# A Novel Collocation Method for Numerical Solution of Hypersingular Integral Equation with Singular Right-Hand Function

Authors: M. R. Elahi; Y. Mahmoudi; A. Salimi Shamloo; M. Jahangiri Rad
Journal: Advances in Mathematical Physics
 (2023)

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## Abstract

In this study, the Fredholm hypersingular integral equation of the first kind with a singular right-hand function on the interval−1,1 is solved. The discontinuous solution on the domain −1,1 is approximated by a piecewise polynomial, and a collocation method is introduced to evaluate the unknown coefficients. This method, which can be applied to both linear and nonlinear integral equations, is very simple and straightforward. The presented illustrations relate that the results are very accurate compared to the other methods in the literature.


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## Body

## 1. Introduction

Hypersingular integral equations (HSIEs) are widely presented in different fields of physics and engineering such as crack problems in fracture mechanics [1], radiation problems involving thin-submerged plates [2–4], antenna theory [5], aerodynamics [6, 7], elliptic wing case in Prandtl’s equation [8, 9], electrodynamics, and quantum physics [6].Many two-dimensional boundary value problems in mathematical physics can be formulated as HSIEs. It is investigated that how the problems such as two-dimensional flow post a rigid plate in an infinite fluid, acoustic scattering by a hard plate, water wave interaction with thin impermeable barriers, stress fields around cracks, certain specific problems involving cracks in an elastic medium and scattering of surface water waves by obstacles in the form of thin barriers, and some other issues give rise to HSIEs (see [7]). There are two strategies to solve these problems. The direct methods in which the problem is solved using the method of solving differential equations are finite difference methods [10] and Runge-Kutta-embedded methods [11]. Another robust tool is indirect methods in which the differential equation is transformed into an equivalent equation such as a HSIE. The numerical solutions for the second category is often less complicated.In certain cases, there exist some analytical methods for solving HSIEs, but in general, there is no known analytical method. Thus, different numerical methods are introduced and developed for solving HSIEs [8, 9, 12–26].The following first kind HSIE(1)∫−11utt−x2dt=fx,−1≤x≤1,with various justifications on the boundaries x=±1 was studied in [21] by using different types of Chebyshev polynomials. In Equation (1), the right-hand function fx is considered a smooth function, and the hypersingular integral is understood as Hadamard finite part as follows:
(2)∫−11utt−x2dt=limδ⟶0∫−1x−δ+∫x+δ1utt−x2dt−ux+δ+ux−δδ.Recently, a special class of Equation (1) was introduced under the assumption that the right-hand function fx possesses singularities at some points c1<c2<⋯<cm−1 of the interval −1,1 [5, 6, 27]. In this case, the solution is not smooth in the integration domain. It was shown in [21] that if the right-hand function fx in Equation (1) belongs to a Hölder class, then, under some additional conditions on the boundaries, the solution can be represented as ux=1−x2±1/2ϕx or ux=1+x/1−x±1/2ϕx, where ϕx is a smooth function. While, for the singular right-hand function fx, the class of solutions for Equation (1) is not determined yet.Authors in [28, 29] introduced a variant of the generalized Euler-Maclaurin summation formula for the product integration rule and applied it for approximating weakly singular integrals. Also, with the expansion of the theory of fractional calculus in recent years, many attempts have been made to use fractional calculus to solve singular integral equations. These methods are suitable for solving weakly singular integral equations, but they cannot be used for a stronger singular integral equations such as HSIEs.Recently, an iterative projection method is presented for solving linear and nonlinear HSIEs with non-Riemann integrable functions on the right-hand side [25]. They applied the continuous method and presented the convergence of the proposed method based on the Lyapunov stability theory. Not many papers have been published to solve the Equation (1) with singular right-hand function yet. To avoid the complexity of the method presented in [25] in the present study, we consider a simple collocation method for the numerical solution of HSIE (Equation (1)) with the singular right-hand function. The most important advantage of our method is its simplicity and adaptability to different states of the right-hand function. The rest of this paper is organized as follows: in Section 2, the piecewise polynomial collocation method is introduced and applied to the proposed linear and nonlinear hypersingular integral equation. The numerical illustrations are presented in Section 3, and finally, a brief conclusion appeared in Section 4.


