Correspondence should be addressed to M. Ali Fariborzi Araghi; fariborzi.araghi@gmail.com
Received 11 November 2016; Revised 7 February 2017; Accepted 8 February 2017; Published 16 March 2017 Academic Editor: Haipeng Peng
Copyright © 2017 Sedigheh Farzaneh Javan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A new approach based on the **Reproducing** **Kernel** **Hilbert** **Space** **Method** is proposed to approximate the solution of the second- kind **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.

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the **reproducing** **kernel** **space** was presented by Du and Cui [8, 9], solution of a system of the lin- ear Volterra **integral** **equations** was discussed by Yang et al. [10], solvability of a **class** of Volterra **integral** **equations** with weakly singular **kernel** us- ing RKM was investigated in [11, 12, 13], Geng [14] explained how to solve a Fredholm **integral** equation of the third kind in the **reproducing** ker- nel **space**, and Ketabchi et al. [7] obtained some error estimates for **solving** Volterra **integral** equa- tions using RKM. In [7] and some other places, a general technique for **solving** Volterra **integral** **equations** was discussed in the **reproducing** ker- nel **space**. This general technique is based on the Gram-Schmidt (GS) orthogonalization pro- cess. In this study, we aim to explain how to construct a **reproducing** **kernel** **method** without using this process. For this purpose, we consider the following **nonlinear** Volterra **integral** equation

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Many authors considered methods for **solving** equation (1.1) includ- ing the Clenshaw-Curtis quadrature **method**, Clenshaw-Curtis-Rational **method** and so on [10, 11, 12, 13, 14]. In this study, a new **method** of solv- ing solution is proposed in a **reproducing** **kernel** **Hilbert** **space**(RKHS). It is called **reproducing** **kernel** **method**. The rest of the paper is orga- nized as follows. In section next, the **reproducing** **kernel** **Hilbert** **space** for **solving** (1.1) is introduced. In section 3, we discuss **reproducing** **kernel** **method** for (1.1). We transform (1.1) into **integral** equation of finite interval by substituting the variables t and s by t = α(1 1+τ − τ) , and s = α(1 1+z − z) respectively:

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The main goal of this article is to solve second-order FVIDEs in the **Hilbert** **space** W 2 3 [a, b] ⊕ W 2 .3 [a, b] under the assumption of strongly generalized diﬀerentiability. More precisely, we provide a numerical approximate solution for fuzzy Volterra integro- diﬀerential equation of the general form

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The mathematical aspects and properties of **reproducing** **kernel** in **Hilbert** **space** (RKHS) is explored in this research to understand basic facts and the importance of RKHS that contribute to the **kernel** **method** and its **application** in statistics are being reviewed. It is known that **kernel** methods provide a framework for **solving** several profound issues in the theories of machine learning. A combination of **kernel** techniques, machine learning theory, and optimization algorithms contribute to the development of **kernel**-based learning methods. Some **reproducing** kernels used in survival analysis will be introduced to show the importance of **reproducing** **kernel** **method** in the area of science and statistics.The mathematical concepts of Newton- Raphson **method** and the numerical methods for function optimization in statistics will be discussed. The function f(x) of the representer theorem that involves the **reproducing** kernels is obtained by generating the mathematical process behind this **method**. The process of finding the solution to the regularised least-squares problem via a system of linear **equations** is illustrated to explain the procedures to find the values of parameters involved in the **kernel** **method**.

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. (7)
3. Smoothness properties of the solution
In this part, the analytical solution of Eq. (1) is briefly discussed. In the case of complex Banach spaces, the operator K is analytic in Ω , if it is Frechet differentiable at each point of Ω . Having analytic **integral** operator gives us analytical solution to Eq. (1) [14]. Ref. [21] includes conditions in which the **nonlinear** operators are Frechet differentiable. But in the case of a real Banach **space**, determination of an analytical solution to Eq. (1) is generally difficult. Atkinson [4] has introduced a special **class** of **nonlinear** **integral** equation. This **class** has been denoted by g 1 (η, µ) . In this notation, η and µ are related to the continuity order of partial derivatives of the **kernel** of **integral** equation with respect to the third variable.

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where g is a function of its variables, A, B, C, D are real constants and unknown function y x ( ) is continuous on the interval [0, 1]. These problems arise in many areas of applied mathematics, physics and engineering, such as fluid mechanics, gas dynamics, reaction diffusion process, nuclear physics, chemical reactor theory, geo- physics, studies of atomic structures and etc. Several numerical techniques such as finite difference approxi- mation [1], cubic splines [2] [3], B-splines [4], Adomian decomposition **method** [5], differential transformation **method** [6] and others [7] [8] have been proposed to obtain approximate solution of these problems by some authors. The **application** of RKM in linear and **nonlinear** problems has been developed by many researchers [9]-[12]. The RKM has been treated singular linear two-point boundary value problem, singular **nonlinear** two- point periodic boundary value problem, **nonlinear** system of boundary value problem, singular **integral** **equations**, **nonlinear** partial differential **equations** and etc. in recent years in [13]-[17].

