Construction of stable explicit finite-difference schemes for Schroedinger type differential equations

A family of conditionally stable, forward Euler finite difference equations can be constructed for the simplest equation of Schroedinger type, namely u sub t - iu sub xx. Generalization of this result to physically realistic Schroedinger type equations is presented

Nonlinear-damped Duffing oscillators having finite time dynamics

A class of modified Duffing oscillator differential equations, having nonlinear damping forces, are shown to have finite time dynamics, i.e., the solutions oscillate with only a finite number of cycles, and, thereafter, the motion is zero. The relevance of this feature is briefly discussed in relationship to the mathematical modeling, analysis, and estimation of parameters for the vibrations of carbon nano-tubes and graphene sheets, and macroscopic beams and plates.Comment: 15 page

Systems Exhibiting Alternative Futures

We construct an explicit example of a physical system having alternative futures (AFs). Several other such systems are also introduced and characterized, but not discussed in detail. Our major goal is to use these results to demonstrate the existence and meaning of the concept of counterfactual histories (CFHs). We find that any system having AFs will also exhibit the phenomenon of CFHs

Art Is I, Science Is We

The expression of the title has been used for some time to produce a concise summary of the major distinction between “art” and “science.” Our goal is to give a fuller and deeper understanding of this statement by discussing its meaning and interpretation within the context of a precise definition of science. We conclude that “Art is I, science is we,” captures accurately the fundamental difference between these two disciplines

A fast, stable and accurate numerical method for the Black-Scholes equation of American options

In this work we improve the algorithm of Han and Wu (SIAM J. Numer. Anal. 41 (2003), 2081-2095) for American Options with respect to stability, accuracy and order of computational effort. We derive an exact discrete artificial boundary condition (ABC) for the Crank-Nicolson scheme for solving the Black-Scholes equation for the valuation of American options. To ensure stability and to avoid any numerical reflections we derive the ABC on a purely discrete level. Since the exact discrete ABC includes a convolution with respect to time with a weakly decaying kernel, its numerical evaluation becomes very costly for large-time simulations. As a remedy we construct approximate ABCs with a kernel having the form of a finite sum-of-exponentials, which can be evaluated in a very efficient recursion. We prove a simple stability criteria for the approximated artificial boundary conditions. Finally, we illustrate the efficiency and accuracy of the proposed method on several benchmark examples and compare it to previously obtained discretized ABCs of Mayfield and Han and Wu

Discrete Models for the Cube-Root Differential Equation

Our main purpose is to construct one standard and three nonstandard finite difference schemes for the cube-root differential equation. After an analysis of the general qualitative features of the solutions to this equation and a calculation of the exact period, we study the stability of the numerical solutions for the four discretization schemes. Our general conclusion is that the standard forward-Euler method gives unstable numerical solutions, while the three nonstandard schemes provide suitable integration procedures

Mathematical assessment of the role of vector insecticide resistance and feeding/resting behavior on malaria transmission dynamics: Optimal control analysis

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.The large-scale use of insecticide-treated bednets (ITNs) and indoor residual spraying (IRS), over the last two decades, has resulted in a dramatic reduction of malaria incidence globally. However, the effectiveness of these interventions is now being threatened by numerous factors, such as resistance to insecticide in the mosquito vector and their preference to feed and rest outdoors or early in the evening (when humans are not protected by the bednets). This study presents a new deterministic model for assessing the population-level impact of mosquito insecticide resistance on malaria transmission dynamics. A notable feature of the model is that it stratifies the mosquito population in terms of type (wild or resistant to insecticides) and feeding preference (indoor or outdoor). The model is rigorously analysed to gain insight into the existence and asymptotic stability properties of the various disease-free equilibria of the model namely the trivial disease-free equilibrium, the non-trivial resistant-only boundary disease-free equilibrium and a non-trivial disease-free equlibrium where both the wild and resistant mosquito geneotypes co-exist). Simulations of the model, using data relevant to malaria transmission dynamics in Ethiopia (a malaria-endemic nation), show that the use of optimal ITNs alone, or in combination with optimal IRS, is more effective than the singular implementation of an optimal IRS-only strategy. Further, when the effect of the fitness cost of insecticide resistance with respect to fecundity (i.e., assuming a decrease in the baseline birth rate of new resistant-type adult female mosquitoes) is accounted for, numerical simulations of the model show that the combined optimal ITNs-IRS strategy could lead to the effective control of the disease, and insecticide resistance effectively managed during the first 8 years of the 15-year implementation period of the insecticides-based anti-malaria control measures in the community.National Institute for Mathematical and Biological SynthesisNSF Award # EF-0832858The University of Tennessee, Knoxvill

Positivity-preserving nonstandard finite difference schemes for cross-diffusion equations in biosciences

We design nonstandard finite difference (NSFD) schemes which are unconditionally dynamically consistent with respect to the positivity property of solutions of cross-diffusion equations in biosciences. This settles a problem that was open for quite some time. The study is done in the setting of three concrete and highly relevant cross-diffusion systems regarding tumor growth, convective predator–prey pursuit and evasion model and reaction–diffusion–chemotaxis model. It is shown that NSFD schemes used for classical reaction–diffusion equations, such as the Fisher equation, for which the solutions enjoy the positivity property, are not appropriate for cross-diffusion systems. The reliable NSFD schemes are therefore obtained by considering a suitable implementation on the crossdiffusive term of Mickens’ rule of nonlocal approximation of nonlinear terms, apart from his rule of complex denominator function of discrete derivatives. We provide numerical experiments that support the theory as well as the power of the NSFD schemes over the standard ones. In the case of the cancer growth model, we demonstrate computationally that our NSFD schemes replicate the property of traveling wave solutions of developing shocks observed in Marchant et al. (2000).South African Research Chairs Initiative of the Department of Science and Technology and National Research Foundation : SARChI Chair in Mathematical Models and Methods in Bioengineering and Biosciences.http://www.elsevier.com/locate/camwa2015-11-30hb201