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Modified homotopy perturbation method for solving hypersingular integral equations of the first kind
SpringerPlus volume 5, Article number: 1473 (2016)
Abstract
Modified homotopy perturbation method (HPM) was used to solve the hypersingular integral equations (HSIEs) of the first kind on the interval [−1,1] with the assumption that the kernel of the hypersingular integral is constant on the diagonal of the domain. Existence of inverse of hypersingular integral operator leads to the convergence of HPM in certain cases. Modified HPM and its norm convergence are obtained in Hilbert space. Comparisons between modified HPM, standard HPM, Bernstein polynomials approach Mandal and Bhattacharya (Appl Math Comput 190:1707−1716, 2007), Chebyshev expansion method Mahiub et al. (Int J Pure Appl Math 69(3):265–274, 2011) and reproducing kernel Chen and Zhou (Appl Math Lett 24:636–641, 2011) are made by solving five examples. Theoretical and practical examples revealed that the modified HPM dominates the standard HPM and others. Finally, it is found that the modified HPM is exact, if the solution of the problem is a product of weights and polynomial functions. For rational solution the absolute error decreases very fast by increasing the number of collocation points.
Background
Hypersingular integral equations (HSIEs) arise a variety of mixed boundary value problems in mathematical physics such as water wave scattering (Kanoria and Mandal 2002), radiation problems involving thin submerged plates (Parsons and Martin 1994) and fracture mechanics (Chan et al. 2003; Nik Long and Eshkuvatov 2009). Chen and Zhou (2011) have solved HSIE using the improvement of reproducing kernel method. Golberg (1987) obtained the approximate solution of HSIEs using Galerkin and collacation method and discuss their convergence. Spline collocations method has also been used to solve linear HSIE of the first kind and nonlinear HSIE of the second kind in Boykov et al. (2010, 2014) respectively. Projection method with Chebyshev polynomials were discussed to solve the singular and hypersingular integral equation of the first kind in Eshkuvatov et al. (2009), Mahiub et al. (2011) respectively.
Homotopy perturbation method (HPM) has been used for a wide range of problems He (1999, 2000), Khan and Wu (2011), Madani et al. (2011), Ramos (2008), Słota (2010), Jafari et al. (2010), Golbabai and Javidi (2007), Dehghan and Shakeri (2008), Ghasemi et al. (2007), Panda et al. (2015), Okayama et al. (2011), Panda (2013), Javidi and Golbabai (2009), Ghorbani and SaberiNadjafi (2006), Mohamad Nor et al. (2013). Particularly, He (1999, 2000) was pioneer of establishing HPM and used it to solve the linear and nonlinear differential equations. Khan and Wu (2011) used He’s polynomials to solve nonlinear problems. Madani et al. (2011) employed HPM together with Laplace transform for solving onedimensional nonhomogeneous partial differential equations with a variable coefficients. Other usage of HPM were finding the exact and approximate solutions of nonlinear ordinary differential equations (ODEs) (Ramos 2008), onephase inverse Stefan problem (Słota 2010), linear and nonlinear integral equations (Jafari et al. 2010), the integrodifferential equations (Golbabai and Javidi 2007; Dehghan and Shakeri 2008) and nonlinear Volterra–Fredholm integral equations Ghasemi et al. (2007). In Panda et al. (2015), a modified Lagrange approach is presented to obtain approximate numerical solutions of Fredholm integral equations of the second kind. The error bound is explained by the aid of several illustrative examples. In Okayama et al. (2011), two improved versions of the Sinccollocation scheme are presented. The first version is obtained by improving the scheme so that it becomes more practical, and natural from a theoretical view point. In the second version, the variable transformation employed in the original scheme, the tanh transformation, is replaced with the double exponential transformation. It is proved that the replacement improves the convergence rate drastically. Numerical examples which support the theoretical results are also given. In Panda (2013), some recently developed analytical methods namely; homotopy analysis method, homotopy perturbation method and modified homotopy perturbation method are applied successfully for solving strongly nonlinear oscillators. The analytical results obtained by using HAM are compared with those of HPM, mHPM.
