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To overcome the weakness of a total variation based model for image restoration, various high order (typically second order) regularization models have been proposed and studied recently. In this paper we analyze and test a fractional-order…
Fractional differential equations (FDEs) are an extension of the theory of fractional calculus. However, due to the difficulty in finding analytical solutions, there have not been extensive applications of FDEs until recent decades. With…
In this work we investigate an inverse problem of recovering a time-dependent potential in a semilinear subdiffusion model from an integral measurement of the solution over the domain. The model involves the Djrbashian--Caputo fractional…
In the recent literature, the g-subdiffusion equation involving Caputo fractional derivatives with respect to another function has been studied in relation to anomalous diffusions with a continuous transition between different subdiffusive…
The fundamental solution of the fractional diffusion equation of distributed order in time (usually adopted for modelling sub-diffusion processes) is obtained based on its Mellin-Barnes integral representation. Such solution is proved to be…
This article concerns second-order time discretization of subdiffusion equations with time-dependent diffusion coefficients. High-order differentiability and regularity estimates are established for subdiffusion equations with…
This article deals with the uniqueness in identifying multiple parameters simultaneously in the one-dimensional time-fractional diffusion-wave equation of fractional time-derivative order $\in (0,2)$ with the zero Robin boundary condition.…
We investigate the behavior of the time derivatives of the solution to a linear time-fractional, advection-diffusion-reaction equation, allowing space- and time-dependent coefficients as well as initial data that may have low regularity.…
This paper establishes explicit solutions for fractional diffusion problems on bounded domains. It also gives stochastic solutions, in terms of Markov processes time-changed by an inverse stable subordinator whose index equals the order of…
In this paper we present numerical methods - finite differences and finite elements - for solution of partial differential equation of fractional order in time for one-dimensional space. This equation describes anomalous diffusion which is…
In this work, an inverse problem in the fractional diffusion equation with random source is considered. The measurements used are the statistical moments of the realizations of single point data $u(x_0,t,\omega).$ We build the…
We study the inverse problem of recovering the order and the diffusion coefficient of an elliptic fractional partial differential equation from a finite number of noisy observations of the solution. We work in a Bayesian framework and show…
In this paper, a high-order approximation to Caputo-type time-fractional diffusion equations involving an initial-time singularity of the solution is proposed. At first, we employ a numerical algorithm based on the Lagrange polynomial…
In this work, we propose an observation system based on the available data which solution is one-be-one mapping to the forward problem(with the unknown initial function) solution. It implies their solutions share the same linear structure…
In this paper, a higher order finite difference scheme is proposed for Generalized Fractional Diffusion Equations (GFDEs). The fractional diffusion equation is considered in terms of the generalized fractional derivatives (GFDs) which uses…
In this paper the inverse problem of determining the fractional orders in mixed-type equations is considered. In one part of the domain the considered equation is the subdiffusion equation with a fractional derivative in the sense of…
This article is in continuation of our earlier article [37] in which computational solution of an unified reaction-diffusion equation of distributed order associated with Caputo derivatives as the time-derivative and Riesz-Feller derivative…
In the present article, we study the diffusion equations with fractional time derivatives. The aim of this paper is to investigate the best possible regularity for the initial value/boundary value problems with non-homogeneous Dirichlet…
The solution of a Caputo time fractional diffusion equation of order $0<\alpha<1$ is expressed in terms of the solution of a corresponding integer order diffusion equation. We demonstrate a linear time mapping between these solutions that…
In this paper, we study an inverse problem for identifying the initial value in a space-time fractional diffusion equation from the final time data. We show the identifiability of this inverse problem by proving the existence of its unique…