-t0.5dt=0,where u(t) is an unknown function and f(x) is a given function. This equation is an example of a nonhomogeneous Volterra equation of first kind with weak singularity. Abel obtained this equation while studying the motion of a particle on a smooth curve lying on a vertical plane. The physical depiction of this condition is given in [21] as pursues. Abel thought about the issue in traditional mechanics, which is that of deciding the time a molecule brings to slide openly down a smooth settled bend in a verticalxy-plane (in Figure 1), from any settled point X,Y on the bend to its absolute bottom (the starting point 0). If m means the mass of the molecule and xp(y) signifies the condition of the smooth bend where x is a differentiable function of y, at that point we acquire the vitality protection condition as(2)12mv2+mgy=mgY,where v is the speed of the molecule at the position (x,y) at time t, if the molecule tumbles from rest at time t=0 from the point (x,y), and g represents acceleration due to gravity. The connection (2) can be expressed as(3)dsdt=-2g(Y-y1/2by utilizing the arc-length s(t), estimated from the starting point to the point(x,y), where a less sign has been utilized in the square root since s diminishes with time t amid the fall of the molecule. Using the formula(4)dsdy=1+p′2y0.5we can compose(5)dydt=dydsdsdt=-2gY-y1+p′2y0.5By integrating both sides of (5), we obtain (6)∫Y01+p′2y2gY-y0.5dy=-∫0Tdy=-T,where T is the total time of fall of the particle, from the point (x,y) to the origin (0,0). Therefore, we have(7)∫0YuyY-y0.5dy=T=fY,uy=1+p′2y2g0.5,0<Y<awhere f0=0. In this way, we can find that the time of descent of the particle,T, can be resolved totally by utilizing the recipe (7), if the state of the curve xp(y), and consequently the function u(y) is known. On the off chance that we consider, on the other hand, the issue of assurance of the state of the bend, when the time of fall T is known, which is the historic Abel’s problem, then the relation (7) is an integral equation for the unknown function u(y), which is known as Abel’s integral equation.Figure 1
Abel’s problem.The most general form of Abel’s integral equation is given by(8)∫0xuthx-htαdt=fx,x>0,f0=0,0<α<1,where h(x) is a monotonically expanding function. We have picked it as h(x). Also, a general form of SVIE of second kind is given as(9)ux=fx+λ∫0xutx-tαdt,0<α<1,where f(x) is in L2(R) on the interim t≤T,0≤x and λ is a steady parameter.We utilize the HOBW method for determining the approximation solution of SNVIE of the shape given by(10)yx=fx+λ∫axkx,tFytx-t1-αdt,0≤x≤1.where f(x),k(x,t) are continuous functions, while 0<α<1 and y(x) is the unknown function to be determined.This paper is organized as follows. Initially the basic formulation of the HOBW method and some properties of HOBW are defined in Section2. In Section 3, we determine the HOBW implementation matrix of integration. While in Section 4, we summarize the process of dissolving weakly singular-Volterra integral equations based on the HOBW implementation matrix method. In Section 5, we consider two examples which demonstrate the validity of this method. Finally, the concluding remarks are demonstrated.


## 2. The HOBW Method and Operational Matrix of the Integration

### 2.1. Wavelets and the HOBW Method

Wavelets constitute a group of functions constructed from dilation and translation of a single functionψ(x) called the mother wavelet. In which parameter of dilation a and parameter of translation b vary continuously.(11)ψa,bt=a-1/2ψt-ba,a,b∈R,a≠0By letting a and b be discrete values such as a=a0-k,b=nb0a0-k,a0>1, b0>0,wheren and k are positive integers, we attain the family of discrete wavelets:(12)ψk,nt=a0k/2ψa0kt-nb0,n,k∈Z+Then we see that ψk,n(t) forms a wavelet basis for L2(R). In particular, when a0=2, b0=1, then ψk,n(t) forms an orthonormal basis. Here, HOBWi,j(t)=HOBW(k,i,j,t) involves four arguments, i=1,…,2k-1,k is to be any positive integer, j is the degree of the Bernstein polynomials, and t is the normalized time. HOBWi,j(t) are defined on [0, 1) as [12, 13](13)HOBWi,jt=2k-1/2nj2k-1x-i+1j1-2k-1x-i+1n-ji-12k-1≤t<i2k-10otherewisewhere i=1,2,…,2k-1, j=0,1,…,M-1 and k is a positive integer. Thus, we attain our new basis as {HOBW1,0,HOBW1,1,…,HOBW2k-1,M-1} and any function is truncated with them.TheHOBWdetect orthonormal basis is given by (14)HOBWijx,HOBWi/j/x=1i,j=i/,j/0i,j≠i/,j/where (·,·) is called the inner product in L2[0,1). The HOBW has compact support [i-1/2k-1,i/2k-1],  i=1,…,2k-1.