To improve the efficiency of the HPM, a few modifications have been made by many researches. For instance, Javidi and Golbabai (2009) added the accelerating parameter to the perturbation equation for obtaining the approximate solution for nonlinear Fredholm integral equation. Ghorbani and SaberiNadjafi (2006) added a series of parameter and selective functions to HPM to find the semianalytical solutions of nonlinear Fredholm and Volterra integral equations. Mohamad Nor et al. (2013) developed the new homotopy function using De Casteljau algorithms to solve the algebraic nonlinear problems.
Consider HSIE of the first kind
where \(\varphi (x)\) is the unknown function of x to be determined, K(s, t) and \(L_1(s,t)\) are the square integrable kernels on \(D=\{(s,t)\in \mathbb {R}^21\le s, t\le 1\}\). Assume that K(s, t) is constant on the diagonal of the region, i.e.
where \(c_0\) is a nonzero constant and \(K_1(x,t)\) is square integrable kernel of the form
Q(x) is smooth function and \(Q_1(x,t)\) is square integrable kernel.
The main objective is to find the bounded solution of Eq. (1). We search a solution in the form
Substituting Eqs. (2) and (3) into (1) yields
where \(L(x,t)=Q_1(x,t)+L_1(x,t)\).
Let us rewrite Eq. (4) in operator form
where
In this paper, the standard (convex) HPM and the modification of improved HPM (in short modified HPM) are utilized to find the bounded approximate solution of HSIEs (4). Norm convergence for both HPM and modified HPM are proved.
The structure of this paper is arranged as follows. In “Hilbert spaces and operators” section, related information regarding to the Hilbert spaces and operators theory are given. Description of standard HPM and modified HPM are presented in “HPM and modified HPM for HSIEs” section. Norm convergence of both standard HPM and modified HPM are proved in “Convergence of the methods” section. Implementation of modified HPM and its comparisons with others are shown in “Numerical examples” section. Finally, “Conclusion” section is for the conclusion.
Hilbert spaces and operators
Let us consider some well known facts concerning the operator H in Eq. (5). Let
denote the Chebyshev polynomials of the second kind, and
is normalized Chebyshev polynomials of the second kind
It is well known that the hypersingular operator \(H_g\) can be considered as the differential Cauchy operator i.e.,
as well as acting operator \(C_g\) for Chebyshev polynomials of second kind yields
where \(T_{n+1}(x)\) is the Chebyshev polynomial of the first kind.
It can easily be shown from (7), (8), (9) and \(T_{n+1}'(x)=(n+1)U_n(x)\) that
where \(\phi _{1}(x)=0\). Note that Eqs. (10) and (11) are crucial to the rest of our analysis.
Let \(L(\rho )\) denotes the space of square integrable real valued function with respect to \(\rho (x)=\sqrt{1x^2}\). The inner product on \(L(\rho )\) is given by
and \(\Vert u\Vert _\rho =\sqrt{\langle u,v \rangle _\rho }\) denotes the norm.
The set \(\{\phi _k\}_{k=0}^{\infty }\) is a complete orthonormal basis for \(L(\rho )\), so that if \(u\in L(\rho )\) then
where the sum converges in \(L(\rho )\). In addition, the norm of u satisfies the Parseval’s equality
We will need the subspace of \(L(\rho )\) which is consisting of all u such that
All functions satisfying (12) is denoted by \(L_1(\rho )\) and it can be made into Hilbert space if the inner product of \(u \in L_1(\rho )\) and \(v \in L_1(\rho )\) are defined by
The norm of \(u \in L_1(\rho )\) is given by
We extend the operator H defined by (5) as a bounded operator from \(L_1(\rho )\) to \(L(\rho )\) by defining
and observe that
It is not hard to show that \(H^{1}:L_1(\rho )\rightarrow L(\rho )\) exist and is given by
hence H is invertible Golberg (1987).
Lemma 1
The norm of operator \(H^{1}: L_1(\rho )\rightarrow L(\rho )\) is
Proof
Assume that \(H^{1}u=v\). On the other hand
Since \(v\in L_1(\rho )\) and due to (16) we have
Therefore
By the norm definition of operator, we obtain
\(\square\)
Some facts from operators theory
Lemma 2
Let A, B be operators acting in Hilbert space. If A is bounded and B is compact then the products AB and BA are compact.
