scholarly journals Finite eigenfuction approximations for continuous spectrum operators

1993 ◽  
Vol 16 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Robert M. Kauffman
Keyword(s):  
Differential Equations ◽  
Continuous Spectrum ◽  
Discrete Spectrum ◽  
Operator Norm ◽  
Abstract Theory ◽  
Eigenfunction Expansions ◽  
Multiplicity One ◽  
Finite Sums ◽  
New Formulation ◽  
Adjoint Operators

In this paper, we introduce a new formulation of the theory of continuous spectrum eigenfunction expansions for self-adjoint operators and analyze the question of when operators may be approximated in an operator norm by finite sums of multiples of eigenprojections of multiplicity one. The theory is designed for application to ordinary and partial differential equations; relationships between the abstract theory and differential equations are worked out in the paper. One motivation for the study is the question of whether these expansions are susceptible to computation on a computer, as is known to be the case for many examples in the discrete spectrum case. The point of the paper is that continuous and discrete spectrum eigenfunction expansions are treated by the same formalism; both are limits in an operator norm of finite sums.

scholarly journals Frederick Gerard Friedlander. 25 December 1917 — 20 May 2001

2017 ◽  
Vol 63 ◽  
pp. 273-307
Author(s):  
D. E. Edmunds ◽  
L. E. Fraenkel ◽  
M. Pemberton
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
General Theory ◽  
Differential Equations ◽  
Trinity College ◽  
Shielding Effect ◽  
Abstract Theory ◽  
Curved Space ◽  
Applied Mathematician ◽  
Civil Defence

Gerard Friedlander was the son of Austrian communist intellectuals, who divorced when he was four. From the age of two he was raised by grandparents in Vienna, while his mother lived in Berlin as a communist organizer. Hitler came to power in 1933; Friedlander was sent to England, aged 16, in 1934; two years later, he won a scholarship to Trinity College, Cambridge. By 1940 he was a fully fledged applied mathematician who came to embrace both the European and British traditions of that subject. His work was marked by profound originality, by the importance of its applications and by the mathematical rigour of his treatment. The applications of his work changed over the years. The first papers (written between 1939 and 1941, but published only in 1946 for security reasons) were a contribution to Civil Defence: they presented entirely new and explicit results on the shielding effect of a wall from a distant bomb blast. The late papers were contributions to the general, more abstract theory of partial differential equations, but, characteristically, with concrete examples that illuminated obscure aspects of the general theory. Between these two, the middle years brought a flowering of results about the wave equation (including results for a curved space-time), of importance to both physicists and mathematicians.

Applying Incompatible Meshes for Modeling Structural-Acoustic Domains in Energy Finite Element Analysis

2014 ◽  
Author(s):  
S. N. Medyanik ◽  
N. Vlahopoulos
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Differential Equations ◽  
Computationally Efficient ◽  
Exact Matching ◽  
Element Analysis ◽  
Energy Finite Element Analysis ◽  
Accuracy Of Results ◽  
New Formulation ◽  
Acoustic Interface

The Energy Finite Element Analysis (EFEA) has been developed for modeling coupled structural-acoustic systems at mid-to-high frequencies when conventional finite element methods are no longer computationally efficient because they require very fine meshes. In standard Finite Element Analysis (FEA) approach, governing differential equations are formulated in terms of displacements which vary harmonically with space. This requires larger numbers of elements at higher frequencies when wavelengths become smaller. In the EFEA, governing differential equations are formulated in terms of energy density that is spatially averaged over a wavelength and time averaged over a period. The resulting solutions vary exponentially with space which makes them smooth and allows for using much coarser meshes. However, current EFEA formulations require exact matching between the meshes at the boundaries between structural and acoustic domains. This creates practical inconveniences in applying the method as well as limits its use to only fully compatible meshes. In this paper, a new formulation is presented that allows for using incompatible meshes in EFEA modeling, when shapes and/or sizes of elements at structural-acoustic interfaces do not match. In the main EFEA procedure, joints formulations between structural and acoustic domains have been changed in order to deal with non-matching elements. In addition, the new Pre-EFEA procedure which allows for automatic searching and formation of the new types of joints is developed for models with incompatible meshes. The new method is tested using a spherical shaped structural-acoustic interface. Results for incompatible meshes are validated by comparing to solutions obtained using regular compatible meshes. The effects of mesh incompatibility on the accuracy of results are discussed.

scholarly journals Scaling laws and warning signs for bifurcations of SPDEs

2018 ◽  
Vol 30 (5) ◽  
pp. 853-868
Author(s):  
CHRISTIAN KUEHN ◽  
FRANCESCO ROMANO
Keyword(s):  
Differential Equations ◽  
Scaling Laws ◽  
Scaling Law ◽  
Warning Signs ◽  
Covariance Operator ◽  
Large Classes ◽  
Practical Applications ◽  
Critical Transitions ◽  
Degenerate Eigenvalues ◽  
Adjoint Operators

Critical transitions (or tipping points) are drastic sudden changes observed in many dynamical systems. Large classes of critical transitions are associated with systems, which drift slowly towards a bifurcation point. In the context of stochastic ordinary differential equations, there are results on growth of variance and autocorrelation before a transition, which can be used as possible warning signs in applications. A similar theory has recently been developed in the simplest setting for stochastic partial differential equations (SPDEs) for self-adjoint operators in the drift term. This setting leads to real discrete spectrum and growth of the covariance operator via a certain scaling law. In this paper, we develop this theory substantially further. We cover the cases of complex eigenvalues, degenerate eigenvalues as well as continuous spectrum. This provides a fairly comprehensive theory for most practical applications of warning signs for SPDE bifurcations.

Numerical Modeling of Optical Fibers Using the Finite Element Method and an Exact Non-reflecting Boundary Condition

2018 ◽  
Vol 18 (4) ◽  
pp. 581-601
Author(s):  
Rafail Z. Dautov ◽  
Evgenii M. Karchevskii
Keyword(s):  
Finite Element Method ◽  
Boundary Condition ◽  
Finite Element ◽  
Numerical Modeling ◽  
Optical Fibers ◽  
Original Problem ◽  
Approximate Solutions ◽  
The Finite Element Method ◽  
New Formulation ◽  
Adjoint Operators

AbstractThe original problem for eigenwaves of weakly guiding optical fibers formulated on the plane is reduced to a convenient for numerical solution linear parametric eigenvalue problem posed in a disk. The study of the solvability of this problem is based on the spectral theory of compact self-adjoint operators. Properties of dispersion curves are investigated for the new formulation of the problem. An efficient numerical method based on FEM approximations is developed. Error estimates for approximate solutions are derived. The rate of convergence for the presented algorithm is investigated numerically.

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