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Mathematics
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Applied Mathematics and Mechanics 2017
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Applied Mathematics and Mechanics 2017

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: closed (31 December 2017) | Viewed by 3417

Special Issue Editors

Dr. Reza Abedi

E-Mail Website
Guest Editor
Department of Mechanical Aerospace and Biomedical Engineering, University of Tennessee Space Institute, Tullahoma, TN 37388, USA
Interests: spacetime discontinuous Galerkin; computational mechanics; fracture mechanics; computational electromagnetics
Special Issues, Collections and Topics in MDPI journals
Dr. Raj Kumar Pal

E-Mail Website
Guest Editor
Daniel Guggenhiem School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA
Interests: wave propagation; lattices; metamaterials; granular media; computational mechanics; solid mechanics

Special Issue Information

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

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

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

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Published Papers (1 paper)

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Research

18 pages, 1049 KiB  
Article
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 2867
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 ∈ Z + , 1 2 T ∫ − T T [ f ( t ) − ( S K f ) ( t ) ] 2 d t 1 / 2 ≤ 1 + 1 K 1 2 T ∫ | t | > T f ( t ) 2 d t 1 / 2 + 1 2 T ∫ | ω | > N | f ^ ( ω ) | 2 d ω 1 / 2 + 1 K 1 2 T ∫ | t | ≤ T f N ( t ) 2 d t 1 / 2 + 1 π 1 + 1 2 K S a ( K , T ) , in which S K f is the K-th partial sum of the Hermite series of f , f ^ is the Fourier transform of f, N = 2 K + 1 + 2 K + 3 2 and f N = ( f ^ χ ( − N , N ) ) ∨ ( t ) = 1 π ∫ − ∞ ∞ 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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