A note on fractional Bessel equation and its asymptotics

2013 ◽  
Vol 16 (3) ◽  
Author(s):  
Wojciech Okrasiński ◽  
Łukasz Płociniczak
Keyword(s):  
Power Series ◽  
Asymptotic Formula ◽  
Asymptotic Analysis ◽  
Fractional Derivative ◽  
Order Term ◽  
High Accuracy ◽  
Fast Convergence ◽  
Leading Order ◽  
Bessel Equation ◽  
Integer Order

AbstractIn this note we propose a fractional generalization of the classical modified Bessel equation. Instead of the integer-order derivatives we use the Riemann-Liouville version. Next, we solve the fractional modified Bessel equation in terms of the power series and provide an asymptotic analysis of its solution for large arguments. We find a leading-order term of the asymptotic formula for the solution to the considered equation. This behavior is verified numerically and shows high accuracy and fast convergence. Our results reduce to the classical formulas when the order of the fractional derivative goes to integer values.

Adaptive Intensity Transformation based Phase-Retrieval with High Accuracy and Fast Convergence

2021 ◽  
Author(s):  
Songnian Fu ◽  
Meng Xiang ◽  
Peijian Zhou ◽  
Ye Bolin ◽  
Ou Xu ◽  
...  
Keyword(s):  
Phase Retrieval ◽  
High Accuracy ◽  
Fast Convergence

Derivation of the maximum voltage drop in power grids of integrated circuits with an array bonding package

2012 ◽  
Vol 23 (6) ◽  
pp. 797-819
Author(s):  
M. AGUARELES ◽  
J. de HARO
Keyword(s):  
Integrated Circuits ◽  
Iterative Scheme ◽  
Voltage Drop ◽  
Order Term ◽  
Singular Limit ◽  
Power Grids ◽  
Leading Order ◽  
Conformal Maps ◽  
Maximum Value ◽  
Aligned Array

In this work we derive a formula for the maximum value of the voltage drop that takes place in power grids of integrated circuits with an array bonding package. We consider a simplified model for the power grid where the voltage is represented as the solution of the Poisson's equation in an infinite planar domain with a regularly aligned array of small square pads where the voltage is set to be zero. We study the singular limit where these pads' size tends to be zero and we derive an asymptotic formula in terms of a power series in ε, being 2ε the side of the square. We also discuss pads of more general shapes, for which we provide an expression for the leading order term. To deduce all these formulae we use the method of matched asymptotic expansions using an iterative scheme, along with conformal maps to compare this problem with the corresponding one when the pads are circular.

A Comparative Study of Integer and Noninteger Order Wavelets for Fractional Nonlinear Fredholm Integro-Differential Equations

2018 ◽  
Vol 13 (8) ◽  
Author(s):  
F. Mohammadi ◽  
J. A. Tenreiro Machado
Keyword(s):  
Differential Equations ◽  
Comparative Study ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Efficient Algorithm ◽  
Operational Matrix ◽  
Tau Method ◽  
Legendre Wavelets ◽  
Numerical Examples ◽  
Integer Order

This paper compares the performance of Legendre wavelets (LWs) with integer and noninteger orders for solving fractional nonlinear Fredholm integro-differential equations (FNFIDEs). The generalized fractional-order Legendre wavelets (FLWs) are formulated and the operational matrix of fractional derivative in the Caputo sense is obtained. Based on the FLWs, the operational matrix and the Tau method an efficient algorithm is developed for FNFIDEs. The FLWs basis leads to more efficient and accurate solutions of the FNFIDE than the integer-order Legendre wavelets. Numerical examples confirm the superior accuracy of the proposed method.

scholarly journals Material barriers to diffusive and stochastic transport

2018 ◽  
Vol 115 (37) ◽  
pp. 9074-9079 ◽  
Author(s):  
George Haller ◽  
Daniel Karrasch ◽  
Florian Kogelbauer
Keyword(s):  
Coherent Structures ◽  
Fluid Transport ◽  
Order Term ◽  
Velocity Fields ◽  
Leading Order ◽  
Transport Barrier ◽  
Material Surfaces ◽  
Stochastic Transport ◽  
Transport Barriers ◽  
Enhancer Detection

