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Journal articles on the topic 'Dirac delta'

Author: Grafiati

Published: 3 June 2025

Last updated: 23 June 2025

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1

Dhanuk, B. B., K. Pudasainee, H. P. Lamichhane, and R. P. Adhikari. "Dirac Delta Function from Closure Relation of Orthonormal Basis and its Use in Expanding Analytic Functions." Journal of Nepal Physical Society 6, no. 2 (2020): 158–63. http://dx.doi.org/10.3126/jnphyssoc.v6i2.34872.

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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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2

Feng, Zaiyong, Linghua Ye, and Yi Zhang. "On the Fractional Derivative of Dirac Delta Function and Its Application." Advances in Mathematical Physics 2020 (October 14, 2020): 1–7. http://dx.doi.org/10.1155/2020/1842945.

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The Dirac delta function and its integer-order derivative are widely used to solve integer-order differential/integral equation and integer-order system in related fields. On the other hand, the fractional-order system gets more and more attention. This paper investigates the fractional derivative of the Dirac delta function and its Laplace transform to explore the solution for fractional-order system. The paper presents the Riemann-Liouville and the Caputo fractional derivative of the Dirac delta function, and their analytic expression. The Laplace transform of the fractional derivative of the Dirac delta function is given later. The proposed fractional derivative of the Dirac delta function and its Laplace transform are effectively used to solve fractional-order integral equation and fractional-order system, the correctness of each solution is also verified.

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3

Ferrando, Juan Carlos. "A Dirac Delta Operator." Mathematics and Statistics 9, no. 2 (2021): 179–87. http://dx.doi.org/10.13189/ms.2021.090213.

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4

Bulnes, J. D., M. A. I. Travassos, and J. López-Bonilla. "On the non-separability of the Lanczos-Dirac Delta and a function presenting a property of this Delta." Journal de Ciencia e Ingeniería 16, no. 1 (2024): 1–4. http://dx.doi.org/10.46571/jci.2024.1.1.

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In this article, we show two independent proofs, which use of different properties of the Lanczos-Dirac Delta, of which, the Delta cannot be separable. Furthermore, the conditions that should be imposed on an ordinary function are identified so that it presents the translation property of the Lanczos-Dirac Delta.

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5

Andino, Fernando, Marlon Recarte, and Michael Spilsbury. "La Función Delta de Dirac." Revista de la Escuela de Física 2, no. 1 (2019): 55–61. http://dx.doi.org/10.5377/ref.v2i1.8292.

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La función de Dirac presenta propiedades verdaderamente útiles para modelar problemas de la física – matemática. Se presenta un resumen detallado de las principales características y propiedades de dicha ”función”, argumentándose la validez de las mismas.

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6

Ahmed, Zafar, Sachin Kumar, Mayank Sharma, and Vibhu Sharma. "Revisiting double Dirac delta potential." European Journal of Physics 37, no. 4 (2016): 045406. http://dx.doi.org/10.1088/0143-0807/37/4/045406.

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7

Schmeelk, John. "Two-dimensional dirac delta reconsidered." Foundations of Physics Letters 7, no. 4 (1994): 315–32. http://dx.doi.org/10.1007/bf02186682.

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8

Katz, Mikhail G., and David Tall. "A Cauchy-Dirac Delta Function." Foundations of Science 18, no. 1 (2012): 107–23. http://dx.doi.org/10.1007/s10699-012-9289-4.

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9

Khairi, Fathul, and Malahayati. "Penerapan Fungsi Green dari Persamaan Poisson pada Elektrostatika." Quadratic: Journal of Innovation and Technology in Mathematics and Mathematics Education 1, no. 1 (2021): 56–79. http://dx.doi.org/10.14421/quadratic.2021.011-08.

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The Dirac delta function is a function that mathematically does not meet the criteria as a function, this is because the function has an infinite value at a point. However, in physics the Dirac Delta function is an important construction, one of which is in constructing the Green function. This research constructs the Green function by utilizing the Dirac Delta function and Green identity. Furthermore, the construction is directed at the Green function of the Poisson's equation which is equipped with the Dirichlet boundary condition. After the form of the Green function solution from the Poisson's equation is obtained, the Green function is determined by means of the expansion of the eigen functions in the Poisson's equation. These results are used to analyze the application of the Poisson equation in electrostatic.

