scholarly journals A Note on the Generalized Relativistic Diffusion Equation

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1009
Author(s):  
Luisa Beghin ◽  
Roberto Garra
Keyword(s):  
Fourier Transform ◽  
Fundamental Solution ◽  
Diffusion Equation ◽  
Fractional Derivatives ◽  
Stable Process ◽  
Caputo Fractional Derivatives ◽  
The Fourier Transform ◽  
Derivatives Of

We study here a generalization of the time-fractional relativistic diffusion equation based on the application of Caputo fractional derivatives of a function with respect to another function. We find the Fourier transform of the fundamental solution and discuss the probabilistic meaning of the results obtained in relation to the time-scaled fractional relativistic stable process. We briefly consider also the application of fractional derivatives of a function with respect to another function in order to generalize fractional Riesz-Bessel equations, suggesting their stochastic meaning.

Fast rigid-body refinement for molecular-replacement techniques

1992 ◽  
Vol 25 (2) ◽  
pp. 281-284 ◽  
Author(s):  
E. E. Castellano ◽  
G. Oliva ◽  
J. Navaza
Keyword(s):  
Fourier Transform ◽  
Rigid Body ◽  
Least Squares ◽  
Fourier Coefficients ◽  
Molecular Replacement ◽  
Present Formulation ◽  
Judicious Choice ◽  
The Fourier Transform ◽  
Derivatives Of ◽  
Patterson Search

A method for the least-squares rigid-body refinement of a general electron density model is described. The present formulation is different from a previously reported one in the computation of the derivatives of the calculated Fourier coefficients, which are derived analytically here. This, together with a judicious choice of the Fourier transform search arrays, makes the procedure extremely fast and sufficiently accurate. Although originally designed simply to optimize the values of the positional parameters obtained by Patterson search techniques, the method proved to be extremely efficient as an aid for evaluation of the correctness of potential molecular-replacement solutions.

scholarly journals On fractional boundary value problems involving fractional derivatives with Mittag-Leffler kernel and nonlinear integral conditions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mohammed S. Abdo ◽  
Thabet Abdeljawad ◽  
Saeed M. Ali ◽  
Kamal Shah
Keyword(s):  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Fractional Integral ◽  
Fixed Point Theorems ◽  
Boundary Value ◽  
Integral Conditions ◽  
Caputo Fractional Derivatives ◽  
Derivatives Of ◽  
Fractional Boundary Value Problems

AbstractIn this paper, we consider two classes of boundary value problems for nonlinear implicit differential equations with nonlinear integral conditions involving Atangana–Baleanu–Caputo fractional derivatives of orders $0<\vartheta \leq 1$ 0 < ϑ ≤ 1 and $1<\vartheta \leq 2$ 1 < ϑ ≤ 2 . We structure the equivalent fractional integral equations of the proposed problems. Further, the existence and uniqueness theorems are proved with the aid of fixed point theorems of Krasnoselskii and Banach. Lastly, the paper includes pertinent examples to justify the validity of the results.

scholarly journals On the General Solutions of Some Non-Homogeneous Div-Curl Systems with Riemann–Liouville and Caputo Fractional Derivatives

2021 ◽  
Vol 5 (3) ◽  
pp. 117
Author(s):  
Briceyda B. Delgado ◽  
Jorge E. Macías-Díaz
Keyword(s):  
Fractional Order ◽  
Fractional Derivatives ◽  
Decomposition Theorem ◽  
Helmholtz Decomposition ◽  
Fractional Operators ◽  
Gradient Vector ◽  
Caputo Fractional Derivatives ◽  
General Solutions ◽  
Derivatives Of ◽  
Fractional Space

In this work, we investigate analytically the solutions of a nonlinear div-curl system with fractional derivatives of the Riemann–Liouville or Caputo types. To this end, the fractional-order vector operators of divergence, curl and gradient are identified as components of the fractional Dirac operator in quaternionic form. As one of the most important results of this manuscript, we derive general solutions of some non-homogeneous div-curl systems that consider the presence of fractional-order derivatives of the Riemann–Liouville or Caputo types. A fractional analogous to the Teodorescu transform is presented in this work, and we employ some properties of its component operators, developed in this work to establish a generalization of the Helmholtz decomposition theorem in fractional space. Additionally, right inverses of the fractional-order curl, divergence and gradient vector operators are obtained using Riemann–Liouville and Caputo fractional operators. Finally, some consequences of these results are provided as applications at the end of this work.

Linear Dynamic System Identification in the Frequency Domain Using Fractional Derivatives

2010 ◽  
Vol 17 (2) ◽  
pp. 279-288 ◽  
Author(s):  
Tomasz Janiczek ◽  
Janusz Janiczek
Keyword(s):  
Fourier Transform ◽  
Dynamic System ◽  
Frequency Domain ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Linear Dynamic System ◽  
Transform Method ◽  
Linear Dynamic ◽  
Fractional Differential ◽  
The Fourier Transform

Linear Dynamic System Identification in the Frequency Domain Using Fractional DerivativesThis paper presents a study of the Fourier transform method for parameter identification of a linear dynamic system in the frequency domain using fractional differential equations. Fundamental definitions of fractional differential equations are briefly outlined. The Fourier transform method of identification and their algorithms are generalized so that they include fractional derivatives and integrals.

Temporal windowing and inverse transform of the wavefield in the Laplace-Fourier domain

Geophysics ◽  
2013 ◽  
Vol 78 (5) ◽  
pp. R207-R222 ◽  
Author(s):  
Sangmin Kwak ◽  
Hyunggu Jun ◽  
Wansoo Ha ◽  
Changsoo Shin
Keyword(s):  
Fourier Transform ◽  
Time Window ◽  
Waveform Inversion ◽  
Velocity Model ◽  
Fourier Domain ◽  
Gain Function ◽  
The Fourier Transform ◽  
The Time Domain ◽  
Derivatives Of ◽  
Complex Damping

Temporal windowing is a valuable process, which can help us to focus on a specific event in a seismogram. However, applying the time window is difficult outside the time domain. We suggest a windowing method which is applicable in the Laplace-Fourier domain. The window function we adopt is defined as a product of a gain function and an exponential damping function. The Fourier transform of a seismogram windowed by this function is equivalent to the partial derivative of the Laplace-Fourier domain wavefield with respect to the complex damping constant. Therefore, we can obtain a windowed seismogram using the partial derivatives of the Laplace-Fourier domain wavefield. We exploit the time-windowed wavefield, which is modeled directly in the Laplace-Fourier domain, to reconstruct subsurface velocity model by waveform inversion in the Laplace-Fourier domain. We present the windowed seismograms by introducing an inverse Laplace-Fourier transform technique and demonstrate the effect of temporal windowing in a synthetic Laplace-Fourier domain waveform inversion example.

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