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# Modified homotopy perturbation method for solving hypersingular integral equations of the first kind

- Z. K. Eshkuvatov
^{1, 3}Email author, - F. S. Zulkarnain
^{2}, - N. M. A. Nik Long
^{2}and - Z. Muminov
^{4}

**Received:**23 April 2016**Accepted:**12 August 2016**Published:**1 September 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.

## Keywords

- Homotopy perturbation method
- Hypersingular integral equation
- Integral equation

## Mathematics Subject Classification

- 45E05 (Integral equations with kernels of Cauchy type )
- 30E20 (Integration, integrals of Cauchy type, integral representations of analytic functions)

## 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 Saberi-Nadjafi (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 one-dimensional non-homogeneous 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), one-phase inverse Stefan problem (Słota 2010), linear and nonlinear integral equations (Jafari et al. 2010), the integro-differential 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 Sinc-collocation 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 Saberi-Nadjafi (2006) added a series of parameter and selective functions to HPM to find the semi-analytical 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.

*x*to be determined,

*K*(

*s*,

*t*) and \(L_1(s,t)\) are the square integrable kernels on \(D=\{(s,t)\in \mathbb {R}^2|-1\le s, t\le 1\}\). Assume that

*K*(

*s*,

*t*) is constant on the diagonal of the region, i.e.

*Q*(

*x*) is smooth function and \(Q_1(x,t)\) is square integrable kernel.

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

*H*in Eq. (5). Let

*u*satisfies the Parseval’s equality

*u*such that

*H*defined by (5) as a bounded operator from \(L_1(\rho )\) to \(L(\rho )\) by defining

*H*is invertible Golberg (1987).

###
**Lemma 1**

*The norm of operator*\(H^{-1}: L_1(\rho )\rightarrow L(\rho )\)

*is*

###
*Proof*

### 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*

*Q*(

*x*) is a continues function on the closed interval \([-1,1]\) and \(T_r-T_l\) 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*

*H*is invertible we get the relation

###
**Assumption 6**

*H*,

*C*,

*L*are defined by (5) is an invertible operator such that

*N*(

*A*) is a nullspace of

*A*.

###
**Lemma 7**

*If*(20)

*holds then*

*and*\(\tilde{S}=(H+C+L_n)\)

*exist and invertible operator*.

###
*Proof*

*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

## HPM and modified HPM for HSIEs

### HPM for HSIE

*p*in Eq. (23), leads to the following iterations

### Modified HPM for HSIEs

*p*gives

###
*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

###
**Theorem 8**

*Let*\( \displaystyle K(x,t)=c_0+(t-x)K_1(x,t)\)

*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*

###
*Remark 9*

###
**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*

### Convergence of modified HPM

*k*then

###
**Theorem 11**

*Let*\( \displaystyle K(x,t)=c_0+(t-x)K_1(x,t)\)

*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*

###
*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*

*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

- 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 aswhich coincides with exact solution.$$\begin{aligned} \varphi (x)=\sqrt{1-x^2}(v_0(x)+v_1(x))=-\sqrt{1-x^2}, \end{aligned}$$(47)

###
*Example 2*

*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).