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10

Murugan, A. R., C. Ganesa Moorthy, and C. T. Ramasamy. "A DEFINITION OF DIRAC DELTA FUNCTIONS." Advances in Mathematics: Scientific Journal 9, no. 3 (2020): 1205–12. http://dx.doi.org/10.37418/amsj.9.3.46.

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11

Meghwal, Ram Swaroop. "Fundamental Study of Dirac Delta Function." International Journal for Research in Applied Science and Engineering Technology 8, no. 12 (2020): 918–24. http://dx.doi.org/10.22214/ijraset.2020.32665.

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12

Chu, J. L., and Sanboh Lee. "Dirac delta function — an alternative expression." International Journal of Mathematical Education in Science and Technology 27, no. 5 (1996): 753–83. http://dx.doi.org/10.1080/0020739960270515.

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13

Simons, Stuart. "90.41 Introducing the Dirac delta function." Mathematical Gazette 90, no. 518 (2006): 292–93. http://dx.doi.org/10.1017/s0025557200179756.

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14

Balakrishnan, V. "All about the dirac delta function(?)." Resonance 8, no. 8 (2003): 48–58. http://dx.doi.org/10.1007/bf02866759.

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15

George, Kernel, and Carlos Imaz. "La delta de Dirac como función." Educación matemática 7, no. 3 (1995): 48–57. http://dx.doi.org/10.24844/em0703.03.

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16

Heltai, Luca, and Wenyu Lei. "A priori error estimates of regularized elliptic problems." Numerische Mathematik 146, no. 3 (2020): 571–96. http://dx.doi.org/10.1007/s00211-020-01152-w.

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Abstract Approximations of the Dirac delta distribution are commonly used to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution. In this work we show a-priori rates of convergence of this approximation process in standard Sobolev norms, with minimal regularity assumptions on the approximation of the Dirac delta distribution. The application of these estimates to the numerical solution of elliptic problems with singularly supported forcing terms allows us to provide sharp $$H^1$$ H 1 and $$L^2$$ L 2 error estimates for the corresponding regularized problem. As an application, we show how finite element approximations of a regularized immersed interface method results in the same rates of convergence of its non-regularized counterpart, provided that the support of the Dirac delta approximation is set to a multiple of the mesh size, at a fraction of the implementation complexity. Numerical experiments are provided to support our theories.

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17

Wang, Xinwei, and Chunhua Jin. "Differential Quadrature Analysis of Moving Load Problems." Advances in Applied Mathematics and Mechanics 8, no. 4 (2016): 536–55. http://dx.doi.org/10.4208/aamm.2014.m844.

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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.

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18

Kapoor, Taniya, Hongrui Wang, Alfredo Núñez, and Rolf Dollevoet. "Physics-informed machine learning for moving load problems." Journal of Physics: Conference Series 2647, no. 15 (2024): 152003. http://dx.doi.org/10.1088/1742-6596/2647/15/152003.

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Abstract This paper presents a new approach to simulate forward and inverse problems of moving loads using physics-informed machine learning (PIML). Physics-informed neural networks (PINNs) utilize the underlying physics of moving load problems and aim to predict the deflection of beams and the magnitude of the loads. The mathematical representation of the moving load considered involves a Dirac delta function, to capture the effect of the load moving across the structure. Approximating the Dirac delta function with PINNs is challenging because of its instantaneous change of output at a single point, causing difficulty in the convergence of the loss function. We propose to approximate the Dirac delta function with a Gaussian function. The incorporated Gaussian function physical equations are used in the physics-informed neural architecture to simulate beam deflections and to predict the magnitude of the load. Numerical results show that PIML is an effective method for simulating the forward and inverse problems for the considered model of a moving load.

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19

Jeong, Darae, Seokjun Ham, Junxiang Yang, et al. "Numerical Study of an Indicator Function for Front-Tracking Methods." Mathematical Problems in Engineering 2022 (July 31, 2022): 1–16. http://dx.doi.org/10.1155/2022/7381115.