Lemma 2 is proven in Reed and Simon (1980, Theorem VI.12, pp. 200).
Lemma 3
The operators \(C: L_{1}(\rho ) \rightarrow L(\rho )\) and \(H^{1}C: L_{1}(\rho ) \rightarrow L_{1}(\rho )\) are compact.
Proof
Let us define operators \(T_r, \ \ T_l: \, L_{1}(\rho ) \rightarrow L(\rho )\) as
These operators are bounded from \(L(\rho ) \rightarrow L(\rho )\). Moreover, boundedness and the compactly embeddability of \(T_r\) and \(T_l\) from \(L_{1}(\rho )\) to \(L(\rho )\) (Berthold Berthold et al. (1992, Conclusion 2.3)) implies the compactness of \(T_r\) and \(T_l\). From (11) and (18) it follows that
Since operators \(T_r\) and \(T_l\) are compact, its linear combinations is also compact i.e. \(T_rT_l: \, L_{1}(\rho ) \rightarrow L(\rho )\). As we know Q(x) is a continues function on the closed interval \([1,1]\) and \(T_rT_l\) is compact, their product C is also compact by Lemma 2. On the other hand \(H^{1}\) is unitary and C is compact due to Lemma 2. Hence, operator \(H^{1}C\) is compact.\(\square\)
Since operators C and L are compact then \(C+L:L_{1}(\rho ) \rightarrow L_{1}(\rho )\) is also compact. We know that \( H^{1}(C+L):L_1(\rho )\rightarrow L_1(\rho ) \) is a compact operator. Due to the Fredholm theorem Reed and Simon (1980, Theorem VI.14) the inverse operator \((I+\lambda H^{1}(C+L) )^{1}\) of the operator function \(I+\lambda H^{1}(C+L), \lambda \in \mathbb C\), exists for all \(\lambda \) in \(\mathbb C\setminus C_1\), where \(C_1\) is a discrete subset of \(\mathbb C\) (i.e. a set \(C_1\) has no limit points in \(\mathbb C\)) and for \(\lambda \in C_1\) the null space \(N(I+\lambda H^{1}(C+L))\) is finite, that is \(z=\lambda ^{1}\) is the eigenvalue of \(H^{1}(C+L)\) with finite multiplicity. These facts allows us to suppose the following
Assumption 4
\(\lambda =1\) does not belong to \(C_1\), i.e. \(N(I+\lambda H^{1}(C+L))=\{0\}\).
Lemma 5
Let the Assumption 4 is satisfied, then the operator \(H+C+L\) is invertible, and the main Eq. (5) has a unique solution.
Proof
Since H is invertible we get the relation
which gives us the fact that \(H+C+L\) is invertible iff \(I+\lambda H^{1}(C+L)\) is invertible. Then due to Assumption 4 the operator \(H+C+L\) is invertible.\(\square\)
Assumption 6
Assume that \(S=H+C+L\) where H, C, L are defined by (5) is an invertible operator such that
where N(A) is a nullspace of A.
Let \(P_n:L(\rho )\rightarrow L(\rho )\) be the orthogonal projection onto the subspace spaned by \(\{ \phi _0,\phi _1,\,\ldots,\,\phi _n\}\) and
Lemma 7
If (20) holds then
and \(\tilde{S}=(H+C+L_n)\) exist and invertible operator.
Proof
Since L is compact operator then \(\tilde{L}\underset{n\rightarrow \infty }{\longrightarrow } 0\) i.e. for \(\forall \varepsilon >0\), there exists \(n_0\) such that \(n\ge n_0\) implies
Due to invertibility of \(S=H+C+L\) and \(\tilde{L}=LL_n \underset{n\rightarrow \infty }{\longrightarrow } 0\) we obtain
Hence \(I(H+C+L)^{1}\tilde{L}\) is invertible and
due to Lemma 2 operator \(\tilde{S}=H+C+L_n\) is invertible.\(\square\)
HPM and modified HPM for HSIEs
HPM for HSIE
We present the application of standard HPM for solving hypersingular integral equations of the first kind (5). The perturbation scheme in convex homotopy form is given by
where \(p\in [0,1] \) is homotopy parameter. For \(p = 0\) the solution of the operator equation \(H^*(v, 0) = 0\) is equivalent to the solution of a trivial problem \(Hv(x)u_{0}(x)=0\). For \(p=1\) the equation \(H^*(v,1)=0\) leads to the solution of Eq. (5).
