Asymptotics of Elliptic and Parabolic PDEs

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.

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 Poisson Nernst Planck Equations in a Ball. - 11 Reconstruction of Surface Diffusion from Projected Data. - 12 Asymptotic Formulas in Molecular and Cellular Biology. - Bibliography. - Index.