- 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 \(\tilde{L}u=Lu-L_nu \equiv 0\). Since \(Cu\equiv 0, \ \ \tilde{L}u \equiv 0\) then \(\tilde{S}=H+L\). From (30) it follows that$$\begin{aligned} (H+L)v_0&=\sum _{j=0}^{m}\alpha _j\,\phi _j(x),\nonumber \\ (H+L)v_1&=f-\sum _{j=0}^{m}\alpha _j\,\phi _j(x). \nonumber \\ (H+L)v_k&=-\tilde{L}(v_{k-1}) \equiv 0, \quad k=2,3, \ldots \end{aligned}$$(49)
- 2.Let \(v_o=u_0=\sum _{j=0}^{m}\alpha _j\,\phi _j(x)\), then from the first equation of (49) we defineFrom the 2nd equation of (49) we obtain$$\begin{aligned} H+L=I. \end{aligned}$$(50)Approximating \(\sin (x)\) by Chebyshev polynomials$$\begin{aligned} \sum _{j=0}^{m}\alpha _j\,\phi _j(x)=f(x). \end{aligned}$$(51)and using first equation of (49) and taking account of (51) we get$$\begin{aligned} \sin (x)\approx \sqrt{\frac{\pi }{2}}\left( \frac{11}{24}\phi _1(x)-\frac{1}{48}\phi _3(x)\right) \end{aligned}$$(52)Comparing the base of Chebyshev polynomial from the both sides of Eq. (53) the solutions are$$\begin{aligned} (H+L)\sum _{j=0}^{m}\alpha _j\,\phi _j(x)&= \sqrt{\frac{\pi }{2}}\left( -5 \phi _4(x)-\frac{11}{24}\phi _1(x)+\frac{1}{48} \phi _3(x)\right) . \end{aligned}$$(53)$$\begin{aligned} \alpha _4 = \sqrt{\frac{\pi }{2}}, \quad \alpha _1=\alpha _2=\alpha _3=0. \end{aligned}$$(54)
- 3.

###
*Example 3*

*N*is a number of iteration, are given in Table 3.

Errors of solutions for Eq. (56) solved by HPM

| \(N=5\) | \(N=10\) |
---|---|---|

−0.9999 | 2.424502288 × 10 | 2.312185561 × 10 |

−0.901 | 6.701859769 × 10 | 6.391391532 × 10 |

−0.725 | 8.561715506 × 10 | 8.165088181 × 10 |

−0.436 | 6.727677283 × 10 | 6.416013033 × 10 |

−0.015 | 2.571605139 × 10 | 2.452473774 × 10 |

0.015 | 2.571605139 × 10 | 2.452473774 × 10 |

0.436 | 6.727677283 × 10 | 6.416013033 × 10 |

0.725 | 8.561715506 × 10 | 8.165088181 × 10 |

0.901 | 6.701859769 × 10 | 6.391391532 × 10 |

0.9999 | 2.424502288 × 10 | 2.312185561 × 10 |

- 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 \(\tilde{L}u=Lu-L_nu \equiv 0\). Since \(Cu\equiv 0, \ \ \tilde{L}u \equiv 0\) then \(\tilde{S}=H+L\). Using (30) we have$$\begin{aligned} (H+L)v_0&=\sum _{j=0}^{m}\alpha _j\,\phi _j(x),\nonumber \\ (H+L)v_1&=f-\sum _{j=0}^{m}\alpha _j\,\phi _j(x). \nonumber \\ (H+L)v_k&=-\tilde{L}(v_{k-1}) \equiv 0, \, k=2,3, \ldots \end{aligned}$$(57)
- 2.Again \(v_k \equiv 0, \quad\, k=2,3, \ldots \) and by equating \(v_1=0\) we haveusing 1st equation of (57) and taking into account (58) yields$$\begin{aligned} \sum _{j=0}^{m}\alpha _j\,\phi _j(x)=f(x). \end{aligned}$$(58)Comparing the base of Chebyshev polynomial from the both sides of Eq. (59) produce a system. Solutions of the system are$$\begin{aligned} (H+L)\sum _{j=0}^{m}\alpha _j\,\phi _j(x) = \sqrt{\frac{\pi }{2}}\left( -\phi _3(x)-\frac{15}{16}\phi _1(x)-\phi _0(x)\right) . \end{aligned}$$(59)$$\begin{aligned} \alpha _0=\sqrt{\frac{\pi }{2}},\quad\; \alpha _1=\frac{1}{2}\sqrt{\frac{\pi }{2}},\quad\;\alpha _2=0,\quad\;\alpha _3=\frac{1}{4}\sqrt{\frac{\pi }{2}}. \end{aligned}$$(60)
- 3.

