Integral Transforms

Arfken, George B.

doi:10.1016/b978-0-12-059820-5.50023-9

Cited by 152 publications

(232 citation statements)

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“…Partial wave Parseval relation as the probability conservation law. Parseval relation is energy conservation law 12 in Fraunhofer diffraction and hence becomes probability conservation law in crystal diffraction. For a unitcell, we may replace integration with summation over discrete locations of atoms and equivalently write as,…”

Section: Violation Ofmentioning

confidence: 99%

The Partial Wave Sub-Amplitude Fourier Transform with Born Probability Interpretation in Crystallography

Bhanumoorthy

2023

Preprint

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We identify Maxwell-Schrödinger equation as the governing partial differential equation. We name the solution as unitcell function such that Born probability rule interprets electron density as modulus square of the unitcell function. The existing traditional full-amplitude Fourier transform relation with electron density as directly the inverse Fourier transform of structure factors therefore violates Born probability rule. We derive the partial wave Parse-val relation as the probability conservation law using Rayleigh expansion. We subdivide the experimental full-amplitudes into partial wave sub-amplitudes, which become amplitudes of spherical waves. The subscript on a sub-amplitude represents localization within the unit-cell and therefore spherical waves become localized wave packets. As a probability measure, the amplitude of a localized wave packet is exact and merits over the amplitude of a non-localized plane wave. We measure angle dependence as the precision angles data. We derive the partial wave sub-amplitude Fourier transform relation to determine orbital structures of individual atoms using the precision angles data. The partial wave sub-amplitude Fourier transform relation alone obeys Born probability rule and hence becomes the only acceptable Fourier transform relation in quantum crystallography.

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Section: Violation Ofmentioning

confidence: 99%

The Partial Wave Sub-Amplitude Fourier Transform with Born Probability Interpretation in Crystallography

Bhanumoorthy

2023

Preprint

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“…There are three possible types of boundary conditions for a second-order differential equation [35,36], namely: #a) Eq. ( 5) is called the Dirichlet's condition, if α≠0 and β=0; #b) Eq.…”

Section: Materials Of the Studymentioning

confidence: 99%

PDE-Based Mathematical Models to Diagnose the Temperature Changes Phenomena on the Single Rectangular Plate-Fin

Goeritno¹,

Prayudyanto²,

Eosina³

et al. 2023

MMEP

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The ideas of the studies in this article are back-grounded by the work diligently and carefully to relate the influence of two independent variables, e.g., distance and time on one dependent variable, i.e., temperature changes which are solved by a partial differential equation. In this article, we explain the particular features of a naturally physical system that it seeks to understand. The diagnosis of the temperature changes in a metal rod to consider a conduction phenomenon can be covered by all-natural physical phenomena through conduction. By making the mathematical equations based on partial differential equations (PDEs), it is shown that the temperature change is influenced by distance and time. The objectives of this study are (i) to make a prototype of the problem based on the one-dimensional heat conduction equation, (ii) to process the solution of the mathematical equation based on the parabolic partial differential equation (PDE) using the separation of variables, and (iii) to display the phenomena of the temperature changes as a function of the length in the copper bar and the time. Achieving the research objectives takes several stages in the research methods which include (a) making the prototype of the problem, (b) processing the solution of a mathematical equation, and (c) displaying the temperature changes phenomenon in the form of a curve. The results are (i) a mass balance is developed for a finite segment ∆𝑧 along the tank's longitudinal axis in order to derive a differential equation, (ii) the final complete solution in the form of a Fourier sine series, and (iii) a three-dimensional curve as an indication of the existence of the phenomenon of temperature changes. In general, it can be concluded that the making of a mathematical model based on partial differential equations with the method of separating variables as an analytical solution can be used to diagnose the phenomenon of temperature changes as a function of distance and time.

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“…This relation is helpful since the Bernoulli numbers are well documented in the literature. The interested reader may refer to [81][82][83][84] for more informations on Genocchi and Bernoulli numbers. Some relations between Taylor expansions of the LB scheme and the Bernoulli polynomials have already been noticed in the literature [49], but no demonstration could be provided to the authors' knowledge.…”

Section: General Taylor Expansion Of the Bgk Modelmentioning

confidence: 99%

Hydrodynamic limits and numerical errors of isothermal lattice Boltzmann schemes

Wissocq

Sagaut

2022

Journal of Computational Physics

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Integral Transforms

Cited by 152 publications

References 0 publications

The Partial Wave Sub-Amplitude Fourier Transform with Born Probability Interpretation in Crystallography

The Partial Wave Sub-Amplitude Fourier Transform with Born Probability Interpretation in Crystallography

PDE-Based Mathematical Models to Diagnose the Temperature Changes Phenomena on the Single Rectangular Plate-Fin

Hydrodynamic limits and numerical errors of isothermal lattice Boltzmann schemes

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