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Full-Text Articles in Physical Sciences and Mathematics
Radial Basis Function Finite Difference Approximations Of The Laplace-Beltrami Operator, Sage Byron Shaw
Radial Basis Function Finite Difference Approximations Of The Laplace-Beltrami Operator, Sage Byron Shaw
Boise State University Theses and Dissertations
Partial differential equations (PDEs) are used throughout science and engineering for modeling various phenomena. Solutions to PDEs cannot generally be represented analytically, and therefore must be approximated using numerical techniques; this is especially true for geometrically complex domains. Radial basis function generated finite differences (RBF-FD) is a recently developed mesh-free method for numerically solving PDEs that is robust, accurate, computationally efficient, and geometrically flexible. In the past seven years, RBF-FD methods have been developed for solving PDEs on surfaces, which have applications in biology, chemistry, geophysics, and computer graphics. These methods are advantageous, as they are mesh-free, operate on arbitrary …
Radial Basis Function Differential Quadrature Method For The Numerical Solution Of Partial Differential Equations, Daniel Watson
Radial Basis Function Differential Quadrature Method For The Numerical Solution Of Partial Differential Equations, Daniel Watson
Dissertations
In the numerical solution of partial differential equations (PDEs), there is a need for solving large scale problems. The Radial Basis Function Differential Quadrature (RBFDQ) method and local RBF-DQ method are applied for the solutions of boundary value problems in annular domains governed by the Poisson equation, inhomogeneous biharmonic equation, and the inhomogeneous Cauchy-Navier equations of elasticity. By choosing the collocation points properly, linear systems can be obtained so that the coefficient matrices have block circulant structures. The resulting systems can be efficiently solved using matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs). For the local RBFDQ method, the …
A Stable Algorithm For Divergence-Free And Curl-Free Radial Basis Functions In The Flat Limit, Kathryn Primrose Drake
A Stable Algorithm For Divergence-Free And Curl-Free Radial Basis Functions In The Flat Limit, Kathryn Primrose Drake
Boise State University Theses and Dissertations
Radial basis functions (RBFs) were originally developed in the 1970s for interpolating scattered topographic data. Since then they have become increasingly popular for other applications involving the approximation of scattered, scalar-valued data in two and higher dimensions, especially data collected on the surface of a sphere. In the late 2000s, matrix-valued RBFs were introduced for approximating divergence-free and curl-free vector fields on the surface of a sphere from scattered samples, which arise naturally in atmospheric and oceanic sciences. The intriguing property of these RBFs is that the resulting vector-valued approximations analytically preserve the divergence-free or curl-free properties of the field. …