Barbara Zubik-Kowal

Waveform Relaxation for Functional-Differential Equations

The convergence of waveform relaxation techniques for solving functional-differential equations is studied. New error estimates are derived that hold under linear and nonlinear conditions for the right-hand side of the equation. Sharp error bounds are obtained under generalized time-dependent Lipschitz conditions. The convergence of the waveform method and the quality of the a priori error bounds are illustrated by means of extensive numerical data obtained by applying the method of lines to three partial functional-differential equations.

Stability in the Numerical Solution of Linear Parabolic Equations with a Delay Term

This paper is concerned with the stability of numerical processes that arise after semi-discretization of linear parabolic equations wit a delay term. These numerical processes are obtained by applying step-by-step methods to the resulting systems of ordinary delay differential equations. Under the assumption that the semi-discretization matrix is normal we establish upper bounds for the growth of errors in the numerical processes under consideration, and thus arrive at conclusions about their stability. More detailed upper bounds are obtained for θ-methods under the additional assumption that the eigenvalues of the semi-discretization matrix are real and negative. In particular, we derive contractivity properties in this case. Contractivity properties are also obtained for the θ-methods applied to the one-dimensional test equation with real coefficients and a delay term. Numerical experiments confirming the derived contractivity properties for parabolic equations with a delay term are presented.

Correlation Between Animal and Mathematical Models for Prostate Cancer Progression

This work demonstrates that prostate tumour progression in vivo can be analysed by using solutions of a mathematical model supplemented by initial conditions chosen according to growth rates of cell lines in vitro. The mathematical model is investigated and solved numerically. Its numerical solutions are compared with experimental data from animal models. The numerical results confirm the experimental results with the growth rates in vivo.

Numerical Solutions for a Model of Tissue Invasion and Migration of Tumour Cells

The goal of this paper is to construct a new algorithm for the numerical simulations of the evolution of tumour invasion and metastasis. By means of mathematical model equations and their numerical solutions we investigate how cancer cells can produce and secrete matrix degradative enzymes, degrade extracellular matrix, and invade due to diffusion and haptotactic migration. For the numerical simulations of the interactions between the tumour cells and the surrounding tissue, we apply numerical approximations, which are spectrally accurate and based on small amounts of grid-points. Our numerical experiments illustrate the metastatic ability of tumour cells.

The time domain integral equation for a straight thin-wire antenna with the reduced kernel is not well-posed

We show that the Pocklington integral equation for time-domain scattering from thin-wire antennas is not mathematically well-posed. This has considerable implications for numerical solution schemes. In particular, our argument explains the observed occurrence of rapidly oscillating errors in numerical solutions as the numerical grid sizes are reduced.

Numerical Solution of a Model for Brain Cancer Progression After Therapy

Abstract We present a numerical scheme used to investigate a mathematical model of tumor growth which incorporates multiple disparate timescales. We simulate the model with different initial data. The initial conditions explored herein correspond to a small remnant of tumor tissue left after surgical resection. Our results indicate that tumor regrowth begins at the pre‐surgery tumor‐healthy tissue interface and penetrates back into the original tumor area. This growth is rate‐limited by the reformation of the tumor vascular network.

Numerical Algorithm for the Growth of Human Tumor Cells and Their Responses to Therapy

Abstract We investigate a system of delay partial differential equations that models the growth of human tumor cells and their responses to therapy. The model includes unknown parameters that need to be estimated according to experimental data. We introduce a numerical algorithm, which shortens the computational time for solving the model equations and estimating their parameters. Numerical results demonstrate the efficiency of our algorithm and show correspondence between predicted and experimental data.

Spectral Versus Pseudospectral Solutions of the Wave Equation by Waveform Relaxation Methods

We examine spectral and pseudospectral methods as well as waveform relaxation methods for the wave equation in one space dimension. Our goal is to study block Gauss–Jacobi waveform relaxation schemes which can be efficiently implemented in a parallel computing environment. These schemes are applied to semidiscrete systems written in terms of sparse or dense matrices. It is demonstrated that the spectral formulations lead to the implicit system of ordinary differential equations Wã′ = Sã + g(t) w, with sparse matrices W and S which can be effectively solved by direct application of any Runge–Kutta method. We also examine waveform relaxation iterations based on splittings W = W1−W2 and S = S1 + S2 and demonstrate that these iterations are only linearly convergent on finite time windows. Waveform relaxation methods applied to the explicit system ã′ = W−1Sã + g(t) W−1w are somewhat faster but less convenient to implement since the matrix W−1S is no longer sparse. The pseudospectral methods lead to the system Ũ′ = DŨ + g(t) w with a differentiation matrix D of order one and the corresponding waveform relaxation iterations are much faster than the iterations corresponding to the spectral cases (both implicit and explicit).

Numerical approximation of time domain electromagnetic scattering from a thin wire

We present two new stable schemes for computing the current induced on the surface of a thin wire by an incident time-dependent electromagnetic field. The problem involves solving a retarded potential integral equation (RPIE). One algorithm solves the previously studied reduced kernel RPIE problem, and the other solves the more complicated exact kernel RPIE problem (for which there are no previous numerical results). Both algorithms behave stably for arbitrarily chosen values of the mesh size. Test experiments and numerically computed values of the induced current are presented.

NUMERICAL VERSUS EXPERIMENTAL DATA FOR PROSTATE TUMOUR GROWTH

The goal of this paper is to solve mathematical model equations on solid tumour growth and compute their parameter values by applying growth rates of prostate cancer cell lines in vivo. For these computations, we investigate previously developed C3(1)/Tag transgenic models of prostate cancer. To make the computations fast, we have constructed an algorithm, which is based on small amounts of spatial grid-points and obtained a correspondence between the in vivo growth of tumours and the solutions of the model equations.