Asymptotic finite-dimensional approximations for a class of extensible elastic systems

<abstract><p>We consider the equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_{tt}+\delta u_t+A^2u+{\lVert{A^{\theta/2} u}\rVert}^2A^\theta u = g $\end{document} </tex-math></disp-formula></p> <p>where $ A^2 $ is a diagonal, self-adjoint and positive-definite operator and $ \theta \in [0, 1] $ and we study some finite-dimensional approximations of the problem. First, we analyze the dynamics in the case when the forcing term $ g $ is a combination of a finite number of modes. Next, we estimate the error we commit by neglecting the modes larger than a given $ N $. We then prove, for a particular class of forcing terms, a theoretical result allowing to study the distribution of the energy among the modes and, with this background, we refine the results. Some generalizations and applications to the study of the stability of suspension bridges are given.</p></abstract>

A survey on numerical methods for spectral Space-Fractional diffusion problems

The survey is devoted to numerical solution of the equation $ {\mathcal A}^\alpha u=f $, 0 < α<1, where $ {\mathcal A} $ is a symmetric positive definite operator corresponding to a second order elliptic boundary value problem in a bounded domain Ω in ℝd. The fractional power $ {\mathcal A}^\alpha $ is a non-local operator and is defined though the spectrum of $ {\mathcal A} $. Due to growing interest and demand in applications of sub-diffusion models to physics and engineering, in the last decade, several numerical approaches have been proposed, studied, and tested. We consider discretizations of the elliptic operator $ {\mathcal A} $ by using an N-dimensional finite element space Vh or finite differences over a uniform mesh with N points. In the case of finite element approximation we get a symmetric and positive definite operator $ {\mathcal A}_h: V_h \to V_h $, which results in an operator equation $ {\mathcal A}_h^{\alpha} u_h = f_h $ for uh ∈ Vh.The numerical solution of this equation is based on the following three equivalent representations of the solution: (1) Dunford-Taylor integral formula (or its equivalent Balakrishnan formula, (2.5), (2) extension of the a second order elliptic problem in Ω × (0, ∞)⊂ ℝd+1 [17,55] (with a local operator) or as a pseudo-parabolic equation in the cylinder (x, t) ∈ Ω × (0, 1), [70, 29], (3) spectral representation (2.6) and the best uniform rational approximation (BURA) of zα on [0, 1], [37,40].Though substantially different in origin and their analysis, these methods can be interpreted as some rational approximation of $ {\mathcal A}_h^{-\alpha} $. In this paper we present the main ideas of these methods and the corresponding algorithms, discuss their accuracy, computational complexity and compare their efficiency and robustness.

Implicit Approximation Scheme for the Solution ofK-Positive Definite Operator Equation

We construct an implicit sequence suitable for the approximation of solutions ofK-positive definite operator equations in real Banach spaces. Furthermore, implicit error estimate is obtained and the convergence is shown to be faster in comparsion to the explicit error estimate obtained by Osilike and Udomene (2001).

The Solution by Iteration of a Composed K-Positive Definite Operator Equation in a Banach Space

The equation , where , with being a K-positive definite operator and being a linear operator, is solved in a Banach space. Our scheme provides a generalization to the so-called method of moments studied in a Hilbert space by Petryshyn (1962), as well as Lax and Milgram (1954). Furthermore, an application of the inverse function theorem provides simultaneously a general solution to this equation in some neighborhood of a point , where is Fréchet differentiable and an iterative scheme which converges strongly to the unique solution of this equation.