On the solutions of fractional differential equations with modified Mittag-Leffler kernel and Dirac Delta function: Analytical results and numerical simulations. by Mohammed Al-Refai, Dumitru Baleanu, A K Alomari

“…We also investigate the modified derivatives for the Dirac delta functions, and study related fractional differential equations. …”
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On the solutions of fractional differential equations with modified Mittag-Leffler kernel and Dirac Delta function: Analytical results and numerical simulations by Mohammed Al-Refai, Dumitru Baleanu, A.K. Alomari

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Generating the generalized distributions by using fractional calculus by Masoud Ganji, Fatemeh Gharari

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Efficiency of noise reduction by a road speed bump by Rufin MAKAREWICZ, Piotr KOKOWSKI

“…The bump is modeled by the Dirac delta function. The linear density of sound energy (in Joules per meter) is used for the calculation of sound energies, ec and e, emitted by a vehicle on a road without- and with a bump, respectively. …”
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Bending of a Viscoelastic Timoshenko Cracked Beam Based on Equivalent Viscoelastic Spring Models by Chao Fu, Xiao Yang

“…Considering the transverse crack as a massless viscoelastic rotational spring, the equivalent stiffness of the viscoelastic cracked beam is derived by Laplace transform and the generalized Dirac delta function. Using the standard linear solid constitutive equation and the inverse Laplace transform, the analytical expressions of the deflection and rotation angle of the viscoelastic Timoshenko beam with an arbitrary number of open cracks are obtained in the time domain. …”
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On the Composition of Distributions x−sln|x| and |x|μ by Biljana Jolevska-Tuneska, Emin Özça¯g

“…The distribution F(f) is defined as the neutrix limit of the sequence {Fn(f)}, where Fn(x)=F(x)*δn(x) and {δn(x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function δ(x). The composition of the distributions x−sIn|x| and |x|μ is evaluated for s=1,2,…,μ>0 and μs≠1,2,….…”
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A General Total Variation Minimization Theorem for Compressed Sensing Based Interior Tomography by Weimin Han, Hengyong Yu, Ge Wang

“…Our major mathematical tool to prove this result is functional analysis without involving the Dirac delta function, which was heuristically used by Yu and Wang (2009).…”
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Modified Cramer-Rao lower bound for symbol width estimation from a phase-shift-keying signal by DENG Zhen-miao, LIU Yu

“…The modified Cramer-Rao lower bound（MCRB） on symbol width estimation of a phase-shift-keying signal was derived.When calculating the integration of the square of dirac delta function δ（t ）, Parseval’s theorem was used to transform the computation from time-domain to frequency-domain.Then the closed-form analytical solution of MCRB for symbol width estimation was derived when the pulse shaping function was the rectangle pulse.However when the pulse shaping function was the raised cosine（RC） pulse, the MCRB is evaluated by numerical simulation.The results show that the MCRB of rectangle pulse shape is higher than that of RC pulse with the roll-off factor to be 0.5 and the dif-ference was about 1 dB.…”
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NEW FUNCTIONAL RELATIONS FOR LINEAR RHEOLOGICAL MODELS OF MAXWELL AND KELVIN-VOIGT by V. D. Karlov

“…In calculations, the Heaviside step function, the Dirac delta function, and the solution of the Cauchy problem proposed by E.M. …”
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Unified Approach to Fractional Calculus Images Involving the Pathway Transform of Extended k-Gamma Function and Applications by Asifa Tassaddiq, Kwara Nantomah

“…Pδ transform of Dirac delta function is obtained which proved useful to achieve the purpose. …”
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On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse Functions by Brian Fisher, Adem Kılıçman

“…The composition F(f(x)) of F and f is said to exist and be equal to the distribution h(x) if the limit of the sequence {Fn(f(x))} is equal to h(x), where Fn(x)=F(x)*δn(x) for n=1,2,… and {δn(x)} is a certain regular sequence converging to the Dirac delta function. It is proved that the neutrix composition δ(rs-1)((tanhx+)1/r) exists and δ(rs-1)((tanhx+)1/r)=∑k=0s-1∑i=0Kk((-1)kcs-2i-1,k(rs)!…”
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A simplified approach for simulating pollutant transport in small rivers with dead zones using convolution by Szymkiewicz Romuald

“…The impulse response function is obtained as an analytical solution of the linear advection-diffusion equation for the Dirac delta function imposed as the boundary condition at the upstream end. …”
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Finite Element Method for Solving the Screened Poisson Equation with a Delta Function by Liang Tang, Yuhao Tang

“…This paper presents a Finite Element Method (FEM) framework for solving the screened Poisson equation with a Dirac delta function as the forcing term. The singularity introduced by the delta function poses challenges for standard numerical methods, particularly in higher dimensions. …”
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Steady-State Dynamical Response of a Strongly Nonlinear System with Impact and Coulomb Friction Subjected to Gaussian White Noise Excitation by Guidong Yang, Dongmei Huang, Wei Li, Meng Su

“…The Zhuravlev nonsmooth transformation of the state variables combined with the Dirac delta function is utilized to simplify the original system to one without velocity jump. …”
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Singularly perturbed rank one linear operators by M.E. Dudkin, O. Yu. Dyuzhenkova

“…The theory of singularly perturbed self-adjoint operators arose from the need to consider differential expressions in such terms as the Dirac $\delta$-function. Since it is important to consider expressions given not only by symmetric operators, the generalization (transfer) of the basic principles of the theory of singularly perturbed self-adjoint operators in the case of non-symmetric ones is important problem. …”
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Towards an RFI Mitigation Algorithm in Synthetic Aperture Interferometer Radiometry: Application to the SMOS Space Mission by Ali Khazaal

“…A recent RFI mitigation algorithm was implemented in the SMOS operational processor, modeling each source as a Dirac delta function whose exact position and intensity are estimated to subtract its contribution from the visibilities. …”
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Discontinuity method for evaluating scattering amplitudes within the worldline formalism by Victor Miguel Banda Guzmán

“…In the three-point and four-point on-shell cases, this computation is particularly tractable, as the discontinuities involve Dirac delta functions arising from the Sokhotski-Plemelj formula, which simplify the resulting integrals. …”
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Connecting scalar amplitudes using the positive tropical Grassmannian by Freddy Cachazo, Bruno Giménez Umbert

“…The same limiting procedure leads to an integral representation of ϕ 4 amplitudes where indicator functions turn into Dirac delta functions. Their support decomposes into C n/2−1 regions, with C q the q th-Catalan number. …”
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