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

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.

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 (1999He ( , 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 (1999He ( , 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.
Consider HSIE of the first kind where ϕ(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) ∈ R 2 | − 1 ≤ s, t ≤ 1}. Assume that K(s, t) is constant on the diagonal of the region, i.e. (1) 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) 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 (4) 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 φ −1 (x) = 0. Note that Eqs. (10) and (11) are crucial to the rest of our analysis. Let L(ρ) denotes the space of square integrable real valued function with respect to ρ(x) = √ 1 − x 2 . The inner product on L(ρ) is given by and �u� ρ = �u, v� ρ denotes the norm. The set {φ k } ∞ k=0 is a complete orthonormal basis for L(ρ), so that if u ∈ L(ρ) then where the sum converges in L(ρ). In addition, the norm of u satisfies the Parseval's equality We will need the subspace of L(ρ) which is consisting of all u such that All functions satisfying (12) is denoted by L 1 (ρ) and it can be made into Hilbert space if the inner product of u ∈ L 1 (ρ) and v ∈ L 1 (ρ) are defined by The norm of u ∈ L 1 (ρ) is given by We extend the operator H defined by (5) as a bounded operator from L 1 (ρ) to L(ρ) by defining and observe that It is not hard to show that H −1 : L 1 (ρ) → L(ρ) exist and is given by hence H is invertible Golberg (1987).

Lemma 1 The norm of operator
On the other hand Since v ∈ L 1 (ρ) and due to (16) we have

Therefore
By the norm definition of operator, we obtain

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
These operators are bounded from L(ρ) → L(ρ). Moreover, boundedness and the compactly embeddability of T r and T l from L 1 (ρ) to L(ρ) (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 r − T l : L 1 (ρ) → L(ρ). As we know 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.
Since operators C and L are compact then C + L : L 1 (ρ) → L 1 (ρ) is also compact. We know that H −1 (C + L) : L 1 (ρ) → L 1 (ρ) is a compact operator. Due to the Fredholm theorem Reed and Simon (1980, Theorem VI.14) the inverse operator (I + H −1 (C + L)) −1 of the operator function I + H −1 (C + L), ∈ C, exists for all in C \ C 1 , where C 1 is a discrete subset of C (i.e. a set C 1 has no limit points in C) and for ∈ C 1 the null space N (I + H −1 (C + L)) is finite, that is z = − −1 is the eigenvalue of H −1 (C + L) with finite multiplicity. These facts allows us to suppose the following Assumption 4 = 1 does not belong to C 1 , i.e. N (I + 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 Then due to Assumption 4 the operator H + C + L is invertible.

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  Remark Note that most cases of modified HPM, the unknown coefficients α 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 ≥ 2 which implies two step method. In general, if v 1 � = 0 but v (m) 1 → 0 as m → ∞ then we can compute the next iteration v k , k ≥ 2. It effects to the next iteration k ≥ 2 but the contribution to the solution of the problem will be very small therefore we can neglect it.

Let us consider HSIE (5) by adding parameter 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.  Eshkuvatov et al. SpringerPlus (2016) 5:1473 Assume that γ 1 < 1, then from (22) at p = 1, we obtain Therefore, series (22) converges to the exact solution in the sense of norm || · || 1 .
Remark 9 Note that in our case = 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 → 1 gives the approximate solution of the form Theorem 10 If γ 1 < 1, then the rate of convergence of the approximate solution ṽ N can be estimated by where E N = �v(x) −ṽ N (x)� 1 and B is defined by (36).
Approximate solution of Eq. (4) in series (37) can be estimated as follows.
Theorem 13 Rate of convergence of approximate solution ṽ N can be estimated by where E n = �v(x) −ṽ N (x)� and ε are defined by (39) and γ 2 < 1.

Proof
Remark 14 Since γ 2 < 1, the term γ N 2 1 − γ 2 ε → 0 as N → ∞. Moreover, sufficiently small ε 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.
For application of modified HPM to the Eq. (44), we do this following steps: (44) we can use inverse operator (45). Based on the scheme (31) for m = 2 we obtain, 2. Since v k ≡ 0, k = 2, 3, . . . we can easily find approximate solution as which coincides with exact solution. Mandal and Bhattacharya (2007) consider the Eq. (44) and comparisons with HPM, modified HPM are summarized in Table 1. et al. (2011). Consider HSIE of the form with exact solution ϕ(x) = 1 − x 2 (16x 4 − 12x 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).
Example 3 Chen and Zhou (2011). Consider HSIE in the form Conditions of Theorem 8 are satisfied, therefore for HPM we choose initial guess as u 0 = φ 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) = φ j (x), j = 0, . . . , m and kernel L(x, t) = tx in Eq. (56) be approximated by projection kernel of the form . Again for this case L u = Lu − L n u ≡ 0. Since Cu ≡ 0,Lu ≡ 0 then S = H + L. Using (30) we have 2. Again v k ≡ 0, k = 2, 3, . . . and by equating v 1 = 0 we have    Chen and Zhou (2011) shown in Table 4.
To obtain the approximate solutions of Eq. (62) by modified HPM (30), we do the following steps: 1. Approximate L(x, t) = e 2x t 3 2 by Chebyshev polynomials therefore L n ≡ 0. Choose selective functions g j (x) = φ j (x), then from (30), we have In this case L � = 0, therefore the scheme (30) has the form To solve Eq. (70), we choose the collocation points, x i as the roots of φ(x) which is Errors of ϕ(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).