## 2. Method of Solution

In this section, we consider a new collocation method for solving the following linear and nonlinear HSIE of the first kind:(3)∫−11uptt−x2dt=fx,p=1,2,3,⋯−1≤x≤1,where the right-hand function fx=f1x+f2x+f3x+⋯ has different types of singularities at the points c1<c2<⋯<cm−1 of the interval −1,1. The singularities of fx may have the following structure:
(4)f1x=gx∏i=1m−11x−ci,f2x=∑i=1m−1gix1x−ci,f3x=∑i=1m−1gixlnx−ci,⋮or any other singular type. In Equation (4), gx and gix, i=1,2,⋯,m−1 are considered smooth functions. To introduce the new method, we need the following definitions.Definition 1.
We define the Cauchy and Hadamard integrals, respectively, as follows(5)Ckx,t=∫tkt−xdt,k=0,1,2,⋯,(6)Hkx,t=∫tkt−x2dt,k=0,1,2,⋯.Theorem 2.
The Cauchy and Hadamard integrals defined in Equations (5) and (6) are calculated as follows:
(7)Ckx,t=∑i=0k−1xitk−ik−i+xklnt−x,(8)Hkx,t=∑i=0k−1xiCk−i−1x,t−xkt−x.Proof.
By the definition of Cauchy integral (Equation (5)), we have
(9)Ckx,t=∫tk−xk+xkt−xdt=∫tk−1+tk−2x+⋯+txk−2+xk−1dt+xk∫dtt−x=∑i=0k−1xitk−ik−i+xklnt−x.
Also, by the definition of Hadamard integral (Equation (6)), we have
(10)Hkx,t=∫tk−xk+xkt−x2dt=∫tk−1+tk−2x+⋯+txk−2+xk−1t−xdt+xk∫dtt−x2=∑i=0k−1xiCk−i−1x,t−xkt−x.Now, we intend to solve Equation (3) by representing the solution as a piecewise continuous polynomial. To do this, we set c0=−1 and cm=1, and then, we seek a solution for Equation (3) on the interval −1,1. According to the structure of the function on the right hand, the solution may be discontinuous at the points ci, i=1,2,⋯,m−1. Therefore, we consider the solution in the following form:
(11)uNx=∑i=0Na1ixi,c0<x<c1⋮∑i=0Najixi,cj−1<x<cj⋮∑i=0Namixi,cm−1<x<cm,where N is a given integer.


### 2.1. Application to Linear HSIE

Consider Equation (3) with p=1. We rewrite Equation (3) as follows:
(12)∑j=1m∫cj−1cjutt−x2dt=fx,−1≤x≤1.Substituting Equation (11) in Equation (3), we get
(13)∑j=1m∑i=0Naji∫cj−1cjtit−x2dt=fx,which gives
(14)∑j=1m∑i=0NajiHix,cj−Hix,cj−1=fx.We selectN+1 distinct points xr0<xr1<⋯<xrN on the jth subinterval cr−1,cr, r=1,2,⋯m.Now, we collocate (14) as follows:
(15)∑j=1m∑i=0NajiHixrs,cj−Hixrs,cj−1=fxrs,r=1,2,⋯,m,s=0,1,⋯,N.Equation (15) represents a system of mN+1 linear equations of mN+1 unknowns aji, j=1,2,⋯,m, i=0,1,⋯,N, which can be converted to matrix format as
(16)AX=F,where
(17)X=a10,⋯,a1N,⋯aj0,⋯,ajN,⋯am0,⋯,amNT,F=fx10,⋯,fx1N,⋯fxj0,⋯,fxjN,⋯fxm0,⋯,fxmNT,and A is a mN+1 square matrix. The elements of A are as follows:
(18)Ar−1N+s,j−1N+i=Hixrs,cj−Hixrs,cj−1,r,j=1,2,⋯,m,s,i=0,1,⋯,N.Remark 3.
Due to Definition1, the Hadamard integrals Hix,t, i=0,1,⋯N are linearly independent functions. Also, the points xrs,cj,r,j=1,2,⋯,m,s=0,1,⋯,N are distinct and xrs≠cj. Then, from Haar property for independent functions, we conclude that A≠0. This means that the linear system (Equation (16)) has a unique solution.


