A discrete eigenfunctions method for computing mixed hyperbolic problems based on an implicit difference scheme

In this paper numerical solutions of mixed hyperbolic problems are computed using a discrete eigenfunctions method combined with an implicit difference scheme. This new numerical technique preserves the qualitative properties of the analytic solution due to the Sturm-Liouville structure of the underlying discrete linear boundary-value problem and has computational stability advantages vs other methods. Illustrative examples are included.     

Polynomial approximation of nonlinear differential systems with prefixed accuracy

This paper presents a method for constructing polynomial approximations of the solutions of nonlinear initial value systems of differential equations. Given an a priori chosen accuracy, the degree of the vector polynomial can be adapted so that the approximate solution has the required precision. The method is based on the AI-method of Dzyadyk developed for the scalar case, and the computational cost is shown to be competitive with other methods.     

Numerical solution of random differential initial value problems: Multistep methods

This paper deals with the construction of numerical methods of random initial value problems. Random linear multistep methods are presented and sufficient conditions for their mean square convergence are established. Main statistical properties of the approximations processes are computed in several illustrative examples. Copyright © 2010 John Wiley & Sons, Ltd.     

ORTHOGONAL MATRIX POLYNOMIALS WITH RESPECT TO A CONJUGATE BILINEAR MATRIX MOMENT FUNCTIONAL: BASIC THEORY

In this paper basic results for a theory of orthogonal matrix polynomials with respect to a conjugate bilinear matrix moment functional are proposed. Properties of orthogonal matrix polynomial sequences including a three term matrix relationship are given. Positive definite conjugate bilinear matrix moment functionals are introduced and a characterization of positive definiteness in terms of a block Haenkel moment matrix is established. For each positive definite conjugate bilinear matrix moment functional an associated matrix inner product is defined.     

Demonstration of image-zooming capability for diffractive axicons

Diffractive axicons are optical components producing achromatic nondiffracting beams. They thus produce a focal line rather than a focal point for classical lenses. This gives the interesting property of a long focal depth. We show that this property can be used to design a simple imaging system with a linear variable zoom by using and translating a diffractive axicon as the only optical component.     

A novel one-step efficient method for the synthesis of tetrahydroindoles from 1-(1-pyrrolidino)cyclohexene and chloropyruvates

A highly efficient, one-step, versatile method for the synthesis of tetrahydroindoles has been developed on the basis of new ring formation in the reactions of 1-(1-pyrrolidino)cyclohexene with chloropyruvates.     

An efficient one-step method for the synthesis of 2-(indolizin-2-yl)benzimidazoles from quinoxalinones and α-picoline via a novel rearrangement

A highly efficient one-step and versatile method for the synthesis of 2-(indolizin-2-yl)benzimidazoles has been developed on the basis of the novel ring contraction of 3-arylchloromethyl- and alkylchloromethylquinoxalin-2-ones with α-picoline.     

Piecewise finite series solutions of seasonal diseases models using multistage Adomian method

The aim of this paper is to apply the multistage Adomian Decomposition Method MADM to solve systems of nonautonomous nonlinear differential equations that describe several epidemic models with periodic behavior. Here the concept of the MADM is introduced and then it is employed to obtain a piecewise finite series solution. The MADM is used here as a hybrid analytical–numerical technique for approximating the solutions of the epidemic models. In order to show the efficiency of the method, the obtained numerical results are compared with the fourth-order Runge–Kutta method solutions. Numerical comparisons show that the MADM is accurate, easy to apply and the calculated solutions preserve the periodic behavior of the continuous models. Moreover, the method has the advantage of giving a functional form of the solution for any time interval. Furthermore, it is shown that if the truncation order and the time step size are not properly chosen large computational work is required and inaccurate solutions may be obtained.     

Solving Boundary Value Problems for the Matrix Equation $X^{(2)}(t)-AX(t)=F(t)$

In this paper we present a method for solving the matrix differential equation $X^{(2)}(t)-AX(t)=F(t)$, without increasing the dimension of the problem. By introducing the concept of co-square root of a matrix, existence and uniqueness conditions for solutions of boundary value problems related to the equation as well as explicit solutions of these solutions are given, even for the case where the matrix $A$ has no square roots.