Tobin A. Driscoll

Bio: Tobin A. Driscoll is an academic researcher from University of Delaware. The author has contributed to research in topics: Interpolation & Boundary value problem. The author has an hindex of 32, co-authored 74 publications receiving 5086 citations. Previous affiliations of Tobin A. Driscoll include University of Colorado Boulder & Wichita State University.

Hydrodynamic Stability Without Eigenvalues

Abstract: Fluid flows that are smooth at low speeds become unstable and then turbulent at higher speeds. This phenomenon has traditionally been investigated by linearizing the equations of flow and testing for unstable eigenvalues of the linearized problem, but the results of such investigations agree poorly in many cases with experiments. Nevertheless, linear effects play a central role in hydrodynamic instability. A reconciliation of these findings with the traditional analysis is presented based on the "pseudospectra" of the linearized problem, which imply that small perturbations to the smooth flow may be amplified by factors on the order of 105 by a linear mechanism even though all the eigenmodes decay monotonically. The methods suggested here apply also to other problems in the mathematical sciences that involve nonorthogonal eigenfunctions.

Schwarz-Christoffel mapping

Abstract: This book provides a comprehensive look at the Schwarz-Christoffel transformation, including its history and foundations, practical computation, common and less common variations, and many applications in fields such as electromagnetism, fluid flow, design and inverse problems, and the solution of linear systems of equations. It is an accessible resource for engineers, scientists, and applied mathematicians who seek more experience with theoretical or computational conformal mapping techniques. The most important theoretical results are stated and proved, but the emphasis throughout remains on concrete understanding and implementation, as evidenced by the 76 figures based on quantitatively correct illustrative examples. There are over 150 classical and modern reference works cited for readers needing more details. There is also a brief appendix illustrating the use of the Schwarz-Christoffel Toolbox for MATLAB, a package for computation of these maps.

Interpolation in the limit of increasingly flat radial basis functions

Abstract: Many types of radial basis functions, such as multiquadrics, contain a free parameter In the limit where the basis functions become increasingly flat, the linear system to solve becomes highly ill-conditioned, and the expansion coefficients diverge Nevertheless, we find in this study that limiting interpolants often exist and take the form of polynomials In the 1-D case, we prove that with simple conditions on the basis function, the interpolants converge to the Lagrange interpolating polynomial Hence, differentiation of this limit is equivalent to the standard finite difference method We also summarize some preliminary observations regarding the limit in 2-D

Algorithm 756: a MATLAB toolbox for Schwarz-Christoffel mapping

Abstract: The Schwarz-Christoffel transformation and its variations yield formulas for conformal maps from standard regions to the interiors or exteriors of possibly unbounded polygons. Computations involving these maps generally require a computer, and although the numerical aspects of these transformations have been studied, there are few software implementations that are widely available and suited for general use. The Schwarz-Christoffel Toolbox for MATLAB is a new implementation of Schwarz-Christoffel formulas for maps from the disk, half-plane, strip, and rectangle domains to polygon interiors, and from the disk to polygon exteriors. The toolbox, written entirely in the MATLAB script language, exploits the high-level functions, interactive environment, visualization tools, and graphical user interface elements supplied by current versions of MATLAB, and is suitable for use both as a standalone tool and as a library for applications written in MATLAB, Fortran, or C. Several examples and simple applications are presented to demonstrate the toolbox's capabilities.

Observations on the behavior of radial basis function approximations near boundaries

Abstract: RBF approximations would appear to be very attractive for approximating spatial derivatives in numerical simulations of PDEs. RBFs allow arbitrarily scattered data, generalize easily to several space dimensions, and can be spectrally accurate. However, accuracy degradations near boundaries in many cases severely limit the utility of this approach. With that as motivation, this study aims at gaining a better understanding of the properties of RBF approximations near the ends of an interval in 1-D and towards edges in 2-D.

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

Spectral Methods in MATLAB

Abstract: Preface 1 Differentiation matrices 2 Unbounded grids: the semidiscrete Fourier transform 3 Periodic grids: the DFT and FFT 4 Smoothness and spectral accuracy 5 Polynomial interpolation and clustered grids 6 Chebyshev differentiation matrices 7 Boundary value problems 8 Chebyshev series and the FFT 9 Eigenvalues and pseudospectra 10 Time-stepping and stability regions 11 Polar coordinates 12 Integrals and quadrature formulas 13 More about boundary conditions 14 Fourth-order problems Afterword Bibliography Index

Principles Of Fluorescence Spectroscopy

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Solving Ordinary Differential Equations

Abstract: Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.