arXiv subjects

Katherine Baker

Publications and source records attributed to Katherine Baker.

4 recordsLinked to original sources

arXiv2022

Numerical analysis of a time-stepping method for the Westervelt equation with time-fractional damping

We develop a numerical method for the Westervelt equation, an important equation in nonlinear acoustics, in the form where the attenuation is represented by a class of non-local in time operators. A semi-discretisation in time based on the trapezoidal rule and A-stable convolution quadrature is stated and analysed. Existence and regularity analysis of the continuous equations informs the stability and error analysis of the semi-discrete system. The error analysis includes the consideration of the singularity at $t = 0$ which is addressed by the use of a correction in the numerical scheme. Extensive numerical experiments confirm the theory.

math.NA↗

arXiv2022

Principal subspaces of basic modules for twisted affine Lie algebras, $q$-series multisums, and Nandi's identities

We provide an observation relating several known and conjectured $q$-series identities to the theory of principal subspaces of basic modules for twisted affine Lie algebras. We also state and prove two new families of $q$-series identities. The first family provides quadruple sum representations for Nandi's identities, including a manifestly positive representation for the first identity. The second is a family of new mod 10 identities connected with principal characters of level 4 integrable, highest-weight modules of $\mathrm{D}_4^{(3)}$.

math.CO↗

arXiv2021

Numerical analysis of a wave equation for lossy media obeying a frequency power law

We study a wave equation with a nonlocal time fractional damping term that models the effects of acoustic attenuation characterized by a frequency dependence power law. First we prove existence of a unique solution to this equation with particular attention paid to the handling of the fractional derivative. Then we derive an explicit time stepping scheme based on the finite element method in space and a combination of convolution quadrature and second order central differences in time. We conduct a full error analysis of the mixed time discretization and in turn the fully space time discretized scheme. Error estimates are given for both smooth solutions and solutions with a singularity at $t = 0$ of a type that is typical for equations involving fractional time-derivatives. A number of numerical results are presented to support the error analysis.

math.NA↗

arXiv2015

Investigating the Selectivity of KcsA Channel by an Image Charge Solvation Method (ICSM) in Molecular Dynamics Simulations

In this paper, we study the selectivity of the potassium channel KcsA by a recently developed image-charge solvation method(ICSM) combined with molecular dynamics simulations. The hybrid solvation model in the ICSM is able to demonstrate atomistically the function of the selectivity filter of the KcsA channel when potassium and sodium ions are considered and their distributions inside the filter are simulated. Our study also shows that the reaction field effect, explicitly accounted for through image charge approximation in the ICSM model, is necessary in reproducing the correct selectivity property of the potassium channels.

q-bio.BM↗