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# Aveiro Discretization Method in Mathematics: A New Discretization Principle

Chapter

## 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

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## Copyright information

© Springer Science+Business Media, LLC 2014

## Authors and Affiliations

• L.P. Castro
• 2
• H. Fujiwara
• 3
• M.M. Rodrigues
• 2
• S. Saitoh
• 1
Email author
• V.K. Tuan
• 4
1. 1.Department of MathematicsInstitute of Reproducing KernelsKiryuJapan
2. 2.CIDMA–Center for Research and Development in Mathematics and Applications, Department of MathematicsUniversity of AveiroAveiroPortugal
3. 3.Graduate School of InformaticsKyoto UniversityKyotoJapan
4. 4.Department of MathematicsUniversity of West GeorgiaCarrolltonUSA

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