From power laws to fractional diffusion processes with and without external forces, the non direct way
Abstract
In this paper, a wide view on the theory of the continuous time random walk (CTRW) and its relations to the space–time fractional diffusion process is given. We begin from the basic model of CTRW (Montroll and Weiss [19], 1965) that also can be considered as a compound renewal process. We are interested in studying the random walks in which the probability distributions of the waiting times and jumps have fat tails characterized by power laws with exponent between 0 and 1 for the waiting times, between 0 and 2 for the jumps. We prove the relation between the integral equation of the CTRW having the above fat tails waiting and jump width distributions and the space–time fractional diffusion equations in the Laplace–Fourier domain. The space–time fractional Fokker–Planck equation could also be driven from the discrete Ehren–Fest model and is represented by the theory of CTRW. These space–time fractional diffusion processes are getting increasing popularity in applications in physics, chemistry, finance, biology, medicine and many other fields. The asymptotic behavior of the Mittag–Leffler function plays a significant rule on simulating these models. The behaviors of the studied CTRW models are well approximated and visualized by simulating various types of random walks by using the Monte Carlo method.
Acknowledgements
This paper is written for the memory of Prof. Rudolf Gorenflo. He was my Ph.D. Supervisor (2001–2004). I began studying the simulation of the CTRW under his supervision and he gave me a great help and knowledge. In this paper, I continue developing the simulation of the CTRW models that are mathematically modelled by the space–time fractional diffusion processes with and without external forces.
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© 2019 Diogenes Co., Sofia
Articles in the same Issue
- Frontmatter
- Editorial Note
- FCAA related news, events and books (FCAA–volume 22–1–2019)
- Survey Paper
- Ranking the scientific output of researchers in fractional calculus
- A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications
- Research Paper
- From power laws to fractional diffusion processes with and without external forces, the non direct way
- Homogeneous robin boundary conditions and discrete spectrum of fractional eigenvalue problem
- The numerical algorithms for discrete Mittag-Leffler functions approximation
- New interpretation of fractional potential fields for robust path planning
- Stable distributions and green’s functions for fractional diffusions
- Fractional calculus in economic growth modelling of the group of seven
- Relationship between controllability and observability of standard and fractional different orders discrete-time linear system
- Optimal control of linear systems of arbitrary fractional order
- Fractional impulsive differential equations: Exact solutions, integral equations and short memory case
- Fractional-order modelling and parameter identification of electrical coils
- Fractional-order value identification of the discrete integrator from a noised signal. part I
Articles in the same Issue
- Frontmatter
- Editorial Note
- FCAA related news, events and books (FCAA–volume 22–1–2019)
- Survey Paper
- Ranking the scientific output of researchers in fractional calculus
- A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications
- Research Paper
- From power laws to fractional diffusion processes with and without external forces, the non direct way
- Homogeneous robin boundary conditions and discrete spectrum of fractional eigenvalue problem
- The numerical algorithms for discrete Mittag-Leffler functions approximation
- New interpretation of fractional potential fields for robust path planning
- Stable distributions and green’s functions for fractional diffusions
- Fractional calculus in economic growth modelling of the group of seven
- Relationship between controllability and observability of standard and fractional different orders discrete-time linear system
- Optimal control of linear systems of arbitrary fractional order
- Fractional impulsive differential equations: Exact solutions, integral equations and short memory case
- Fractional-order modelling and parameter identification of electrical coils
- Fractional-order value identification of the discrete integrator from a noised signal. part I
