Structural design synthesis approach to filamentary composites

Structural design methods for analysis of multilayer or laminated filamentary composite

Minimum weight design of symmetrically stiffened orthotropic cylinders under axial compression

Minimum weight design of symmetrically stiffened orthotropic cylinders under axial compressio

Minimum weight design aspects of stiffened cylinders under compression

Survey on minimum weight design aspects of stiffened cylinders under compressio

The Natural Logarithm on Time Scales

We define an appropriate logarithm function on time scales and present its main properties. This gives answer to a question posed by M. Bohner in [J. Difference Equ. Appl. {\bf 11} (2005), no. 15, 1305--1306].Comment: 6 page

On approximate solutions of semilinear evolution equations

A general framework is presented to discuss the approximate solutions of an evolution equation in a Banach space, with a linear part generating a semigroup and a sufficiently smooth nonlinear part. A theorem is presented, allowing to infer from an approximate solution the existence of an exact solution. According to this theorem, the interval of existence of the exact solution and the distance of the latter from the approximate solution can be evaluated solving a one-dimensional "control" integral equation, where the unknown gives a bound on the previous distance as a function of time. For example, the control equation can be applied to the approximation methods based on the reduction of the evolution equation to finite-dimensional manifolds: among them, the Galerkin method is discussed in detail. To illustrate this framework, the nonlinear heat equation is considered. In this case the control equation is used to evaluate the error of the Galerkin approximation; depending on the initial datum, this approach either grants global existence of the solution or gives fairly accurate bounds on the blow up time.Comment: 33 pages, 10 figures. To appear in Rev. Math. Phys. (Shortened version; the proof of Prop. 3.4. has been simplified

Lyapunov functions and strict stability of Caputo fractional differential equations

One of the main properties studied in the qualitative theory of differential equations is the stability of solutions. The stability of fractional order systems is quite recent. There are several approaches in the literature to study stability, one of which is the Lyapunov approach. However, the Lyapunov approach to fractional differential equations causes many difficulties. In this paper a new definition (based on the Caputo fractional Dini derivative) for the derivative of Lyapunov functions to study a nonlinear Caputo fractional differential equation is introduced. Comparison results using this definition and scalar fractional differential equations are presented, and sufficient conditions for strict stability and uniform strict stability are given. Examples are presented to illustrate the theory

Euler-Lagrange equations for composition functionals in calculus of variations on time scales

In this paper we consider the problem of the calculus of variations for a functional which is the composition of a certain scalar function $H$ with the delta integral of a vector valued field $f$, i.e., of the form $H(\int_{a}^{b}f(t,x^{\sigma}(t),x^{\Delta}(t))\Delta t)$. Euler-Lagrange equations, natural boundary conditions for such problems as well as a necessary optimality condition for isoperimetric problems, on a general time scale, are given. A number of corollaries are obtained, and several examples illustrating the new results are discussed in detail.Comment: Submitted 10-May-2009 to Discrete and Continuous Dynamical Systems (DCDS-B); revised 10-March-2010; accepted 04-July-201

Monotone iterative procedure and systems of a finite number of nonlinear fractional differential equations

The aim of the paper is to present a nontrivial and natural extension of the comparison result and the monotone iterative procedure based on upper and lower solutions, which were recently established in (Wang et al. in Appl. Math. Lett. 25:1019-1024, 2012), to the case of any finite number of nonlinear fractional differential equations.The author is very grateful to the reviewers for the remarks, which improved the final version of the manuscript. This article was financially supported by University of Łódź as a part of donation for the research activities aimed at the development of young scientists, grant no. 545/1117