Analysis of a hidden memory variably distributed-order space-fractional diffusion equation
type: Item
instance of: scholarly article
title: Analysis of a hidden memory variably distributed-order space-fractional diffusion equation
author: Jinhong Jia
author: Hong Wang
published in: Applied Mathematics Letters
publication date: 2021-12-13T00:00:00Z
review text: The wellposedness of a space-fractional diffusion equation governed by a hidden memory variably distributed-order fractional derivative was considered in the work. Recall that according to J. Hadamard, a fractional diffusion equation is wellposed if it satisfies the following conditions: i) a solution to the equation exists (existence problem); ii) the solution is uniqueness (uniqueness problem); and iii) the solution's behaviour changes continuously with respect to the boundary conditions (stability problem). The authors made two assumptions on the probability density function (\(\omega\)): \begin{itemize} \item[(i)] \(\operatorname{supp}\omega (\cdot, x) \in [0,\bar{\gamma}(x)]\) for \(0 \leq \bar{\gamma}(x) \leq \gamma *\) with \(0<\gamma * <1\), \(x\in [0,1]\). \item[(ii)] \(\bar{\gamma} \in C^1[0,\gamma ^*]\) and \(\omega, \ \omega _x \in C([0,\gamma ^*];[0,1])\). \end{itemize} The hidden memory variably distributed-order space-fractional diffusion equation was first transformed into into a Volterra integral equation before integrating in order to its solution. To address the uniqueness and stability problems, the authors stated and proved Theorem 2.1 and Theorem 2.2. In Theorem 2.1, which lays the foundation for the uniqueness of the solution, the authors defined a functional sequence \(\{v_n(x)\}_{n=0}^{\infty}\) on \([0,1]\) by \(v_{n+1}(x):=v_0(x)-d(x) _0I_{x}^{\omega}v_n(x)\), \(\geq 0\), \(v_0(x):=-f(x)\) and showed that this sequence converges uniformly on \([0,1]\). The uniqueness follows from this result, although the proof is skipped by the authors. Theorem 2.2 shows that the solution is uniformly continuous, thereby addressing the stability problem.
reviewed by: Ogbu F. Imaga
Mathematics Subject Classification ID: 34A08
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 45D05
zbMATH DE Number: 7443282
zbMATH Keywords: space-fractional diffusion equation
zbMATH Keywords: hidden memory
zbMATH Keywords: variably distributed-order fractional derivative
zbMATH Keywords: wellposedness
MaRDI profile type: Publication
full work available at URL: j.aml.2021.107617
OpenAlex ID: W3195365920
cites work: A fast finite difference method for distributed-order space-fractional partial differential equations on convex domains
cites work: Preconditioned Iterative Methods for Two-Dimensional Space-Fractional Diffusion Equations
cites work: Variable order and distributed order fractional operators
cites work: The random walk's guide to anomalous diffusion: A fractional dynamics approach
cites work: Q4233638
cites work: Adaptive finite element method for fractional differential equations using hierarchical matrices
cites work: Numerical analysis for distributed-order differential equations
cites work: Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density
cites work: Analyticity of solutions to a distributed order time-fractional diffusion equation and its application to an inverse problem
cites work: A Generalized Spectral Collocation Method with Tunable Accuracy for Variable-Order Fractional Differential Equations
cites work: Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data
cites work: Well-posedness of general Caputo-type fractional differential equations
cites work: Error Analysis of a Finite Difference Method on Graded Meshes for a Time-Fractional Diffusion Equation
cites work: Wellposedness and regularity of the variable-order time-fractional diffusion equations
cites work: A variably distributed-order time-fractional diffusion equation: analysis and approximation
cites work: A Hidden-Memory Variable-Order Time-Fractional Optimal Control Model: Analysis and Approximation
cites work: An Error Estimate of a Numerical Approximation to a Hidden-Memory Variable-Order Space-Time Fractional Diffusion Equation
cites work: Weakly singular Gronwall inequalities and applications to fractional differential equations
DOI: 10.1016/J.AML.2021.107617
description: scientific article; zbMATH DE number 7443282