James P. Fink

On the Discretization Error of Parametrized Nonlinear Equations

Many applications lead to nonlinear, parameter dependent equations

Asymptotic Estimates of Feynman Integrals

H(y,t) = y_0

A geometric framework for the numerical study of singular points

, where

Solution manifolds and submanifolds of parametrized equations and their discretization errors

H:Y \times T \to Y

A convergent two-time method for periodic differential equations

,

Perturbation Expansions for some Nonlinear Wave Equations

y_0 \in {\operatorname{rge}}H

Methods of Local and Global Differential Geometry in General Relativity

, and the state space Y is infinite-dimensional while the parameter space T has finite dimension. The case

One last visit to the capillarity correction for free surface flow

\dim T = 1

Entrainment of frequency in evolution equations

is of special interest in connection with continuation methods. For this case, a general theory is developed which provides for the existence of solution paths of a rather general class of such equations and of their finite-dimensional approximations, and which allows for an assessment of the error between these paths. A principal tool in this analysis is the theory of nonlinear Fredholm operators. The results cover a more general class of operators than the mildly nonlinear mappings to which other approaches appear to be restricted.

Folds on the solution manifold of a parametrized equation

In this paper, we consider the problem of determining logarithmic, as well as polynomial, asymptotic estimates for certain convergent integrals containing parameters. We state and prove an asymptotic theorem which gives the logarithmic asymptotic behavior of a convergent integral where any subset of the parameters becomes large while the remaining parameters remain bounded. This theorem is then applied to the photon and electron self‐energy graphs of quantum electrodynamics.