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Dear Colleagues,

The advances in technology and material science have required constitutive modelling of modern materials and the formulation of computational tools necessary for their analyses. For example, many new designs, such as microelectromechanical and nanoelectromechanical systems (MEMS and NEMS), smart materials and multi-functional materials, are inherently multiphysic and require rigorous constitutive modelling. Successful experimental demonstration of negative electrical permittivity, magnetic permeability, effective elastic modulus, and mass density in metamaterials and extreme solids are other examples that emphasize the importance of classical applied mechanics fields such as continuum mechanics in recent years. Of particular importance have been multiscale and homogenization approaches, given the role of specific microstructural designs on the response of modern materials. There has also been a greater emphasize in nondeterministic approaches, given the higher sensitivity of the aforementioned materials to design deviations and the importance on the stochastic distribution on small scale features in overall response for example in fracture mechanics and turbulence. Such advances have, in turn, necessitated the formulation of computational methods capable of efficient and accurate rendering of these material models. Theoretical and computational tools, including but not limited to multiscale and high-order methods, rigorous analysis of numerical errors and efficiency, homogenization schemes, and efficient approaches for the solution of discrete lattices, periodic media, ordinary, partial and stochastic partial differential equations are a few of the relevant topics.

Prof. Reza Abedi
Dr. Raj Kumar Pal
Guest Editors

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Keywords

Solid mechanics
Fluid mechanics
Thermodynamics
Fracture mechanics
Continuum mechanics
Constitutive models for modern materials
Multiphysics problems
Homogenization
Multiscale methods
Stochastic partial differential equations
Computational mechanics including error and efficiency analysis
Finite element methods
Metamaterials
Wave propagation
Granular media
Instabilities in solids
Acoustics and ultrasonics
Smart materials

Published Papers (1 paper)

Research

18 pages, 1049 KiB

Open AccessArticle

An Estimate of the Root Mean Square Error Incurred When Approximating an f ∈ L²(ℝ) by a Partial Sum of Its Hermite Series

by Mei Ling Huang, Ron Kerman and Susanna Spektor

Mathematics 2018, 6(4), 64; https://doi.org/10.3390/math6040064 - 23 Apr 2018

Viewed by 2603

Abstract

Let f be a band-limited function in

L^{2} (R)

. Fix

T > 0

, and suppose

f^{'}

exists and is integrable on

[- T, T]

. This paper gives a concrete estimate of the error [...] Read more.

Let f be a band-limited function in

L^{2} (R)

. Fix

T > 0

, and suppose

f^{'}

exists and is integrable on

[- T, T]

. This paper gives a concrete estimate of the error incurred when approximating f in the root mean square by a partial sum of its Hermite series. Specifically, we show, that for

K = 2 n, n \in Z_{+},

{[\frac{1}{2 T}, \int_{- T}^{T}, {[f (t) - (S_{K} f) (t)]}^{2}, d, t]}^{1 / 2} \leq (1 + \frac{1}{K}) ({[\frac{1}{2 T}, \int_{| t | > T}, f, {(t)}^{2}, d, t]}^{1 / 2} + {[\frac{1}{2 T}, \int_{| ω | > N}, {| \hat{f} (ω) |}^{2}, d, ω]}^{1 / 2}) + \frac{1}{K} {[\frac{1}{2 T}, \int_{| t | \leq T}, f_{N}, {(t)}^{2}, d, t]}^{1 / 2} + \frac{1}{π} (1 + \frac{1}{2 K}) S_{a} (K, T),

in which

S_{K} f

is the K-th partial sum of the Hermite series of

f, \hat{f}

is the Fourier transform of f,

N = \frac{\sqrt{2 K + 1} + \sqrt{2 K + 3}}{2}

and

f_{N} = {(\hat{f} χ_{(- N, N)})}^{\lor} (t) = \frac{1}{π} \int_{- \infty}^{\infty} \frac{sin (N (t - s))}{t - s} f (s) d s

. An explicit upper bound is obtained for

S_{a} (K, T)

. Full article

(This article belongs to the Special Issue Applied Mathematics and Mechanics 2017)

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