The solution of operator equation \(H^*(v,p)=0\) is searched in the form of power series
We assume that the series (22) possesses a radius of convergence not smaller than 1. Substituting (22) into (5) yelds
Existence of \(H^{1}\) and equating the coefficients of like powers of p in Eq. (23), leads to the following iterations
By computing the iterations \(v_k\) in Eq. (24), we can find semianalytical solution as follows
Approximate solution can be computed by
where
Modified HPM for HSIEs
Let us rewrite Eq. (5) in the equivalent form
where
Modified HPM for Eq. (28) is constructed as
Equating \(H^*(v,p)=0\) leads to
Substituting series solution (22) into (29) yields
Equating both sides to the like power of p gives
Since \(\tilde{S}\) is invertible by Lemma 7, we have
Semianalytical solution of Eq. (5) can be computed by (25).
Remark
Note that most cases of modified HPM, the unknown coefficients \(\alpha _j\) of \(v_0\) in the first equation of (31) are defined by equating the next iteration \(v_1\) to be zero and it leads to \(v_k=0, \ \ k \ge 2\) which implies two step method. In general, if \(v_1\not = 0\) but \(v_1^{(m)}\rightarrow 0\) as \(m\rightarrow \infty \) then we can compute the next iteration \(v_k, k\ge 2\). It effects to the next iteration \(k\ge 2\) but the contribution to the solution of the problem will be very small therefore we can neglect it.
Convergence of the methods
Convergence of HPM
Let us consider HSIE (5) by adding parameter \(\lambda \) of the form
Standard HPM for Eq. (32) has the scheme
Since \(H^{1}\) exists, Eq. (33) is computable. The convergence of the method is given in the following theorem.
Theorem 8
Let \( \displaystyle K(x,t)=c_0+(tx)K_1(x,t)\) and \(K_1(x,t), \ \ L(s,t) \in C(D) \) and \( f \in C[1,1]\) be continuous functions. In addition, if the following inequality
holds and initial guess \(u_0(t)\) is chosen as a continuous function for \(t \in [1, 1]\), then the series (22) is norm convergent to the exact solution u on the interval \([1,1] \) for each \(p =[0,1] \).
Proof
Let \(\Vert u_0\Vert _\rho =M\) and \(\Vert f\Vert _\rho =M_1\). Based on (33) and Lemma 1 we have
where
Assume that \(\gamma _1<1\), then from (22) at \(p=1\), we obtain
Therefore, series (22) converges to the exact solution in the sense of norm \(\cdot _1\).\(\square\)
Remark 9
Note that in our case \(\lambda = 1\) and the convergence of HPM can be established if and only if
It implies that HPM converges to the solution of HSIEs (5) in rare cases.
The first \(N+1\) terms of series (22) as \(p\rightarrow 1\) gives the approximate solution of the form
Theorem 10
If \(\gamma _1<1\), then the rate of convergence of the approximate solution \(\tilde{v}_N\) can be estimated by
where \(E_N=\Vert v(x)\tilde{v}_N(x)\Vert _1\) and B is defined by (36).
