Subdiffusion equation with fractional Caputo time derivative with respect to another function in modeling transition from ordinary subdiffusion to superdiffusion

Abstract

We use a subdiffusion equation with fractional Caputo time derivative with respect to another function $g$ ($g$--subdiffusion equation) to describe a smooth transition from ordinary subdiffusion to superdiffusion. Ordinary subdiffusion is described by the equation with the ``ordinary'' fractional Caputo time derivative, superdiffusion is described by the equation with a fractional Riesz type spatial derivative. We find the function $g$ for which the solution (Green's function, GF) to the $g$--subdiffusion equation takes the form of GF for ordinary subdiffusion in the limit of small time and GF for superdiffusion in the limit of long time. To solve the $g$--subdiffusion equation we use the $g$--Laplace transform method. It is shown that the scaling properties of the GF for $g$--subdiffusion and the GF for superdiffusion are the same in the long time limit. We conclude that for a sufficiently long time the $g$--subdiffusion equation describes superdiffusion well, despite a different stochastic interpretation of the processes. Then, paradoxically, a subdiffusion equation with a fractional time derivative describes superdiffusion. The superdiffusive effect is achieved here not by making anomalously long jumps by a diffusing particle, but by greatly increasing the particle jump frequency which is derived by means of the $g$--continuous time random walk model. The $g$--subdiffusion equation is shown to be quite general, it can be used in modeling of processes in which a kind of diffusion change continuously over time. In addition, some methods used in modeling of ordinary subdiffusion processes, such as the derivation of local boundary conditions at a thin partially permeable membrane, can be used to model $g$--subdiffusion processes, even if this process is interpreted as superdiffusion.

arXiv comment18 pagres, 8 figures

References in corpus (55)

• Active Particles in Complex and Crowded Environments
• Self-motile colloidal particles: from directed propulsion to random walk
• A Caputo fractional derivative of a function with respect to another function
• In vivo anomalous diffusion and weak ergodicity breaking of lipid granules
• From Diffusion to Anomalous Diffusion: A Century after Einstein's Brownian Motion
• Retarding Sub- and Accelerating Super-Diffusion Governed by Distributed Order Fractional Diffusion Equations
• Brownian yet non-Gaussian diffusion: from superstatistics to subordination of diffusing diffusivities
• Levy Flights in External Force Fields: Langevin and Fractional Fokker-Planck Equations, and their Solutions
• Anomalous dynamics of cell migration
• Enhanced diffusion due to active swimmers at a solid surface
• A practical guide to Prabhakar fractional calculus
• First passage and arrival time densities for Lévy flights and the failure of the method of images
• Fractional Fokker-Planck Equation for Ultraslow Kinetics
• Paradigm shift in diffusion-mediated surface phenomena
• Thermodynamics and Fractional Fokker-Planck Equations
• Is diffusion anomalous in two-dimensional Yukawa liquids?
• How to Measure Subdiffusion Parameters
• Fractional Langevin Equation of Distributed Order
• First-passage and first-hitting times of Levy flights and Levy walks
• Lévy flights versus Lévy walks in bounded domains
• Brownian Motion of Boomerang Colloidal Particles
• Cytoskeletal network morphology regulates intracellular transport dynamics
• Paradoxical diffusion: Discriminating between normal and anomalous random walks
• Paradoxes of Subdiffusive Infiltration in Disordered Systems
• Subdiffusion equation with Caputo fractional derivative with respect to another function
• Continuous-time random-walk model for anomalous diffusion in expanding media
• On relation between generalized diffusion equations and subordination schemes
• From the solutions of diffusion equation to the solutions of subdiffusive one
• Diffusion of antibiotics through a biofilm in the presence of diffusion and absorption barriers
• Characteristic time scales for diffusion processes through layers and across interfaces
• Subordinated diffusion and CTRW asymptotics
• The fractional Dodson diffusion equation: a new approach
• The Unequal Twins - Probability Distributions Aren't Everything
• Asymptotic behavior of the solution of the space dependent variable order fractional diffusion equation: ultra-slow anomalous aggregation
• Fractional Fokker-Planck equation from non-singular kernel operators
• Models for characterizing the transition among anomalous diffusions with different diffusion exponents
• Continuous-time random walk with a superheavy-tailed distribution of waiting times
• Accelerating and Retarding Anomalous Diffusion
• Random walk model of subdiffusion in a system with a thin membrane
• Cattaneo--type subdiffusion--reaction equation
• Active Matter in Lateral Parabolic Confinement: From Subdiffusion to Superdiffusion
• Model of anomalous diffusion--absorption process in a system consisting of two different media separated by a thin membrane
• Application of the Efros theorem to the function represented by the inverse Laplace transform of $s^{-\mu }\exp (-s^{\nu })$
• Anomalous diffusion and generalized Sparre-Andersen scaling
• Probability distribution functions of sub- and super-diffusive systems
• Subdiffusion--reaction process with $A⟶B$ reactions versus subdiffusion--reaction process with $A+B⟶B$ reactions
• Space-Time Duality and High-Order Fractional Diffusion
• Distributed order reaction-diffusion systems associated with Caputo derivatives
• Stochastic foundations of $g$-subdiffusion process
• Subdiffusion equation with Caputo fractional derivative with respect to another function in modelling diffusion in a complex system consisting of matrix and channels
• Boundary conditions at a thin membrane for normal diffusion equation which generate subdiffusion
• Wright functions of the second kind and Whittaker functions
• Inhomogeneous parametric scaling and variable-order fractional diffusion equations
• First passage time for $g$--subdiffusion process of vanishing particles
• Subdiffusion in a system with a partially permeable partially absorbing wall

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