The novelty and innovativeness of this paper are the combination of reproducing kernel theory and spline, this leads to a new simple but effective numerical method for solving variable-order anomalous sub-diffusion equation successfully. This combination overcomes the weaknesses of piecewise polynomials that can not be used to solve differential equations directly because of lack of the smoothn...

In this paper, a new partition of unity ± the synchronized reproducing kernel (SRK) interpolant ± is derived. It is a class of meshless shape functions that exhibit synchronized convergence phenomenon: the convergence rate of the interpolation error of the higher order derivatives of the shape function can be tuned to be that of the shape function itself. This newly designed synchronized reprod...

We introduce a vector differential operator P and a vector boundary operator B to derive a reproducing kernel along with its associated Hilbert space which is shown to be embedded in a classical Sobolev space. This reproducing kernel is a Green kernel of differential operator L := P∗T P with homogeneous or nonhomogeneous boundary conditions given by B, where we ensure that the distributional ad...

A differential reproducing kernel (DRK) approximation-based collocation method is developed for solving ordinary and partial differential equations governing the oneand two-dimensional problems of elastic bodies, respectively. In the conventional reproducing kernel (RK) approximation, the shape functions for the derivatives of RK approximants are determined by directly differentiating the RK ap...

Aims: In this paper an approximate method for the solution of third-order differential equation with two and three point boundary condition is developed using iterative reproducing kernel method. Methods: The third order boundary value problem is converted into integro-differential equation of second order two point boundary value problem. The reproducing kernel method which takes the form of a...

A semi-parametric spatial model for spatial dependence is proposed in Poisson regressions to study the effects of risk factors on incidence outcomes. The spatial model is constructed through an application of reproducing kernels. A Bayesian framework is proposed to infer the unknown parameters. Simulations are performed to compare the reproducing kernel-based method with several commonly used a...

In this paper and its sequels, we introduce and analyze a new class of methods, collectively called the reproducing kernel element method (RKEM). The central idea in the development of the new method is to combine the strengths of both finite element methods (FEM) and meshfree methods. Two distinguished features of RKEM are: the arbitrarily high order smoothness and the interpolation property o...

In this paper we propose a new approach for the modelling of the multi-variable systems (MIMO) on the Reproducing Kernel Hilbert Space (RKHS). The proposed approach considers the MIMO system as a set of MISO processes modelled in RKHS space. We propose also a comparative study of three identification kernel methods of nonlinear systems modelled in Reproducing Kernel Hilbert Space (RKHS), where ...