1 research outputs found
A reproducing kernel Hilbert space approach in meshless collocation method
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
multidimensional domains. The differentiation matrices are constructed and also
we drive pointwise error estimate of applying them. In addition we prove the
nonsingularity of collocation matrix. The proposed method is truly meshless and
can be applied conveniently and accurately for high order and also
multidimensional 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 metho