Barbara Szyszka

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.

The central difference interval method for solving the wave equation

A way of constructing the interval method of second order for solving one dimensional wave equation is presented in the paper. The central difference interval method for the hyperbolic Partial Differential Equation is taken into consideration. The suitable Dirichlet and Cauchy conditions are satisfied for the string with fixed endpoints. The estimations of discretization errors are proposed. The method of floating-point interval arithmetic is studied. The numerical experiment is presented.

A central-backward difference interval method for solving the wave equation

The paper is devoted to an interval difference method for solving one dimensional wave equation with the initial-boundary value problem. The method is an adaptation of the well-known central and backward difference methods with respect to discretization errors of the methods. The approximation of an initial condition is derived on the basis of expansion of a third-degree Taylor polynomial. The initial condition is also written in the interval form with respect to a discretization error. Therefore, the presented interval method includes all approximation errors (of the wave equation and the initial condition). The floating-point interval arithmetic is used. It allows to obtain interval solutions which contain all calculations errors. Moreover, it is indicated that an exact solution belongs to the interval solution obtained.

Central Difference Interval Method for Solving the Wave Equation

This paper presents path of construction the interval method of second order for solving the wave equation. Taken into consideration is the central difference interval method for one‐dimensional partial differential equation. Numerical results, obtained by two presented algorithms, in floating‐point interval arithmetic are considered.

Interval versions of central-difference method for solving the poisson equation in proper and directed interval arithmetic

Abstract To study the Poisson equation, the central-difference method is often used. This method has the local truncation error of order O(h2 +k2), where h and k are mesh constants. Using this method in conventional floating-point arithmetic, we get solutions including the method, representation and rounding errors. Therefore, we propose interval versions of the central-difference method in proper and directed interval arithmetic. Applying such methods in floating-point interval arithmetic allows one to obtain solutions including all possible numerical errors. We present numerical examples from which it follows that the presented interval method in directed interval arithmetic is a little bit better than the one in proper interval arithmetic, i.e. the intervals of solutions are smaller. It appears that applying both proper and directed interval arithmetic the exact solutions belong to the interval solutions obtained.

Mathematical Modeling of Secondary Timber Processing

The paper presents the way of optimizing wood yield without defects for semi‐products. The range of research presented in this work is mainly based on the application of numerical methods and optimization, but it is an issue of widely apprehended industrial mathematics. There were introduced the mathematical conditions, which described relationships between wood and their efficiency. The effect of the research is a computer program called TARPAK2, by means of which one can make simulations whether and out of what material the order will be executed, so as to minimize waste material and multiply the sawmill profits. The program is such the first product in Poland to assist research of secondary timber processing.

Mathematical Modeling of Primary Wood Processing

This work presents a way of optimizing wood logs’ conversion into semi‐products. Calculating algorithms have been used in order to choose the cutting patterns and the number of logs needed to realize an order, including task specification. What makes it possible for the author’s computer program TARPAK1 to be written is the visualization of the results, the generation pattern of wood logs’ conversion for given entry parameters and prediction of sawn timber manufacture. This program has been created with the intention of being introduced to small and medium sawmills in Poland. The Project has been financed from government resources and written by workers of the Institute of Mathematics (Poznan University of Technology) and the Department of Mechanical Wood Technology (Poznan University of Life Sciences).