Fractional Diffusion Equations And Anomalous Diffusion
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Fractional Diffusion Equations And Anomalous Diffusion
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Author : Luiz Roberto Evangelista
language : en
Publisher: Cambridge University Press
Release Date : 2018-01-25
Fractional Diffusion Equations And Anomalous Diffusion written by Luiz Roberto Evangelista and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-01-25 with Mathematics categories.
Presents a unified treatment of anomalous diffusion problems using fractional calculus in a wide range of applications across scientific and technological disciplines.
High Accuracy Algorithm For The Differential Equations Governing Anomalous Diffusion Algorithm And Models For Anomalous Diffusion
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Author : Weihua Deng
language : en
Publisher: World Scientific
Release Date : 2019-01-22
High Accuracy Algorithm For The Differential Equations Governing Anomalous Diffusion Algorithm And Models For Anomalous Diffusion written by Weihua Deng and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-01-22 with Mathematics categories.
The aim of this book is to extend the application field of 'anomalous diffusion', and describe the newly built models and the simulation techniques to the models.The book first introduces 'anomalous diffusion' from the statistical physics point of view, then discusses the models characterizing anomalous diffusion and its applications, including the Fokker-Planck equation, the Feymann-Kac equations describing the functional distribution of the anomalous trajectories of the particles, and also the microscopic model — Langevin type equation. The second main part focuses on providing the high accuracy schemes for these kinds of models, and the corresponding convergence and stability analysis.
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.
The Langevin Equation
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Author : William Coffey
language : en
Publisher: World Scientific
Release Date : 2012
The Langevin Equation written by William Coffey and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012 with Mathematics categories.
This volume is the third edition of the first-ever elementary book on the Langevin equation method for the solution of problems involving the translational and rotational Brownian motion of particles and spins in a potential highlighting modern applications in physics, chemistry, electrical engineering, and so on. In order to improve the presentation, to accommodate all the new developments, and to appeal to the specialized interests of the various communities involved, the book has been extensively rewritten and a very large amount of new material has been added. This has been done in order to present a comprehensive overview of the subject emphasizing via a synergetic approach that seemingly unrelated physical problems involving random noise may be described using virtually identical mathematical methods in the spirit of the founders of the subject, viz., Einstein, Langevin, Smoluchowski, Kramers, The book has been written in such a way that all the material should be accessible both to an advanced researcher and a beginning graduate student. It draws together, in a coherent fashion, a variety of results which have hitherto been available only in the form of scattered research papers and review articles.
Fractional Diffusion
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Author : Nirupama Bhattacharya
language : en
Publisher:
Release Date : 2014
Fractional Diffusion written by Nirupama Bhattacharya and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014 with categories.
In biological contexts, experimental evidence suggests that classical diffusion is not the best description in instances of complex biophysical transport. Instead, anomalous diffusion has been shown to occur in various circumstances, potentially caused by such underlying mechanisms as active transport, macromolecular crowding in a complex and tortuous extracellular or intracellular environment, or complex media geometry. Elegant ways of simulating these complicated transport processes are to connect the spatial characteristics of a medium (porosity or tortuosity of a complex extracellular environment), to fractional order operators. Some approaches include special random walk models representing crowded or disordered media; at the continuum limit, these random walk models approach fractional differential equations (FDEs), including variations of the fractional diffusion equation. Fractional differential equations are an extension of classical integer-order differential equations, and in recent decades have been increasingly used to model the dynamics of complex systems in a wide variety of fields including science, engineering, and finance. However, finding tractable and closed form analytical solutions to FDEs, including the fractional diffusion equation and its variants, is generally extremely difficult and often not feasible, and especially so when integrating these equations into more complex physical models with multiple other components; therefore, the development of stable and accurate numerical methods is vital. In this thesis we explore the topic of anomalous diffusion and the fractional diffusion equation from multiple perspectives. We begin by connecting the micro-molecular behavior of diffusing particles undergoing anomalous diffusion, to the general derivation of the fractional diffusion equation. We then develop numerical approaches to efficiently solve the time-fractional diffusion equation, and characterize these methods in terms of accuracy, stability, and algorithmic complexity. We then make use of these numerical methods by applying fractional diffusion to a model of the signaling events leading up the induction of long-term depression (LTD). We leverage the fact that the fractional diffusion equation can capture the complex geometry in which diffusing particles travel, and use this to simplify an existing model of LTD induction; furthermore, we show that our modified model is capable of retaining the most important functionality of the original model.
Spectral And High Order Methods For Partial Differential Equations Icosahom 2020 1
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Author : Jens M. Melenk
language : en
Publisher: Springer Nature
Release Date : 2023-06-30
Spectral And High Order Methods For Partial Differential Equations Icosahom 2020 1 written by Jens M. Melenk and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-06-30 with Mathematics categories.
The volume features high-quality papers based on the presentations at the ICOSAHOM 2020+1 on spectral and high order methods. The carefully reviewed articles cover state of the art topics in high order discretizations of partial differential equations. The volume presents a wide range of topics including the design and analysis of high order methods, the development of fast solvers on modern computer architecture, and the application of these methods in fluid and structural mechanics computations.
Langevin Equation The With Applications To Stochastic Problems In Physics Chemistry And Electrical Engineering 3rd Edition
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Author : Yuri P Kalmykov
language : en
Publisher: World Scientific
Release Date : 2012-07-31
Langevin Equation The With Applications To Stochastic Problems In Physics Chemistry And Electrical Engineering 3rd Edition written by Yuri P Kalmykov and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-07-31 with Science categories.
This volume is the third edition of the first-ever elementary book on the Langevin equation method for the solution of problems involving the translational and rotational Brownian motion of particles and spins in a potential highlighting modern applications in physics, chemistry, electrical engineering, and so on. In order to improve the presentation, to accommodate all the new developments, and to appeal to the specialized interests of the various communities involved, the book has been extensively rewritten and a very large amount of new material has been added. This has been done in order to present a comprehensive overview of the subject emphasizing via a synergetic approach that seemingly unrelated physical problems involving random noise may be described using virtually identical mathematical methods in the spirit of the founders of the subject, viz., Einstein, Langevin, Smoluchowski, Kramers, etc. The book has been written in such a way that all the material should be accessible both to an advanced researcher and a beginning graduate student. It draws together, in a coherent fashion, a variety of results which have hitherto been available only in the form of scattered research papers and review articles.
Langevin Equation The With Applications To Stochastic Problems In Physics Chemistry And Electrical Engineering 2nd Edition
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Author : William T Coffey
language : en
Publisher: World Scientific
Release Date : 2004-03-03
Langevin Equation The With Applications To Stochastic Problems In Physics Chemistry And Electrical Engineering 2nd Edition written by William T Coffey and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004-03-03 with Science categories.
This volume is the second edition of the first-ever elementary book on the Langevin equation method for the solution of problems involving the Brownian motion in a potential, with emphasis on modern applications in the natural sciences, electrical engineering and so on. It has been substantially enlarged to cover in a succinct manner a number of new topics, such as anomalous diffusion, continuous time random walks, stochastic resonance etc, which are of major current interest in view of the large number of disparate physical systems exhibiting these phenomena. The book has been written in such a way that all the material should be accessible to an advanced undergraduate or beginning graduate student. It draws together, in a coherent fashion, a variety of results which have hitherto been available only in the form of research papers or scattered review articles.
Fractals Diffusion And Relaxation In Disordered Complex Systems
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Author : Yuri P. Kalmykov
language : en
Publisher: John Wiley & Sons
Release Date : 2006-07-21
Fractals Diffusion And Relaxation In Disordered Complex Systems written by Yuri P. Kalmykov 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 2006-07-21 with Science categories.
Fractals, Diffusion and Relaxation in Disordered Complex Systems is a special guest-edited, two-part volume of Advances in Chemical Physics that continues to report recent advances with significant, up-to-date chapters by internationally recognized researchers.
Langevin Equation The With Applications To Stochastic Problems In Physics Chemistry And Electrical Engineering Fourth Edition
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Author : William T Coffey
language : en
Publisher: World Scientific
Release Date : 2017-03-22
Langevin Equation The With Applications To Stochastic Problems In Physics Chemistry And Electrical Engineering Fourth Edition written by William T Coffey and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-03-22 with Science categories.
Our original objective in writing this book was to demonstrate how the concept of the equation of motion of a Brownian particle — the Langevin equation or Newtonian-like evolution equation of the random phase space variables describing the motion — first formulated by Langevin in 1908 — so making him inter alia the founder of the subject of stochastic differential equations, may be extended to solve the nonlinear problems arising from the Brownian motion in a potential. Such problems appear under various guises in many diverse applications in physics, chemistry, biology, electrical engineering, etc. However, they have been invariably treated (following the original approach of Einstein and Smoluchowski) via the Fokker-Planck equation for the evolution of the probability density function in phase space. Thus the more simple direct dynamical approach of Langevin which we use and extend here, has been virtually ignored as far as the Brownian motion in a potential is concerned. In addition two other considerations have driven us to write this new edition of The Langevin Equation. First, more than five years have elapsed since the publication of the third edition and following many suggestions and comments of our colleagues and other interested readers, it became increasingly evident to us that the book should be revised in order to give a better presentation of the contents. In particular, several chapters appearing in the third edition have been rewritten so as to provide a more direct appeal to the particular community involved and at the same time to emphasize via a synergetic approach how seemingly unrelated physical problems all involving random noise may be described using virtually identical mathematical methods. Secondly, in that period many new and exciting developments have occurred in the application of the Langevin equation to Brownian motion. Consequently, in order to accommodate all these, a very large amount of new material has been added so as to present a comprehensive overview of the subject.