A reproducing kernel Hilbert space approach in meshless collocation method

Abstract

In this paper we combine the theory of reproducing kernel Hilbert spaces with the field of collocation methods to solve boundary value problems with special emphasis on reproducing property of kernels. From the reproducing property of kernels we proposed a new efficient algorithm to obtain the cardinal functions of a reproducing kernel Hilbert space which can be apply conveniently for multi-dimensional domains. The differentiation matrices are constructed and also we drive pointwise error estimate of applying them. In addition we prove the non-singularity of collocation matrix. The proposed method is truly meshless and can be applied conveniently and accurately for high order and also multi-dimensional problems. Numerical results are presented for the several problems such as second and fifth order two point boundary value problems, one and two dimensional unsteady Burgers equations and a parabolic partial differential equation in three dimensions. Also we compare the numerical results with those reported in the literature to show the high accuracy and efficiency of the proposed method.

On n-dimension Extended cubic B-splines collocation algorithms

“…By substitution from ( 4)-( 6) into ( 29) with (31) we obtain the numerical results as in the next table: Let 15 × 15 grid points, we compar between the results of the proposed method and the results of using different methods that shown in Table 6 [18,[33][34][35][36]. The fourth test problem: [18] We take the fourth test problem in the 3-dimensional in this form:…”

“…Cubic B-splines, quasi B-splines, quartic B-splines, quintic B-splines, and so on are utilized in combination with the collocation strategy in [22][23][24][25][26][27] for managing with different straight and nonlinear boundary esteem issues. Strategies like Haar wavelet collocation strategy [30], a slope replicating part collocation strategy [31] and Newton premise capacities collocation strategy [32] are moreover picking up ubiquity to illuminate differential conditions.…”

A new structure to n-dimensional trigonometric cubic B-spline functions for solving n-dimensional partial differential equations

“…To solve various straight and nonlinear boundary esteem problems, cubic B-splines, quasi-B-splines, quartic B-splines, quintic B-splines, and other forms of B-splines are used in conjunction with the collocation technique [20][21][22][23][24][25]. Collocation strategies such as Haar wavelet collocation technique [28], a slope replicating component collocation technique [29], and Newton premise capacities collocation technique [30] are also gaining popularity for illuminating various conditions.…”

A new application of the reproducing kernel method

“…We investigate difference equations by reproducing kernel method in this work. A reproducing kernel Hilbert space approach in meshless collocation method has been investigated in [6]. Piecewise reproducing kernel method for linear impulsive delay differential equations with piecewise constant arguments has been worked in [14].…”