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Asymptotics of Elliptic and Parabolic PDEs

and their Applications in Statistical Physics, Computational Neuroscience, and Biophysics

  • David Holcman
  • Zeev Schuss

Part of the Applied Mathematical Sciences book series (AMS, volume 199)

Table of contents

  1. Front Matter
    Pages i-xxiii
  2. Singular Perturbations of Elliptic Boundary Problems

    1. Front Matter
      Pages 1-1
    2. David Holcman, Zeev Schuss
      Pages 11-48
    3. David Holcman, Zeev Schuss
      Pages 49-113
    4. David Holcman, Zeev Schuss
      Pages 115-158
    5. David Holcman, Zeev Schuss
      Pages 159-187
  3. Mixed Boundary Conditions for Elliptic and Parabolic Equations

    1. Front Matter
      Pages 189-189
    2. David Holcman, Zeev Schuss
      Pages 191-200
    3. David Holcman, Zeev Schuss
      Pages 201-256
    4. David Holcman, Zeev Schuss
      Pages 257-309
    5. David Holcman, Zeev Schuss
      Pages 341-383
    6. David Holcman, Zeev Schuss
      Pages 385-402
    7. David Holcman, Zeev Schuss
      Pages 403-419
  4. Back Matter
    Pages 421-444

About this book

Introduction

This is a monograph on the emerging branch of mathematical biophysics combining asymptotic analysis with numerical and stochastic methods to analyze partial differential equations arising in biological and physical sciences.

In more detail, the book presents the analytic methods and tools for approximating solutions of mixed boundary value problems, with particular emphasis on the narrow escape problem. Informed throughout by real-world applications, the book includes topics such as the Fokker-Planck equation, boundary layer analysis, WKB approximation, applications of spectral theory, as well as recent results in narrow escape theory. Numerical and stochastic aspects, including mean first passage time and extreme statistics, are discussed in detail and relevant applications are presented in parallel with the theory.

Including background on the classical asymptotic theory of differential equations, this book is written for scientists of various backgrounds interested in deriving solutions to real-world problems from first principles.

Keywords

boundary value problems Poisson-Nernst-Planck extreme statistics narrow escape first passage times asymptotic formula partial differential equations WKB eigenvalues matched asymptotics non-self adjoint operators short-time asymptotics long-time asymptotics Green’s functions Neumann’s function applied conformal transformation Eikonal equation Ray method Helmholtz equation integral equations

Authors and affiliations

  • David Holcman
    • 1
  • Zeev Schuss
    • 2
  1. 1.Group of Applied Mathematics and Computational BiologyÉcole Normale Supérieure ParisFrance
  2. 2.Department of Mathematics, Tel Aviv School of Mathematical ScienceTel Aviv University Tel AvivIsrael

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-76895-3
  • Copyright Information Springer International Publishing AG, part of Springer Nature 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-76894-6
  • Online ISBN 978-3-319-76895-3
  • Series Print ISSN 0066-5452
  • Series Online ISSN 2196-968X
  • Buy this book on publisher's site