Proof
Since \(\gamma _1<1\), the norm
whenever \(n\rightarrow \infty \).\(\square\)
Convergence of modified HPM
Let us consider HSIE in the form of Eq. (28). If \(v_1=0\) in (30), then \(v_0\) satisfies \(\tilde{S}v_0+\tilde{L}v_0=f\) and coincides with the exact solution. If \(v_1 \ne 0\) then \(v_1^{(m)}\rightarrow 0\) i.e. for any \(\varepsilon \), there exists \(m_0\), such that \(m>m_0\) implies
Let \(\left\ \sum _{j=0}^{m}\alpha _j\,g_j(x))\right\ _\rho =M_2\) and \(\Vert f\Vert _\rho =M_1\), then due to (31) and existence of \(S^{1}\) (Lemma 6) we obtain
where \(\gamma _2=\Bigl \Vert \tilde{S}^{1}\Bigr \Vert \,\Bigl \Vert \tilde{L}\Bigr \Vert \) and by continuing these procedure
Due to \(\Vert \tilde{L}\Vert =\Vert LL_n\Vert \underset{n\rightarrow \infty }{\longrightarrow } 0\) it can be easily shown that \(\gamma _2 <1\) for large enough k then
Thus, we have proved the following theorem
Theorem 11
Let \( \displaystyle K(x,t)=c_0+(tx)K_1(x,t)\) and \(K_1(x,t), \ \ L(s,t) \in C(D) \) and \( f \in C[1,1]\) be continuous functions. In addition, if the following inequality
holds and selective functions \(g_j(x), \ \ j=0,\ldots ,N\) are chosen as a continuous function on the interval \([1, 1]\), then the series solution (22) is norm convergent to the exact solution \(\varphi (x)\) on the interval \([1,1] \) for each \(p =[0,1] \).
Remark 12
Theorems 8 and 11 show the fact that the exact solution u belongs to \(L_1(\rho )\). Then due to Berthold et al. (1992, Theorem 2.13) the function u belongs to \(C^{(1)}(1,1)\).
Approximate solution of Eq. (4) in series (37) can be estimated as follows.
Theorem 13
Rate of convergence of approximate solution \(\tilde{v}_N\) can be estimated by
where \(E_n=\Vert v(x)\tilde{v}_N(x)\Vert \) and \(\varepsilon \) are defined by (39) and \(\gamma _2<1\).
Proof
\(\square\)
Remark 14
Since \(\gamma _2 < 1\), the term \(\displaystyle \frac{\gamma _2^N}{1\gamma _2}\varepsilon \rightarrow 0\) as \( N\rightarrow \infty \). Moreover, sufficiently small \(\varepsilon \) gives the smaller error rate for \(E_N\) in (43) than error \(E_N\) in (38). This fact shows that the modified HPM is dominates the standard HPM.
Numerical examples
Example 1
(Mandal and Bhattacharya 2007). Consider HSIE (1) of the form
The exact solution of Eq. (44) is \(\varphi (x)=\sqrt{1x^2}\) and \(c_0=1, f(x)=1\).
Solution It is easy to find that Eq. (44) satisfied all conditions in Theorem 8. To apply HPM, we choose the initial guess as \(u_0=\phi _0(x)\). Since \(Cu=Lu\equiv 0\) and from (14) we can easily get
Referring to (24) and using (45), we obtain successive functions
Since \(v_2(x)=v_3(x)=\cdots =0\), the approximate solution of Eq. (44) is
which is identical with exact solution.
For application of modified HPM to the Eq. (44), we do this following steps:

1.
Let selective functions \(g_j(x)=\phi _j(x), j=0, \ldots m\). Since \(\tilde{S}=H, \ \ \tilde{L}\equiv 0\) for (44) we can use inverse operator (45). Based on the scheme (31) for \(m=2\) we obtain,
$$\begin{aligned} v_0(x)&=H^{1}(\alpha _0\,\phi _0+\alpha _1\,\phi _1+\alpha _2\,\phi _2) = \alpha _0\phi _0\frac{\alpha _1}{2}\, \phi _1\frac{\alpha _2}{3}\, \phi _2, \nonumber \\ v_1(x)&=H^{1}(1 \alpha _0\,\phi _0 \alpha _1\,\phi _1 \alpha _2\,\phi _2) = 1 + \alpha _0\phi _0 +\frac{\alpha _1}{2}\,\phi _1 +\frac{\alpha _2}{3}\,\phi _2, \nonumber \\ v_k&= H^{1}(0) \equiv 0, \, k=2,3, \ldots \end{aligned}$$ 
2.
Since \(v_k \equiv 0, \, k=2,3, \ldots \) we can easily find approximate solution as
$$\begin{aligned} \varphi (x)=\sqrt{1x^2}(v_0(x)+v_1(x))=\sqrt{1x^2}, \end{aligned}$$(47)which coincides with exact solution.
Mandal and Bhattacharya (2007) consider the Eq. (44) and comparisons with HPM, modified HPM are summarized in Table 1.
