TEMPERED FRACTIONAL CALCULUS

By Farzad Sabzikar, Mark M. Meerschaert, Jinghua Chen. Journal of Computational Physics, 2015

422 citations recorded by the source corpus.

Fractional derivatives and integrals are convolutions with a power law. Multiplying by an exponential factor leads to tempered fractional derivatives and integrals. Tempered fractional diffusion equations, where the usual second derivative in space is replaced by a tempered fractional derivative, govern the limits of random walk models with an exponentially tempered power law jump distribution. The limiting tempered stable probability densities exhibit semi-heavy tails, which are commonly observed in finance. Tempered power law waiting times lead to tempered fractional time derivatives, which have proven useful in geophysics. The tempered fractional derivative or integral of a Brownian motion, called a tempered fractional Brownian motion, can exhibit semi-long range dependence. The increments of this process, called tempered fractional Gaussian noise, provide a useful new stochastic model for wind speed data. A tempered difference forms the basis for numerical methods to solve tempered fractional diffusion equations, and it also provides a useful new correlation model in time series.

Subfields: Economics and Econometrics, Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/1ef6935b088f41df03f295d239de58af8066f336

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Economics and Econometrics

1. Arran Fernandez
2. G. Em Karniadakis
3. Mark M. Meerschaert
4. Mamikon A. Gulian
5. Hafiz Muhammad Fahad

In Finance

1. Arran Fernandez
2. G. Em Karniadakis
3. Dumitru Băleanu
4. Mark M. Meerschaert
5. Mamikon A. Gulian

In Modeling and Simulation

1. Arran Fernandez
2. G. Em Karniadakis
3. Dumitru Băleanu
4. Guofei Pang
5. Lijing Zhao
6. Zhiping Mao
7. Weihua Deng
8. Hafiz Muhammad Fahad

Citations in the corpus

• Weighted Fractional Calculus: A General Class of Operators (2022, background): “1, are precisely those of tempered fractional calculus, defined [17, 18] as follows:”
• On Fractional Operators and Their Classifications (2019, background): “• A model known as tempered fractional calculus [44,45], utilises an integral transform with the product of a power function and an exponential function in the kernel:”
• Well-posedness and numerical algorithm for the tempered fractional differential equations (2019, background): “Then the space fractional operator will be replaced by the spatially tempered fractional operator in the corresponding models [9, 10, 39].”
• What is the fractional Laplacian? A comparative review with new results (2018, background): “Recently, tempered fractional calculus has been developed to avoid this issue [42].”