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In this paper, we present a detailed derivation and numerical investigation of an indicator function for front-tracking methods. We use the discrete Dirac delta function to construct an indicator function from a set of Lagrangian points and solve the resulting discrete Poisson equation with the zero Dirichlet boundary condition using an iterative method. We present several computational tests to investigate the effect of parameters such as distance between points, uniformity of the distance, and types of the Dirac delta functions on the indicator function.

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20

Shutovskyi, Arsen. "Some applied aspects of the Dirac delta function." Ukrainian Mathematical Bulletin 20, no. 3 (2023): 442–53. http://dx.doi.org/10.37069/1810-3200-2023-20-3-7.

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The study is devoted to some applied aspects of the Dirac delta function. On the basis of this function, an integral representation was found for the deviation of the functions of the Holder class ${H}^{\alpha }$ ($0<\alpha <1$) from their Poisson integrals in the upper half-plane. In the current research, exact equalities of the upper bounds for the deviations of the functions of the Holder class ${H}^{\alpha }$ from the Poisson operators in the upper half-plane were found by applying the known properties of the Dirac delta function.

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21

Kovacevic, Milan S., Miroslav R. Jovanovic, and Marko M. Milosevic. "On the calculus of Dirac delta function with some applications in classical electrodynamics." Revista Mexicana de Física E 18, no. 2 Jul-Dec (2021): 020205. http://dx.doi.org/10.31349/revmexfise.18.020205.

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The Dirac delta function is a concept that is useful throughout physics as a standard mathematical tool that appears repeatedly in the undergraduate physics curriculum including electrodynamics, optics, and quantum mechanics. Our analysis was guided by an analytical framework focusing on how students activate, construct, execute, and reflect on the Dirac delta function in the context of classical electrodynamics problems solving. It’s applications in solving the charge density associated with a point charge as well as electrostatic point dipole field, for more advanced situations to describe the charge density of hydrogen atom were presented.

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22

Domínguez, V., M. L. Rapún, and F. J. Sayas. "Dirac delta methods for Helmholtz transmission problems." Advances in Computational Mathematics 28, no. 2 (2006): 119–39. http://dx.doi.org/10.1007/s10444-006-9015-2.

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23

Hosseini, Bamdad, Nilima Nigam, and John M. Stockie. "On regularizations of the Dirac delta distribution." Journal of Computational Physics 305 (January 2016): 423–47. http://dx.doi.org/10.1016/j.jcp.2015.10.054.

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24

Chicurel-Uziel, Enrique. "Dirac delta representation by exact parametric equations." Journal of Sound and Vibration 305, no. 1-2 (2007): 134–50. http://dx.doi.org/10.1016/j.jsv.2007.03.087.

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25

Makris, Nicos. "The Fractional Derivative of the Dirac Delta Function and Additional Results on the Inverse Laplace Transform of Irrational Functions." Fractal and Fractional 5, no. 1 (2021): 18. http://dx.doi.org/10.3390/fractalfract5010018.

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Motivated from studies on anomalous relaxation and diffusion, we show that the memory function M(t) of complex materials, that their creep compliance follows a power law, J(t)∼tq with q∈R+, is proportional to the fractional derivative of the Dirac delta function, dqδ(t−0)dtq with q∈R+. This leads to the finding that the inverse Laplace transform of sq for any q∈R+ is the fractional derivative of the Dirac delta function, dqδ(t−0)dtq. This result, in association with the convolution theorem, makes possible the calculation of the inverse Laplace transform of sqsα∓λ where α<q∈R+, which is the fractional derivative of order q of the Rabotnov function εα−1(±λ,t)=tα−1Eα,α(±λtα). The fractional derivative of order q∈R+ of the Rabotnov function, εα−1(±λ,t) produces singularities that are extracted with a finite number of fractional derivatives of the Dirac delta function depending on the strength of q in association with the recurrence formula of the two-parameter Mittag–Leffler function.

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26

Lemos, Nivaldo A. "Três mitos sobre a "função" delta de Dirac." Revista Brasileira de Ensino de Física 32, no. 4 (2010): 4701–1. http://dx.doi.org/10.1590/s1806-11172010000400014.

