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Asymptotics of elliptic and parabolic PDEs: and their applications in statistical physics, computational neuroscience, and biophysics

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"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  Read more...

Abstract: "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"--Page 4 of cover

Author(s): Holcman, David; Schuss, Zeev
Series: Applied mathematical sciences (Springer-Verlag New York Inc.) 199
Publisher: Springer
Year: 2018

Language: English
Pages: 444
Tags: Differential equations, Partial -- Asymptotic theory.;Differential equations, Elliptic -- Asymptotic theory.;Differential equations, Parabolic -- Asymptotic theory.

Content: Singular perturbations of elliptic boundary problems --
Second-order 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 first-order initial value problem --
Matched asymptotic expansions --
An Application to stochastic differential equations : the exit problem in R1 --
Asymptotics of a second-order boundary value problem --
Asymptotics of a homogeneous second-order 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 fokker-planck operator --
A diffusion model of random signals --
Loss of lock in a first-order phase-locked loop in phase-modulated radio signals --
Annotations --
Singular perturbations in higher dimensions --
Introduction --
The WKB method --
The eikonal equation --
The transport equation --
The characteristic equations --
Boundary layers at non-characteristic boundaries --
Boundary layers at characteristic boundaries in the plane --
The boundary value problem with non-characteristic boundaries --
The boundary value problem in planar domains with characteristic boundaries --
Loss of lock in a second-order phase-locked loop --
The phase plane of the reduced problem --
The mean time to lose lock --
The boundary layer structure of ... --
Asymptotic solution of the stationary fokker-planck 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 fokker-planck equation near the limit cycle --
The boundary value problem for the fokker-planck equation in ... --
Annotations --
Eigenvalues of a non-self-adjoint 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 --
Higher-order eigenvalues --
Oscillatory escape time --
Spontaneous activity in the cerebral cortex --
Numerical study of oscillatory decay --
A model of up-state 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 two-term approximation of the exit-time 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 --
Short-time asymptotics of the heat kernel --
The one-dimensional 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 short-time asymptotic of the fokker-planck 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 --
Higher-order asymptotics in the unit ball --
The narrow escape time through multiple absorbing windows --
Annotations --
The mixed boundary value problem in R2 --
A neumann-dirichlet 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 dumbbell-shaped 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 --
Second-order asymptotics for a circular window --
The first eigenvalue for two small dirichlet windows --
Multiple absorbing windows --
Higher-order 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 funnel-shaped 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 --
Short-time 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 --
Poissonian-like approximation --
Pr ... of N Brownian i.i.d. Trajectories in a segment --
Applications of the FAT in biophysics --
Annotations --
The poisson-nernst-planck equations in a ball --
Introduction --
Synopsis of results --
Poisson-nernst-planck equations in a ball --
The steady-state 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 voltage-clamped dendritic spine --
Appendix 1 : reverse liouville-gelfand-bratu equation --
Small a expansion of ... (x) --
Numerical scheme for the solution of (10.13) --
Steady solution in a ball with a cusp-shaped funnel --
Reduced equations in an uncharged cusp-shaped funnel --
Asymptotics of voltage between funnel and center --
Poisson-nernst-planck solutions in a 3d cusp-shaped funnel --
Asymptotic analysis of the pnp equations in a cusp-shaped 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 three-dimensions --
Cusp-shaped funnel : hidden targets control rates in R ... --
DNA repair in a confined chromatin structure in R ... --
Asymmetric dumbbell-shaped cell division --
Annotations --
Bibliography --
Index.