Distributed order fractional diffusion equation with fractional Laplacian in axisymmetric cylindrical configuration

Asymptotic analysis of fundamental solution of multi-dimensional distributed-order time-fractional diffusion equation with unit density function

“…Here, we intend to represent the fractional moment of the fundamental solution of (1.1) in the Laplace domain. Let us start with the relation (1.4) and follow the approach of [6,50] for defining the fractional moment by the Mellin transform of ū(|x n |, s) as…”

“…is a modification of sub-diffusion and has attracted a lot attention by many researchers in the last two decades. In the literature, this equation relates to the initial works [13,14,[20][21][22][23][24], and provides new mathematical and theoretical aspects for studying of the physical and natural sciences in different cases of b(σ) and fractional operators using the analytical and numerical methods [6,8,9,18,25,27,34,35,44,45,47,49,50].…”

Numerical solution of multidimensional time‐space fractional differential equations of distributed order with Riesz derivative

“…To fill this gap, the distributed order FDEs are introduced by Caputo [19] to define the processes getting more anomalous. Because of the reasons mentioned above, there has been a big interest of using FDEs with distributed order in control system, physics, and signal processing [20][21][22]. Especially, examples of areas for the use of distributed order fractional derivatives include applications in anomalous diffusion and relaxation phenomena, in which diffusion exponent varies with time [23].…”

An efficient numerical method for the time-fractional distributed order nonlinear Klein–Gordon equation with shifted fractional Gegenbauer multi-wavelets method

“…Therefore, it is absolutely necessary to pay attention to the use of numerical methods. Many authors and researchers have been proposed different numerical methods for the numerical solution of partial differential equations with fractional distributed order operators, such as standard quadrature method [15], implicit finite difference method [16], compact difference method [17], implicit numerical method [18], weighted and shifted Grünwald difference method [19], finite element method [20], Chebyshev collocation method [21], Petrov-Galerkin and spectral collocation methods [22], mid-point quadrature method [23], Legendre wavelets method [24], improved meshless method [25], finite volume method [26] and combination of alternating direction implicit difference, Laplace transform and Hankel transform [27], matrix transfer technique [28], fourth-order compact difference scheme [29], Chebyshev cardinal polynomials [30], and extrapolation method [31]. Recently, the use of wavelets has led to better numerical methods in fractional calculus.…”