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27

Nedeljkov, Marko, and Michael Oberguggenberger. "Ordinary differential equations with delta function terms." Publications de l'Institut Math?matique (Belgrade) 91, no. 105 (2012): 125–35. http://dx.doi.org/10.2298/pim1205125n.

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This article is devoted to nonlinear ordinary differential equations with additive or multiplicative terms consisting of Dirac delta functions or derivatives thereof. Regularizing the delta function terms produces a family of smooth solutions. Conditions on the nonlinear terms, relating to the order of the derivatives of the delta function part, are established so that the regularized solutions converge to a limiting distribution.

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28

Delfan, Nasibeh, Amir Pishkoo, Mahdi Azhini, and Maslina Darus. "Using Fractal Calculus to Express Electric Potential and Electric Field in Terms of Staircase and Characteristic Functions." European Journal of Pure and Applied Mathematics 13, no. 1 (2020): 19–32. http://dx.doi.org/10.29020/nybg.ejpam.v13i1.3609.

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The Dirac Delta function is usually used to express the discrete distribution of electric charges in electrostatic problems. The integration of the product of the Dirac Delta function and the Green functions can calculate the electric potential and the electric field. Using fractal calculus, characteristic function, $\chi_{C_{n}}(x)$, as an alternative for dirac delta function is used to describe Cantor set charge distribution which is typical example of a discrete set. In these cases we deal with $F^{\alpha}$-integration and $F^{\alpha}$-derivative of the product of characteristic function and function of staircase function, namely $f(S^{\alpha}_{C_{n}}(x))$, which lead to calculation of electric potential and electric field. Recently, a calculus based fractals, called F$^{\alpha}$-calculus, has been developed which involve F$^{\alpha}$-integral and F$^{\alpha}$-derivative, of orders $\alpha$, $0<\alpha<1$, where $\alpha$ is dimension of $F$. In F$^{\alpha}$-calculus the staircase function and characteristic function have special roles. Finally, using COMSOL Multiphysics software we solve ordinary Laplace's equation (not fractional) in the fractal region with Koch snowflake boundary which is non-differentiable fractal, and give their graphs for the three first iterations.

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29

Delfan, Nasibeh, Amir Pishkoo, Mahdi Azhini, and Maslina Darus. "Using Fractal Calculus to Express Electric Potential and Electric Field in Terms of Staircase and Characteristic Functions." European Journal of Pure and Applied Mathematics 13, no. 1 (2020): 19–32. http://dx.doi.org/10.29020/nybg.ejpam.v1i1.3609.

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The Dirac Delta function is usually used to express the discrete distribution of electric charges in electrostatic problems. The integration of the product of the Dirac Delta function and the Green functions can calculate the electric potential and the electric field. Using fractal calculus, characteristic function, $\chi_{C_{n}}(x)$, as an alternative for dirac delta function is used to describe Cantor set charge distribution which is typical example of a discrete set. In these cases we deal with $F^{\alpha}$-integration and $F^{\alpha}$-derivative of the product of characteristic function and function of staircase function, namely $f(S^{\alpha}_{C_{n}}(x))$, which lead to calculation of electric potential and electric field. Recently, a calculus based fractals, called F$^{\alpha}$-calculus, has been developed which involve F$^{\alpha}$-integral and F$^{\alpha}$-derivative, of orders $\alpha$, $0<\alpha<1$, where $\alpha$ is dimension of $F$. In F$^{\alpha}$-calculus the staircase function and characteristic function have special roles. Finally, using COMSOL Multiphysics software we solve ordinary Laplace's equation (not fractional) in the fractal region with Koch snowflake boundary which is non-differentiable fractal, and give their graphs for the three first iterations.

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30

Ege, Inci. "Some Results on the Composition of Distributions." Sarajevo Journal of Mathematics 8, no. 1 (2024): 119–31. http://dx.doi.org/10.5644/sjm.08.1.09.

