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

Determining the source in complete parabolic equations

The problem of determining the source has been analyzed during the last years in different areas of ap plied mathematics and has received considerable attention in many current research, as it has applications in fields such as driving of heat, crack identification, electromagnetic theory, geophysical prospecting, the detection of contaminants and detection of tumor cells, among others.Fil: Umbricht, Guillermo Federico. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Determining the source in complete parabolic equations

The problem of determining the source has been analyzed during the last years in different areas of ap plied mathematics and has received considerable attention in many current research, as it has applications in fields such as driving of heat, crack identification, electromagnetic theory, geophysical prospecting, the detection of contaminants and detection of tumor cells, among others.Fil: Umbricht, Guillermo Federico. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

A numerical treatment of block nuclear magnetic resonance flow equation

The time-dependent Bloch nuclear magnetic resonance flow equation in one dimensional space is investigated numerically. To investigate some physiological and biological properties of living tissues NMR plays pivotal role. In this paper, an applicable approach is used to solve the proposed equation with appropriate initial and boundary conditions. This method is a kind of regularization approaches based on the finite difference and mollification methods. The numerical algorithm is well supported with stability and convergence results and the numerical results for two test problems confirm the ability of the numerical method.Publisher's Versio

Solución de un problema inverso de advección-dispersión con derivada temporal fraccionaria por medio de molificación discreta

Consideramos un problema inverso para una ecuación de advección-dispersión con derivada temporal fraccionaria, en una configuración unidimensional. La derivada fraccionaria se interpreta en el sentido de Caputo y las coeficientes de advección y de dispersión son constantes. El problema inverso involucra la reconstruccción simultánea de la concentración de soluto y del flujo de dispersión en una de las fronteras del dominio físico, a partir de lecturas de datos perturbados en un punto interior del dominio. Mostramos que el problema inverso es mal condicionado y por tanto una solución numérica del problema requiere de alguna técnica de regularización. Proponemos un esquema de diferencias finitas de marcha en el espacio, que utiliza molifocación discreta como técnica de regularización. Se incluyen estimativos de error y ejemplos numéricos ilustrativos.We consider an inverse problem for a time fractional advection-dispersion equation in a 1-D semi-infinite setting. The fractional derivative is interpreted in the sense of Caputo and advection and dispersion coefficients are constant. The inverse problem consists on the recovery of the boundary distribution of solute concentration and dispersion flux from measured (noisy) data known at an interior location. This inverse problem is ill-posed and thus the numerical solution must include some regularization technique. Our approach is a finite difference space marching scheme enhanced by adaptive discrete mollification. Error estimates and illustrative numerical examples are provided

Inverse Problems for Fractional Diffusion Equations

By Fick’s laws of diffusion, in the classical diffusion process, the mean square path ‹x2› is proportional to the time t as t →∞,. However, in practice, some anomalous diffusion processes may occur, in which the relation ‹x2› t, ≠ 1 holds. To describe such processes, we need to add the fractional derivative on the time t, which forms the fractional diffusion equation, and we call it FDE for short. This dissertation contains some inverse problems in FDEs. Specifically, the recovery of unknown conditions of coefficients from additional data on the solution u will be considered. The results of fractional inverse problems are totally different from the ones of the classical case. For instance, the degree of ill-posedness. This is due to the polynomial asymptotic behavior of the Mittag-Leffler function, which consists of the fundamental solution of FDE. This difference leads to new physics and we can ask a question that do similar things always occur? The short answer is not always and the slightly longer version is the analysis is always more complex. This makes the research on inverse problems in FDEs both challenging and interesting. For each inverse problem in this dissertation, at first it was necessary to extend existing results about the direct problem, namely the situation where all parameters in the equation are known and we must recover u(x, t). This includes the existence, uniqueness and regularity estimates of the solution. Then for the inverse problem, the initial step in many of these situations is to use the equation structure to obtain an operator K one of whose fixed points is the unknown function we seek. With this K; the key step is proving the monotonicity of the operator in a suitable partially ordered space and then showing uniqueness of its fixed points. In conclusion, the monotonicity property and the domain of the operator K will lead to an iterative reconstruction algorithm and some numerical results are reproduced to verify the theoretical conclusions

Numerical schemes for reconstructing profiles of moving sources in (time-fractional) evolution equations (Analysis of inverse problems through partial differential equations and related topics)

This article is concerned with the derivation of numerical reconstruction schemes for the inverse moving source problem on determining source profiles in (time-fractional) evolution equations. As a continuation of the theoretical uniqueness, we adopt a minimization procedure with regularization to construct iterative thresholding schemes for the reduced backward problems on recovering one or two unknown initial value(s). Moreover, an elliptic approach is proposed to solving a convection equation in the case of two profiles

An optimal control approach for solving an inverse heat source problem applying shifted Legendre polynomials

This study addresses the inverse issue of identifying the space-dependent heat source of the heat equation, which is stated using the optimal con-trol framework. For the numerical solution of this class of problems, an approach based on shifted Legendre polynomials and the associated oper-ational matrix is presented. The approach turns the primary problem into the solution of a system of nonlinear algebraic equations. To do this, the temperature and heat source variables are enlarged in terms of the shifted Legendre polynomials with unknown coefficients employed in the objectivefunction, inverse problem, and initial and Neumann boundary conditions. When paired with their operational matrix, these basis functions provide a quadratic optimization problem with linear constraints, which is then solved using the Lagrange multipliers approach. To assess the method’s efficacy and precision, two examples are provided