We seek transport barriers and transport enhancers as material surfaces across which the transport of diffusive tracers is minimal or maximal in a general, unsteady flow. We find that such surfaces are extremizers of a universal, nondimensional transport functional whose leading-order term in the diffusivity can be computed directly from the flow velocity. The most observable (uniform) transport extremizers are explicitly computable as null surfaces of an objective transport tensor. Even in the limit of vanishing diffusivity, these surfaces differ from all previously identified coherent structures for purely advective fluid transport. Our results extend directly to stochastic velocity fields and hence enable transport barrier and enhancer detection under uncertainties.

Large-Reynolds-number asymptotic analysis of viscous centre modes in vortices

2007 ◽  
Vol 585 ◽  
pp. 153-180 ◽  
Author(s):  
STÉPHANE LE DIZÈS ◽  
DAVID FABRE
Keyword(s):  
Reynolds Number ◽  
Spatial Structure ◽  
Asymptotic Analysis ◽  
Layer Structure ◽  
Axial Flow ◽  
Multiple Scale ◽  
Large Reynolds Number ◽  
Leading Order ◽  
Vortex Model ◽  
Theoretical Predictions

This paper presents a large-Reynolds-number asymptotic analysis of viscous centre modes on an arbitrary axisymmetrical vortex with an axial jet. For any azimuthal wavenumber m and axial wavenumber k, the frequency of these modes is given at leading order by ω0 = mΩ0 + kW0 where Ω0 and W0 are the angular and axial velocities of the vortex at its centre. These modes possess a multi-layer structure localized in an O(Re−1/6) neighbourhood of the vortex. By a multiple-scale matching analysis, we demonstrate the existence of three different families of viscous centre modes whose frequency expands as ω(n) ∼ ω0 + Re−1/3ω1 + Re−1/2ω(n)2. One of these families is shown to have unstable eigenmodes when H0 = 2Ω0k(2kΩ0 − mW2) < 0 where W2 is the second radial derivative of the axial flow in the centre. The growth rate of these modes is given at leading order by σ ∼ (3/2)(H0/4)1/3Re−1/3. Our results prove that any vortex with a jet (or jet with swirl) such that Ω0W2 ≠ 0 is unstable if the Reynolds number is sufficiently large. The spatial structure of the viscous centre modes is obtained and simple approximations which capture the main feature of the eigenmodes are also provided.The theoretical predictions are compared with numerical results for the q-vortex model (or Batchelor vortex) for Re ≥ 105. For all modes, a good agreement is demonstrated for both the frequency and the spatial structure.

scholarly journals Extracting Hawking radiation near the horizon of AdS black holes

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Krishan Saraswat ◽  
Niayesh Afshordi
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Evaporation Rate ◽  
Order Term ◽  
Energy Condition ◽  
Null Energy Condition ◽  
Leading Order ◽  
Proper Distance ◽  
Ads Black Holes ◽  
A Cell

Abstract We study how the evaporation rate of spherically symmetric black holes is affected through the extraction of radiation close to the horizon. We adopt a model of extraction that involves a perfectly absorptive screen placed close to the horizon and show that the evaporation rate can be changed depending on how close to the horizon the screen is placed. We apply our results to show that the scrambling time defined by the Hayden-Preskill decoding criterion, which is derived in Pennington’s work (arXiv:1905.08255) through entanglement wedge reconstruction is modified. The modifications appear as logarithmic corrections to Pennington’s time scale which depend on where the absorptive screen is placed. By fixing the proper distance between the horizon and screen we show that for small AdS black holes the leading order term in the scrambling time is consistent with Pennington’s scrambling time. However, for large AdS black holes the leading order Log contains the Bekenstein-Hawking entropy of a cell of characteristic length equal to the AdS radius rather than the entropy of the full horizon. Furthermore, using the correspondence between the radial null energy condition (NEC) and the holographic c-theorem, we argue that the screen cannot be arbitrarily close to the horizon. This leads to a holographic argument that black hole mining using a screen cannot significantly alter the lifetime of a black hole.

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