Reproducing Kernel Space Research Articles

Random sampling and reconstruction of concentrated signals in a reproducing kernel space

In this paper, we consider (random) sampling of signals concentrated on a bounded Corkscrew domain Ω of a metric measure space, and reconstructing concentrated signals approximately from their (un)corrupted sampling data taken on a sampling set contained in Ω. We establish a weighted stability of bi-Lipschitz type for a (random) sampling scheme on the set of concentrated signals in a reproducing kernel space. The weighted stability of bi-Lipschitz type provides a weak robustness to the sampling scheme, however due to the nonconvexity of the set of concentrated signals, it does not imply the unique signal reconstruction. From (un)corrupted samples taken on a finite sampling set contained in Ω, we propose an algorithm to find approximations to signals concentrated on a bounded Corkscrew domain Ω. Random sampling is a sampling scheme where sampling positions are randomly taken according to a probability distribution. Next we show that, with high probability, signals concentrated on a bounded Corkscrew domain Ω can be reconstructed approximately from their uncorrupted (or randomly corrupted) samples taken at i.i.d. random positions drawn on Ω, provided that the sampling size is at least of the order μ(Ω)ln⁡(μ(Ω)), where μ(Ω) is the measure of the concentrated domain Ω. Finally, we demonstrate the performance of proposed approximations to the original concentrated signals when the sampling procedure is taken either with large density or randomly with large size.

A new reproducing kernel method for Duffing equations

Reproducing kernel theories have attracted much attention for solving various problems. However, discussions of stability and convergence order are very difficult in the traditional reproducing kernel method by orthogonal expansion because of the randomness of Schmidt's orthogonalization coefficients. Later, the convergence order of the reproducing kernel method is estimated by polynomial interpolation of residuals. But the convergence order is not high. In this paper, taking Duffing equation as an example, noting that the reproducing kernel function is a spline function, a new scheme with much higher convergence order is proposed by constructing spline bases function in the reproducing kernel space and combining polynomial interpolation of residuals. Numerical examples verify the convergence order theories proposed in this paper. It is worth to say that our main results could be applied to construct approximate solutions of various equations.

A meshless method in reproducing kernel space for solving variable-order time fractional advection–diffusion equations on arbitrary domain

In this paper, a meshless method in reproducing kernel space is proposed for solving VOTFA-DE on arbitrary domain. Advantages of the meshless method proposed could avoid effectively difficulties of constructing shape functions using known Mercer kernel and deal with arbitrary domains. And the accuracy is verified by two examples.

Solving a linear fractional equation with nonlocal boundary conditions based on multiscale orthonormal bases method in the reproducing kernel space

Abstract In recent years, the study of fractional differential equations has become a hot spot. It is more difficult to solve fractional differential equations with nonlocal boundary conditions. In this article, we propose a multiscale orthonormal bases collocation method for linear fractional-order nonlocal boundary value problems. In algorithm construction, the solution is expanded by the multiscale orthonormal bases of a reproducing kernel space. The nonlocal boundary conditions are transformed into operator equations, which are involved in finding the collocation coefficients as constrain conditions. In theory, the convergent order and stability analysis of the proposed method are presented rigorously. Finally, numerical examples show the stability, accuracy and effectiveness of the method.

A stable minimal search method for solving multi-order fractional differential equations based on reproducing kernel space

In this paper, a stable minimal search method based on reproducing kernel space is proposed for solving multi-order fractional differential equations. The existence and uniqueness of solution of the considered equation is proved and the smoothness of the solution is studied. Based on orthonormal bases, we give smooth transformation and a method for obtaining the e-approximate solution by searching the minimum value. Subsequently, error estimation and stability analysis of the method are obtained. The final several numerical experiments are presented to illustrate the correctness of the theory and the effectiveness of the method.

