Continuous time random walks and space-time fractional differential equations
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Mark Meerschaert
Abstract
The continuous time random walk is a model from statistical physics that elucidates the physical interpretation of the space-time fractional diffusion equation. In this model, each step in the random walk is separated by a random waiting time. The long-time limit of this model is governed by a fractional diffusion equation. If the step length of the random walk follows a power law, we get a space fractional diffusion equation. If the waiting times also follow a power law, we get a space-time fractional diffusion equation. The index of the power law equals the order of the fractional derivative. If the waiting times and jumps are dependent random variables, the governing equation involves coupled space-time fractional derivatives.
Abstract
The continuous time random walk is a model from statistical physics that elucidates the physical interpretation of the space-time fractional diffusion equation. In this model, each step in the random walk is separated by a random waiting time. The long-time limit of this model is governed by a fractional diffusion equation. If the step length of the random walk follows a power law, we get a space fractional diffusion equation. If the waiting times also follow a power law, we get a space-time fractional diffusion equation. The index of the power law equals the order of the fractional derivative. If the waiting times and jumps are dependent random variables, the governing equation involves coupled space-time fractional derivatives.
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents VII
- Recent history of the fractional calculus: data and statistics 1
- Basic FC operators and their properties 23
- Mathematical and physical interpretations of fractional derivatives and integrals 47
- Generalized fractional calculus operators with special functions 87
- General fractional calculus 111
- Multiple Erdélyi–Kober integrals and derivatives as operators of generalized fractional calculus 127
- Fractional Laplace operator and its properties 159
- Applications of the Mellin integral transform technique in fractional calculus 195
- Fractional Fourier transform 225
- The Wright function and its applications 241
- Mittag-Leffler function: properties and applications 269
- Asymptotics of the special functions of fractional calculus 297
- Analysis of fractional integro-differential equations of thermistor type 327
- A survey on fractional variational calculus 347
- Variational principles with fractional derivatives 361
- Continuous time random walks and space-time fractional differential equations 385
- Inverse subordinators and time fractional equations 407
- Spectral theory of fractional order integration operators, their direct sums, and similarity problem to these operators of their weak perturbations 427
- Fractional differentiation in p-adic analysis 461
- Index 473
Kapitel in diesem Buch
- Frontmatter I
- Preface V
- Contents VII
- Recent history of the fractional calculus: data and statistics 1
- Basic FC operators and their properties 23
- Mathematical and physical interpretations of fractional derivatives and integrals 47
- Generalized fractional calculus operators with special functions 87
- General fractional calculus 111
- Multiple Erdélyi–Kober integrals and derivatives as operators of generalized fractional calculus 127
- Fractional Laplace operator and its properties 159
- Applications of the Mellin integral transform technique in fractional calculus 195
- Fractional Fourier transform 225
- The Wright function and its applications 241
- Mittag-Leffler function: properties and applications 269
- Asymptotics of the special functions of fractional calculus 297
- Analysis of fractional integro-differential equations of thermistor type 327
- A survey on fractional variational calculus 347
- Variational principles with fractional derivatives 361
- Continuous time random walks and space-time fractional differential equations 385
- Inverse subordinators and time fractional equations 407
- Spectral theory of fractional order integration operators, their direct sums, and similarity problem to these operators of their weak perturbations 427
- Fractional differentiation in p-adic analysis 461
- Index 473
