Modeling river delta formation

A new model to simulate the time evolution of river delta formation process is presented. It is based on the continuity equation for water and sediment flow and a phenomenological sedimentation/ erosion law. Different delta types are reproduced using different parameters and erosion rules. The structures of the calculated patterns are analyzed in space and time and compared with real data patterns. Furthermore our model is capable to simulate the rich dynamics related to the switching of the mouth of the river delta. The simulation results are then compared with geological records for the Mississippi river

Numerical Results for the Generalized Mittag-Leffler Function

Mathematics Subject Classification: 33E12, 33FXX PACS (Physics Abstracts Classification Scheme): 02.30.Gp, 02.60.GfResults of extensive calculations for the generalized Mittag-Leffler function E0.8,0.9(z) are presented in the region −8 ≤ Re z ≤ 5 and −10 ≤ Im z ≤ 10 of the complex plane. This function is related to the eigenfunction of a fractional derivative of order α = 0.8 and type β = 0.5

Computation of the generalized Mittag-Leffler function and its inverse in the complex plane

The generalized Mittag-Leffler function E α,β (z) has been studied for arbitrary complex argument z ∈ C and parameters α ∈ R + and β ∈ R. This function plays a fundamental role in the theory of fractional differential equations and numerous applications in physics. The Mittag-Leffler function interpolates smoothly between exponential and algebraic functional behaviour. A numerical algorithm for its evaluation has been developed. The algorithm is based on integral representations and exponential asymptotics. Results of extensive numerical calculations for E α,β (z) in the complex z-plane are reported here. We find that all complex zeros emerge from the point z = 1 for small α. They diverge towards −∞ + (2k − 1)πi for α → 1 − and towards −∞ + 2kπi for α → 1 + (k ∈ Z). All the complex zeros collapse pairwise onto the negative real axis for α → 2. We introduce and study also the inverse generalized Mittag-Leffler function L α,β (z) defined as the solution of the equation L α,β (E α,β (z)) = z. We determine its principal branch numerically

Path selection in the growth of rivers

River networks exhibit a complex ramified structure that has inspired decades of studies. However, an understanding of the propagation of a single stream remains elusive. Here we invoke a criterion for path selection from fracture mechanics and apply it to the growth of streams in a diffusion field. We show that, as it cuts through the landscape, a stream maintains a symmetric groundwater flow around its tip. The local flow conditions therefore determine the growth of the drainage network. We use this principle to reconstruct the history of a network and to find a growth law associated with it. Our results show that the deterministic growth of a single channel based on its local environment can be used to characterize the structure of river networks.United States. Department of Energy (Grant FG02-99ER15004

Galerkin FEM for fractional order parabolic equations with initial data in H−s, 0<s≤1H^{-s},~0 < s \le 1

We investigate semi-discrete numerical schemes based on the standard Galerkin and lumped mass Galerkin finite element methods for an initial-boundary value problem for homogeneous fractional diffusion problems with non-smooth initial data. We assume that $\Omega\subset \mathbb{R}^d$, $d=1,2,3$ is a convex polygonal (polyhedral) domain. We theoretically justify optimal order error estimates in $L_2$- and $H^1$-norms for initial data in $H^{-s}(\Omega),~0\le s \le 1$. We confirm our theoretical findings with a number of numerical tests that include initial data $v$ being a Dirac $\delta$-function supported on a $(d-1)$-dimensional manifold.Comment: 13 pages, 3 figure