A new algorithm for fractional differential equation based on fractional order reproducing kernel space

This paper develops an effective and new method to solve a class of fractional differential equations. The method is based on a fractional order reproducing kernel space. First, depending on some theories, a fractional order reproducing kernel spaceWα[0, 1] is constructed. The fractional order reproducing kernel space is a very suitable space to solve a class of fractional differential equations. Then, we calculate the reproducing kernelRy(x) of the spaceWα[0, 1] skilfully in §3. And convergence order and time complexity of this algorithm are discussed. We prove that the approximate solutionunof (1.1) converges to its exact solutionuis not less than the second order. The time complexity of the algorithm is equal to the polynomial time of the third degree. Finally, three experiments support the algorithm strongly from the aspect of theory and technique.

A new algorithm based on improved Legendre orthonormal basis for solving second-order BVPs

In this article, a new algorithm based on improved Legendre orthonormal basis for solving second-order BVPs is constructed. The method of this article is an improved Legendre orthogonal polynomial algorithm, and the ε-best approximate solution is constructed skillfully in the reproducing kernel space. Compared with other methods, this method has the advantages of high approximation, m-order convergence and strong stability. Some numerical examples show the effectiveness and stability of the algorithm.

A high order convergence collocation method based on the reproducing kernel for general interface problems

By utilizing the Legendre polynomials, we first establish a broken reproducing kernel space which is provided with piecewise high order polynomials. Then a new efficient collocation method based on the reproducing kernel is proposed for solving the general 1-D interface problems. By considering a system of algebraic linear equations, we obtain an approximate solution. Our main contribution is that the presented method has higher accuracy than some existing methods. Through a series of benchmark examples, the robustness and high precision of the algorithm are verified.

Reproducing kernel method to solve parabolic partial differential equations with nonlocal conditions

AbstractIn this study, the parabolic partial differential equations with nonlocal conditions are solved. To this end, we use the reproducing kernel method (RKM) that is obtained from the combining fundamental concepts of the Galerkin method, and the complete system of reproducing kernel Hilbert space that was first introduced by Wang et al. who implemented RKM without Gram–Schmidt orthogonalization process. In this method, first the reproducing kernel spaces and their kernels such that satisfy the nonlocal conditions are constructed, and then the RKM without Gram–Schmidt orthogonalization process on the considered problem is implemented. Moreover, convergence theorem, error analysis theorems, and stability theorem are provided in detail. To show the high accuracy of the present method several numerical examples are solved.

A stable least residue method in reproducing kernel space for solving a nonlinear fractional integro-differential equation with a weakly singular kernel

A stable least residue method for solving a nonlinear fractional integro-differential equation with a weakly singular kernel in the reproducing kernel space is proposed. To solve the equation, the multiwavelets bases in the reproducing kernel space are constructed based on the cubic Legendre wavelets in L2[0,1]. The best approximate solution could be obtained by solving the normal equation. Meanwhile, we provide the adaptive convergence order and the stability proof of the scheme. It is encouraging that the new method is stable and the accuracy of the method is preserved even in the case if the solution has fast oscillations. Four examples strongly demonstrate the validity of the scheme.

Numerical Solution of Fractional Order Burgers’ Equation with Dirichlet and Neumann Boundary Conditions by Reproducing Kernel Method

In this research, obtaining of approximate solution for fractional-order Burgers’ equation will be presented in reproducing kernel Hilbert space (RKHS). Some special reproducing kernel spaces are identified according to inner products and norms. Then an iterative approach is constructed by using kernel functions. The convergence of this approach and its error estimates are given. The numerical algorithm of the method is presented. Furthermore, numerical outcomes are shown with tables and graphics for some examples. These outcomes demonstrate that the proposed method is convenient and effective.

On the extension of positive definite kernels to topological algebras

We define an extension of operator-valued positive definite functions from the real or complex setting to topological algebras and describe their associated reproducing kernel spaces. The case of entire functions is of special interest, and we give a precise meaning to some power series expansions of analytic functions that appears in many algebras.

Random sampling in reproducing kernel subspaces of [formula omitted

In this paper, we study random sampling in a reproducing kernel space V, which is the range of an idempotent integral operator. Under certain decay condition on the integral kernel, we show that any element in V can be approximated by an element in a finite-dimensional subspace of V. Moreover, we prove with overwhelming probability that random points uniformly distributed over a cube C is a stable set of sampling for the set of functions concentrated on C. Further, we discuss a reconstruction algorithm for functions in a finite-dimensional subspace of V from its random samples.

