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

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

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

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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A New Approach for Solving Volterra Integral Equations Using The Reproducing Kernel ‎Method

A New Approach for Solving Volterra Integral Equations Using The Reproducing Kernel ‎Method

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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Reproducing kernel method for solving Wiener-Hopf equations of the second kind

Reproducing kernel method for solving Wiener-Hopf equations of the second kind

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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Application of reproducing kernel Hilbert space method for solving second order fuzzy Volterra integro differential equations

Application of reproducing kernel Hilbert space method for solving second order fuzzy Volterra integro differential equations

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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Reproducing kernel Hilbert space method for cox proportional hazard model

Reproducing kernel Hilbert space method for cox proportional hazard model

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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Application of Sinc-collocation method for solving a class of nonlinear Fredholm integral equations

Application of Sinc-collocation method for solving a class of nonlinear Fredholm integral equations

 . (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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New Implementation of Reproducing Kernel Method for Solving Functional Differential Equations

New Implementation of Reproducing Kernel Method for Solving Functional Differential Equations

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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The reproducing kernel Hilbert space method for solving Troesch’s problem

The reproducing kernel Hilbert space method for solving Troesch’s problem

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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Reproducing kernel method of solving singular integral equation with cosecant kernel

Reproducing kernel method of solving singular integral equation with cosecant kernel

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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A NUMERICAL APPROACH BASED ON THE REPRODUCING KERNEL HILBERT METHOD ON NON-UNIFORM GIRDS FOR SOLVING SYSTEM OF FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS

A NUMERICAL APPROACH BASED ON THE REPRODUCING KERNEL HILBERT METHOD ON NON-UNIFORM GIRDS FOR SOLVING SYSTEM OF FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS

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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A computational method for solving weakly singular Fredholm integral equation in reproducing kernel spaces

A computational method for solving weakly singular Fredholm integral equation in reproducing kernel spaces

method (PHPM), for solving integro-differential equation with weakly sin- gular kernel. In [3] authors solved (1), using Taylor series of the unknown function u to remove singularity, and then Taylor expansion of k together with Legendre polynomials as bases to implement Galerkin method. The Sinc-collocation method is studied by Maleknejad, Mollapourasl, and Os- tadi, to solve nonlinear Fredholm integral equations with weakly singular kernel [15]. Beyrami, Lotfi, and Mahdiani solved Fredholm integral equa- tion of the second kind with Cauchy kernel [6]; they removed singularity by smooth transform and used reproducing kernel Hilbert space (RKHS) method to solve problem in W 3
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Solving multi-order fractional differential equations by reproducing kernel Hilbert space method

Solving multi-order fractional differential equations by reproducing kernel Hilbert space method

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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New implementation of reproducing kernel Hilbert space method for solving a class of functional integral equations

New implementation of reproducing kernel Hilbert space method for solving a class of functional integral equations

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 Reproducing Kernel Hilbert Space Method for Solving System          of Linear Weakly Singular Volterra Integral Equations

The Reproducing Kernel Hilbert Space Method for Solving System of Linear Weakly Singular Volterra Integral Equations

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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Solving a Volterra integral equation with weakly singular kernel in the reproducing kernel space

Solving a Volterra integral equation with weakly singular kernel in the reproducing kernel space

[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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Wind power probabilistic forecast in the reproducing Kernel Hilbert space

Wind power probabilistic forecast in the reproducing Kernel Hilbert space

[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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Reproducing Kernel Hilbert space regression with notes on the Wasserstein Distance

Reproducing Kernel Hilbert space regression with notes on the Wasserstein Distance

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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Reproducing kernel Hilbert spaces

Reproducing kernel Hilbert spaces

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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Model-free Variable Selection in Reproducing Kernel Hilbert Space

Model-free Variable Selection in Reproducing Kernel Hilbert Space

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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Numerical Method for solving Volterra Integral Equations with a Convolution Kernel

Numerical Method for solving Volterra Integral Equations with a Convolution Kernel

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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