scholarly journals A Differential Quadrature Procedure with Regularization of the Dirac-delta Function for Numerical Solution of Moving Load Problem

2015 ◽  
Vol 12 (7) ◽  
pp. 1241-1265 ◽  
Author(s):  
S. A. Eftekhari
Keyword(s):  
Numerical Solution ◽  
Moving Load ◽  
Delta Function ◽  
Differential Quadrature ◽  
Dirac Delta Function ◽  
Dirac Delta

Differential Quadrature Analysis of Moving Load Problems

2016 ◽  
Vol 8 (4) ◽  
pp. 536-555 ◽  
Author(s):  
Xinwei Wang ◽  
Chunhua Jin
Keyword(s):  
Differential Equations ◽  
Differential Quadrature Method ◽  
Moving Load ◽  
Delta Function ◽  
Point Load ◽  
Differential Quadrature ◽  
Dirac Delta Function ◽  
Dirac Delta ◽  
Singular Functions ◽  
Grid Points

AbstractThe differential quadrature method (DQM) has been successfully used in a variety of fields. Similar to the conventional point discrete methods such as the collocation method and finite difference method, however, the DQM has some difficulty in dealing with singular functions like the Dirac-delta function. In this paper, two modifications are introduced to overcome the difficulty encountered in solving differential equations with Dirac-delta functions by using the DQM. The moving point load is work-equivalent to loads applied at all grid points and the governing equation is numerically integrated before it is discretized in terms of the differential quadrature. With these modifications, static behavior and forced vibration of beams under a stationary or a moving point load are successfully analyzed by directly using the DQM. It is demonstrated that the modified DQM can yield very accurate solutions. The compactness and computational efficiency of the DQM are retained in solving the partial differential equations with a time dependent Dirac-delta function.

A modified differential quadrature procedure for numerical solution of moving load problem

2015 ◽  
Vol 230 (5) ◽  
pp. 715-731 ◽  
Author(s):  
SA Eftekhari
Keyword(s):  
Differential Equations ◽  
Differential Quadrature Method ◽  
Moving Load ◽  
Finite Difference Methods ◽  
Delta Function ◽  
Differential Quadrature ◽  
Quadrature Method ◽  
Dirac Delta ◽  
Singular Functions ◽  
Grid Points

The differential quadrature method is a powerful numerical method for the solution of partial differential equations that arise in various fields of engineering, mathematics, and physics. It is easy to use and also straightforward to implement. However, similar to the conventional point discretization methods like the collocation and finite difference methods, the differential quadrature method has some difficulty in solving differential equations involving singular functions like the Dirac-delta function. This is due to complexities introduced by the singular functions to the discretization process of the problem region. To overcome this difficulty, this paper presents a combined differential quadrature–integral quadrature procedure in which such singular functions are simply handled. The mixed scheme can be easily applied to the problems in which the location of the singular point coincides with one of the differential quadrature grid points. However, for problems in which such condition is not fulfilled (i.e. for the case of arbitrary arranged grid points), especially for moving load class of problems, the coupled approach may fail to produce accurate solutions. To solve this drawback, we also introduce two simple approximations and show that they can yield accurate results. The reliability and applicability of the proposed method are demonstrated herein through the solution of some illustrative problems, including the moving load problems of Euler–Bernoulli and Timoshenko beams. The results generated by the proposed method are compared with analytical and numerical results available in the literature and excellent agreement is achieved.

Dirac delta regularization

2020 ◽  
Author(s):  
Matheus Pereira Lobo
Keyword(s):  
Delta Function ◽  
Dirac Delta Function ◽  
Dirac Delta ◽  
Finite Result

I present a finite result for the Dirac delta "function."

scholarly journals Electromagnetic Waves Through Disordered Systems: Comparison of Intensity, Transmission and Conductance

2001 ◽  
Vol 694 ◽  
Author(s):  
Fredy R Zypman ◽  
Gabriel Cwilich
Keyword(s):  
Probability Distribution ◽  
Disordered Systems ◽  
Electromagnetic Waves ◽  
Delta Function ◽  
Rayleigh Distribution ◽  
Dirac Delta Function ◽  
Universal Form ◽  
Dirac Delta ◽  
Diffusive Regime ◽  
Analytical Expressions

AbstractWe obtain the statistics of the intensity, transmission and conductance for scalar electromagnetic waves propagating through a disordered collection of scatterers. Our results show that the probability distribution for these quantities x, follow a universal form, YU(x) = xne−xμ. This family of functions includes the Rayleigh distribution (when α=0, μ=1) and the Dirac delta function (α →+ ∞), which are the expressions for intensity and transmission in the diffusive regime neglecting correlations. Finally, we find simple analytical expressions for the nth moment of the distributions and for to the ratio of the moments of the intensity and transmission, which generalizes the n! result valid in the previous case.

All about the dirac delta function(?)

Resonance ◽  
2003 ◽  
Vol 8 (8) ◽  
pp. 48-58 ◽  
Author(s):  
V Balakrishnan
Keyword(s):  
Delta Function ◽  
Dirac Delta Function ◽  
Dirac Delta

Dirac Delta Function from Closure Relation of Orthonormal Basis and its Use in Expanding Analytic Functions

2020 ◽  
Vol 6 (2) ◽  
pp. 158-163
Author(s):  
B. B. Dhanuk ◽  
K. Pudasainee ◽  
H. P. Lamichhane ◽  
R. P. Adhikari
Keyword(s):  
Analytic Functions ◽  
Orthonormal Basis ◽  
Special Functions ◽  
Single Point ◽  
Functional Space ◽  
Delta Function ◽  
Hamiltonian Operator ◽  
Dirac Delta Function ◽  
Closure Relation ◽  
Dirac Delta

One of revealing and widely used concepts in Physics and mathematics is the Dirac delta function. The Dirac delta function is a distribution on real lines which is zero everywhere except at a single point, where it is infinite. Dirac delta function has vital role in solving inhomogeneous differential equations. In addition, the Dirac delta functions can be used to explore harmonic information’s imbedded in the physical signals, various forms of Dirac delta function and can be constructed from the closure relation of orthonormal basis functions of functional space. Among many special functions, we have chosen the set of eigen functions of the Hamiltonian operator of harmonic oscillator and angular momentum operators for orthonormal basis. The closure relation of orthonormal functions  used to construct the generator of Dirac delta function which is used to expand analytic functions log(x + 2),exp(-x2) and x within the valid region of arguments.

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