###
*Example 4*

*Solution* For this example, the conditions of Theorem 8 does not hold. Therefore, HPM is not a reliable method to solve Eq. (62).

- 1.Approximate \(L(x,t)=\dfrac{e^{2x}t^3}{2}\) by Chebyshev polynomialstherefore \(\tilde{L}_n\equiv 0\). Choose selective functions \(g_j(x)=\phi _j(x)\), then from (30), we have$$\begin{aligned} L_n(x,t)=\frac{e^{2x}}{16}(\phi _3(t)-2\phi _1(t))=L(x,t), \end{aligned}$$(63)$$\begin{aligned} (H+C)v_0&=\sum _{j=0}^{m}\alpha _j\,\phi _j(x), \end{aligned}$$(64)$$\begin{aligned} (H+C)v_1&=f-\sum _{j=0}^{m}\alpha _j\,\phi _j(x). \end{aligned}$$(65)with \(v_0=\sum _{j=0}^{m} b_j\,\phi _j(x)\).$$\begin{aligned} (H+C)v_{k}&=-\tilde{L}(v_{k-1})\equiv 0, \end{aligned}$$(66)
- 2.Since \(v_k\equiv 0\), for \(k\ge 2\) and approximating \(e^{2x}\) into Chebyshev polynomials with 4 basesthen substituting (67) into (65) and equating \(v_1=0\) for \(m=4\) yields$$\begin{aligned} e^{2x}\simeq \sqrt{\frac{\pi }{2}}\quad \left( \frac{19}{12}\phi _0(x)+\frac{4}{3}\phi _1(x)+\frac{5}{8}\phi _2(x) +\frac{1}{6}\phi _3(x)-\frac{1}{24}\phi _4(x) \right) , \end{aligned}$$(67)$$\begin{aligned} b_0=\frac{19}{384}\,\sqrt{\frac{\pi }{2}},\quad b_1=\frac{25}{24}\,\sqrt{\frac{\pi }{2}},\quad b_2=-\frac{507}{256}\,\sqrt{\frac{\pi }{2}}, \quad b_3 =-\frac{959}{192}\,\sqrt{\frac{\pi }{2}},\quad b_4=-\frac{767}{768}\,\sqrt{\frac{\pi }{2}}. \end{aligned}$$(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 haveThus, we obtain the approximate solution in the form$$\begin{aligned} v_0=\sum _{k=0}^{4}\alpha _k \phi _k(x)=\sqrt{\frac{\pi }{2}}\phi _3(x)=8x^3+4x^3-4x-1. \end{aligned}$$which is same as exact solution. Modified HPM has zero error for solving Eq. (62).$$\begin{aligned} \varphi (x)=\sqrt{1-x^2}(8x^3+4x^3-4x-1) \end{aligned}$$(69)

###
*Example 5*

The exact solution of Eq. (70) is \(\varphi (x)=\displaystyle \sqrt{1-x^2}\frac{10}{x+2}\).

*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

Errors of solution for Eq. (70) solved by modified HPM

| Modified HPM, \(m=n=6\) | Modified HPM, \(m=n=6\) |
---|---|---|

−0.9999 | 1.0851729 × 10 | 1.4040446 × 10 |

−0.901 | 3.5501594 × 10 | 1.8203460 × 10 |

−0.725 | 1.8899796 × 10 | 6.1943369 × 10 |

−0.436 | 3.0648319 × 10 | 2.6221447 × 10 |

−0.015 | 8.6784464 × 10 | 1.7385004 × 10 |

0.015 | 1.9992451 × 10 | 1.8524990 × 10 |

0.436 | 3.3916634 × 10 | 1.9700974 × 10 |

0.725 | 6.3326189 × 10 | 2.3479494 × 10 |

0.901 | 1.2745747 × 10 | 1.3403888 × 10 |

0.9999 | 3.3327100 × 10 | 3.5400390 × 10 |

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).

## Declarations

### 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 *GP-i(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.

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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