Malgorzata A. Jankowska

A survey of interval Runge-Kutta and multistep methods for solving the initial value problem

The paper is dealt with a number of one- and multistep interval methods developed by our team during the last decade. We present implicit interval methods of Runge-Kutta type, interval versions of symplectic Runge-Kutta methods and interval multistep methods of Adams-Bashforth, Adams-Moulton, Nystrm and Milne-Simpson types.

Experimental and constitutive modeling approaches for a study of biomechanical properties of human coronary arteries

The study concerns the determination of mechanical properties of human coronary arterial walls with both experimental and constitutive modeling approaches. The research material was harvested from 18 patients (range 50-84 years). On the basis of hospital records and visual observation, each tissue sample was classified according to the stage (0, I, II, III) of atherosclerosis development (SAD). Then, strip samples considered as a membrane with the shape of rectangular parallelepiped were preconditioned and subjected to uniaxial tensile tests in longitudinal (n=27) and circumferential (n=4) direction. With experimental data obtained, the stress-strain characteristics were prepared. Furthermore, tensile strengths and related strains, stiffness coefficients and tangent modules of elasticity were computed. For a constitutive model of passive mechanical behavior of coronary arteries, values of material parameters were computed. The studies led to the following conclusions. Most importantly, the atherosclerotic changes affect all the mechanical properties of arterial walls. A progress of arteriosclerosis contributes to an increase of vascular stiffness. The highest values of the stiffness coefficients are obtained for the tissues in the advanced stage of the disease. We were also able to observe that gradual calcification, progression of atherosclerosis and degradation of collagen in the tissue caused a decrease of tensile strengths and related strains. Finally, a comparison made for the tissues with the advanced SAD showed that the tensile strengths and strains were much higher in the case of the samples with the circumferential orientation rather than those with the longitudinal one.

Remarks on algorithms implemented in some C++ libraries for floating-point conversions and interval arithmetic

The main aim of the paper is to give a short presentation of selected conversion functions developed by the author. They are included in two C++ libraries. The FloatingPointConversion library is dedicated for conversions in the area of floating-point numbers and the second one, the IntervalArithmetic library, carries out the similar task for interval values as well as supports computations in the floating-point interval arithmetic with a suitable CInterval class. The functions considered are all intended to be used with the Intel Architectures (i.e. the IA-32 and the IA-64) and dedicated for C++ compilers that specify 10 bytes for the long double data type.

An interval finite difference method of crank-nicolson type for solving the one-dimensional heat conduction equation with mixed boundary conditions

In the paper an interval method for solving the one-dimensio-nal heat conduction equation with mixed boundary conditions is considered. The idea of the interval method is based on the finite difference scheme of the conventional Crank-Nicolson method adapted to the mixed boundary conditions. The interval method given in the form presented in the paper includes the error term of the conventional method.

Interval finite difference method for solving the one-dimensional heat conduction problem with heat sources

The one-dimensional heat conduction equation with the term concerning some heat sources, together with the mixed boundary conditions is considered. Such problems occur in the area of the bioheat transfer and their well-known example is given by the Pennes equation. The paper deals with some interval finite difference method based on the Crank-Nicolson finite difference scheme. In the approach presented, the local truncation error of the conventional method is bounded by some interval values. A method of approximation of such error term intervals is also presented.

An interval backward finite difference method for solving the diffusion equation with the position dependent diffusion coefficient

The paper deals with the interval backward finite difference method for solving the one-dimensional diffusion equation with the position dependent diffusion coefficient and the boundary conditions of the first type. The interval method considered is based on the conventional backward finite difference method. Moreover, it takes into account a formula of a local truncation error of the method. Such local truncation error of the conventional method is bounded by the appropriate interval values. In most scientific applications we cannot find the endpoints of such intervals exactly and it is of great importance to approximate them in the most accurate way. The paper presents a method of such approximation.

Kansa-RBF algorithms for elliptic problems in regular polygonal domains

We propose matrix decomposition algorithms for the efficient solution of the linear systems arising from Kansa radial basis function discretizations of elliptic boundary value problems in regular polygonal domains. These algorithms exploit the symmetry of the domains of the problems under consideration which lead to coefficient matrices possessing block circulant structures. In particular, we consider the Poisson equation, the inhomogeneous biharmonic equation, and the inhomogeneous Cauchy-Navier equations of elasticity. Numerical examples demonstrating the applicability of the proposed algorithms are presented.

On interval predictor-corrector methods

One can approximate numerically the solution of the initial value problem using single or multistep methods. Linear multistep methods are used very often, especially combinations of explicit and implicit methods. In floating-point arithmetic from an explicit method (a predictor), we can get the first approximation to the solution obtained from an implicit method (a corrector). We can do the same with interval multistep methods. Realizing such interval methods in floating-point interval arithmetic, we compute solutions in the form of intervals which contain all possible errors. In this paper, we propose interval predictor-corrector methods based on conventional Adams-Bashforth-Moulton and Nyström-Milne-Simpson methods. In numerical examples, these methods are compared with interval methods of Runge-Kutta type and methods based on high-order Taylor series. It appears that the presented methods yield comparable approximations to the solutions.

Interval versions of Milne’s multistep methods

The paper presents explicit interval multistep methods of Milne type, which may be considered as alternative methods to other known explicit interval multistep methods (of Adams-Bashforth and Nyström). It is proved that enclosures of solutions (in the form of intervals) obtained by these methods contain the exact solutions of the initial value problem. Numerical examples show that the widths of intervals obtained by proposed methods are smaller than those obtained by explicit interval multistep methods known so far.

Interval Nine-Point Finite Difference Method for Solving the Laplace Equation with the Dirichlet Boundary Conditions

An interval version of the conventional nine-point finite difference method for solving the two-dimensional Laplace equation with the Dirichlet boundary conditions is proposed. This interval scheme is interesting due to the fact that the local truncation error of the conventional method is of the high (fourth) order, but it becomes of the sixth order for square mesh. In the theoretical approach presented, this error is bounded by some interval values and we can prove that the exact solution belongs to the interval solutions obtained.