Asymptotics of elliptic and parabolic PDEs : and their applications in statistical physics, computational neuroscience, and biophysics
 Responsibility
 David Holcman, Zeev Schuss.
 Publication
 Cham : Springer, [2018]
 Physical description
 xxiii, 444 pages : illustrations (some color) ; 25 cm.
 Series
 Applied mathematical sciences (SpringerVerlag New York Inc.) ; v. 199.
Access
Available online
Science Library (Li and Ma)
Stacks
Call number  Status 

QA377 .H65 2018  Unknown 
More options
Creators/Contributors
 Author/Creator
 Holcman, David, author.
 Contributor
 Schuss, Zeev, 1937 author.
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 421437) and index.
 Contents

 Singular perturbations of elliptic boundary problems
 Secondorder elliptic boundary value problems with a small leading part
 Introduction
 Application to stochastic differential equations
 The survival probability and the eigenvalue problem
 Discussion
 A primer of asymptotics for ODEs
 The laplace expansion of integrals
 The asymptotics of a firstorder initial value problem
 Matched asymptotic expansions
 An Application to stochastic differential equations : the exit problem in R1
 Asymptotics of a secondorder boundary value problem
 Asymptotics of a homogeneous secondorder boundary value problem
 Asymptotics of the inhomogeneous boundary value problem
 Examples and applications to stochastic equations
 Small diffusion with the flow : the homogeneous boundary value problem
 Small diffusion against the flow
 Small diffusion against the flow : the inhomogeneous boundary value problem
 The boundary value problem with a sharp potential barrier
 The problem for a smooth potential barrier at the boundary
 The second eigenvalue of the fokkerplanck operator
 A diffusion model of random signals
 Loss of lock in a firstorder phaselocked loop in phasemodulated radio signals
 Annotations
 Singular perturbations in higher dimensions
 Introduction
 The WKB method
 The eikonal equation
 The transport equation
 The characteristic equations
 Boundary layers at noncharacteristic boundaries
 Boundary layers at characteristic boundaries in the plane
 The boundary value problem with noncharacteristic boundaries
 The boundary value problem in planar domains with characteristic boundaries
 Loss of lock in a secondorder phaselocked loop
 The phase plane of the reduced problem
 The mean time to lose lock
 The boundary layer structure of ...
 Asymptotic solution of the stationary fokkerplanck equation
 The eikonal equation for (3.136)
 The eikonal on the separatrix
 The transport equation
 Derivation of (3.114)
 Green's function for the boundary value problem is the exit density
 Annotations
 An attractor inside an unstable limit cycle
 The reduced equation : an underdamped forced pendulum
 Asymptotics of the fokkerplanck equation near the limit cycle
 The boundary value problem for the fokkerplanck equation in ...
 Annotations
 Eigenvalues of a nonselfadjoint elliptic operator
 Introduction
 Eigenvalues and the survival probability
 The principal eigenvalue and the structure of the field a(x)
 The precise WKB structure of the principal eigenfunction
 The eikonal equation
 The transport equation
 The boundary layer equation
 The first eigenfunction of the adjoint problem
 Higherorder eigenvalues
 Oscillatory escape time
 Spontaneous activity in the cerebral cortex
 Numerical study of oscillatory decay
 A model of upstate dynamics in a neuronal network
 The phase space of the model
 Brownian simulations of oscillation phenomena in (4.80)
 The exit density from a focus near a limit cycle
 The mean first passage time ...(x)
 Numerical study of the eikonal equation
 Normal flux on ... : the exit time density
 Computation of the second eigenvalue
 Brownian dynamics simulations
 A twoterm approximation of the exittime density
 Exit time densities in three ranges of noise amplitude
 Appendices
 The density of exit points
 Expansion of the field near the boundary and no cycling
 The lacobian of b... at ...
 The real part ...