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Let $F$ be a distribution, $f$ be a locally summable function and $F_{n} = (F * \delta_{n})(x)$, where $\delta_{n}(x)$ is a certain sequence converging to the Dirac delta-function $\delta(x)$. The distribution $F(f)$ is defined as the neutrix limit of the sequence $\{ F_n(f) \}$, provided its limit $h$ exists in the sense that $$ \Nn \int_{-\infty}^{\infty} F_{n}(f(x))\varphi (x) dx = \la h(x) , \varphi (x) \ra $$ for all test functions $\varphi(x)$ in ${\mathcal D}(a,b)$. The composition of the distributions $x_{-}^{-s}\ln x_{-}$ and $x_{+}^{r}$ is evaluated for $r=0,1, \ldots$ and the composition of the distributions $x_{-}^{-s}\ln^{m} x_{-}$ and $H(x)$ is evaluated for $s,m=1,2,\ldots\,$. 2000 Mathematics Subject Classification. 46F10

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31

Yakhno, V., and D. Ozdek. "Computing the Green's Function of the Initial Boundary Value Problem for the Wave Equation in a Radially Layered Cylinder." International Journal of Computational Methods 12, no. 05 (2015): 1550027. http://dx.doi.org/10.1142/s0219876215500279.

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In this paper, a method for construction of the time-dependent approximate Green's function for the initial boundary value problem in a radially multilayered cylinder is suggested. This method is based on determination of the eigenvalues and the orthogonal set of the eigenfunctions; regularization of the Dirac delta function in the form of the Fourier series with a finite number of terms; expansion of the unknown Green's function in the form of Fourier series with unknown coefficients and computation of a finite number of unknown Fourier coefficients. Computational experiment confirms the robustness of the method for the approximate computation of the Dirac delta function and Green's function.

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32

Tonidandel, D. A. V., and A. E. A. Araújo. "A função delta revisitada: De Heaviside a Dirac." Revista Brasileira de Ensino de Física 37, no. 3 (2015): 3306–1. http://dx.doi.org/10.1590/s1806-11173731851.

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A função delta (δ), nomeada após o trabalho pioneiro do físico inglês Paul Adrien Maurice Dirac (⋆ 1902, †1984) em 1927, tornou-se ferramenta primordial para a ciência e engenharia atuais, com aplicações que vão da teoria quântica até o controle de processos industriais. Ela tem a capacidade facilitar a obtenção de inúmeros resultados que, de outra forma, necessitariam de complicados argumentos. Não obstante, suas definições na literatura são, frequentemente, apresentadas com pouco significado, mesmo que tenham sido corretamente aplicadas na solução de problemas. Este artigo tentará mostrar o engenhoso e inventivo caminho de desenvolvimento desta extraordinária ferramenta, que, além de Dirac, teve a contribuição de outros nomes, como o matemático francês Laurent Moise Schwartz (⋆ 1915, † 2002), com a teoria das distribuições, e do excêntrico físico-matemático e autodidata inglês chamado Oliver Heaviside (⋆ 1850, † 1925).

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33

Fischer, Jens V., and Rudolf L. Stens. "On Inverses of the Dirac Comb." Mathematics 7, no. 12 (2019): 1196. http://dx.doi.org/10.3390/math7121196.

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We determine tempered distributions which convolved with a Dirac comb yield unity and tempered distributions, which multiplied with a Dirac comb, yield a Dirac delta. Solutions of these equations have numerous applications. They allow the reversal of discretizations and periodizations applied to tempered distributions. One of the difficulties is the fact that Dirac combs cannot be multiplied or convolved with arbitrary functions or distributions. We use a theorem of Laurent Schwartz to overcome this difficulty and variants of Lighthill’s unitary functions to solve these equations. The theorem we prove states that double-sided (time/frequency) smooth partitions of unity are required to neutralize discretizations and periodizations on tempered distributions.

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34

SOURROUILLE, LUCAS. "SINGULAR SOLITON SOLUTION IN THE CHERN–SIMONS–CP(1) MODEL." Modern Physics Letters A 26, no. 33 (2011): 2523–31. http://dx.doi.org/10.1142/s021773231103684x.

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35

Lee, Wanho, and Seunggyu Lee. "Immersed Boundary Method for Simulating Interfacial Problems." Mathematics 8, no. 11 (2020): 1982. http://dx.doi.org/10.3390/math8111982.