### 2.2. Function Approximation by Using the HOBW Functions

Any functiony(t), which is integrable in [0,1), is truncated by using the HOBW method as follows:(15)yt=∑i=1∞∑j=0∞cijHOBWijt,i=1,2,…,∞,j=0,1,2,…,∞,t∈0,1,where the HOBW coefficients cij can be calculated as given below:(16)cij=yt,HOBWijtHOBWijt,HOBWijtWe approximate y(t) by a truncated series as follows:(17)yt=∑i=12k-1∑j=0M-1cijHOBWijt=CTHOBWtwhere HOBW(t) and C are 2k-1M×1 vectors given by(18)HOBWt=HOBW10,HOBW11,…,HOBW1M-1,HOBW20,HOBW21,…,HOBW2M-1,…,HOBW2k-10,…,HOBW2k-1M-1T.and(19)C=c10,c11,…,c1M-1,c20,c21,…,c2M-1,…,c2k-10,…,c2k-1M-1T.We define the HOBW matrix Φ2k-1M×2k-1M as follows:(20)Φ2k-1M×2k-1M=HOBW12·2k-1M,HOBW32·2k-1M,…,HOBW2·2k-1M-12·2k-1M,The series in (17) contains an infinite number of terms for a smooth function y(t). Therefore, we have(21)CTHOBWt,HOBWt=yt,HOBWtso that(22)C=D-1yt,HOBWt,where(23)D=HOBWt,HOBWt,(24)=∫01HOBWt.HOBWTtdt(25)=D10⋯00D2⋯0⋮⋱000⋯DMThen, by using (14), Di(i=1,2,…,2k-1) is defined as follows:(26)Dni+1,j+1=∫i-1/2k-1i/2k-1HOBWi,n2k-1t-i+1HOBWj,n2k-1t-i+1dt(27)Dni+1,j+1=12k-1∫01HOBWi,ntHOBWj,ntdt(28)Dni+1,j+1=ninj2k-12n+12ni+jWe can also approximate the function k(x,t)∈L[0,1] as follows:(29)kx,t≈HOBWTxKHOBWt,where K is an 2k-1M×2k-1M matrix that we attain as follows:(30)K=D-1HOBWxkx,t,HOBWtD-1


### 2.3. Multiplication of the Hybrid Functions

We can evaluateHOBW(2k-1M×1)(t)HOBWT(2k-1M×1)(t) for VIE of the second kind via the HOBW functions as detailed below.Let the product ofHOBW(2k-1M×1)(t) and HOBWT(2k-1M×1)(t) be given by(31)HOBW2k-1M×1tHOBWT2k-1M×1t≅M2k-1M×2k-1Mtwhere(32)M2k-1M×2k-1Mt=HOBW10tHOBW10tHOBW10tHOBW20t⋯HOBW10tHOBW2k-1,MtHOBW20tHOBW10tHOBW20tHOBW20t⋯HOBW20tHOBW2k-1,MtHOBW30tHOBW10tHOBW30tHOBW20t⋯HOBW30tHOBW2k-1,Mt⋮⋮⋯⋮HOBW2k-1,MtHOBW10tHOBW2k-1,MtHOBW20t⋯HOBW2k-1,MtHOBW2k-1,MtWith the recursive formulas, we calculate M(2k-1M×2k-1M)(t) for any k and M.The matrixM(2k-1M×2k-1M)(t) in (23) satisfies the following relation:(33)M2k-1M×2k-1Mtc2k-1M×1=C~2k-1M×2k-1MHOBW2k-1M×1twhere c(2k-1M) is defined in (33) and C~(2k-1M×2k-1M) is the matrix coefficient. We consider the case when k=3 and M=4. Thus, we have(34)M16×16)t=HOBW10tHOBW10tHOBW10tHOBH20t⋯HOBW10tHOBW43tHOBW20tHOBW10tHOBW20tHOBH20t⋯HOBW20tHOBW43tHOBW30tHOBW10tHOBW30tHOBH20t⋯HOBW30tHOBW43t⋮⋮⋯⋮HOBW43tHOBW10tHOBW43tOBH20t⋯HOBW43tHOBW43tThe coefficient matrix C~(2k-1M×2k-1M) in (33) is determined by(35)C~2k-1M×2k-1M=C00000C10000C20000C3where Ci,i=0,1,2,3 are 4×4 matrices given by(36)Ci2k-1M×2k-1M=73c1i+523c2i1116c1i+311c2i322c1i+516c2i-319c1i+119c2i-1103c3i-7201c4i+521c3i+1101c4i-144c3i+1103c4i+3209c3i-3209c4i22144c1i+123c2i2223c1i+14123c2i419c1i+213c2i-123c1i+119c2i-17143c3i-13209c4i+1547c3i+2175c4i-281c3i+2147c4i+1190c3i-3105c4i914c1i+719c2i2425c1i+711c2i415c1i+24132c2i-19142c1i+319c2i-13132c3i-11120c4i+734c3i+3107c4i-571c3i+3107c4i+3201c3i-3208c4i1113c1i+120c2i2223c1i+1327c2i723c1i+213c2i-323c1i+325c2i-3103c3i-1209c4i+737c3i+2103c4i-190c3i+3103c4i+1213c3i-1201c4i


