The aim of this study is to present a good modernistic strategy for solving some well-known classes of Lane-Emden type singular differential equations. The proposed approach is based on the reproducing kernel Hilbert space (RKHS) and introducing the reproducing kernel properties in which the initial conditions of the problem are satisfied. The analytical solution that obtained involves in the form of a convergent series with easily computable terms in its reproducing kernel space. The approximation solution is expressed by n-term summation of reproducing kernel functions and it is converge to the analytical solution. Our investigations indicate that there is excellent agreement between the numerical results and the RKHS method, which is applied to some examples to illustrate the accuracy, efficiency, and applicability of the method. The present work shows the potential of the RKHS technique in solving such nonlinear singular initial value problems.
In this article, a new reproducing kernel approach is developed for obtaining numerical solution of nonlinear three-point boundary value problems with fractional order. This approach is based on reproducing kernel which is constructed by shifted Legendre polynomials. In considered problem, fractional derivatives with respect to $α$ and $β$ are defined in Caputo sense. This method has been applied to some examples which have exact solutions. In order to shows the robustness of the proposed method, some numerical results are given in tabulated forms.
Fitted Reproducing Kernel Method for Solving a Class of Third-Order Periodic Boundary Value Problems
Ajloun College, Al Balqa Applied University（阿勒琼学院，巴尔卡应用大学）；Faculty of Science, Zarqa University（扎尔卡大学理学院）；University of Malakand（马拉坎德大学）；Faculty of Engineering Technology, Al-Balqa Applied University（巴尔卡应用大学工程技术学院）
In this paper we analyze a greedy procedure to approximate a linear functional defined in a Reproducing Kernel Hilbert Space by nodal values. This procedure computes a quadrature rule which can be applied to general functionals, including integration functionals. For a large class of functionals, we prove convergence results for the approximation by means of uniform and greedy points which generalize in various ways several known results. A perturbation analysis of the weights and node computation is also discussed. Beyond the theoretical investigations, we demonstrate numerically that our alg…
This note consists of two largely independent parts. In the first part we give conditions on the kernel $k: Ω\times Ω\rightarrow \mathbb{R}$ of a reproducing kernel Hilbert space $H$ continuously embedded via the identity mapping into $L^2(Ω, μ),$ which are equivalent to the fact that $H$ is even compactly embedded into $L^2(Ω, μ).$ In the second part we consider a scenario from infinite-variate $L^2$-approximation. Suppose that the embedding of a reproducing kernel Hilbert space of univariate functions with reproducing kernel $1+k$ into $L^2(Ω, μ)$ is compact. We provide a simple criterion fo…
Kernel methods, being supported by a well-developed theory and coming with efficient algorithms, are among the most popular and successful machine learning techniques. From a mathematical point of view, these methods rest on the concept of kernels and function spaces generated by kernels, so called reproducing kernel Hilbert spaces. Motivated by recent developments of learning approaches in the context of interacting particle systems, we investigate kernel methods acting on data with many measurement variables. We show the rigorous mean field limit of kernels and provide a detailed analysis of…
We show that polynomials do not belong to the reproducing kernel Hilbert space of infinitely differentiable translation-invariant kernels whose spectral measures have moments corresponding to a determinate moment problem. Our proof is based on relating this question to the problem of best linear estimation in continuous time one-parameter regression models with a stationary error process defined by the kernel. In particular, we show that the existence of a sequence of estimators with variances converging to $0$ implies that the regression function cannot be an element of the reproducing kernel…
In this paper we consider the problem of reconstructing an image that is downsampled in the space of its $SE(2)$ wavelet transform, which is motivated by classical models of simple cells receptive fields and feature preference maps in primary visual cortex. We prove that, whenever the problem is solvable, the reconstruction can be obtained by an elementary project and replace iterative scheme based on the reproducing kernel arising from the group structure, and show numerical results on real images.
