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

2021 ◽  
Vol 4 (4) ◽  
pp. 1-36
Author(s):  
Matteo Fogato ◽  
Keyword(s):  
Theoretical Result ◽  
Finite Number ◽  
Positive Definite ◽  
Suspension Bridges ◽  
Forcing Term ◽  
Finite Dimensional ◽  
Definite Operator ◽  
Elastic Systems ◽  
The Stability ◽  
Positive Definite Operator

<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>

Inequalities between means of positive operators

1978 ◽  
Vol 83 (3) ◽  
pp. 393-401 ◽  
Author(s):  
K. V. Bhagwat ◽  
R. Subramanian
Keyword(s):  
Hilbert Space ◽  
Dimensional Space ◽  
Adjoint Operator ◽  
Positive Definite ◽  
Inner Product ◽  
Finite Dimensional Space ◽  
Finite Dimensional ◽  
Definite Operator ◽  
Unit Operator ◽  
Adjoint Operators

One of the most fruitful – and natural – ways of introducing a partial order in the set of bounded self-adjoint operators in a Hilbert space is through the concept of a positive operator. A bounded self-adjoint operator A denned on is called positive – and one writes A ≥ 0 - if the inner product (ψ, Aψ) ≥ 0 for every ψ ∈ . If, in addition, (ψ, Aψ) = 0 only if ψ = 0, then A is called positive-definite and one writes A > 0. Further, if there exists a real number γ > 0 such that A — γI ≥ 0, I being the unit operator, then A is called strictly positive (in symbols, A ≫ 0). In a finite dimensional space, a positive-definite operator is also strictly positive.

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

2010 ◽  
Vol 2010 ◽  
pp. 1-7
Author(s):  
S. J. Aneke
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Method Of Moments ◽  
Iterative Scheme ◽  
Inverse Function ◽  
Positive Definite ◽  
Inverse Function Theorem ◽  
Definite Operator ◽  
Fréchet Differentiable ◽  
Positive Definite Operator

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.

scholarly journals Iterative solutions ofK-positive definite operator equations in real uniformly smooth Banach spaces

2001 ◽  
Vol 27 (3) ◽  
pp. 155-160
Author(s):  
Zeqing Liu ◽  
Shin Min Kang ◽  
Jeong Sheok Ume
Keyword(s):  
Banach Space ◽  
Iterative Method ◽  
Banach Spaces ◽  
Positive Definite ◽  
Operator Equations ◽  
Uniformly Smooth ◽  
Definite Operator ◽  
Iterative Solutions ◽  
Uniformly Smooth Banach Spaces ◽  
Positive Definite Operator

LetXbe a real uniformly smooth Banach space and letT:D(T)⫅X→Xbe aK-positive definite operator. Under suitable conditions we establish that the iterative method by Bai (1999) converges strongly to the unique solution of the equationTx=f,f∈X. The results presented in this paper generalize the corresponding results of Bai (1999), Chidume and Aneke (1993), and Chidume and Osilike (1997).

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

2020 ◽  
Vol 23 (6) ◽  
pp. 1605-1646
Author(s):  
Stanislav Harizanov ◽  
Raytcho Lazarov ◽  
Svetozar Margenov
Keyword(s):  
Finite Element ◽  
Numerical Solution ◽  
Rational Approximation ◽  
Positive Definite ◽  
Second Order ◽  
Local Operator ◽  
Uniform Mesh ◽  
Element Approximation ◽  
Definite Operator ◽  
Positive Definite Operator

AbstractThe 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.

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