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In this work we investigate an inverse problem of recovering point sources and their time-dependent strengths from {a posteriori} partial internal measurements in a subdiffusion model which involves a Caputo fractional derivative in time…
A fractional derivative is a temporally nonlocal operation which is computationally intensive due to inclusion of the accumulated contribution of function values at past times. In order to lessen the computational load while maintaining the…
In this work, we consider the numerical recovery of a spatially dependent diffusion coefficient in a subdiffusion model from distributed observations. The subdiffusion model involves a Caputo fractional derivative of order $\alpha\in(0,1)$…
Recovering high-dimensional signals from corrupted measurements is a central challenge in inverse problems. Recent advances in generative diffusion models have shown remarkable empirical success in providing strong data-driven priors, but…
We present a numerical procedure of solving the subdiffusion equation with Caputo fractional time derivative. On the basis of few examples we show that the subdiffusion is a 'long time memory' process and the short memory principle should…
In this work, we investigate the recovery of a parameter in a diffusion process given by the order of derivation in time for a class of diffusion type equations, including both classical and time-fractional diffusion equations, from the…
An adaptive finite difference scheme for variable-order fractional-time subdiffusion equations in the Caputo form is studied. The fractional time derivative is discretized by the L1 procedure but using nonhomogeneous timesteps. The size of…
We introduced and analyzed robust recovery-based a posteriori error estimators for various lower order finite element approximations to interface problems in [9, 10], where the recoveries of the flux and/or gradient are implicit (i.e.,…
In this paper, we are interested in the study of a problem with fractional derivatives having boundary conditions of integral types. The problem represents a Caputo type advection-diffusion equation where the fractional order derivative…
An implicit finite difference method with non-uniform timesteps for solving the fractional diffusion equation in the Caputo form is proposed. The method allows one to build adaptive methods where the size of the timesteps is adjusted to the…
Recently it has been shown that using diffusion models for inverse problems can lead to remarkable results. However, these approaches require a closed-form expression of the degradation model and can not support complex degradations. To…
The reconstruction of unsteady flow fields from limited measurements is a challenging and crucial task for many engineering applications. Machine learning models are gaining popularity for solving this problem due to their ability to learn…
In this work, we study the inverse problem of determining a potential coefficient in an abstract wave equation that includes a lower-order term. The equation incorporates a time-fractional derivative in the Caputo sense, as well as a…
In this article we study inverse problems of recovering a space-time dependent source component from the lateral boundary observation in a subidffusion model. The mathematical model involves a Djrbashian-Caputo fractional derivative of…
The subdiffusion equations with a Caputo fractional derivative of order $\alpha \in (0,1)$ arise in a wide variety of practical problems, which is describing the transport processes, in the force-free limit, slower than Brownian diffusion.…
We investigate the regional gradient observability of fractional sub-diffusion equations involving the Caputo derivative. The problem consists of describing a method to find and recover the initial gradient vector in the desired region,…
We present an image reconstruction algorithm for the Inverse Conductivity Problem based on reformulating the problem in terms of integral equations. We use as data the values of injected electric currents and of the corresponding induced…
Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately…
In this paper, we develop a numerical algorithm for an inverse problem on determining fractional orders of time derivatives simultaneously in a coupled subdiffusion system. Following the theoretical uniqueness, we reformulate the order…
In this note we analyse the propagation of a small density perturbation in a one-dimensional compressible fluid by means of fractional calculus modelling, replacing thus the ordinary time derivative with the Caputo fractional derivative in…