The Kohlrausch functions $\exp (- {t}^{\beta } )$, with $\beta \in (0, 1)$, which are important in a wide range of physical, chemical and biological applications, correspond to specific realizations of completely monotone functions. In this paper, using nonuniform grids and midpoint estimates, constructive procedures are formulated and analysed for the Kohlrausch functions. Sharper estimates are discussed to improve the approximation results. Numerical results and representative approximations are presented to illustrate the effectiveness of the proposed method.

Numerical solutions of a non-Fickian diffusion equation belonging to a hyperbolic type are presented in one space dimension. The Brownian particle modelled by this diffusion equation is subjected to a symmetric periodic potential whose spatial shape can be varied by a single parameter. We consider a numerical method which consists of applying Laplace transform in time; we then obtain an elliptic diffusion equation which is discretized using a finite difference method. We analyze some aspects of the convergence of the method. Numerical results for particle density, flux and mean-square-displacement (covering both inertial and diffusive regimes) are presented.

In this paper, the collocation methods are used to solve the boundary integral equations of the first kind on the polygon. By means of Sidi’s periodic transformation and domain decomposition, the errors are proved to possess the multi-parameter asymptotic expansion at the interior point with the powers (i = 1,...,d), which means that the approximations of higher accuracy and a posteriori estimation of the errors can be obtained by splitting extrapolations. Numerical experiments are carried out to show that the methods are very efficient.

In earlier work we have studied a method for discretization in time of a parabolic problem, which consists of representing the exact solution as an integral in the complex plane and then applying a quadrature formula to this integral. In application to a spatially semidiscrete finite-element version of the parabolic problem, at each quadrature point one then needs to solve a linear algebraic system having a positive-definite matrix with a complex shift. We study iterative methods for such systems, considering the basic and preconditioned versions of first the Richardson algorithm and then a conjugate gradient method.

The immersed boundary (IB) method is an approach to problems of fluid-structure interaction in which an elastic structure is immersed in a viscous incompressible fluid. The IB formulation of such problems uses a Lagrangian description of the structure and an Eulerian description of the fluid. It is well known that some versions of the IB method can suffer from poor volume conservation. Methods have been introduced to improve the volume-conservation properties of the IB method, but they either have been fairly specialized, or have used complex, nonstandard Eulerian finite-difference discretizations. In this paper, we use quasi-static and dynamic benchmark problems to investigate the effect of the choice of Eulerian discretization on the volume-conservation properties of a formally second-order accurate IB method. We consider both collocated and staggered-grid discretization methods. For the tests considered herein, the staggered-grid IB scheme generally yields at least a modest improvement in volume conservation when compared to cell-centered methods, and in many cases considered in this work, the spurious volume changes exhibited by the staggered-grid IB method are more than an order of magnitude smaller than those of the collocated schemes. We also compare the performance of cell-centered schemes that use either exact or approximate projection methods. We find that the volume-conservation properties of approximate projection IB methods depend strongly on the formulation of the projection method. When used with the IB method, we find that pressure-free approximate projection methods can yield extremely poor volume conservation, whereas pressure-increment approximate projection methods yield volume conservation that is nearly identical to that of a cell-centered exact projection method.

The mean time to extinction of a critical Galton-Watson process with initial population size k is shown to be asymptotically equivalent to two integral transforms: one involving the kth iterate of the probability generating function and one involving the generating function itself. Relating the growth of these transforms to the regular variation of their arguments, immediately connects statements involving the regular variation of the probability generating function, its iterates at 0, the quasistationary measures, their partial sums, and the limiting distribution of the time to extinction. In the critical case of finite variance we also give the growth of the mean time to extinction, conditioned on extinction occurring by time n.

We analyse the role of Euler summation in a numerical inversion algorithm for Laplace transforms due to Abate and Whitt called the EULER algorithm. Euler summation is shown to accelerate convergence of a slowly converging truncated Fourier series; an explicit bound for the approximation error is derived that generalizes a result given by O'Cinneide. An enhanced inversion algorithm called EULER-GPS is developed using a new variant of Euler summation. The algorithm EULER-GPS makes it possible to accurately invert transforms of functions with discontinuities at arbitrary locations. The effectiveness of the algorithm is verified through numerical experiments. Besides numerical transform inversion, the enhanced algorithm is applicable to a wide range of other problems where the goal is to recover point values of a piecewise-smooth function from the Fourier series.