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obtained the representation of the exact solution for the nonlin- ear Volterra–Fredholm **integral** **equations** by using the repro- ducing **kernel** **space** **method**. Wu and Li (2010) applied the iterative **reproducing** **kernel** **method** to obtain the analytical approximate solution of a **nonlinear** oscillator with discontinu- ties. Recently, the **method** was applied to the fractional partial differential **equations** and multi-point boundary value problems ( Jiang and Lin, 2011; Mohammadi and Mokhtari, 2011 ). For more details about RKHSM and the modiﬁed forms and its effectiveness, see ( Cui and Deng, 1986; Yao and Lin, 2011 ) and the references therein.

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In the paper, a **reproducing** **kernel** **method** of **solving** singular **integral** **equations** (SIE) with cosecant **kernel** is proposed. For **solving** SIE, diﬃculties lie in its singular term. In order to remove singular term of SIE, an equivalent transformation is made. Compared with known investigations, its advantages are that the representation of exact solution is obtained in a **reproducing** **kernel** **Hilbert** **space** and accuracy in numerical computation is higher. On the other hand, the representation of **reproducing** **kernel** becomes simple by improving the deﬁnition of traditional inner product and requirements for image **space** of operators are weakened comparing with traditional **reproducing** **kernel** **method**. The ﬁnal numerical experiments illustrate the **method** is eﬃcient.

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The integro-differential equation arises in many physical applications, such as potential theory and Dirichlet problems, electrostatics, mathe- matical problems of radiative equilibrium, the particle transport prob- lems of astrophysics and reactor theory, and radiative heat transfer prob- lems. Recently, a huge amount of research work has been motivated by the concept of a system of integro-differential **equations**. Several power- ful mathematical methods such as Galerkin **method** [6], Petrov Galerkin **method** [7], Tau **method** [8], collocation **method** [9], block pulse functions **method** [10], Chebyshev polynomial **method** [11], Legendre wavelets [12], Taylor series [13], Adomain’s **method** [14], He’s homotopy perturbation **method** [15] and others [16–22] have been proposed to obtain exact and approximate solution of linear Fredholm integro-differential **equations** system. The **application** of RKHSM in linear and **nonlinear** problems has been developed by many researchers [23–25]. This **method** obtains the exact solution in series form and provides approximate solution with high precision [26–32].

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Abstract In this paper, we propose a relatively new semi-analytical technique to approximate the solution of **nonlinear** multi-order fractional diﬀerential **equations** (FDEs). We present some results concerning to the uniqueness of solution of **nonlinear** multi- order FDEs and discuss the existence of solution for **nonlinear** multi-order FDEs in **reproducing** **kernel** **Hilbert** **space** (RKHS). We further give an error analysis for the proposed technique in diﬀerent **reproducing** **kernel** **Hilbert** spaces and present some useful results. The accuracy of the proposed technique is examined by comparing with the exact solution of some test examples.

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Keywords: Fredholm **integral** equation, Volterra **integral** equation, **Reproducing** **kernel** **Hilbert** **space** **method**.
1 Introduction
**Integral** **equations** (IEs) have an important role in the fields of science and engineering [1, 2, 3]. Some boundary value problems arising in electromagnetic theory lead to the problem of **solving** functional IEs [4]. Functional IEs arise in solid state physics, plasma physics, quantum mechanics, astrophysics, fluid dynamics, cell kinetics, chemical kinetics, the theory of gases, mathematical economics, hereditary phenomena in biology. Some analytical and nu- merical methods have been developed for obtaining approximate solutions to IEs. For instance we can mention the following works. Babolian et al. [5] applied a numerical **method** for **solving** a **class** of functional and two dimensional **integral** **equations**, Abbasbandy [6] used Hes homotopy perturbation **method** for **solving** functional **integral** **equations**, Rashed [7] used Lagrange interpolation and Chebyshev interpolation for obtaining numerical solution of functional differential, **integral** and integro-differential **equations**.