A new multiscale algorithm for solving second order boundary value problems

In this study, a new multiscale algorithm was proposed to solve the boundary value problems of second order differential equations. A multiscale basis consisting of two sets of multiscale functions was constructed in the reproducing kernel space, and the proposed multiscale basis was proved to be orthonormal. The ε− approximate solution was defined, and then it was proved to be the optimal solution. In addition, the stability, convergence and complexity of this algorithm were discussed and illustrated in this study. Numerical examples verify the effectiveness and feasibility of the algorithm, and the results show that the proposed intelligent multiscale algorithm has advantages in accuracy and stability compared with other methods.

A generalized collocation method in reproducing kernel space for solving a weakly singular Fredholm integro-differential equations

A generalized collocation method for solving a weakly singular Fredholm integro-differential equation with Kalman kernel is proposed in reproducing kernel space. To obtain the generalized collocation method, the multiwaves basis in reproducing kernel space Wn+1[0,b] is constructed based on Legendre multiwaves in L2[0,1]. Using the multiwaves basis, we propose ε-approximate solutions and use the method of searching the minimum to obtain the best approximate solution of the equation. Meanwhile, convergence order and stability of the generalized collocation method are studied. It is worth to show that the generalized collocation method proposed in the paper is stable and could be applied to solve other integral equations or differential equations.

A Novel Method for Nonlinear Impulsive Differential Equations in Broken Reproducing Kernel Space

In this article, a new algorithm is presented to solve the nonlinear impulsive dif- ferential equations. In the first time, this article combines the reproducing kernel method with the least squares method to solve the second-order nonlinear impulsive differential equations. Then, the uniform convergence of the numerical solution is proved, and the time consuming Schmidt orthogonalization process is avoided. The algorithm is employed successfully on some numerical examples.

Numerical Solution of a Class of Time-Fractional Order Diffusion Equations in a New Reproducing Kernel Space

In this paper, we structure some new reproducing kernel spaces based on Jacobi polynomial and give a numerical solution of a class of time fractional order diffusion equations using piecewise reproducing kernel method (RKM). Compared with other methods, numerical results show the reliability of the present method.

A new reproducing kernel method with higher convergence order for solving a Volterra–Fredholm integral equation

In the paper, linear Volterra–Fredholm integral equations of the second kind are considered in a reproducing kernel space. The approximate solution is gotten by constructing a uniquely solvable system of linear algebraic equations. Our important contributions are to provide convergence order theorems of the scheme using spline function theories and propose a new scheme with high convergence order for solving the approximate solutions to oscillation and non-oscillation of exact solutions.

On the MHD boundary layer flow with diffusion and chemical reaction over a porous flat plate with suction/blowing: two reliable methods

In this paper, a Lie-group integrator based on $$GL_4(\mathbb {R})$$ and the reproducing kernel functions has been constructed to investigate the flow characteristics in an electrically conducting second-grade fluid over a stretching sheet. Accurate initial values can be achieved when the target equation is matched precisely, and then, we can apply the group preserving scheme (GPS) to get a rather accurate results. On the other hand, the reproducing kernel method (RKM) is successfully applied to the underlying equation with convergence analysis. We show exact and approximate solutions by series in the reproducing kernel space. We use a bounded linear operator in the reproducing kernel space to get the solutions by the reproducing kernel method. Comparison of these two methods demonstrates the power and reliability. Finally, effects of magnetic parameter, viscoelastic parameter, stagnation-point flow, and stretching of the sheet parameters are illustrated.

A reproducing kernel method for solving heat conduction equations with delay

We attempt to propose a numerical algorithm for the one-dimensional conduction equations with delay. Firstly, a novel reproducing kernel space satisfying the delay condition is constructed, then an approximating solution space is derived by using the orthogonal projection. In fact, the proposed scheme is a collocation method. The unique solvability as well as uniform convergence of the new scheme is discussed. Two numerical experiments are provided to illustrate our theory.