Aveiro Discretization Method in Mathematics: A New Discretization Principle

L.P. Castro (), H. Fujiwara (), M.M. Rodrigues (), S. Saitoh () and V.K. Tuan ()
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L.P. Castro: University of Aveiro, CIDMA–Center for Research and Development in Mathematics and Applications, Department of Mathematics
H. Fujiwara: Kyoto University, Graduate School of Informatics
M.M. Rodrigues: University of Aveiro, CIDMA–Center for Research and Development in Mathematics and Applications, Department of Mathematics
S. Saitoh: Institute of Reproducing Kernels, Department of Mathematics
V.K. Tuan: University of West Georgia, Department of Mathematics

A chapter in Mathematics Without Boundaries, 2014, pp 37-92 from Springer

Abstract: Abstract We found a very general discretization method for solving wide classes of mathematical problems by applying the theory of reproducing kernels. An illustration of the generality of the method is here performed by considering several distinct classes of problems to which the method is applied. In fact, one of the advantages of the present method—in comparison to other well-known and well-established methods—is its global nature and no need of special or very particular data conditions. Numerical experiments have been made, and consequent results are here exhibited. Due to the powerful results which arise from the application of the present method, we consider that this method has everything to become one of the next-generation methods of solving general analytical problems by using computers. In particular, we would like to point out that we will be able to solve very global linear partial differential equations satisfying very general boundary conditions or initial values (and in a somehow independent way of the boundary and domain). Furthermore, we will be able to give an ultimate sampling theory and an ultimate realization of the consequent general reproducing kernel Hilbert spaces. The general theory is here presented in a constructive way, and contains some related historical and concrete examples.

Keywords: Reproducing kernel; Discretization; Computer; Numerical; PDE; ODE; Integral equation; Numerical experiment; Generalized inverse; Tikhonov regularization; Real inversion of the Laplace transform; Matrix; Convolution; Singular integral equation; Sampling theory; Analyticity; Smoothness of function (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/978-1-4939-1106-6_3

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