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  • Book
  • © 2018

Asymptotics of Elliptic and Parabolic PDEs

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

  • Discusses asymptotic formulae in the context of the life sciences
  • Presents applications in molecular and cellular biology, biophysics, as well as computational neuroscience
  • Contains over 100 figures
  • Includes bibliographical notes

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

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Table of contents (12 chapters)

  1. Front Matter

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

    1. Front Matter

      Pages 1-1
    2. A Primer of Asymptotics for ODEs

      • David Holcman, Zeev Schuss
      Pages 11-48
    3. Singular Perturbations in Higher Dimensions

      • David Holcman, Zeev Schuss
      Pages 49-113
    4. Eigenvalues of a Non-self-adjoint Elliptic Operator

      • David Holcman, Zeev Schuss
      Pages 115-158
    5. Short-Time Asymptotics of the Heat Kernel

      • David Holcman, Zeev Schuss
      Pages 159-187
  3. Mixed Boundary Conditions for Elliptic and Parabolic Equations

    1. Front Matter

      Pages 189-189
    2. The Mixed Boundary Value Problem

      • David Holcman, Zeev Schuss
      Pages 191-200
    3. The Mixed Boundary Value Problem in \(\mathbb R^2\)

      • David Holcman, Zeev Schuss
      Pages 201-256
    4. Narrow Escape in \(\mathbb R^3\)

      • David Holcman, Zeev Schuss
      Pages 257-309
    5. The Poisson–Nernst–Planck Equations in a Ball

      • David Holcman, Zeev Schuss
      Pages 341-383
    6. Reconstruction of Surface Diffusion from Projected Data

      • David Holcman, Zeev Schuss
      Pages 385-402
    7. Asymptotic Formulas in Molecular and Cellular Biology

      • David Holcman, Zeev Schuss
      Pages 403-419
  4. Back Matter

    Pages 421-444

About this book

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 inderiving solutions to real-world problems from first principles.

Reviews

“The monograph under review deals with asymptotic methods for the construction of solutions to boundary value problems for partial differential equations arising in applications, as molecular and cellular biology and biophysics. … The monograph is well written, interesting, and surely recommended to applied mathematicians, engineers, physicists, chemists, and neuroscientists interested into analytical methods for the asymptotic analysis of elliptic and parabolic PDEs of relevance in applications.” (Paolo Musolino, zbMATH 1402.35004, 2019)

Authors and Affiliations

  • Group of Applied Mathematics and Computational Biology, École Normale Supérieure , Paris, France

    David Holcman

  • Department of Mathematics, Tel Aviv School of Mathematical Science, Tel Aviv University , Tel Aviv, Israel

    Zeev Schuss

About the authors

David Holcman is an applied mathematician and computational biologist. He developed mathematical modeling and simulations of molecular dynamics in micro-compartments in cell biology using stochastic processes and PDEs. He has derived physical principles of physiology at various scales, including diffusion laws in dendritic spines, potential wells hidden in super-resolution single particle trajectories or first looping time in polymer models. Together with Zeev Schuss, he developed the Narrow escape and Dire strait time theory.


Zeev Schuss is an applied mathematician who significantly shaped the field of modern asymptotics in PDEs with applications to first passage time problems. Methods developed have been applied to various fields, including signal processing, statistical physics, and molecular biophysics.

Bibliographic Information

Buy it now

Buying options

eBook USD 99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 129.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book USD 129.00
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access