Application of Reproducing Kernel Hilbert Space Method for Solving a Class of Nonlinear Integral Equations

A new approach based on the Reproducing Kernel Hilbert Space Method is proposed to approximate the solution of the secondkind nonlinear integral equations. In this case, the Gram-Schmidt process is substituted by another process so that a satisfactory result is obtained. In this method, the solution is expressed in the form of a series. Furthermore, the convergence of the proposed technique is proved. In order to illustrate the effectiveness and efficiency of the method, four sample integral equations arising in electromagnetics are solved via the given algorithm.


Introduction
Electromagnetics is the phenomenon associated with electric and magnetic fields and their interactions which is generally one of the most important sciences.Exterior calculus is given in [1,2] inside some textbooks.A way to teach electromagnetics can be approached via the use of differential forms which is given in [3].According to electromagnetic field problems from many years ago, some solutions via linear and nonlinear integral equations (NIE) have been given which can be useful in the field.In those methods like block-pulse functions (BPFs), Galerkin, and collocation, the most important ways are basic functions and appropriate projection.Based on the Reproducing Kernel Hilbert Space method, an approach has been found to solve some electromagnetic issues.
Nonlinear integral equations are encountered in different fields of science and numerous applications as elasticity, plasticity, heat and mass transfer, oscillation theory, fluid dynamics, filtration theory, electrostatics, electrodynamics, biomechanics, game theory, control, queuing theory, electrical engineering, economics, and medicine, among others.There are different types of NIE usually which cannot be worked out explicitly, so it should be approached approximately.
Therefore, many researchers studied and focused on different numerical techniques which can work out these integral equations.For instance, in [4,5], the authors presented the homotopy analysis method to solve the second kind of nonlinear Fredholm and Volterra integral equations.The linear multistep techniques were applied in [6], to obtain the numerical solution of a singular nonlinear Volterra integral equation.In [7], an asymptotic technique to approach numerically the nonlinear Abel-Volterra integral equation was applied.
Consider the following nonlinear integral equation: where ,  are real constants,  ∈  .The existence and uniqueness conditions of the solution for (1) were discussed in [19][20][21][22][23].We assume that the solution of ( 1) is unique.Over several decades, numerical methods in electromagnetic problems have been one of the most important subjects of extensive researches [1][2][3][4].On the other hand, many problems in electromagnetics can be modeled by integral equations mentioned in [24][25][26], for example, electric field integral equation (EFIE) and magnetic field integral equation (MFIE).In recent years, several numerical methods for solving linear and nonlinear integral equations have been presented.Applicable equations of electromagnetics have been implied in the presented paper.
In previous works like [13][14][15], the Gram-Schmidt orthogonalization process has been considered to implement RKHSM.Since this process is unstable numerically and it may take a lot of time to run the algorithm, here, we put away this process and act with another way.Our approach combines the methods mentioned in [13][14][15][16][17].More specifically, on the contrary to [13][14][15], without use of the orthogonalization process, the RKSHM is applied successfully to solve the nonlinear problem (1).
The structure of this paper would be described as follows.In Section 2, the basic definitions, assumptions, and preliminaries of RKHS are described.The main idea and convergence of the proposed scheme are discussed in Section 3. Section 4 contains the numerical experiments.Finally, Section 5 is dedicated to a brief conclusion.

Preliminaries
In this section, some basic definitions and important properties of Reproducing Kernel Hilbert Spaces (RKHS) are mentioned [8,9,[27][28][29].Definition 1.A Hilbert Space  is an inner product space that is complete and separable with respect to the norm defined by the inner product.Completeness of the space  holds provided that every Cauchy sequence of points in  that has a limit that is also in  and separable of  admits a countable orthonormal basis of it.Definition 2. For an abstract set X, let H be a Hilbert Space of real or complex-valued functions on set X. We say H is a Reproducing Kernel Hilbert Space if there exist a linear and bounded evaluation functional   over H, or, equivalently,   :   →  () ∀ ∈ . ( Riesz Representation Theorem implies that for all  in X there exists a unique function   of H with the reproducing property, The inner product and the norm in   2 [, ] are of forms Lemma 4 (see [9,29]).Functional space   2 [, ] is inner space.
Theorem 6 (see [9,29]).Functional space   2 [, ] is Reproducing Kernel Hilbert Space.Now, it is taken away that expression form of the Reproducing Kernel function   () ∈   2 [, ].Based on essay, it is easy to prove that   () is the answer of the following generalized differential equation [9,29]: where  is Dirac's delta function.While  ̸ = ,   () is the answer of the following constant linear homogeneous differential equation with 2 order: with the boundary conditions Equation ( 7) is characteristic  2 = 0. Then the general solution of Equation ( 7) is where coefficients   () and   (),  = 1, 2, . . ., 2, could be calculated by solving the following linear equations: Subsequently, the representation of the Reproducing Kernel of  1 2 [, ] is provided by

Main Idea and Theoretical Discussion
The uniqueness conditions for nonlinear problems exist in [21][22][23].The unique solution of ( 1) is assumed in this paper.The solution of ( 1) is given in  1 2 [, ] space.We consider (1) as where is the complete function system of the space  1 2 [, ] and   () =     ()| =  , where the subscript t in the operator L indicates that the operator L applies to the function of .
Proof.We have Clearly Assume that {  } ∞ =1 is dense on [, ] and so (L)() = 0.It follows that  ≡ 0 from the existence of L −1 .Now, the theorem is proved.
=1 is dense on [, ] and the solution of ( 12) is unique, then the solution of ( 12) is Proof.Using (13), we have On the other hand, () ∈  1 2 [, ] and () = ∑ ∞ =0 â   (), â = ⟨(),   ()⟩, are the Fourier series expansion about normal orthogonal system {  ()} ∞ =1 and  1 2 [, ] is the Hilbert Space.Thus the series ∑ ∞ =0 â   () is convergent in the sense of ‖ ⋅ ‖  1  2 and the proof would be complete.Now the approximate solution   () can be obtained by the -term intercept of the exact solution () and In the sequel, a new iterative method to achieve the solution of ( 12) is presented.If then ( 16) can be written as Now suppose, for some   , (  ) is known.There is no problem if we assume  = 1.We put  0 ( 1 ) = ( 1 ) and define the -term approximation to () by where In the following, it would be proven that the approximate solution   () in the iterative ( 22) is convergent to the exact solution of ( 12) uniformly.
Proof.First of all, the convergence of   () from ( 22) would be proven.We infer Subsequence {  ()} ∞ =1 is orthogonal, and it yields that It is obvious that the sequence is bounded and this implies that =  2  → 0 as  → ∞.To prove the completeness of  1 2 [, ] it requires û, where û ∈  1 2 [, ] that   → û as  → ∞.Now we can prove û is the solution of (12).
If we take limit from ( 22), we will have
Algorithm 14.The following steps exist for approximating the solution without applying Gram-Schmidt orthogonal process: Step 1. Fix  ≤  and  ≤ .
Step 8.If  <  then set  =  + 1 and go to step 6.

Else stop.
Algorithm 15.The following steps exist for approximating the solution by applying Gram-Schmidt orthogonal process: Step 1. Fix  ≤  and  ≤ .
Step 9.If  <  then set  =  + 1 and go to step 7.
Else stop.

Numerical Experiments
In this part, four numerical examples are solved for potency and utility of the present method.All computations are performed by MAPLE package.Results which are taken by this method show a proper agreement with the exact solution.A comprehensive applicability of this method is given the stability and consistence of the presented method.The reliability of the method and increasing of the accuracy cause this method to be more applicable.
Example 1.For first applicable instance, we offer nonlinear Fredholm integral equations [26,30]: The absolute errors comparison between the proposed approach and method [26].
Example 2. For second applicable example, an electromagnetic problem is solved via the presented method.It is simulated to nonlinear Volterra integral equations model [26,30]: The exact solution of this equation is () = sin().
According to (39) we can assume an initial approximation  0 (0) = (0) = 0. Numerical results are given in Table 2 by taking   = (−1)/(−1),  = 1, 2, . . .,  and  = 128.In Table 2 a comparison between the absolute errors of the proposed method and the BPFs method [26] is given.Figure 2 shows the approximate solution and its errors.In Table 3, a comparison execution time between Algorithms 14 and 15 is given.
Example 3.An electromagnetic problem is solved via our method for another applicable example.It is simulated to nonlinear Fredholm integral equations model [26,30]: The exact solution of this equation is () =  2 .According to (40), the initial approximation  0 (0) = (0) = 0 is chosen.Numerical results are given in   between the absolute errors of the proposed method and the BPFs method [26] is given.Figure 3 shows the approximate solution and its error.In Table 5 a comparison execution time between Algorithms 14 and 15 is given.
Numerical results are given in Table 6 by taking   = ( − 1)/(−1),  = 1, 2, . . .,  and  = 128.In Table 6 a comparison between the absolute errors of the proposed method and the Haar wavelet method [30] is given.Figure 4 shows the approximate solution and its error.

Conclusion
According to this essay, supplementary of iterative Reproducing Kernel Hilbert Space Method was introduced and applied to acquire the approximate solution of some nonlinear integral equation.In this method, unlike other similar methods, orthogonal process is not used.However the time is increasing, and accuracy is also increasing.The main point which is mentioned in this paper is that Algorithm 14 has higher execution time in comparison with Algorithm 15, but the approximate solution in Algorithm 14 is more accurate than Algorithm 15.Current uniform convergence method is stated and proved.The obtained numerical results confirm that it is a good candidate for solving the nonlinear integral equation.x Figure 4: The absolute errors comparison between the proposed approach and method [30].

Figure 3 :
Figure3: The absolute errors comparison between the proposed approach and method[26].

Table 3 :
The execution time (seconds) comparison between Algorithms 14 and 15.

Table 5 :
The execution time (seconds) comparison between Algorithms 14 and 15.