8 citations · 17 across the 12 of their papers we have counts for
16 papers
Integration of an arbitrary linear ODE
Peter C. Gibson
The standard text book theory of ODEs lacks a general method to solve linear equations having variable coefficients, providing instead a collection of special techniques for partic…
Solution of the scalar Riccati equation
Peter C. Gibson
The scalar Riccati equation is a prototypical nonlinear ODE having diverse mathematical connections. In the centuries since its initial formulation, a standard textbook theory has…
Explicit solution of the 1D Schrödinger equation
Peter Gibson
Evaluation of a product integral with values in the Lie group SU(1,1) yields the explicit solution to the impedance form of the Schrödinger equation. Explicit formulas for the tran…
Scattering on the line via singular approximation
Peter C. Gibson
Motivated by applications to acoustic imaging, the present work establishes a framework to analyze scattering for the one-dimensional wave, Helmholtz, Schrödinger and Riccati equat…
Inverse scattering for the one-dimensional Helmholtz equation with piecewise constant wave speed
Sophia Bugarija, Peter C. Gibson, Guanghui Hu +2
This paper analyzes inverse scattering for the one-dimensional Helmholtz equation in the case where the wave speed is piecewise constant. Scattering data recorded for an arbitraril…
The refined impedance transform for 1D acoustic reflection data
Peter C. Gibson
The one dimensional wave equation serves as a basic model for imaging modalities such as seismic which utilize acoustic data reflected back from a layered medium. In 1955 Peterson…