Parabolic and Hyperbolic Contours for Computing the Bromwich Integral

Some of the most effective methods for the numerical inversion of the Laplace transform are based on the approximation of the Bromwich contour integral. The accuracy of these methods often hinges on a good choice of contour, and several such contours have been proposed in the literature. Here we analyze two recently proposed contours, namely a parabola and a hyperbola. Using a representative model problem, we determine estimates for the optimal parameters that define these contours. An application to a fractional diffusion equation is presented.\ud \ud JACW was supported by the National Research Foundation in South Africa under grant FA200503230001

From Diffusion to Anomalous Diffusion: A Century after Einstein's Brownian Motion

Einstein's explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian property and exhibit to anomalous diffusion. We consider here the case of subdiffusive processes, which are semi-Markovian and correspond to continuous-time random walks in which the waiting time for a step is given by a probability distribution with a diverging mean value. Such a process can be considered as a process subordinated to normal diffusion under operational time which depends on this pathological waiting-time distribution. We derive two different but equivalent forms of kinetic equations, which reduce to know fractional diffusion or Fokker-Planck equations for waiting-time distributions following a power-law. For waiting time distributions which are not pure power laws one or the other form of the kinetic equation is advantageous, depending on whether the process slows down or accelerates in the course of time

Fast Method of Particular Solutions for Solving Partial Differential Equations

Method of particular solutions (MPS) has been implemented in many science and engineering problems but obtaining the closed-form particular solutions, the selection of the good shape parameter for various radial basis functions (RBFs) and simulation of the large-scale problems are some of the challenges which need to overcome. In this dissertation, we have used several techniques to overcome such challenges. The closed-form particular solutions for the Matérn and Gaussian RBFs were not known yet. With the help of the symbolic computational tools, we have derived the closed-form particular solutions of the Matérn and Gaussian RBFs for the Laplace and biharmonic operators in 2D and 3D. These derived particular solutions play an important role in solving inhomogeneous problems using MPS and boundary methods such as boundary element methods or boundary meshless methods. In this dissertation, to select the good shape parameter, various existing variable shape parameter strategies and some well-known global optimization algorithms have also been applied. These good shape parameters provide high accurate solutions in many RBFs collocation methods. Fast method of particular solutions (FMPS) has been developed for the simulation of the large-scale problems. FMPS is based on the global version of the MPS. In this method, partial differential equations are discretized by the usual MPS and the determination of the unknown coefficients is accelerated using a fast technique. Numerical results confirm the efficiency of the proposed technique for the PDEs with a large number of computational points in both two and three dimensions. We have also solved the time fractional diffusion equations by using MPS and FMPS

From the area under the Bessel excursion to anomalous diffusion of cold atoms

Levy flights are random walks in which the probability distribution of the step sizes is fat-tailed. Levy spatial diffusion has been observed for a collection of ultra-cold Rb atoms and single Mg+ ions in an optical lattice. Using the semiclassical theory of Sisyphus cooling, we treat the problem as a coupled Levy walk, with correlations between the length and duration of the excursions. The problem is related to the area under Bessel excursions, overdamped Langevin motions that start and end at the origin, constrained to remain positive, in the presence of an external logarithmic potential. In the limit of a weak potential, the Airy distribution describing the areal distribution of the Brownian excursion is found. Three distinct phases of the dynamics are studied: normal diffusion, Levy diffusion and, below a certain critical depth of the optical potential, x~ t^{3/2} scaling. The focus of the paper is the analytical calculation of the joint probability density function from a newly developed theory of the area under the Bessel excursion. The latter describes the spatiotemporal correlations in the problem and is the microscopic input needed to characterize the spatial diffusion of the atomic cloud. A modified Montroll-Weiss (MW) equation for the density is obtained, which depends on the statistics of velocity excursions and meanders. The meander, a random walk in velocity space which starts at the origin and does not cross it, describes the last jump event in the sequence. In the anomalous phases, the statistics of meanders and excursions are essential for the calculation of the mean square displacement, showing that our correction to the MW equation is crucial, and points to the sensitivity of the transport on a single jump event. Our work provides relations between the statistics of velocity excursions and meanders and that of the diffusivity.Comment: Supersedes arXiv: 1305.008