Barbara Zubik-Kowal

Convergence of the method of lines for parabolic differential-functional equations

Barbara Zubik-Kowal — Sun, 05 Feb 2012 20:09:01 PST

Parabolic differential-functional equations with initial-boundary conditions of the Dirichlet type are studied. Spatial derivatives occurring in the original problems are replaced by suitable differences and the problem is transformed into an initial-boundary value problem for a system of ordinary differential-functional equations. The Perron type estimation for the right hand side of the original equation with respect to the functional argument is assumed.

Numerical solutions of thalamo-cortical systems

Barbara Zubik-Kowal et al. — Sun, 05 Feb 2012 19:54:46 PST

Thalamocortical problems, which are written as systems of Volterra integro-differential equations, are studied. Waveform relaxation, which can be efficiently implemented in parallel computing environments, is applied to the systems. A new error bound is derived for waveform relaxation applied to the thalamocortical problems. Exact values of the kernel of the Volterra integro-differential equations are used in the derivation. Fast convergence of waveform relaxation applied to the problems is illustrated by numerical experiments.

Numerical experiments with model equations of cancer invasion of tissue

Barbara Zubik-Kowal et al. — Sun, 05 Feb 2012 19:31:34 PST

In this paper we investigate a mathematical model of cancer invasion of tissue, which incorporates haptotaxis, chemotaxis, proliferation and degradation rates for cancer cells and the extracellular matrix, kinetics of urokinase receptor, and urokinase plasminogen activator cycle. We solve the model using spectrally accurate approximations and compare its numerical solutions with laboratory data. The spectral accuracy allows to use low-dimensional matrices and vectors, which speeds up the computations of the numerical solutions and thus to estimate the parameter values for the model equations. Our numerical results demonstrate correlations between numerical data computed from the mathematical model and in vivo tumour growth rates from prostate cell lines.

Numerical solution of Volterra integro-differential equations modeling thalamo-cortical systems

Barbara Zubik-Kowal et al. — Sun, 05 Feb 2012 19:09:38 PST

Our study concerns thalamo-cortical systems which are modelled by nonlinear systems of Volterra integro-differential equa- tions of convolution type. The thalamo-cortical systems describe a new architecture for a neurocomputer. Such a computer employs principles of human brain. It consists of oscillators which have different frequencies and are weakly connected via a common medium forced by an external input. Since a neurocomputer consists of many interconnected oscillators (referred also as neurons), the thalamo-cortical systems include large numbers of Volterra integro-differential equations. Solving such systems numerically is expensive not only because of their large dimensions but also because of many kernel evaluations which are needed over the whole interval from the initial point, where the initial condition is imposed, up to the present point, where the computations are currently executed. Moreover, the whole computed history of the solution has to be stored in the memory of the computing machine. Therefore, robust and efficient numerical algorithms are needed for computer simulations for the solutions to the thalamo- cortical systems. In this paper, we illustrate an iteration technique to solve the thalamo-cortical systems. The proposed successive iterates are vector functions of time, which change the original problems into systems of easier and separated equations. Such separated equations can then be solved in parallel computing environments. Results of numerical experiments are presented for large numbers of oscillators.

Numerical Experiments For Mammary Adenocarcinoma Cell Progression

Barbara Zubik-Kowal et al. — Sun, 05 Feb 2012 18:46:21 PST

An estimated 192,370 women in the United States were diagnosed with breast cancer during 2009. Of these, over 40,000 will succumb to the disease as a result of widespread metastasis to secondary organs such as bone, lungs, and liver. These statistics suggest the need for improved early detection techniques and treatment options. The most common types of breast cancer originate either in the milk ducts or in the milk producing glands of the breast. Breast cancer arising from ducts is called invasive ductal carcinoma (IDC), while cancer arising from glandular tissue is called invasive lobular carcinoma (ILC). IDC comprises 70-80% of all breast cancer, while ILC makes up only 8-14% of breast cancer cases. More rare forms of breast cancer include inflammatory breast cancer and Paget's disease of the breast.

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

Mikhail Kolev et al. — Thu, 01 Sep 2011 15:16:48 PDT

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.

Numerical Solution of a Model for Brain Cancer Progression After Therapy

Z. Jackiewicz et al. — Thu, 14 Jul 2011 09:59:30 PDT

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 Versus Experimental Data for Prostate Tumour Growth

Mikhail Kolev et al. — Fri, 13 May 2011 08:42:05 PDT

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.

Correlation Between Animal and Mathematical Models for Prostate Cancer Progression

Z. Jackiewiczy et al. — Wed, 09 Feb 2011 10:10:18 PST

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.

Pseudospectral Iterated Method for Differential Equations with Delay Terms

Jodi Mead et al. — Wed, 09 Feb 2011 10:10:16 PST

New efficient numerical methods for hyperbolic and parabolic partial differential equations with delay terms are investigated. These equations model a development of cancer cells in human bodies. Our goal is to study numerical methods which can be applied in a parallel computing environment. We apply our new numerical method to the delay partial differential equations and analyse the error of the method. Numerical experiments confirm our theoretical results.

Nonlinear Modeling with Mammographic Evidence of Carcinoma

K. Drucis et al. — Mon, 29 Nov 2010 08:29:34 PST

The goal of this paper is to apply oncological data for mathematical modeling of breast cancer progression. The studied model is composed of nonlinear partial integro-differential equations, which are formulated with unknown parameters. It is demonstrated that it is possible to find such parameter values for the nonlinear model so that its solutions correspond to the oncological data, therefore showing the potential and extending the applications of the model to breast cancer. The nonlinear model equations are solved numerically and the numerical results confirm the oncological data.

Fourier stability analysis of a numerical method for time domain electromagnetic scattering from a thin wire

Barbara Zubik-Kowal et al. — Fri, 11 Jun 2010 08:55:03 PDT

Spectral versus pseudospectral solutions of the wave equation by waveform relaxation methods

B. Zubik-Kowal et al. — Fri, 11 Jun 2010 08:53:18 PDT

Error bounds for spatial discretization and waveform relaxation applied to parabolic functional-differential equations

Barbara Zubik-Kowal — Fri, 11 Jun 2010 08:50:55 PDT

On the stability of Radau IIA collocation methods for delay differential equations

B. Zubik-Kowal et al. — Fri, 11 Jun 2010 08:45:08 PDT

The stability of numerical approximations of the time domain current induced on a thin wire and strip antennas

Barbara Zubik-Kowal et al. — Fri, 11 Jun 2010 08:16:37 PDT

Spectral collocation and waveform relaxation methods with Gengenbauer reconstruction for nonlinear conservation laws

Barbara Zubik-Kowal et al. — Fri, 11 Jun 2010 08:14:27 PDT

Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations

B. Zubik-Kowal et al. — Fri, 11 Jun 2010 08:11:11 PDT

Numerical solutions of thalamo-cortical systems

B. Zubik-Kowal et al. — Fri, 11 Jun 2010 08:09:17 PDT

A variant of pseudospectral method for activity-dependent dendritic branch model

Barbara Zubik-Kowal et al. — Fri, 11 Jun 2010 07:42:21 PDT