### 2.2. Application to Nonlinear HSIE

Let us consider Equation (3) with p>1. For solving the nonlinear HSIE (Equation (3)), we rewrite it as follows:
(19)∑j=1m∫cj−1cjuptt−x2dt=fx,−1≤x≤1.Substituting Equation (11) in Equation (19), we get
(20)∑j=1m∫cj−1cj1t−x2∑i=0Najitipdt=fx,−1≤x≤1,after some simplification we get
(21)∑j=1m∑n=0pNAjn∫cj−1cjtnt−x2dt=fx,−1≤x≤1,which gives
(22)∑j=1m∑n=0pNAjnHnx,cj−Hnx,cj−1=fx.−1≤x≤1.We note thatAjn in Equation (21) is the corresponding coefficient of tn in the expansion of uNpt, which is calculated using Adomian polynomials as follows:
(23)Ajn=1n!dndtn∑i=0Najitipt=0.If we collocate Equation (22) at the points xrs, r=1,2,⋯,m,s=0,1,⋯,N, we get the following nonlinear system of equations:
(24)∑j=1m∑n=0pNAjnHnxrs,cj−Hnxrs,cj−1=fxrs,r=1,2,⋯,m,s=0,1,⋯,N.


## 2.1. Application to Linear HSIE

Consider Equation (3) with p=1. We rewrite Equation (3) as follows:
(12)∑j=1m∫cj−1cjutt−x2dt=fx,−1≤x≤1.Substituting Equation (11) in Equation (3), we get
(13)∑j=1m∑i=0Naji∫cj−1cjtit−x2dt=fx,which gives
(14)∑j=1m∑i=0NajiHix,cj−Hix,cj−1=fx.We selectN+1 distinct points xr0<xr1<⋯<xrN on the jth subinterval cr−1,cr, r=1,2,⋯m.Now, we collocate (14) as follows:
(15)∑j=1m∑i=0NajiHixrs,cj−Hixrs,cj−1=fxrs,r=1,2,⋯,m,s=0,1,⋯,N.Equation (15) represents a system of mN+1 linear equations of mN+1 unknowns aji, j=1,2,⋯,m, i=0,1,⋯,N, which can be converted to matrix format as
(16)AX=F,where
(17)X=a10,⋯,a1N,⋯aj0,⋯,ajN,⋯am0,⋯,amNT,F=fx10,⋯,fx1N,⋯fxj0,⋯,fxjN,⋯fxm0,⋯,fxmNT,and A is a mN+1 square matrix. The elements of A are as follows:
(18)Ar−1N+s,j−1N+i=Hixrs,cj−Hixrs,cj−1,r,j=1,2,⋯,m,s,i=0,1,⋯,N.Remark 3.
Due to Definition1, the Hadamard integrals Hix,t, i=0,1,⋯N are linearly independent functions. Also, the points xrs,cj,r,j=1,2,⋯,m,s=0,1,⋯,N are distinct and xrs≠cj. Then, from Haar property for independent functions, we conclude that A≠0. This means that the linear system (Equation (16)) has a unique solution.


## 2.2. Application to Nonlinear HSIE

Let us consider Equation (3) with p>1. For solving the nonlinear HSIE (Equation (3)), we rewrite it as follows:
(19)∑j=1m∫cj−1cjuptt−x2dt=fx,−1≤x≤1.Substituting Equation (11) in Equation (19), we get
(20)∑j=1m∫cj−1cj1t−x2∑i=0Najitipdt=fx,−1≤x≤1,after some simplification we get
(21)∑j=1m∑n=0pNAjn∫cj−1cjtnt−x2dt=fx,−1≤x≤1,which gives
(22)∑j=1m∑n=0pNAjnHnx,cj−Hnx,cj−1=fx.−1≤x≤1.We note thatAjn in Equation (21) is the corresponding coefficient of tn in the expansion of uNpt, which is calculated using Adomian polynomials as follows:
(23)Ajn=1n!dndtn∑i=0Najitipt=0.If we collocate Equation (22) at the points xrs, r=1,2,⋯,m,s=0,1,⋯,N, we get the following nonlinear system of equations:
(24)∑j=1m∑n=0pNAjnHnxrs,cj−Hnxrs,cj−1=fxrs,r=1,2,⋯,m,s=0,1,⋯,N.