Example 2
Mahiub et al. (2011). Consider HSIE of the form
with exact solution \(\varphi (x)=\displaystyle \sqrt{1x^2}\,(16 x^412x^2+1)\).
Solution Conditions of the Theorem 8 does not hold for Example 2. Therefore we did comparisons between modified HPM and method given in Mahiub et al. (2011).
To solve the Eq. (48) by modified HPM we do the following steps:

1.
Let us choose selective functions \(g_j(x)=\phi _j(x), \, j=0,\ldots , m\) and kernel \(L(x,t)=\sin (x)t^4\) in Eq. (48) be approximated by projection kernel \(L_n(x,t)=\sum _{i=1}^{l}b_i(x)\phi _i(t)\). In this case \(\tilde{L}u=LuL_nu \equiv 0\). Since \(Cu\equiv 0, \ \ \tilde{L}u \equiv 0\) then \(\tilde{S}=H+L\). From (30) it follows that
$$\begin{aligned} (H+L)v_0&=\sum _{j=0}^{m}\alpha _j\,\phi _j(x),\nonumber \\ (H+L)v_1&=f\sum _{j=0}^{m}\alpha _j\,\phi _j(x). \nonumber \\ (H+L)v_k&=\tilde{L}(v_{k1}) \equiv 0, \quad k=2,3, \ldots \end{aligned}$$(49) 
2.
Let \(v_o=u_0=\sum _{j=0}^{m}\alpha _j\,\phi _j(x)\), then from the first equation of (49) we define
$$\begin{aligned} H+L=I. \end{aligned}$$(50)From the 2nd equation of (49) we obtain
$$\begin{aligned} \sum _{j=0}^{m}\alpha _j\,\phi _j(x)=f(x). \end{aligned}$$(51)Approximating \(\sin (x)\) by Chebyshev polynomials
$$\begin{aligned} \sin (x)\approx \sqrt{\frac{\pi }{2}}\left( \frac{11}{24}\phi _1(x)\frac{1}{48}\phi _3(x)\right) \end{aligned}$$(52)and using first equation of (49) and taking account of (51) we get
$$\begin{aligned} (H+L)\sum _{j=0}^{m}\alpha _j\,\phi _j(x)&= \sqrt{\frac{\pi }{2}}\left( 5 \phi _4(x)\frac{11}{24}\phi _1(x)+\frac{1}{48} \phi _3(x)\right) . \end{aligned}$$(53)Comparing the base of Chebyshev polynomial from the both sides of Eq. (53) the solutions are
$$\begin{aligned} \alpha _4 = \sqrt{\frac{\pi }{2}}, \quad \alpha _1=\alpha _2=\alpha _3=0. \end{aligned}$$(54) 
3.
Substituting (54) into Eq. (25) yields the exact solution
$$\begin{aligned} \varphi (x)=\sqrt{1x^2}\,(16 x^412x^2+1). \end{aligned}$$(55)
Comparisons of Modified HPM and Chebyshev expansion Mahiub et al. (2011) is given in Table 2 for Eq. (48).
Example 3
Chen and Zhou (2011). Consider HSIE in the form
with exact solution \(\varphi (x)=\displaystyle \sqrt{1x^2}\,(1+2x^3)\).
Conditions of Theorem 8 are satisfied, therefore for HPM we choose initial guess as \(u_0=\phi _1(x)\). Errors of numerical solution, computed for \(N=\{5,10\}\) where N is a number of iteration, are given in Table 3.
To use modified HPM for solving Eq. (56), we do the following steps:

1.
As usual we choose selective functions as \(g_j(x)=\phi _j(x)\), \(j=0,\ldots , m\) and kernel \(L(x,t)=tx\) in Eq. (56) be approximated by projection kernel of the form \(L_n(x,t)=\sum _{i=1}^{l}b_i(x)\phi _i(t)\). Again for this case \(\tilde{L}u=LuL_nu \equiv 0\). Since \(Cu\equiv 0, \ \ \tilde{L}u \equiv 0\) then \(\tilde{S}=H+L\). Using (30) we have
$$\begin{aligned} (H+L)v_0&=\sum _{j=0}^{m}\alpha _j\,\phi _j(x),\nonumber \\ (H+L)v_1&=f\sum _{j=0}^{m}\alpha _j\,\phi _j(x). \nonumber \\ (H+L)v_k&=\tilde{L}(v_{k1}) \equiv 0, \, k=2,3, \ldots \end{aligned}$$(57) 
2.
