scholarly journals Upper bounds of nodal sets for eigenfunctions of eigenvalue problems

2020 ◽  
Author(s):  
Fanghua Lin ◽  
Jiuyi Zhu
Keyword(s):  
Eigenvalue Problems ◽  
Upper Bounds ◽  
Nodal Sets

Convergence of the Ritz Method

1995 ◽  
Vol 48 (11S) ◽  
pp. S90-S95 ◽  
Author(s):  
Arthur W. Leissa ◽  
Samir M. Shihada
Keyword(s):  
Eigenvalue Problems ◽  
Ritz Method ◽  
Equation Of Motion ◽  
Upper Bounds ◽  
Free Vibrations ◽  
Bending Moments ◽  
Shear Forces ◽  
Buckling Loads ◽  
Derivatives Of ◽  
Admissible Functions

The Ritz method is widely used for the solution of problems in structural mechanics, especially eigenvalue problems where the free vibration frequencies or buckling loads are sought. It is well-known that the method yields upper bounds for these eigenvalues, and that convergence to exact eigenvalues will occur if proper admissible functions are used to represent the displacements (eigenfunctions). However, little is known about the convergence of the derivatives of the eigenfunctions. In this paper the method is studied for the problem of free vibrations of a cantilever beam. Convergences of the eigenfunctions and their second and third derivatives (i.e., bending moments and shear forces) are examined, as well as convergence to satisfy the differential equation of motion (which involves fourth derivatives). Another study examines the effects of deliberately omitting one term from the set of admissible displacement functions which would otherwise be complete. It is found that in such cases, when orthogonal polynomials are used to represent the displacement, one eigenvalue (such as the lowest frequency) may be completely missed, and that the others will converge incorrectly. When ordinary polynomials are used, correct convergence is obtained even with a missing term.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

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