## 2.1. Wavelets and the HOBW Method

Wavelets constitute a group of functions constructed from dilation and translation of a single functionψ(x) called the mother wavelet. In which parameter of dilation a and parameter of translation b vary continuously.(11)ψa,bt=a-1/2ψt-ba,a,b∈R,a≠0By letting a and b be discrete values such as a=a0-k,b=nb0a0-k,a0>1, b0>0,wheren and k are positive integers, we attain the family of discrete wavelets:(12)ψk,nt=a0k/2ψa0kt-nb0,n,k∈Z+Then we see that ψk,n(t) forms a wavelet basis for L2(R). In particular, when a0=2, b0=1, then ψk,n(t) forms an orthonormal basis. Here, HOBWi,j(t)=HOBW(k,i,j,t) involves four arguments, i=1,…,2k-1,k is to be any positive integer, j is the degree of the Bernstein polynomials, and t is the normalized time. HOBWi,j(t) are defined on [0, 1) as [12, 13](13)HOBWi,jt=2k-1/2nj2k-1x-i+1j1-2k-1x-i+1n-ji-12k-1≤t<i2k-10otherewisewhere i=1,2,…,2k-1, j=0,1,…,M-1 and k is a positive integer. Thus, we attain our new basis as {HOBW1,0,HOBW1,1,…,HOBW2k-1,M-1} and any function is truncated with them.TheHOBWdetect orthonormal basis is given by (14)HOBWijx,HOBWi/j/x=1i,j=i/,j/0i,j≠i/,j/where (·,·) is called the inner product in L2[0,1). The HOBW has compact support [i-1/2k-1,i/2k-1],  i=1,…,2k-1.


## 2.2. Function Approximation by Using the HOBW Functions

Any functiony(t), which is integrable in [0,1), is truncated by using the HOBW method as follows:(15)yt=∑i=1∞∑j=0∞cijHOBWijt,i=1,2,…,∞,j=0,1,2,…,∞,t∈0,1,where the HOBW coefficients cij can be calculated as given below:(16)cij=yt,HOBWijtHOBWijt,HOBWijtWe approximate y(t) by a truncated series as follows:(17)yt=∑i=12k-1∑j=0M-1cijHOBWijt=CTHOBWtwhere HOBW(t) and C are 2k-1M×1 vectors given by(18)HOBWt=HOBW10,HOBW11,…,HOBW1M-1,HOBW20,HOBW21,…,HOBW2M-1,…,HOBW2k-10,…,HOBW2k-1M-1T.and(19)C=c10,c11,…,c1M-1,c20,c21,…,c2M-1,…,c2k-10,…,c2k-1M-1T.We define the HOBW matrix Φ2k-1M×2k-1M as follows:(20)Φ2k-1M×2k-1M=HOBW12·2k-1M,HOBW32·2k-1M,…,HOBW2·2k-1M-12·2k-1M,The series in (17) contains an infinite number of terms for a smooth function y(t). Therefore, we have(21)CTHOBWt,HOBWt=yt,HOBWtso that(22)C=D-1yt,HOBWt,where(23)D=HOBWt,HOBWt,(24)=∫01HOBWt.HOBWTtdt(25)=D10⋯00D2⋯0⋮⋱000⋯DMThen, by using (14), Di(i=1,2,…,2k-1) is defined as follows:(26)Dni+1,j+1=∫i-1/2k-1i/2k-1HOBWi,n2k-1t-i+1HOBWj,n2k-1t-i+1dt(27)Dni+1,j+1=12k-1∫01HOBWi,ntHOBWj,ntdt(28)Dni+1,j+1=ninj2k-12n+12ni+jWe can also approximate the function k(x,t)∈L[0,1] as follows:(29)kx,t≈HOBWTxKHOBWt,where K is an 2k-1M×2k-1M matrix that we attain as follows:(30)K=D-1HOBWxkx,t,HOBWtD-1