 Annotations
 Shorttime asymptotics of the heat kernel
 The onedimensional case
 The ray method for short time asymptotics of green's function
 The trace
 Simply connected domains
 Multiply connected domains
 Recovering ... from P(t)
 Discussion
 Construction of the shorttime asymptotic of the fokkerplanck equation with a periodic potential
 Annotations
 Mixed boundary conditions for elliptic and parabolic equations
 The mixed boundary value problem
 Introduction
 Formulation of the mixed boundary value problem
 The narrow escape time problem
 A pathological example
 The matched asymptotics approach
 Higherorder asymptotics in the unit ball
 The narrow escape time through multiple absorbing windows
 Annotations
 The mixed boundary value problem in R2
 A neumanndirichlet boundary value problem
 The neumann function
 The mixed boundary value problem on a riemannian manifold in R2
 Exit though several windows
 The helmholtz equation for two windows
 Asymptotic solution of the helmholtz equation
 The mixed boundary value problem for poisson's equation in dire straits
 The case of a bottleneck
 The case of several bottlenecks
 The mixed boundary value problem on a surface of revolution
 A composite domain with a bottleneck
 The narrow escape time from domains in r2 and r3 with bottlenecks
 The principal eigenvalue and bottlenecks
 Connecting head and neck
 The principal eigenvalue in dumbbellshaped domains
 Diffusion of a needle in dire straits
 The diffusion law of a needle in a planar strip
 The turnaround time ...
 Applications of the narrow escape time
 Annotation to the narrow escape time problem
 Narrow escape in R3
 The neumann function in regular domains in R3
 Elliptic absorbing window
 Secondorder asymptotics for a circular window
 The first eigenvalue for two small dirichlet windows
 Multiple absorbing windows
 Higherorder expansion of the net through many small windows on a sphere
 Application to leakage in a conductor of brownian particles
 Activation through a narrow opening
 The neumann function
 Solution of the mixed boundary value problem
 Deep well : a markov chain model
 The mixed boundary value problem in a solid funnelshaped domain
 The mixed boundary value problem with a dirichlet ribbon
 Selected applications in molecular biophysics
 Leakage from a ccylinder
 Applications of the mixed boundary value problem
 Annotations
 Shorttime asymptotics of the heat kernel and extreme statistics of the NET
 Introduction
 The pdf of the first escape time
 The pdf of the first arrival time in an interval
 Asymptotics of the expected shortest time
 Escape from a ray
 Escape from an interval ...
 The FAT in a bounded domain in R...
 Asymptotics in R3
 Asymptotics in R2
 Statistics of the arrival time of the second particle
 Poissonianlike approximation
 Pr... of N Brownian i.i.d. Trajectories in a segment
 Applications of the FAT in biophysics
 Annotations
 The poissonnernstplanck equations in a ball
 Introduction
 Synopsis of results
 Poissonnernstplanck equations in a ball
 The steadystate solution
 Existence of solutions
 The distribution of voltage and charge in a dielectric ball
 Scaling laws for the maximal number of charges
 Ionic flux in a small window at high charge
 Flow through a narrow window at high charge
 Current in a voltageclamped dendritic spine
 Appendix 1 : reverse liouvillegelfandbratu equation
 Small a expansion of ...(x)
 Numerical scheme for the solution of (10.13)
 Steady solution in a ball with a cuspshaped funnel
 Reduced equations in an uncharged cuspshaped funnel
 Asymptotics of voltage between funnel and center
 Poissonnernstplanck solutions in a 3d cuspshaped funnel
 Asymptotic analysis of the pnp equations in a cuspshaped funnel
 The potential drop in ...
 Annotations
 Reconstruction of surface diffusion from projected data
 Projection of diffusion from a curve to a line
 Driftless diffusion on a curve
 The case of diffusion with drift
 Reconstruction of a parabola from projected diffusion data
 Appendix 2
 Reconstruction of projected stochastic dynamics
 Reconstruction of a surface from planar projections of diffusion trajectories
 The drift field
 The reconstruction procedure
 Annotations
 Asymptotic formulas in molecular and cellular biology
 Introduction
 From molecular to cellular description
 Flux through narrow passages identifies cellular compartments
 Examples of asymptotic formulas : fluxes into small targets
 Formulas in two dimensions
 Narrow escape formulas in threedimensions
 Cuspshaped funnel : hidden targets control rates in R...
 DNA repair in a confined chromatin structure in R...
 Asymmetric dumbbellshaped cell division
 Annotations
 Bibliography
 Index.
 Publisher's Summary
 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 realworld applications, the book includes topics such as the FokkerPlanck 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 realworld problems from first principles.
(source: Nielsen Book Data)9783319768946 20180917
Subjects
Bibliographic information
 Publication date
 2018
 Series
 Applied mathematical sciences, 00665452 ; volume 199
 Note
 "ISSN 2196968X (electronic)"Title page verso.
 ISBN
 9783319768946 (hd.bd.)
 9783319768953 (online)
 3319768948 (hardcover)