Fractional Behaviours Modelling

Sabatier, Jocelyn; Farges, Christophe; Tartaglione, Vincent

doi:10.1007/978-3-030-96749-9

Cited by 10 publications

(2 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In recent years, fractional calculus and fractional differential equations (FDEs) can be used to model many nonlocal phenomena in lots of scientific and engineering fields, see [1][2][3][4][5] for examples. However, these nonlocal phenomena are often affected by some stochastic noises.…”

Section: Introductionmentioning

confidence: 99%

Well-posedness and an Euler-Maruyama method for multi-term caputo tempered fractional stochastic differential equations

Huang,

Shao,

Liu

2024

Phys. Scr.

View full text Add to dashboard Cite

In this paper, we first prove the existence, uniqueness and continuity dependence on the initial value of multi-term Caputo tempered fractional stochastic differential equations with initial value. Then, an Euler-Maruyama (EM) method is presented for solving the considered equation. The strong convergence of the presented EM method is strictly investigated with the order to be $\min\{\alpha_m-\alpha_{m-1},\alpha_{m}-0.5\}$ with $0<\alpha_{1}<\alpha_{2}<\cdots<\alpha_{m-1}<\alpha_{m}<1$, $\alpha_{m}>0.5$, and $m$ is a given positive integer. Finally, three numerical examples are provided to support our theoretical findings.

show abstract

Section: Introductionmentioning

confidence: 99%

Well-posedness and an Euler-Maruyama method for multi-term caputo tempered fractional stochastic differential equations

Huang,

Shao,

Liu

2024

Phys. Scr.

View full text Add to dashboard Cite

show abstract

“…The recent publication of the book [1] where it was demonstrated that "fractal models" should not be confounded with "fractal" behaviors, which are seen in many physical phenomena, substantially influenced the creation of this paper. As an illustration of this point of view, several alternative modeling tools are proposed in [2,3] to model real powerlaw behaviors (also denoted "fractional behaviors").…”

Section: Introduction and Formulation Of The Problemmentioning

confidence: 99%

Can Self-Similarity Processes Be Reflected by the Power-Law Dependencies?

Nigmatullin

Sabatier

2023

Algorithms

View full text Add to dashboard Cite

This work was greatly influenced by the opinions of one of the authors (JS), who demonstrated in a recent book that it is important to distinguish between “fractal models” and “fractal” (power-law) behaviors. According to the self-similarity principle (SSP), the authors of this study completely distinguish between independent “fractal” (power-law) behavior and the “fractal models”, which result from the solution of equations incorporating non-integer differentiation/integration operators. It is feasible to demonstrate how many random curves resemble one another and how they can be predicted by functions with real and complex-conjugated power-law exponents. Bellman’s inequality can be used to demonstrate that the generalized geometric mean, not the arithmetic mean, which is typically recognized as the fundamental criterion in the signal processing field, corresponds to the global fitting minimum. To highlight the efficiency of the proposed algorithms, they are applied to two sets of data: one without a clearly expressed power-law behavior, the other containing clear power-law dependence.

show abstract

Initial‐boundary value and interface problems on the real half line for the fractional advection–diffusion‐type equation

Mariano‐Morales,

Vivas‐Cruz,

Taneco‐Hernández

2024

Math Methods in App Sciences

View full text Add to dashboard Cite

We use the unified transform method (UTM) to solve initial‐boundary value problems for the fractional advection–diffusion‐type equation (FADE) on the real half line. We generalize this equation using the modified definition of the Atangana–Baleanu fractional derivative of order in order to satisfy the initial condition. A solution methodology is proposed when the UTM is implemented in fractional differential equations with boundary conditions of Dirichlet and Robin type, in particular when using the modified definitions of fractional operators with non‐singular kernel. In addition, an interface problem is stated and solved in the adjacent domains and , where perfect contact continuity conditions are imposed. The exact solutions obtained include as a particular case, the diffusion and advection–diffusion equations with integer‐order derivatives. Finally, representative curves for the solution are shown by varying the fractional order.

show abstract

Fractional Behaviours Modelling

Cited by 10 publications

References 0 publications

Well-posedness and an Euler-Maruyama method for multi-term caputo tempered fractional stochastic differential equations

Well-posedness and an Euler-Maruyama method for multi-term caputo tempered fractional stochastic differential equations

Can Self-Similarity Processes Be Reflected by the Power-Law Dependencies?

Initial‐boundary value and interface problems on the real half line for the fractional advection–diffusion‐type equation

Contact Info

Product

Resources

About