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We review the immersed boundary (IB) method in order to investigate the fluid-structure interaction problems governed by the Navier–Stokes equation. The configuration is described by the Lagrangian variables, and the velocity and pressure of the fluid are defined in Cartesian coordinates. The interaction between two different coordinates is involved in a discrete Dirac-delta function. We describe the IB method and its numerical implementation. Standard numerical simulations are performed in order to show the effect of the parameters and discrete Dirac-delta functions. Simulations of flow around a cylinder and movement of Caenorhabditis elegans are introduced as rigid and flexible boundary problems, respectively. Furthermore, we provide the MATLAB codes for our simulation.

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36

Bedzhev, Borislav. "ANALYSIS OF THE CONDITIONS FOR SYNTHESIS OF EFFICIENT SIDE-LOBES SUPPRESSION FILTERS FOR PHASE MANIPULATED SIGNALS." Journal Scientific and Applied Research 6, no. 1 (2014): 106–13. http://dx.doi.org/10.46687/jsar.v6i1.146.

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The radio signals, possessing the so-named ideal auto – correlation function, resembling the Dirac delta – function, have a critical role for the wireless communication systems, especially for radars, radio-navigation systems, radio – synchronization and so on. Despite of this at the moment only few classes of such signals are invented. With regard in the paper we analyze in more detail the method for synthesis of pairs of phase manipulated signals and mismatched side-lobe suppression filters (SLSFs), which modify the real periodic auto – correlation function of the signals in the so-named ideal form, resembling the Dirac delta-function. As a result a new necessary condition for synthesis of efficient SLSFs is proven.

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37

Kim, Dohyun, June-Haak Ee, Chaehyun Yu, and Jungil Lee. "Derivation of Jacobian formula with Dirac delta function." European Journal of Physics 42, no. 3 (2021): 035006. http://dx.doi.org/10.1088/1361-6404/abdca9.

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38

Borys, Andrzej, Andrzej Kamiński, and Sławomir Sorek. "Volterra systems and powers of Dirac delta impulses." Integral Transforms and Special Functions 20, no. 3-4 (2009): 301–8. http://dx.doi.org/10.1080/10652460802565214.

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39

Altunkaynak, Bariş, Fatih Erman, and O. Teoman Turgut. "Finitely many Dirac-delta interactions on Riemannian manifolds." Journal of Mathematical Physics 47, no. 8 (2006): 082110. http://dx.doi.org/10.1063/1.2259581.

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40

Chung, Won Sang, and Hassan Hassanabadi. "Doubly superstatistics with bivariate modified Dirac delta distribution." Physica A: Statistical Mechanics and its Applications 554 (September 2020): 124712. http://dx.doi.org/10.1016/j.physa.2020.124712.

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41

Xu, Xiubin. "Nonlinear trigonometric approximation and the Dirac delta function." Journal of Computational and Applied Mathematics 209, no. 2 (2007): 234–45. http://dx.doi.org/10.1016/j.cam.2006.11.001.

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42

Carrillo, Luis Alberto Soriano, and Jose Alexandre Nogueira. "Força de Casimir para potenciais delta de Dirac." Revista Brasileira de Ensino de Física 31, no. 2 (2009): 2311.1–2311.8. http://dx.doi.org/10.1590/s1806-11172009000200012.

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Neste trabalho mostramos explicitamente como determinar as funções de Green para o cálculo da força de Casimir devido a campos escalares entre fronteiras representadas por duas funções delta de Dirac em 1+1 dimensões. Reobtemos os resultados de K.A. Milton (J. Phys. A37, 209 (2004)), porém mostrando de forma detalhada os calculos das funções de Green.

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43

Ee, June-Haak, Jungil Lee, and Chaehyun Yu. "Proof of Cramer’s rule with Dirac delta function." European Journal of Physics 41, no. 6 (2020): 065002. http://dx.doi.org/10.1088/1361-6404/aba455.

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44

Dang, Quang A., and Matthias Ehrhardt. "On Dirac delta sequences and their generating functions." Applied Mathematics Letters 25, no. 12 (2012): 2385–90. http://dx.doi.org/10.1016/j.aml.2012.07.009.

