Fractional Diffusion Equation: Variations and Applications

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (31 December 2022) | Viewed by 3153

Special Issue Editors


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Guest Editor
National Research University “Moscow Power Engineering Institute”, Krasnokazarmennaya St. 14, 111250 Moscow, Russia
Interests: nonlinear dynamics; phase transitions; radiation

Special Issue Information

Dear Colleagues,

Anomalous diffusion is a process of a chaotic motion in which the mean square displacement of a particle is non-linear with time. Anomalous diffusion can be observed in many systems, and this physical phenomenon requires a specific mathematical description. The fractional diffusion equation is the most promising way to mathematically describe anomalous diffusion.

In this Special Issue, we will explore the application of fractional calculus to the diffusion process. A fractional diffusion equation may contain various forms of the fractional operators that can be discussed in this Special Issue. Additionally, the complex transfer process can generally be described with a fractional wave-diffusion equation, which is also a topic of this Special Issue. Papers concerning the use of numerical schemes for fractional wave-diffusion equations are also welcome.

The topics for this Special Issue include (but are not limited to):

  • The derivation of fractional diffusion equation for various problems;
  • Fractional derivative operators, including fractional derivatives of variable order;
  • Analytical and numerical solutions of fractional wave-diffusion equations;
  • The discretization of fractional wave-diffusion equations;
  • The application of fractional diffusion equations in physics and engineering;
  • Origin of anomalous diffusion.

Dr. Denis N. Gerasimov
Prof. Dr. Jordan Hristov
Guest Editors

Manuscript Submission Information

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Keywords

  • fractional wave-diffusion equation
  • fractional derivatives
  • anomalous diffusion

Published Papers (2 papers)

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Research

21 pages, 3973 KiB  
Article
RBF-Based Local Meshless Method for Fractional Diffusion Equations
by Kamran, Muhammad Irfan, Fahad M. Alotaibi, Salma Haque, Nabil Mlaiki and Kamal Shah
Fractal Fract. 2023, 7(2), 143; https://doi.org/10.3390/fractalfract7020143 - 02 Feb 2023
Cited by 16 | Viewed by 1319
Abstract
The fractional diffusion equation is one of the important recent models that can efficiently characterize various complex diffusion processes, such as in inhomogeneous or heterogeneous media or in porous media. This article provides a method for the numerical simulation of time-fractional diffusion equations. [...] Read more.
The fractional diffusion equation is one of the important recent models that can efficiently characterize various complex diffusion processes, such as in inhomogeneous or heterogeneous media or in porous media. This article provides a method for the numerical simulation of time-fractional diffusion equations. The proposed scheme combines the local meshless method based on a radial basis function (RBF) with Laplace transform. This scheme first implements the Laplace transform to reduce the given problem to a time-independent inhomogeneous problem in the Laplace domain, and then the RBF-based local meshless method is utilized to obtain the solution of the reduced problem in the Laplace domain. Finally, Stehfest’s method is utilized to convert the solution from the Laplace domain into the real domain. The proposed method uses Laplace transform to handle the fractional order derivative, which avoids the computation of a convolution integral in a fractional order derivative and overcomes the effect of time-stepping on stability and accuracy. The method is tested using four numerical examples. All the results demonstrate that the proposed method is easy to implement, accurate, efficient and has low computational costs. Full article
(This article belongs to the Special Issue Fractional Diffusion Equation: Variations and Applications)
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25 pages, 390 KiB  
Article
Mild Solution for the Time-Fractional Navier–Stokes Equation Incorporating MHD Effects
by Ramsha Shafqat, Azmat Ullah Khan Niazi, Mehmet Yavuz, Mdi Begum Jeelani and Kiran Saleem
Fractal Fract. 2022, 6(10), 580; https://doi.org/10.3390/fractalfract6100580 - 10 Oct 2022
Cited by 12 | Viewed by 1191
Abstract
The Navier–Stokes (NS) equations involving MHD effects with time-fractional derivatives are discussed in this paper. This paper investigates the local and global existence and uniqueness of the mild solution to the NS equations for the time fractional differential operator. In addition, we work [...] Read more.
The Navier–Stokes (NS) equations involving MHD effects with time-fractional derivatives are discussed in this paper. This paper investigates the local and global existence and uniqueness of the mild solution to the NS equations for the time fractional differential operator. In addition, we work on the regularity effects of such types of equations which are caused by MHD flow. Full article
(This article belongs to the Special Issue Fractional Diffusion Equation: Variations and Applications)
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