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The exact solutions of a system of linear weakly singular Volterra **integral** **equations** (VIE) have been a difficult to find. The aim of this paper is to apply **reproducing** **kernel** **Hilbert** **space** (RKHS) **method** to find the approximate solutions to this type of systems. At first, we used Taylor's expansion to omit the singularity. From an expansion the given system of linear weakly singular VIE is transform into a system of linear ordinary differential **equations** (LODEs). The approximate solutions are represent in the form of series in the **reproducing** **kernel** space𝑊 1 [0,1]. By comparing with the exact solutions of two examples, we saw that RKHS is

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[15] Geng F.Z., Cui M.G. (2007) ”**Solving** singular **nonlinear** second-order periodic boundary value problems in the **reproducing** **kernel** **space**,” Applied Mathematics and Computation, 192, 389-398.
[16] Geng F.Z., Cui M.G. (2009) ”New **method** based on the HPM and RKHSM for **solving** forced Duffing **equations** with **integral** boundary conditions,” Journal of Computational and Applied Mathematics, 233, 165-172.

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[14] J. B. Bremnes, “Probabilistic wind power forecasts using local quantile regression,” Wind Energy, vol. 7, no. 1, pp. 47–54, January/March 2004. [15] H. A. Nielsen, H. Madsen, and T. S. Nielsen, “Using quantile regression to extend an existing wind power forecasting system with probabilistic forecasts,” Wind Energy, vol. 9, no. 1-2, pp. 95–108, January/April 2006. [16] G. Rubio, H. Pomares, L. J. Herrera, and I. Rojas, Computational and Ambient Intelligence: 9th International Work-Conference on Artificial Neural Networks, IWANN 2007, San Sebasti´an, Spain, June 20-22, 2007. Proceedings. Berlin, Heidelberg: Springer Berlin Heidelberg, 2007, ch. **Kernel** Methods Applied to Time Series Forecasting, pp. 782–789. [17] I. Takeuchi, Q. Le, T. Sears, and A. Smola, “Nonparametric quantile

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In nonparametric statistics, it is assumed that the estimand belongs to a very large parameter **space** in order to avoid model misspecification. Such misspecification can lead to large approximation errors and poor estimator performance. However, it is often challenging to produce estimators which are robust against such large parameter spaces. An important tool which allows us to achieve this aim is adaptive estimation. Adaptive estimators behave as if they know the true model from a collection of models, despite being a function of the data. In particular, adaptive estimators can often achieve the same optimal rates of convergence as the best estimators when the true model is known.

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We now turn the tables. Since, by the Riemann mapping theorem, each simply connected domain, which is not equal to C , is mapped conformally onto the open unit disk D , we can find the Bergman **kernel** for an arbitrary simply connected domain Ω in terms of the associated conformal map- ping function, as we will see in the proof of the following theorem.

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square loss function with a sparsity-inducing penalty, leading to sparse representation of the resultant regression function. With the linear regression model, the sparse representation leads to variable selection based on whether the corresponding regression coefficient is zero. The aforementioned variable selection methods have demonstrated superior performance in many real applications. Yet their success largely relies on the validity of the linear model assumption. To relax the model assumption, attempts have been made to extend the variable selection methods to a nonparametric regression context. For example, under the additive regression model assumption, a number of variable selection methods have been developed (Shively et al., 1999; Huang and Yang, 2004; Xue, 2009; Huang et al., 2010). Furthermore, higher-order additive models can be considered, allowing each func- tional component contain more than one variables, such as the component selection and smoothing operator (Cosso) **method** (Lin and Zhang, 2006). While this **method** provides a more flexible and still interpretable model compared to the classical additive models, the number of functional components increases exponentially with the dimension. Another stream of research on variable selection is to conduct screening (Fan et al., 2011; Zhu et al., 2011; Li et al., 2012), which treats each individual variable separately and assures the sure screening properties. To overcome the issue of ignoring interaction effects, a higher- order interaction screening **method** is also developed (Hao and Zhang, 2014). Model-free variable selection has also been approached in the context of sufficient dimension reduction (Li et al., 2005; Bondell and Li, 2009). More recently, Stefanski et al. (2014) introduced a novel measurement-error-model-based variable selection **method** that can be adapted to a nonparametric **kernel** regression.

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k(x − t)u(t)dt, x ∈ [0, T ], (1) where the source function f and the **kernel** function k are given, and u(x) is the unknown function. Several numerical methods are available for approximating the Volterra **integral** equation. In particular, Huang[3] used the Taylor expansion of unknown function and obtained an approximate solution. Yang[4] proposed a **method** for the solution of **integral** equation using the Chebyshev polynomials, while Yousefi[5] presented a numerical **method** for the Abel **integral** equation by Legendre wavelets. Khodabin [6] numerically solved the stochastic Volterra **integral** **equations** using triangular functions and their operational matrix of integration. Kamyad [7] proposed a new algorithm based on the calculus of variations and discretisation **method**, in order to solve linear and **nonlinear** Volterra **integral** **equations**. The Adomian de- composition [8], [9], [10], Homotopy perturbation [10], [11] and the Laplace decomposition methods[12] were proposed for obtaining the approximate analytic solution of the **integral** equation.

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