Reproducing kernel method for a class of weakly singular Fredholm integral equations

ABSTRACT Numerical methods for solving integral equations have been the focus of much research, including reproducing kernel methods. We present a new algorithm to solve weakly singular Fredholm integral equations (WSFIEs). The advantage of this method is that it is possible to pick any point in the interval of integration and also the approximate solution. The advantage of this method is used to remove singularity and reproducing kernel functions are used as a basis. The convergence of approximation solution to the exact solution is also proved. Some examples are displayed to demonstrate that the method is accurate and efficient for WSFIEs.


Introduction
Some important applications of weakly singular Fredholm integral equations (WSFIEs) in the fields of fracture mechanics, the theory of porous filtering, elastic contact problems, combined infrared radiation and molecular conduction were provided in [1]. In recent years, several numerical methods have been done in order to find the solution of singular integral and integrodifferential equations. For instance, Lifanov et al. [2] introduced hypersingular integral equations with their applications and then introduced some new numerical algorithms for solving them. In [3], the discrete Galerkin method has been proposed and analysed for obtaining the numerical solution of these equations. The trapezoidal method was applied for approximating singular or nonsingular integral equations [4][5][6][7]. In [8], a generalization of the Euler-Maclaurin summation formula for solving WSFIEs of the second kind was introduced. One can refer to the methods that were proposed in [9][10][11][12].
Recently, the reproducing kernel method to solve a variety of singular or nonsingular integral equations was presented. Alvandi and Paripour solved nonlinear Abel's integral equations with weakly singular kernel in the reproducing kernel space and removed the singularity of the equation considered [13]. The same authors presented a simple and efficient method to solve linear Volterra integro-differential equations [14], nonlinear Volterra-Fredholm integro-differential equations [15], and see [16][17][18][19][20][21].
In this paper, a reproducing kernel method is used for the solution of WSFIEs.
Consider the following (WSFIE): where the singularity of kernel may be stated in the forms k(x, t) = 1/(x − t) α with the assumption 0 < α < 1.
This paper is organized into five sections including the introduction. In the next section, two different reproducing kernel spaces are presented in order to construct reproducing kernel functions in the space W m [0, 1]. The representation of solutions for WSFIEs is obtained in Section 3. The numerical experiments are given in Section 4. Section 5 ends this paper with a brief conclusion.

Preliminaries
Let us present the definitions of reproducing kernel Hilbert spaces.

Definition of operators
To deal with the system, we consider L : Then Equation (1) can be written as We can assume that L is an invertible bounded linear operator, where L * is the adjoint operator of L.

The exact and approximate solution of Equation (I)
We choose a countable dense subset hence, we have where β ik are orthogonal coefficients.
Proof: The subscript y by the operator L indicates that the operator L applies to the function of y. So, we have is dense on [0, 1]; therefore, Lu(x) = 0. It follows that u(x) = 0 from the existence of L −1 . So, the proof of the theorem is complete.
So, the proof is complete. Now, the approximate solution of Equation (1) can be obtained by the n-term intercept of Equation (13) and Proof: there exists a constant c > 0 such that The proof of the lemma is complete.
Proof: First, we will prove u n (x) convergent. we infer that From the orthogonality of By eliminating B n 2 , it is clear that u n W m 2 ≥ u n−1 W m 2 . Due to the condition that u n W m 2 is bounded and descending, u n W m 2 is convergent as soon as n −→ ∞. Then there exists a constant c such that In [22], it was proved that the space W m [0, 1] is complete. Considering the completeness of W m [0, 1], it has u n (x) It was proved that u n (x) is convergent to u(x). Second, we will prove that u(x) is the solution of Equation (9). From ψ i (x), ϕ j (x) =ψ i (x j ) and Equation (16), it follows . Moreover, it is easy to see by induction that Since That is, u(x) is the solution of Equation (9) and The proof is complete.

Numerical experiments
In this section, some examples will be presented by using the method discussed above. All experiments were performed in MATHEMATICA 8.0. The numerical results are compared with their exact solution and approximate solution. In this regard, we have reported in tables and figures the values of the absolute error function |u n (x) − u(x)|, then the maximum absolute error function of this method is compared with two methods presented in [4,8]. Node Using the present method (RKHS), taking n = 32, x i = .03i, i = 1, 2, . . . , n.  Figure 1, respectively. However, by increasing m, the behaviour improves. The maximum value of the absolute errors in space W 5 [0, 1] and n = 32 in Figure 1       The maximum value of the absolute errors of two other methods, the classic Trapezoidal rule and the Euler-Maclaurin summation formula, is introduced in [8], 2.5231 × 10 −1 and 5.0214 × 10 −6 , respectively. It is obviously seen that the reproducing kernel method is more accurate than the classic Trapezoidal rule and the Euler-Maclaurin summation formula.

Example 4.2:
In this example, we solve the WSFIE of the second kind with the exact solution Using the present method (RKHS), taking n = 32, x i = .03i, i = 1, 2, . . . , n.  Figure 2, respectively. However, by increasing m, the behaviour improves. The maximum value of the absolute errors in space W 5 [0, 1] and n = 32 in Figure 2 is 4.6 × 10 −10 . The maximum value of the absolute errors of two other methods, the classic Trapezoidal rule and the Euler-Maclaurin summation formula, is more than 10 −2 that is introduced in [8], 1.273 × 10 −1 and 5.075 × 10 −5 , respectively. It is obviously seen the reproducing kernel method is more accurate than the classic Trapezoidal rule and the Euler-Maclaurin summation formula.

Example 4.3:
In this example, we solve the WSFIE of the second kind with the exact solution u(t) = √ t Using the present method (RKHS), taking n = 32, x i = .03i, i = 1, 2, . . . , n.  Figure 3, respectively. However, by increasing m, the behaviour improves. The maximum value of the absolute errors in space W 5 [0, 1] and n = 32 in Figure 3 is 5.3 × 10 −7 . The maximum value of the absolute errors of two other methods, the classic Trapezoidal rule and the Euler-Maclaurin summation formula, is introduced in [8], 1.273 × 10 −1 and 5.075 × 10 −5 , respectively. It is obviously seen the reproducing kernel method is more accurate than the classic Trapezoidal rule and the Euler-Maclaurin summation formula.

Concluding remarks
In this paper, we presented the new implementation of the reproducing kernel Hilbert space method to obtain the approximate solution of a class of WSFIEs. We obtain the sequence which is proved to converge to the exact solution uniformly. The comparison of numerical results in two spaces shows that our method is an accurate, efficient and reliable analytical technique for these equations. However, to obtain better results, use of the larger parameter m is recommended.

Disclosure statement
No potential conflict of interest was reported by the authors.