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.
Arvustused
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)
Part I. Singular Perturbations of Elliptic Boundary Problems.- 1
Second-Order Elliptic Boundary Value Problems with a Small Leading Part.- 2 A
Primer of Asymptotics for ODEs.- 3 Singular Perturbations in Higher
Dimensions.- 4 Eigenvalues of a Non-self-adjoint Elliptic Operator.- 5
Short-time Asymptotics of the Heat Kernel.- Part II Mixed Boundary Conditions
for Elliptic and Parabolic Equations.- 6 The Mixed Boundary Value Problem.- 7
THe Mixed Boundary Value Problem in R2.- 8 Narrow Escape in R3.- 9 Short-time
Asymptotics of the Heat Kernel and Extreme Statistics of the NET.- 10 The
PoissonNernstPlanck Equations in a Ball.- 11 Reconstruction of Surface
Diffusion from Projected Data.- 12 Asymptotic Formulas in Molecular and
Cellular Biology.- Bibliography.- Index.
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.
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