Again \(v_k \equiv 0, \quad\, k=2,3, \ldots \) and by equating \(v_1=0\) we have
$$\begin{aligned} \sum _{j=0}^{m}\alpha _j\,\phi _j(x)=f(x). \end{aligned}$$(58)using 1st equation of (57) and taking into account (58) yields
$$\begin{aligned} (H+L)\sum _{j=0}^{m}\alpha _j\,\phi _j(x) = \sqrt{\frac{\pi }{2}}\left( \phi _3(x)\frac{15}{16}\phi _1(x)\phi _0(x)\right) . \end{aligned}$$(59)Comparing the base of Chebyshev polynomial from the both sides of Eq. (59) produce a system. Solutions of the system are
$$\begin{aligned} \alpha _0=\sqrt{\frac{\pi }{2}},\quad\; \alpha _1=\frac{1}{2}\sqrt{\frac{\pi }{2}},\quad\;\alpha _2=0,\quad\;\alpha _3=\frac{1}{4}\sqrt{\frac{\pi }{2}}. \end{aligned}$$(60) 
3.
Substituting (60) into Eq. (25) yields
$$\begin{aligned} \varphi (x)=\sqrt{1x^2}\,(1+2x^3). \end{aligned}$$(61)which is identical with the exact solution.
Results are calculated by taking the maximum of absolute errors for Eq. (56). Comparison of the results between HPM, modified HPM and reproducing kernel Chen and Zhou (2011) shown in Table 4.
Example 4
Solve HSIE of the form
The exact solution of Eq. (62) is \(\varphi =\sqrt{1x^2}(8x^3+4x^34x1)\).
Solution For this example, the conditions of Theorem 8 does not hold. Therefore, HPM is not a reliable method to solve Eq. (62).
To obtain the approximate solutions of Eq. (62) by modified HPM (30), we do the following steps:

1.
Approximate \(L(x,t)=\dfrac{e^{2x}t^3}{2}\) by Chebyshev polynomials
$$\begin{aligned} L_n(x,t)=\frac{e^{2x}}{16}(\phi _3(t)2\phi _1(t))=L(x,t), \end{aligned}$$(63)therefore \(\tilde{L}_n\equiv 0\). Choose selective functions \(g_j(x)=\phi _j(x)\), then from (30), we have
$$\begin{aligned} (H+C)v_0&=\sum _{j=0}^{m}\alpha _j\,\phi _j(x), \end{aligned}$$(64)$$\begin{aligned} (H+C)v_1&=f\sum _{j=0}^{m}\alpha _j\,\phi _j(x). \end{aligned}$$(65)$$\begin{aligned} (H+C)v_{k}&=\tilde{L}(v_{k1})\equiv 0, \end{aligned}$$(66)with \(v_0=\sum _{j=0}^{m} b_j\,\phi _j(x)\).

2.
Since \(v_k\equiv 0\), for \(k\ge 2\) and approximating \(e^{2x}\) into Chebyshev polynomials with 4 bases
$$\begin{aligned} e^{2x}\simeq \sqrt{\frac{\pi }{2}}\quad \left( \frac{19}{12}\phi _0(x)+\frac{4}{3}\phi _1(x)+\frac{5}{8}\phi _2(x) +\frac{1}{6}\phi _3(x)\frac{1}{24}\phi _4(x) \right) , \end{aligned}$$(67)then substituting (67) into (65) and equating \(v_1=0\) for \(m=4\) yields
$$\begin{aligned} b_0=\frac{19}{384}\,\sqrt{\frac{\pi }{2}},\quad b_1=\frac{25}{24}\,\sqrt{\frac{\pi }{2}},\quad b_2=\frac{507}{256}\,\sqrt{\frac{\pi }{2}}, \quad b_3 =\frac{959}{192}\,\sqrt{\frac{\pi }{2}},\quad b_4=\frac{767}{768}\,\sqrt{\frac{\pi }{2}}. \end{aligned}$$(68) 
3.
