20,942 research outputs found
Fractional Cauchy problems on bounded domains: survey of recent results
In a fractional Cauchy problem, the usual first order time derivative is
replaced by a fractional derivative. This problem was first considered by
\citet{nigmatullin}, and \citet{zaslavsky} in for modeling some
physical phenomena.
The fractional derivative models time delays in a diffusion process. We will
give a survey of the recent results on the fractional Cauchy problem and its
generalizations on bounded domains D\subset \rd obtained in \citet{m-n-v-aop,
mnv-2}. We also study the solutions of fractional Cauchy problem where the
first time derivative is replaced with an infinite sum of fractional
derivatives. We point out a connection to eigenvalue problems for the
fractional time operators considered. The solutions to the eigenvalue problems
are expressed by Mittag-Leffler functions and its generalized versions. The
stochastic solution of the eigenvalue problems for the fractional derivatives
are given by inverse subordinators
Inverse problem of determining time-dependent leading coefficient in the time-fractional heat equation
In this paper, we investigate direct and inverse problems for the
time-fractional heat equation with a time-dependent diffusion coefficient for
positive operators. First, we consider the direct problem, and the unique
existence of the generalized solution is established. We also deduce some
regularity results. Here, our proofs are based on the eigenfunction expansion
method. Second, we consider the inverse problem of determining the diffusion
coefficient. The well-posedness of this inverse problem is shown by reducing
the problem to an operator equation for the diffusion coefficient.Comment: arXiv admin note: text overlap with arXiv:1708.07756 by other author
Inverse source problems for degenerate time-fractional PDE
In this paper, we investigate two inverse source problems for degenerate
time-fractional partial differential equation in rectangular domains. The first
problem involves a space-degenerate partial differential equation and the
second one involves a time-degenerate partial differential equation. Solutions
to both problem are expressed in series expansions. For the first problem, we
obtained solutions in the form of Fourier-Legendre series. Convergence and
uniqueness of solutions have been discussed. Solutions to the second problem
are expressed in the form of Fourier-Sine series and they involve a generalized
Mittag- Leffler type function. Moreover, we have established a new estimate for
this generalized Mittag-Leffler type function. The obtained results are
illustrated by providing example solutions using certain given data at the
initial and final time.Comment: 12 pages, 8 figure
A new mathematical formulation for a phase change problem with a memory flux
A mathematical formulation for a one-phase change problem in a form of Stefan problem with a memory flux is obtained. The hypothesis that the integral of weighted backward fluxes is proportional to the gradient of the temperature is considered. The model that arises involves fractional derivatives with respect to time both in the sense of Caputo and of Riemann–Liouville. An integral relation for the free boundary, which is equivalent to the “fractional Stefan condition”, is also obtained.Fil: Roscani, Sabrina Dina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; ArgentinaFil: Bollati, Julieta. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; ArgentinaFil: Tarzia, Domingo Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; Argentin
Inverse Problems of Determining Sources of the Fractional Partial Differential Equations
In this chapter, we mainly review theoretical results on inverse source
problems for diffusion equations with the Caputo time-fractional derivatives of
order . Our survey covers the following types of inverse
problems: 1. determination of time-dependent functions in interior source terms
2. determination of space-dependent functions in interior source terms 3.
determination of time-dependent functions appearing in boundary condition
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