Inverse Problems For Fractional Diffusion Equations
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Inverse Problems For Fractional Diffusion Equations
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Author : Durdimurod K. Durdiev
language : en
Publisher: Springer Nature
Release Date : 2025-06-15
Inverse Problems For Fractional Diffusion Equations written by Durdimurod K. Durdiev and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-06-15 with Mathematics categories.
This book discusses various inverse problems for the time-fractional diffusion equation, such as inverse coefficient problems (nonlinear problems) and inverse problems for determining the right-hand sides of equations and initial functions (linear problems). The study of inverse problems requires a comprehensive investigation of direct problems (such as representation formulas, a priori estimates and differential properties of the solution). This is particularly evident in nonlinear problems, where obtaining solvability theorems necessitates careful tracking of the exact dependence of the differential properties of the solution to the direct problem on the smoothness of the coefficients and other problem data. Therefore, a significant portion of the book is devoted to direct problems, such as initial problems (Cauchy problems) and initial-boundary value problems with various boundary conditions.
Inverse Problems For Fractional Diffusion Equations
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Author : Lihua Zuo
language : en
Publisher:
Release Date : 2013
Inverse Problems For Fractional Diffusion Equations written by Lihua Zuo and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013 with categories.
In recent decades, significant interest, based on physics and engineering applications, has developed on so-called anomalous diffusion processes that possess different spread functions with classical ones. The resulting differential equation whose fundamental solution matches this decay process is best modeled by an equation containing a fractional order derivative. This dissertation mainly focuses on some inverse problems for fractional diffusion equations. After some background introductions and preliminaries in Section 1 and 2, in the third section we consider our first inverse boundary problem. This is where an unknown boundary condition is to be determined from overposed data in a time- fractional diffusion equation. Based upon the fundamental solution in free space, we derive a representation for the unknown parameters as the solution of a nonlinear Volterra integral equation of second kind with a weakly singular kernel. We are able to make physically reasonable assumptions on our constraining functions (initial and given boundary values) to be able to prove a uniqueness and reconstruction result. This is achieved by an iterative process and is an immediate result of applying a certain fixed point theorem. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method. In the fourth section a reaction-diffusion problem with an unknown nonlinear source function, which has to be determined from overposed data, is considered. A uniqueness result is proved and a numerical algorithm including convergence analysis under some physically reasonable assumptions is presented in the one-dimensional case. To show effectiveness of the proposed method, some results of numerical simulations are presented. In Section 5, we also attempted to reconstruct a nonlinear source in a heat equation from a number of known input sources. This represents a new research even for the case of classical diffusion and would be the first step in a solution method for the fractional diffusion case. While analytic work is still in progress on this problem, Newton and Quasi-Newton method are applied to show the feasibility of numerical reconstructions. In conclusion, the fractional diffusion equations have some different properties with the classical ones but there are some similarities between them. The classical tools like integral equations and fixed point theory still hold under slightly different assumptions. Inverse problems for fractional diffusion equations have applications in many engineering and physics areas such as material design, porous media. They are trickier than classical ones but there are also some advantages due to the mildly ill-conditioned singularity caused by the new kernel functions. The electronic version of this dissertation is accessible from http://hdl.handle.net/1969.1/151079
The Theory Of Tikhonov Regularization For Fredholm Equations Of The First Kind
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Author : C. W. Groetsch
language : en
Publisher: Pitman Advanced Publishing Program
Release Date : 1984
The Theory Of Tikhonov Regularization For Fredholm Equations Of The First Kind written by C. W. Groetsch and has been published by Pitman Advanced Publishing Program this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984 with Mathematics categories.
Inverse Problems For Fractional Partial Differential Equations
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Author : Barbara Kaltenbacher
language : en
Publisher: American Mathematical Society
Release Date : 2023-07-13
Inverse Problems For Fractional Partial Differential Equations written by Barbara Kaltenbacher and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-07-13 with Mathematics categories.
As the title of the book indicates, this is primarily a book on partial differential equations (PDEs) with two definite slants: toward inverse problems and to the inclusion of fractional derivatives. The standard paradigm, or direct problem, is to take a PDE, including all coefficients and initial/boundary conditions, and to determine the solution. The inverse problem reverses this approach asking what information about coefficients of the model can be obtained from partial information on the solution. Answering this question requires knowledge of the underlying physical model, including the exact dependence on material parameters. The last feature of the approach taken by the authors is the inclusion of fractional derivatives. This is driven by direct physical applications: a fractional derivative model often allows greater adherence to physical observations than the traditional integer order case. The book also has an extensive historical section and the material that can be called "fractional calculus" and ordinary differential equations with fractional derivatives. This part is accessible to advanced undergraduates with basic knowledge on real and complex analysis. At the other end of the spectrum, lie nonlinear fractional PDEs that require a standard graduate level course on PDEs.