From (64), we obtain the values of \(\alpha _k\), \(k=0,1,\ldots , 4\).
$$\begin{aligned} \alpha _0=\alpha _1=\alpha _4=0, \quad \alpha _2=\alpha _3=\sqrt{\frac{\pi }{2}}. \end{aligned}$$ 
4.
Substitute all values of \(\alpha _i, i=0,\ldots , 4\) into (25), we have
$$\begin{aligned} v_0=\sum _{k=0}^{4}\alpha _k \phi _k(x)=\sqrt{\frac{\pi }{2}}\phi _3(x)=8x^3+4x^34x1. \end{aligned}$$Thus, we obtain the approximate solution in the form
$$\begin{aligned} \varphi (x)=\sqrt{1x^2}(8x^3+4x^34x1) \end{aligned}$$(69)which is same as exact solution. Modified HPM has zero error for solving Eq. (62).
Example 5
Let us rewrite Eq. (4) in the form of
where \(f(x)=\displaystyle \frac{20\sqrt{3}}{2+x^2}\frac{10 x^2}{x+2}(2\sqrt{3}+x)+10\,(2\sqrt{3})x+\frac{10}{3}(2\sqrt{3}3)+\frac{10(2\sqrt{3})}{x+2}\).
The exact solution of Eq. (70) is \(\varphi (x)=\displaystyle \sqrt{1x^2}\frac{10}{x+2}\).
Solution Standard HPM is not suitable for solving the Eq. (70) as it is not satisfies the conditions in Theorem 8. For the modified HPM, we choose the selective functions \(g_j(x)=\phi _j(x), \ \ j=0,\ldots ,m\). Approximating L(x, t) in Chebyshev polynomials form as follows
In this case \(\tilde{L} \not = 0\), therefore the scheme (30) has the form
To solve Eq. (70), we choose the collocation points, \(x_i\) as the roots of \(\phi (x)\) which is
Errors of \(\varphi (x)\) using modified HPM for values of \(m=\{6,26\}\) are presented in Table 5.
From Tables 1, 2 and 4 show the comparison between the past method with HPM and modified HPM. It is clearly seen that Modified HPM gives more accurate results compare to the Chebyshev expansion method Mahiub et al. (2011), Bernstein polynomials approach Mandal and Bhattacharya (2007) and Reproducing Kernel method Chen and Zhou (2011).
Table 5 conclude that the modified HPM converges to the exact solution of Eq. (70) by increasing the number of collocation points n and number of selection functions m. It can also be seen that the convergence is achieved at all singular points x including the one which is close to the end points of the interval \([1,1]\).
Conclusion
In this work, the standard and modified HPM are used to find the approximate solution of the first kind HSIE. The theoretical aspect supported by the same numerical examples have shown the modified HPM gives better approximation than the standard HPM. Based on the examples, the modified HPM ables to handle the problem that can not be solved by standard HPM. Modified HPM is effective and reliable method for solving HSIE of the first kind of the form (4).
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Authors' contributions
ZKE and ZM carried out mainly theoretical investigations (Section 2 and 4) of the HPM and modified HPM for Hypersingular integral equations and norm convergence for both HPM and modified HPM are proved. NMANL carried out mainly in Introduction and literature review which is Section 1. FSZ participated in the derivation of HPM and modified HPM together with numerical results which is Section 3 and 5, moreover she helped to draft the manuscript. All authors read and approved the final manuscript.
Acknowledgements
This work was supported by University Putra Malaysia (UPM) and Universiti Sains Islam Malaysia (USIM) under Research Grands (Research Grand of UPM, project code is GPi(2014) 9442300 and Research Grand of USIM PPP/GP/FST/30/14915). Authors are grateful for sponsor and financial support of the Research Management Center (RMC) of UPM and USIM.
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The authors declare that they have no competing interests.
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Eshkuvatov, Z.K., Zulkarnain, F.S., Nik Long, N.M.A. et al. Modified homotopy perturbation method for solving hypersingular integral equations of the first kind. SpringerPlus 5, 1473 (2016). https://doi.org/10.1186/s400640163070z
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DOI: https://doi.org/10.1186/s400640163070z