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45

Prieto, F., P. B. Lourenço, and C. S. Oliveira. "Impulsive Dirac-delta forces in the rocking motion." Earthquake Engineering & Structural Dynamics 33, no. 7 (2004): 839–57. http://dx.doi.org/10.1002/eqe.381.

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46

Benham, Richard, Chris Mortensen, and Graham Priest. "Chunk and permeate III: the Dirac delta function." Synthese 191, no. 13 (2014): 3057–62. http://dx.doi.org/10.1007/s11229-014-0473-7.

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Lee, Hyun Geun, and Junseok Kim. "Regularized Dirac delta functions for phase field models." International Journal for Numerical Methods in Engineering 91, no. 3 (2012): 269–88. http://dx.doi.org/10.1002/nme.4262.

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48

Lin, Zhenyi, Nan Chen, Yongzhen Fan, Wei Li, Knut Stamnes, and Snorre Stamnes. "New Treatment of Strongly Anisotropic Scattering Phase Functions: The Delta-M+ Method." Journal of the Atmospheric Sciences 75, no. 1 (2018): 327–36. http://dx.doi.org/10.1175/jas-d-17-0233.1.

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Abstract:

The treatment of strongly anisotropic scattering phase functions is still a challenge for accurate radiance computations. The new delta- M+ method resolves this problem by introducing a reliable, fast, accurate, and easy-to-use Legendre expansion of the scattering phase function with modified moments. Delta- M+ is an upgrade of the widely used delta- M method that truncates the forward scattering peak with a Dirac delta function, where the “+” symbol indicates that it essentially matches moments beyond the first M terms. Compared with the original delta- M method, delta- M+ has the same computational efficiency, but for radiance computations, the accuracy and stability have been increased dramatically.

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Koley, Santanu, and Kottala Panduranga. "Convergence of eigenfunction expansions for flexural gravity waves in infinite water depth." Journal of Physics: Conference Series 2070, no. 1 (2021): 012006. http://dx.doi.org/10.1088/1742-6596/2070/1/012006.

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Abstract In the present paper, point-wise convergence of the eigenfunction expansion to the velocity potential associated with the flexural gravity waves problem in water wave theory is established for infinite water depth case. To take into account the hydroelastic boundary condition at the free surface, a flexible membrane is assumed to float in water waves. In this context, firstly the eigenfunction expansion for the velocity potentials is obtained. Thereafter, an appropriate Green’s function is constructed for the associated boundary value problem. Using suitable properties of the Green’s functions, the vertical components of the eigenfunction expansion is written in terms of the Dirac delta function. Finally, using the property of the Dirac delta function, the convergence of the eigenfunction expansion to the velocity potential is shown.

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Zhang, L. H., S. K. Lai, and J. Yang. "A DSC Regularized Dirac-Delta Method for Flexural Vibration of Elastically Supported FG Beams Subjected to a Moving Load." International Journal of Structural Stability and Dynamics 20, no. 03 (2020): 2050039. http://dx.doi.org/10.1142/s021945542050039x.

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This research presents a numerical approach to address the moving load problem of functionally graded (FG) beams with rotational elastic edge constraints, in which the regularized Dirac-delta function is used to describe a time-dependent moving load source. The governing partial differential equations of the system, derived in accordance with the classical Euler–Bernoulli beam theory, are approximated by the discrete singular convolution (DSC) method. The resulting set of algebraic equations can then be solved by the Newmark-β integration scheme. Such a singular Dirac-delta formulation is also employed as the kernel function of the DSC method. In this work, the material properties of FG beams are assumed to be changed in the thickness direction. A convergence study is performed to validate the accuracy and reliability of the numerical results. In addition, the effects of moving load velocity and material distribution on the dynamic behavior of elastically restrained FG beams are also studied to serve as new benchmark solutions. By comparing with the available results in the existing literature, the present results show good agreement. More importantly, the major finding of this work indicates that the DSC regularized Dirac-delta approach is a good candidate for moving load problems, since the equally spaced grid system adopted in the DSC scheme can achieve a preferable representation of moving load sources.

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