Inverse Boundary Spectral Problems
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Author : Alexander Kachalov
language : en
Publisher: CRC Press
Release Date : 2001-07-30
Inverse Boundary Spectral Problems written by Alexander Kachalov and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001-07-30 with Mathematics categories.
Inverse boundary problems are a rapidly developing area of applied mathematics with applications throughout physics and the engineering sciences. However, the mathematical theory of inverse problems remains incomplete and needs further development to aid in the solution of many important practical problems. Inverse Boundary Spectral Problems
Fractional Diffusion And Wave Equations
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Author : Yong Zhou
language : en
Publisher:
Release Date : 2024
Fractional Diffusion And Wave Equations written by Yong Zhou and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024 with Integral equations categories.
Preface -- Introduction -- Preliminaries -- Well-posedness of Fractional Diffusion Equations -- Inverse Problems of Fractional Diffusion Equations -- Well-posedness and Regularity of Fractional Wave Equations -- Inverse Problems of Fractional Wave Equations -- References -- Index.
Anomalous Transport
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Author : Rainer Klages
language : en
Publisher: John Wiley & Sons
Release Date : 2008-09-02
Anomalous Transport written by Rainer Klages and has been published by John Wiley & Sons this book supported file pdf, txt, epub, kindle and other format this book has been release on 2008-09-02 with Science categories.
This multi-author reference work provides a unique introduction to the currently emerging, highly interdisciplinary field of those transport processes that cannot be described by using standard methods of statistical mechanics. It comprehensively summarizes topics ranging from mathematical foundations of anomalous dynamics to the most recent experiments in this field. In so doing, this monograph extracts and emphasizes common principles and methods from many different disciplines while providing up-to-date coverage of this new field of research, considering such diverse applications as plasma physics, glassy material, cell science, and socio-economic aspects. The book will be of interest to both theorists and experimentalists in nonlinear dynamics, statistical physics and stochastic processes. It also forms an ideal starting point for graduate students moving into this area. 18 chapters written by internationally recognized experts in this field provide in-depth introductions to fundamental aspects of anomalous transport.
Mittag Leffler Functions Related Topics And Applications
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Author : Rudolf Gorenflo
language : en
Publisher: Springer
Release Date : 2014-10-16
Mittag Leffler Functions Related Topics And Applications written by Rudolf Gorenflo and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-10-16 with Mathematics categories.
As a result of researchers’ and scientists’ increasing interest in pure as well as applied mathematics in non-conventional models, particularly those using fractional calculus, Mittag-Leffler functions have recently caught the interest of the scientific community. Focusing on the theory of the Mittag-Leffler functions, the present volume offers a self-contained, comprehensive treatment, ranging from rather elementary matters to the latest research results. In addition to the theory the authors devote some sections of the work to the applications, treating various situations and processes in viscoelasticity, physics, hydrodynamics, diffusion and wave phenomena, as well as stochastics. In particular the Mittag-Leffler functions allow us to describe phenomena in processes that progress or decay too slowly to be represented by classical functions like the exponential function and its successors. The book is intended for a broad audience, comprising graduate students, university instructors and scientists in the field of pure and applied mathematics, as well as researchers in applied sciences like mathematical physics, theoretical chemistry, bio-mathematics, theory of control and several other related areas.
Inverse Source Problems
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Author : Victor Isakov
language : en
Publisher: American Mathematical Soc.
Release Date : 1990
Inverse Source Problems written by Victor Isakov and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1990 with Mathematics categories.
A careful exposition of a research field of current interest. This includes a brief survey of the subject and an introduction to recent developments and unsolved problems.
New Trends Of Mathematical Inverse Problems And Applications
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Author : Amine Laghrib
language : en
Publisher: Springer Nature
Release Date : 2023-07-15
New Trends Of Mathematical Inverse Problems And Applications written by Amine Laghrib and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-07-15 with Mathematics categories.
This volume comprises the thoroughly reviewed and revised papers of the First International Conference on New Trends in Applied Mathematics, ICNTAM 2022, which took place in Béni Mellal, Morocco, 19-21 May 2022. The papers deal with the following topics: Inverse Problems, Partial Differential Equations, Mathematical Control, Numerical Analysis and Computer Science. The main interest is in recent trends on Inverse Problems analysis and real applications in Computer Science. The latter is viewed as a dynamic branch on the interface of mathematics and related fields, that has been growing rapidly over the past several decades. However, its mathematical analysis and interpretation still not well-detailed and needs much more clarifications. The main contribution of this book is to give some sufficient mathematical content with expressive results and accurate applications. As a growing field, it is gaining a lot of attention both in media as well as in the industry world, which will attract the interest of readers from different scientist discipline.