523 research outputs found
Stable L\'{e}vy diffusion and related model fitting
A fractional advection-dispersion equation (fADE) has been advocated for
heavy-tailed flows where the usual Brownian diffusion models fail. A stochastic
differential equation (SDE) driven by a stable L\'{e}vy process gives a forward
equation that matches the space-fractional advection-dispersion equation and
thus gives the stochastic framework of particle tracking for heavy-tailed
flows. For constant advection and dispersion coefficient functions, the
solution to such SDE itself is a stable process and can be derived easily by
least square parameter fitting from the observed flow concentration data.
However, in a more generalized scenario, a closed form for the solution to a
stable SDE may not exist. We propose a numerical method for solving/generating
a stable SDE in a general set-up. The method incorporates a discretized finite
volume scheme with the characteristic line to solve the fADE or the forward
equation for the Markov process that solves the stable SDE. Then we use a
numerical scheme to generate the solution to the governing SDE using the fADE
solution. Also, often the functional form of the advection or dispersion
coefficients are not known for a given plume concentration data to start with.
We use a Levenberg--Marquardt (L-M) regularization method to estimate advection
and dispersion coefficient function from the observed data (we present the case
for a linear advection) and proceed with the SDE solution construction
described above.Comment: Published at https://doi.org/10.15559/18-VMSTA106 in the Modern
Stochastics: Theory and Applications (https://vmsta.org/) by VTeX
(http://www.vtex.lt/
Inverse Problems of Determining Coefficients of the Fractional Partial Differential Equations
When considering fractional diffusion equation as model equation in analyzing
anomalous diffusion processes, some important parameters in the model, for
example, the orders of the fractional derivative or the source term, are often
unknown, which requires one to discuss inverse problems to identify these
physical quantities from some additional information that can be observed or
measured practically. This chapter investigates several kinds of inverse
coefficient problems for the fractional diffusion equation
Ecuaciones diferenciales fraccionarias y problemas inversos.
DiagramasOur goal is the study of identification problems in the framework of transport equations with
fractional derivatives. We consider time fractional diffusion equations and space fractional
advection dispersion equations. The majority of inverse problems are ill-posed and require
regularization. In this thesis we implement one and two dimensional discrete mollification
as regularization procedures.
The main original results are located in chapters 4 and 5 but chapter 2 and the appendices
contain other material studied for the thesis, including several original proofs.
The selected software tool is MATLAB and all the routines for numerical examples are
original. Thus, the routines are part of the original results of the thesis.
Chapters 1, 2 and 3 are introductions to the thesis, inverse problems and fractional derivatives
respectively. They are survey chapters written specifically for this thesis.Nuestro objetivo es el estudio de problemas de identificación en el marco de ecuaciones de transporte con derivadas fraccionarias. Consideramos ecuaciones difusivas con derivada temporal fraccionaria y ecuaciones de advección dispersión con derivada espacial fraccionaria. La mayoría de los problemas inversos son mal condicionados y requieren regularización. En esta tesis implementamos procedimientos de regularización basados en molificación discreta en una y dos dimensiones. Los principales resultados originales se encuentran en los capítulos 4 y 5 pero el capítulo 2 y los apéndices contienen material adicional estudiado para la tesis incluídas varias demostra- ciones originales. La herramienta de software escogida es MATLAB y todas las rutinas para los ejemplos numéricos son originales, de manera que las rutinas son parte de los resultados originales de la tesis. Los capítulos 1, 2 y 3 son introductorios a la tesis, a los problemas inversos y a las derivadas fraccionarias respectivamente. Se trata de capítulos monográficos escritos especialmente para esta tesis. (Texto tomado de la fuente)Convocatoria 647 de ColcienciasDoctoradoDoctor en Ciencias - MatemáticasAnálisis Numéric
An inverse Sturm-Liouville problem with a fractional derivative
In this paper, we numerically investigate an inverse problem of recovering
the potential term in a fractional Sturm-Liouville problem from one spectrum.
The qualitative behaviors of the eigenvalues and eigenfunctions are discussed,
and numerical reconstructions of the potential with a Newton method from finite
spectral data are presented. Surprisingly, it allows very satisfactory
reconstructions for both smooth and discontinuous potentials, provided that the
order of fractional derivative is sufficiently away from 2.Comment: 16 pages, 6 figures, accepted for publication in Journal of
Computational Physic
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Subdiffusive Source Sensing by a Regional Detection Method.
Motivated by the fact that the danger may increase if the source of pollution problem remains unknown, in this paper, we study the source sensing problem for subdiffusion processes governed by time fractional diffusion systems based on a limited number of sensor measurements. For this, we first give some preliminary notions such as source, detection and regional spy sensors, etc. Secondly, we investigate the characterizations of regional strategic sensors and regional spy sensors. A regional detection approach on how to solve the source sensing problem of the considered system is then presented by using the Hilbert uniqueness method (HUM). This is to identify the unknown source only in a subregion of the whole domain, which is easier to be implemented and could save a lot of energy resources. Numerical examples are finally included to test our results
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