## 2.3. Multiplication of the Hybrid Functions

We can evaluateHOBW(2k-1M×1)(t)HOBWT(2k-1M×1)(t) for VIE of the second kind via the HOBW functions as detailed below.Let the product ofHOBW(2k-1M×1)(t) and HOBWT(2k-1M×1)(t) be given by(31)HOBW2k-1M×1tHOBWT2k-1M×1t≅M2k-1M×2k-1Mtwhere(32)M2k-1M×2k-1Mt=HOBW10tHOBW10tHOBW10tHOBW20t⋯HOBW10tHOBW2k-1,MtHOBW20tHOBW10tHOBW20tHOBW20t⋯HOBW20tHOBW2k-1,MtHOBW30tHOBW10tHOBW30tHOBW20t⋯HOBW30tHOBW2k-1,Mt⋮⋮⋯⋮HOBW2k-1,MtHOBW10tHOBW2k-1,MtHOBW20t⋯HOBW2k-1,MtHOBW2k-1,MtWith the recursive formulas, we calculate M(2k-1M×2k-1M)(t) for any k and M.The matrixM(2k-1M×2k-1M)(t) in (23) satisfies the following relation:(33)M2k-1M×2k-1Mtc2k-1M×1=C~2k-1M×2k-1MHOBW2k-1M×1twhere c(2k-1M) is defined in (33) and C~(2k-1M×2k-1M) is the matrix coefficient. We consider the case when k=3 and M=4. Thus, we have(34)M16×16)t=HOBW10tHOBW10tHOBW10tHOBH20t⋯HOBW10tHOBW43tHOBW20tHOBW10tHOBW20tHOBH20t⋯HOBW20tHOBW43tHOBW30tHOBW10tHOBW30tHOBH20t⋯HOBW30tHOBW43t⋮⋮⋯⋮HOBW43tHOBW10tHOBW43tOBH20t⋯HOBW43tHOBW43tThe coefficient matrix C~(2k-1M×2k-1M) in (33) is determined by(35)C~2k-1M×2k-1M=C00000C10000C20000C3where Ci,i=0,1,2,3 are 4×4 matrices given by(36)Ci2k-1M×2k-1M=73c1i+523c2i1116c1i+311c2i322c1i+516c2i-319c1i+119c2i-1103c3i-7201c4i+521c3i+1101c4i-144c3i+1103c4i+3209c3i-3209c4i22144c1i+123c2i2223c1i+14123c2i419c1i+213c2i-123c1i+119c2i-17143c3i-13209c4i+1547c3i+2175c4i-281c3i+2147c4i+1190c3i-3105c4i914c1i+719c2i2425c1i+711c2i415c1i+24132c2i-19142c1i+319c2i-13132c3i-11120c4i+734c3i+3107c4i-571c3i+3107c4i+3201c3i-3208c4i1113c1i+120c2i2223c1i+1327c2i723c1i+213c2i-323c1i+325c2i-3103c3i-1209c4i+737c3i+2103c4i-190c3i+3103c4i+1213c3i-1201c4i