## 3. Numerical Illustrations

In this section, we present the effectiveness of the proposed method by solving some HSIEs. All the simulations have been done using Maple software with an accuracy of 20 digits. The first fewHix,t are as follows (see Equation (8)).
(25)H0x,t=−1t−x,H1x,t=lnt−x−xt−x,H2x,t=t+2xlnt−x−x2t−x,H3x,t=12t2+2xt+3x2lnt−x−x3t−x,H4x,t=13t3+xt2+3x2t+4x3lnt−x−x4t−x,H5x,t=14t4+23xt3+32x2t2+4x3t+5x4lnt−x−x5t−x,⋮Example 4.
Suppose the following linear HSIE of the first kind [25],
(26)∫−11utt−x2dt=fx,−1≤x≤1,where
(27)fx=0.2−0.5x−1+0.5x1+x−1.5+0.3x1−x+0.5lnx1+x+0.3ln1−xx.Here, we setc0=−1, c1=0, and c2=1, where fx has singularity. Then, we apply the proposed method with N=3 and get
(28)u3x=1+0.5x,−1≤x≤01.5+0.3x,0<x≤1,which gives the exact solution. In Figure 1, we plotted the graph of exact and approximate solutions for N=2,4 and 6.Example 5.
Consider the following linear HSIE of the first kind,(29)∫−11utt−x2dt=fx,−1≤x≤1,where
(30)fx=0.5−0.5+0.5x−0.5x20.5+x−0.25+0.5x−0.5x20.5−x−0.5x21+x−0.25+0.5x21−x+0.5−xln−0.5+x0.5+x+xln−1+x1+x.Figure 1
The graph of exact solutionux and approximate solutions u2x,u4x,u6x for Example 4.Here, we setc0=−1, c1=−0.5, c2=0.5, and c3=1. Then, we apply the proposed method with N=3 and get
(31)u3x=0.5x2,−1≤x≤−0.50.5+0.5x,−0.5<x≤0.50.25+0.5x2,0.5<x≤1,which is the exact solution to the problem. The graph for the exact and approximate solution for N=3 and 5 is plotted in Figure 2.Example 6.
Suppose the following nonlinear HSIE of the first kind [25],
(32)∫−11u2tt−x2dt=fx,−1≤x≤1,where
(33)fx=2.53+0.09x−0.96x+1+0.5xlnx1+x−1+x+0.25x21+x−1.96−1.12x+0.16x21−x+−1.12+0.32xln1−xx.Figure 2
The graph of exact solutionux and approximate solutions u3x,u5x for Example 5.The right-hand functionfx is singular at three points c0=−1, c1=0, and c2=1. Then, we apply the proposed method with N=3, and we get
(34)ux=1+0.5x,−1≤x≤01.4−0.4x,0<x≤1,which is the exact solution to the problem. The corresponding graphs for the exact and approximate solution are plotted in Figure 3.Example 7.
For the next example, we suppose the following nonlinear HSIE of the first kind,(35)∫−11u2tt−x2dt=fx,−1≤x≤1,where
(36)fx=−1x+4−1+2x+2x21+x−2+2x+2x21−x+2+4xlnx−1x+1.Figure 3
The graph of exact solutionux approximate solution u2x for Example 6.The right-hand functionfx is singular at three points c0=−1, c1=0, and c2=1, while the exact solution is not known. The graph for approximate solutions for different values of N is plotted in Figure 4.Example 8.
Suppose the following nonlinear HSIE of the first kind,(37)∫−11u2tt−x2dt=1x−0.5,−1≤x≤1,where the right-hand function fx is singular at c1=0.5. We solved this problem with different values of N. The graphs for approximate solutions are plotted in Figure 5. Note that the exact solution is not known here.Remark 9.
According to the illustrated examples, we can see that when the solution of the HSIE is polynomial, the present method produces the exact solution of the problem. In case the exact solution is not available, the numerical approximations converge to each other by increasingN (see Examples 7 and 8). Comparison of numerical results with the results of [25] show that the present method is very efficient.Figure 4
The graph of approximate solutionsu2x,u4x,6x for Example 7.Figure 5
The graph of approximate solutionsu4x,u6x,u8x for Example 8.


## 4. Conclusion

In this study, we have developed a novel collocation method for solving the first kind of hypersingular integral equations of the first kind with a singular right-hand function. The solution is considered as a piecewise polynomial on the integration domain, and the unknown coefficients are determined by solving an algebraic system of equations. The new method applies to both linear and nonlinear integral equations. This method covers a wide range of singular integral equations with different types of singularities on the right-hand function. It can also be applied for supersingular integral equations with singular right-hand function.For further research in this field, it is suggested that the class of the solution space of the equation be analyzed analytically based on the type of singularity of the right-hand function.



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