An Introduction to Fractional Powers of Quaternionic Operators and New Fractional Diffusion Processes

2017 ◽  
pp. 101-134 ◽  
Author(s):  
Fabrizio Colombo ◽  
Jonathan Gantner
Keyword(s):  
Diffusion Processes ◽  
Fractional Diffusion ◽  
Fractional Powers

scholarly journals Rational Krylov methods for fractional diffusion problems on graphs

2021 ◽  
Author(s):  
Michele Benzi ◽  
Igor Simunec
Keyword(s):  
Analytic Function ◽  
Diffusion Equation ◽  
Graph Laplacian ◽  
Fractional Diffusion ◽  
Singular Matrix ◽  
Krylov Methods ◽  
Fractional Powers ◽  
Directed Networks ◽  
Rank One ◽  
Diffusion Problems

AbstractIn this paper we propose a method to compute the solution to the fractional diffusion equation on directed networks, which can be expressed in terms of the graph Laplacian L as a product $$f(L^T) \varvec{b}$$ f ( L T ) b , where f is a non-analytic function involving fractional powers and $$\varvec{b}$$ b is a given vector. The graph Laplacian is a singular matrix, causing Krylov methods for $$f(L^T) \varvec{b}$$ f ( L T ) b to converge more slowly. In order to overcome this difficulty and achieve faster convergence, we use rational Krylov methods applied to a desingularized version of the graph Laplacian, obtained with either a rank-one shift or a projection on a subspace.

scholarly journals Integrated Time-Fractional Diffusion Processes for Fractional-Order Chaos-Based Image Encryption

Sensors ◽  
2021 ◽  
Vol 21 (20) ◽  
pp. 6838
Author(s):  
Fudong Ge ◽  
Zufa Qin ◽  
YangQuan Chen
Keyword(s):  
Boundary Conditions ◽  
Image Encryption ◽  
Fractional Order ◽  
Diffusion Processes ◽  
Fractional Diffusion ◽  
Spatiotemporal Chaos ◽  
Diffusion System ◽  
Encryption Algorithm ◽  
Secret Key ◽  
Image Encryption Algorithm

The purpose of this paper is to explore a novel image encryption algorithm that is developed by combining the fractional-order Chua’s system and the 1D time-fractional diffusion system of order α∈(0,1]. To this end, we first discuss basic properties of the fractional-order Chua’s system and the 1D time-fractional diffusion system. After these, a new spatiotemporal chaos-based cryptosystem is proposed by designing the chaotic sequence of the fractional-order Chua’s system as the initial condition and the boundary conditions of the studied time-fractional diffusion system. It is shown that the proposed image encryption algorithm can gain excellent encryption performance with the properties of larger secret key space, higher sensitivity to initial-boundary conditions, better random-like sequence and faster encryption speed. Efficiency and reliability of the given encryption algorithm are finally illustrated by a computer experiment with detailed security analysis.

Regional Analysis of Time-Fractional Diffusion Processes

2018 ◽  
Author(s):  
Fudong Ge ◽  
YangQuan Chen ◽  
Chunhai Kou
Keyword(s):  
Diffusion Processes ◽  
Regional Analysis ◽  
Fractional Diffusion

Estimation of change point for switching fractional diffusion processes

Stochastics ◽  
2013 ◽  
Vol 86 (3) ◽  
pp. 429-449 ◽  
Author(s):  
M.N. Mishra ◽  
B.L.S. Prakasa Rao
Keyword(s):  
Change Point ◽  
Diffusion Processes ◽  
Fractional Diffusion

scholarly journals Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data

2016 ◽  
Vol 19 (1) ◽  
Author(s):  
Bangti Jin ◽  
Raytcho Lazarov ◽  
Dongwoo Sheen ◽  
Zhi Zhou
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Error Estimates ◽  
Diffusion Processes ◽  
Fractional Diffusion ◽  
Galerkin Finite Element Method ◽  
Nonsmooth Data ◽  
Galerkin Finite Element ◽  
Semidiscrete Scheme ◽  
Distributed Order

AbstractIn this work, we consider the numerical solution of a distributed order subdiffusion model, arising in the modeling of ultra-slow diffusion processes. We develop a space semidiscrete scheme based on the Galerkin finite element method, and establish error estimates optimal with respect to data regularity in

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