## 3. HOBW Operational Matrix

Firstly, we review some basic definitions of fractional calculus [22–24], which are required for establishing our results.Definition 1.
The Riemann–Liouville fractional integral operatorI of order α, of a function f∈Cv, v≥-1, is defined as follows:(37)Iαft=1Γα∫0tt-τα-1fτdτ,α>0,t>0,ft,α=0The block-pulse functions (BPFs), an m-set of BPFs on [0, 1), are defined by(38)bit=1,im<t<i+1m,0,otherwise,where i=0,1,2,…,m-1. The BPFs have the orthogonal properties as follows:(39)bitbjt=0,i≠j,bit,i=j,and(40)∫01biτbjτ=0,i≠j,1,i=j,Every function f(t) which is integrable in [0,1) can be truncated with the aid of BPFs series as(41)ft≈∑i=0m-1fibit=FTBmt,where F=[f0,f1,…,fm-1]T,Bm(t)=[b0,b1,…,bm-1]T.
Using the disjointness of BPFs and the matrix ofBm(t) can be gotten by(42)BmtBmTt=b0t0b1t⋱0bm-1tEquation (41) implies that the HOBW method can be truncated into an m-set BPFs as follows:(43)HOBW2k-1M=Φ2k-1M×2k-1MB2k-1Mt.The block-pulse implementation matrix of the fractional integration Fα has been given in [14] as follows:(44)IαB2k-1Mt≈FαB2k-1Mtwhere(45)Fα=12k-1Mα1Γα+21ζ1ζ2ζ3⋯ζm′-101ζ1ζ2⋯ζm′-2001ζ1⋯ζm′-3⋮⋮⋮⋮⋮⋮00⋯01ζ1000⋯01(46)ζk=k+1α+1-2kα+1+k-1α+1.At α=1, Fα is BPF’s implementation matrix of integration.
Let(47)Iαψ2k-1Mt≈P2k-1M×2k-1MαB2k-1Mtwhere the matrix P2k-1M×2k-1Mα is called the HOBW implementation matrix of fractional integration [2, 17]. Using (43) and (44), we have (48)Iαψ2k-1Mt≈IαΦ2k-1M×2k-1MB2k-1Mt=Φ2k-1M×2k-1MIαB2k-1Mt=Φ2k-1M×2k-1MFαB2k-1Mt.From (38) and (39) we can get(49)P2k-1M×2k-1Mαψ2k-1Mt=P2k-1M×2k-1MαΦ2k-1M×2k-1MB2k-1Mt=Φ2k-1M×2k-1MFαB2k-1MtThen the matrix is P2k-1M×2k-1Mα given by(50)P2k-1M×2k-1Mα=Φ2k-1M×2k-1MFαΦ2k-1M×2k-1M-1For example, when α=0.5,M = 2, and k = 3, the operational matrix of the fractional integration P2k-1M×2k-1Mα is expressed as follows:(51)P8×80.5=0.193430.25790.17250.102620.108560.085390.086080.07330-0.06450.5158000000000.193440.257920.172470.102620.108560.0853900-0.042990.343890.248620.099460.120970.0907700000.193440.257920.172470.102620000-0.042990.343890.248640.099460000000.193440.25792000000-0.042990.34389


## 4. Solution of Nonlinear Volterra Integral Equations via the HOBW Method

Consider the following integral equation:(52)yx=fx+λ∫0x(kx,tytqx-t1-αdt,0≤x≤1,(53)yx≈YTHOBWx≈YTΦ2k-1M×2k-1MB2k-1Mxkx,t≈HOBWTxKHOBWt≈Φ2k-1M×2k-1MB2k-1MxTKΦ2k-1M×2k-1MB2k-1Mtfx≈FTHOBWx≈FTΦ2k-1M×2k-1MB2k-1Mxwhere K is 2k-1M×2k-1M matrix with (54)K=HOBWx,kx,t,HOBWt.The functions yq(x) can be truncated into the HOBB functions as(55)y2t=YTHOBWt2=YTHOBWtHOBWtTY=HOBWtTY~Yy3t=YTHOBWtYTHOBWt2=YTHOBWtHOBWtTY~Y=HOBWtTY~Y~Y=Φ2k-1M×2k-1MB2k-1MtTY~2Yyqt=HOBWtTY~q-1Y=Φ2k-1M×2k-1MB2k-1MtTY~q-1YTherefore, upon substituting into (52), we get(56)YTHOBWx=FTHOBWx+λ∫0xx-tα-1HOBWTxKHOBWtHOBWtTY~q-1Ydt(57)YTHOBWx=FTHOBWx+λHOBWTxK∫0xx-tα-1HOBWtHOBWtTY~q-1Ydtwhere (58)∫0xx-tα-1HOBWtHOBWTtY~q-1Ydt=∫0xx-tα-1Y~q~-1YHOBWtdt=ΓαY~q~-1YIαB2k-1Mx=ΓαY~q~-1YFαB2k-1MxWith the aid of the previous equations, (52) becomes (59)YTΦ2k-1M×2k-1MB2k-1Mx=FTΦ2k-1M×2k-1MB2k-1Mx+λΓαY~q~-1YFαΦ2k-1M×2k-1MB2k-1Mxwhere HOBW(x)≈Φ2k-1M×2k-1MB2k-1M(x).To compute the unknown HOBW coefficients, we use the collocation points as follows:(60)ti=2i-12k-1M,i=1,2,…,2k-1MFrom (60), we have a system of 2k-1M nonlinear equations with 2k-1M unknowns. Newton iteration method is used for completing the solution of the resulting nonlinear system, to get the unknown vectors Y. So, the approximated results y(x) can be calculated as (61)yx=Y2k-1M×1THOBW2k-1M×1x.


