Anomalous diffusion modeling by fractal and fractional derivatives
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TLDR
The fundamental solution of the fractal derivative equation for anomalous diffusion is derived, which characterizes a clear power law, and this new model is compared with the corresponding fractional derivative model in terms of computational efficiency, diffusion velocity, and heavy tail property.Abstract:
This paper makes an attempt to develop a fractal derivative model of anomalous diffusion. We also derive the fundamental solution of the fractal derivative equation for anomalous diffusion, which characterizes a clear power law. This new model is compared with the corresponding fractional derivative model in terms of computational efficiency, diffusion velocity, and heavy tail property. The merits and distinctions of these two models of anomalous diffusion are then summarized.read more
Citations
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Fractal-fractional differentiation and integration: Connecting fractal calculus and fractional calculus to predict complex system
TL;DR: In this paper, new operators of differentiation have been introduced, such as convolution of power law, exponential decay law, and generalized Mittag-Leffler law with fractal derivative, referred as fractal-fractional differential and integral operators.
Journal ArticleDOI
Modeling attractors of chaotic dynamical systems with fractal–fractional operators
Abdon Atangana,Sania Qureshi +1 more
TL;DR: In this article, the fractal-fractional derivatives and integrals have been used to predict chaotic behavior of some attractors from applied mathematics, and the general conditions for the existence and the uniqueness of the exact solutions are obtained.
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A new fractional operator of variable order: Application in the description of anomalous diffusion
TL;DR: In this article, a new fractional operator of variable order with the use of the monotonic increasing function is proposed in sense of Caputo type, which is efficient in modeling a class of concentrations in the complex transport process.
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Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations
TL;DR: In this article, a semi-analytical method based on Adomian polynomials and a fractional Taylor series was proposed to investigate chaotic behavior and stability of fractional differential equations within a new generalized Caputo derivative.
References
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Posted Content
The fundamental solution of the space-time fractional diffusion equation
TL;DR: In this paper, the Cauchy problem for the space-time fractional diffusion equation is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation.