Random Walk And The Heat Equation PDF Download
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| Author | : Gregory F. Lawler |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 170 |
| Release | : 2010-11-22 |
| Genre | : Mathematics |
| ISBN | : 0821848291 |
Download Random Walk and the Heat Equation Book in PDF, ePub and Kindle
The heat equation can be derived by averaging over a very large number of particles. Traditionally, the resulting PDE is studied as a deterministic equation, an approach that has brought many significant results and a deep understanding of the equation and its solutions. By studying the heat equation and considering the individual random particles, however, one gains further intuition into the problem. While this is now standard for many researchers, this approach is generally not presented at the undergraduate level. In this book, Lawler introduces the heat equations and the closely related notion of harmonic functions from a probabilistic perspective. The theme of the first two chapters of the book is the relationship between random walks and the heat equation. This first chapter discusses the discrete case, random walk and the heat equation on the integer lattice; and the second chapter discusses the continuous case, Brownian motion and the usual heat equation. Relationships are shown between the two. For example, solving the heat equation in the discrete setting becomes a problem of diagonalization of symmetric matrices, which becomes a problem in Fourier series in the continuous case. Random walk and Brownian motion are introduced and developed from first principles. The latter two chapters discuss different topics: martingales and fractal dimension, with the chapters tied together by one example, a random Cantor set. The idea of this book is to merge probabilistic and deterministic approaches to heat flow. It is also intended as a bridge from undergraduate analysis to graduate and research perspectives. The book is suitable for advanced undergraduates, particularly those considering graduate work in mathematics or related areas.
| Author | : Wolfgang König |
| Publisher | : Birkhäuser |
| Total Pages | : 199 |
| Release | : 2016-06-30 |
| Genre | : Mathematics |
| ISBN | : 3319335960 |
Download The Parabolic Anderson Model Book in PDF, ePub and Kindle
This is a comprehensive survey on the research on the parabolic Anderson model – the heat equation with random potential or the random walk in random potential – of the years 1990 – 2015. The investigation of this model requires a combination of tools from probability (large deviations, extreme-value theory, e.g.) and analysis (spectral theory for the Laplace operator with potential, variational analysis, e.g.). We explain the background, the applications, the questions and the connections with other models and formulate the most relevant results on the long-time behavior of the solution, like quenched and annealed asymptotics for the total mass, intermittency, confinement and concentration properties and mass flow. Furthermore, we explain the most successful proof methods and give a list of open research problems. Proofs are not detailed, but concisely outlined and commented; the formulations of some theorems are slightly simplified for better comprehension.
| Author | : M. T. Barlow |
| Publisher | : Cambridge University Press |
| Total Pages | : 239 |
| Release | : 2017-02-23 |
| Genre | : Mathematics |
| ISBN | : 1107674425 |
Download Random Walks and Heat Kernels on Graphs Book in PDF, ePub and Kindle
Useful but hard-to-find results enrich this introduction to the analytic study of random walks on infinite graphs.
| Author | : Erich Zauderer |
| Publisher | : Wiley-Interscience |
| Total Pages | : 0 |
| Release | : 1998-08-04 |
| Genre | : Mathematics |
| ISBN | : 9780471315162 |
Download Partial Differential Equations of Applied Mathematics Book in PDF, ePub and Kindle
The only comprehensive guide to modeling, characterizing, and solving partial differential equations This classic text by Erich Zauderer provides a comprehensive account of partial differential equations and their applications. Dr. Zauderer develops mathematical models that give rise to partial differential equations and describes classical and modern solution techniques. With an emphasis on practical applications, he makes liberal use of real-world examples, explores both linear and nonlinear problems, and provides approximate as well as exact solutions. He also describes approximation methods for simplifying complicated solutions and for solving linear and nonlinear problems not readily solved by standard methods. The book begins with a demonstration of how the three basic types of equations (parabolic, hyperbolic, and elliptic) can be derived from random walk models. It continues in a less statistical vein to cover an exceptionally broad range of topics, including stabilities, singularities, transform methods, the use of Green's functions, and perturbation and asymptotic treatments. Features that set Partial Differential Equations of Applied Mathematics, Second Edition above all other texts in the field include: Coverage of random walk problems, discontinuous and singular solutions, and perturbation and asymptotic methods More than 800 practice exercises, many of which are fully worked out Numerous up-to-date examples from engineering and the physical sciences Partial Differential Equations of Applied Mathematics, Second Edition is a superior advanced-undergraduate to graduate-level text for students in engineering, the sciences, and applied mathematics. The title is also a valuable working resource for professionals in these fields. Dr. Zauderer received his doctorate in mathematics from the New York University-Courant Institute. Prior to joining the staff of Polytechnic University, he was a Senior Weitzmann Fellow of the Weitzmann Institute of Science in Rehovot, Israel.
| Author | : Karl K. Sabelfeld |
| Publisher | : Walter de Gruyter |
| Total Pages | : 148 |
| Release | : 2013-07-05 |
| Genre | : Mathematics |
| ISBN | : 311094202X |
Download Random Walks on Boundary for Solving PDEs Book in PDF, ePub and Kindle
This monograph presents new probabilistic representations for classical boundary value problems of mathematical physics and is the first book devoted to the walk on boundary algorithms. Compared to the well-known Wiener and diffusion path integrals, the trajectories of random walks in this publication are simlated on the boundary of the domain as Markov chains generated by the kernels of the boundary integral equations equivalent to the original boundary value problem. The book opens with an introduction for solving the interior and exterior boundary values for the Laplace and heat equations, which is followed by applying this method to all main boundary value problems of the potential and elasticity theories.
| Author | : D. V. Widder |
| Publisher | : Academic Press |
| Total Pages | : 285 |
| Release | : 1976-01-22 |
| Genre | : Science |
| ISBN | : 0080873839 |
Download The Heat Equation Book in PDF, ePub and Kindle
The Heat Equation
| Author | : Sandro Salsa |
| Publisher | : Springer |
| Total Pages | : 714 |
| Release | : 2015-04-24 |
| Genre | : Mathematics |
| ISBN | : 3319150936 |
Download Partial Differential Equations in Action Book in PDF, ePub and Kindle
The book is intended as an advanced undergraduate or first-year graduate course for students from various disciplines, including applied mathematics, physics and engineering. It has evolved from courses offered on partial differential equations (PDEs) over the last several years at the Politecnico di Milano. These courses had a twofold purpose: on the one hand, to teach students to appreciate the interplay between theory and modeling in problems arising in the applied sciences, and on the other to provide them with a solid theoretical background in numerical methods, such as finite elements. Accordingly, this textbook is divided into two parts. The first part, chapters 2 to 5, is more elementary in nature and focuses on developing and studying basic problems from the macro-areas of diffusion, propagation and transport, waves and vibrations. In turn the second part, chapters 6 to 11, concentrates on the development of Hilbert spaces methods for the variational formulation and the analysis of (mainly) linear boundary and initial-boundary value problems.
| Author | : José M. Mazón |
| Publisher | : Springer Nature |
| Total Pages | : 396 |
| Release | : 2023-08-04 |
| Genre | : Mathematics |
| ISBN | : 3031335848 |
Download Variational and Diffusion Problems in Random Walk Spaces Book in PDF, ePub and Kindle
This book presents the latest developments in the theory of gradient flows in random walk spaces. A broad framework is established for a wide variety of partial differential equations on nonlocal models and weighted graphs. Within this framework, specific gradient flows that are studied include the heat flow, the total variational flow, and evolution problems of Leray-Lions type with different types of boundary conditions. With many timely applications, this book will serve as an invaluable addition to the literature in this active area of research. Variational and Diffusion Problems in Random Walk Spaces will be of interest to researchers at the interface between analysis, geometry, and probability, as well as to graduate students interested in exploring these areas.
| Author | : Richard Wilson |
| Publisher | : Springer |
| Total Pages | : 604 |
| Release | : 2013-08-17 |
| Genre | : Computers |
| ISBN | : 3642402461 |
Download Computer Analysis of Images and Patterns Book in PDF, ePub and Kindle
The two volume set LNCS 8047 and 8048 constitutes the refereed proceedings of the 15th International Conference on Computer Analysis of Images and Patterns, CAIP 2013, held in York, UK, in August 2013. The 142 papers presented were carefully reviewed and selected from 243 submissions. The scope of the conference spans the following areas: 3D TV, biometrics, color and texture, document analysis, graph-based methods, image and video indexing and database retrieval, image and video processing, image-based modeling, kernel methods, medical imaging, mobile multimedia, model-based vision approaches, motion analysis, natural computation for digital imagery, segmentation and grouping, and shape representation and analysis.
| Author | : Andras Telcs |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 194 |
| Release | : 2006-05-17 |
| Genre | : Mathematics |
| ISBN | : 3540330275 |
Download The Art of Random Walks Book in PDF, ePub and Kindle
Einstein proved that the mean square displacement of Brownian motion is proportional to time. He also proved that the diffusion constant depends on the mass and on the conductivity (sometimes referred to Einstein’s relation). The main aim of this book is to reveal similar connections between the physical and geometric properties of space and diffusion. This is done in the context of random walks in the absence of algebraic structure, local or global spatial symmetry or self-similarity. The author studies the heat diffusion at this general level and discusses the following topics: The multiplicative Einstein relation, Isoperimetric inequalities, Heat kernel estimates Elliptic and parabolic Harnack inequality.