## 5. Numerical Examples

We use the demonstrated technique in this article for finding the numerical results of four weakly singular-Volterra integral equations.Example 1.
Consider the generalized Abel’s integral equation [21].(62)ux=x2+1615x5/2-∫0xutx-tdt,The exact solution is ux=x2.The outcomes demonstrate the high exactness and the effectiveness of the technique. This outcome can be effortlessly confirmed that the strategy yields the desired accuracy only in a few values ofk and M. The results of this example at different values of k and M are presented in Table 1.Table 1
Maximum absolute errors at different values ofk and M for Example 1 via HOBW.
K M=8 M=16 M=32 4 4.21 × 1 0 - 6 4.10 × 1 0 - 10 6.06 × 1 0 - 11 5 3.71 × 1 0 - 7 1.37 × 1 0 - 12 2.92 × 1 0 - 12 6 4.82 × 1 0 - 8 3.27 × 1 0 - 13 3.51 × 1 0 - 14Example 2.
Consider the following WSVIE:(63)ux=πx5csc⁡π5+x1/5-∫0xutx-t0.2dt,The exact solution is ux=x5.Table2 likewise checks all favorable circumstances of the strategy examined in the past examinations. It ought to be noticed that the HOBW additionally effortlessly composes PC code. This is another vital trademark for the numerical calculation. These actualities delineate the HOBW strategy as a quick, dependable, legitimate, and useful asset for understanding WSVIEs.Table 2
The comparison among HOBW, exact, and Chebyshev solutions for Example2.
x i HOBW Exact Solution Absolute error Of HOBW Absolute error Of Chebyshev base atn=8 k = 8 ,   M = 8 k = 8 ,   M = 8 0.1 0.6309573449 0.6309573445 4 × 1 0 - 10 1.4 × 1 0 - 5 0.2 0.7247796642 0.7247796637 5 × 1 0 - 10 2.1 × 1 0 - 5 0.3 0.7860030896 0.7860030856 4 × 1 0 - 9 3.5 × 1 0 - 5 0.4 0.8325532052 0.8325532074 2.2 × 1 0 - 9 5.5 × 1 0 - 5 0.5 0.8705505674 0.8705505633 4.1 × 1 0 - 9 2.4 × 1 0 - 5 0.6 0.9028804552 0.9028804514 3.8 × 1 0 - 9 4.4 × 1 0 - 5 0.7 0.9311499137 0.9311499151 1.4 × 1 0 - 9 5.2 × 1 0 - 5 0.8 0.9563524927 0.9563524998 7.1 × 1 0 - 9 7.1 × 1 0 - 5 0.9 0.9791483643 0.9791483624 1.9 × 1 0 - 9 6.9 × 1 0 - 5Example 3.
Consider the singular kernel Volterra integral equation [25]:(64)yx=fx+∫0xxtx-t0.5y2tdtfx=x3-4096x17/26435The analytic solution of (49) can be detected in [18] as y(x)=x3.The comparison among theHOBW solution and the second Chebyshev wavelet (SCW) solution is shown in Table 3 for k=4 and M=2, which confirms that the HOBW method gives almost the closer loose as the analytic solution. Figure 2 shows the comparison among the HOBW solution and the analytic one for t∈[0,1). Better approximation is expected by the values of k and M as in Table 2.Table 3
The comparison among HOBW, exact, and SCW solutions for Example3.
x i HOBW Exact Solution Absolute error Of HOBW Absolute error Of SCW at k = 4 ,   M = 2 k = 4 ,   M = 2 k = 4 ,   M = 2 0.1 0.0010010729 0.001 1.0729 × 1 0 - 6 9.6679 × 1 0 - 5 0.2 0.0080011451 0.008 1.145146 × 1 0 - 6 4.7192 × 1 0 - 4 0.3 0.0269910936 0.027 8.9064 × 1 0 - 6 7.6218 × 1 0 - 4 0.4 0.0639790239 0.064 2.0976 × 1 0 - 5 5.2959 × 1 0 - 4 0.5 0.1246774895 0.125 3.2251 × 1 0 - 4 4.7042 × 1 0 - 3 0.6 0.2157735888 0.216 2.2641 × 1 0 - 4 4.3467 × 1 0 - 4 0.7 0.3416681741 0.343 1.3318 × 1 0 - 3 2.7024 × 1 0 - 5 0.8 0.5074658635 0.512 4.5341 × 1 0 - 3 5.0051 × 1 0 - 5 0.9 0.7139664754 0.729 1.5033 × 1 0 - 3 7.3236 × 1 0 - 3Figure 2
Comparison of numerical solutions and exact solution of Example1 for k = 4, M = 2.Example 4.
Consider the nonlinear Volterra integral equation with singular kernel [25]:(65)yx=fx+∫0xy4tx-t0.5dtfx=x1515-16x2with the exact solution y(x)=x.The comparison among the HOBW solution and the analytic solution fort∈[0,1) is shown in Table 4 and Figure 3 for k=4 and M=2 and confirms that the HOBW method gives almost the same solution as the analytic method. Better approximation is expected by choosing higher values of k and M.Table 4
The comparison among HOBW, analytic, and SCW solutions for Example4.
x i Absolute error of HOBW Absolute error of SCW at Absolute error of HOBW Absolute error of SCW at k = 4 ,   M = 2 k = 4 ,   M = 2 k = 5 ,   M = 2 k = 5 , M = 2 0.1 4.1161 × 1 0 - 4 6.8039 × 1 0 - 3 2.413 × 1 0 - 4 1.4565 × 1 0 - 3 0.2 7.0253 × 1 0 - 4 1.4873 × 1 0 - 3 3.6508 × 1 0 - 5 1.8367 × 1 0 - 4 0.3 2.1435 × 1 0 - 4 5.2635 × 1 0 - 4 1.1701 × 1 0 - 5 1.2211 × 1 0 - 4 0.4 5.1621 × 1 0 - 5 2.7043 × 1 0 - 4 2.1061 × 1 0 - 4 6.8020 × 1 0 - 5 0.5 9.7003 × 1 0 - 5 8.2247 × 1 0 - 4 7.2708 × 1 0 - 5 2.2140 × 1 0 - 4 0.6 6.4325 × 1 0 - 4 1.6089 × 1 0 - 4 1.1578 × 1 0 - 6 2.6100 × 1 0 - 5 0.7 1.3045 × 1 0 - 4 7.3143 × 1 0 - 5 1.0186 × 1 0 - 6 2.9370 × 1 0 - 5 0.8 3.1324 × 1 0 - 3 4.6887 × 1 0 - 5 9.4106 × 1 0 - 4 2.4144 × 1 0 - 6 0.9 4.0371 × 1 0 - 3 7.3699 × 1 0 - 5 1.0512 × 1 0 - 4 8.3715 × 1 0 - 6Figure 3
Results of Example4 fork= 4,M= 2.


## 6. Conclusion

In this investigation, the combination of orthonormal Bernstein, block-pulse functions, and wavelets is applied for resolving SNVIE. The main purpose of our method is to combine the orthonormal Bernstein and block-pulse functions wavelet method with the definition of the Riemann–Liouville fractional integral with the singular integral. The method depends on reducing the considered system to a set of nonlinear algebraic equations. The generated system just needs sampling of functions and no integration. Wavelets as orthogonal systems have different resolution capability for truncating functions by the increasing of dilation parameterk that can give a good truncation for integral equations without using a polynomial solution. The considered method has its efficiency and simplicity. The matrices D and P are sparse; hence the CPU time and the computer memory will be reduced and at the same time the solution remains accurate. We also noted that when the degree of HOBW is increased, the errors will be decreased to smaller values. When the values of k and M are higher, we get more accurate solutions for the given problems.



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