Narrow Results

The Lie series technique is applied to solve a class of nonlinear differential equations. We show that it includes also the dynamical integration technique. In particular systems with limit cycle behavior are studied. Differential equations with chaotic behavior are also considered with this technique.

Multiphase flows occurring in circular curved pipes exhibit important physical phenomena.They are characterized by a large pressure drop and are composed of different phases. In the past, erosion–corrosion was measured through the use of experimental methods. Today numerical simulation models provide a more in depth look into the problem of erosion. Solid particle erosion is of major concern in the industrial engineering sector. In this study, erosion occurring in a (90)-degree elbow has been simulated. The generated two-dimensional data was done through the use of the Commercial software ANSYS Fluent. The primary idea comes from the petrochemicals industry. To overcome this problem, counter measures are proposed in this paper to the piping setup in order to protect pumps from unwanted excessive sand concentrations. Note that the physical properties of the simulated fluid mixture are taken the same as for the real-studied sample.

The Newton–Raphson method for solving nonlinear equations f(x) = 0 in ℝⁿ is discussed within the context of ordinary differential equations. This framework makes it possible to reformulate the scheme by means of an adaptive step size control procedure that aims at reducing the chaotic behavior of the original method without losing the quadratic convergence close to the roots. The performance of the modified scheme is illustrated with a few low-dimensional examples.

The Hobson–Rogers model is used to price derivative securities under the no-arbitrage condition in a stochastic volatility setting, preserving the completeness of the market. Here we are studying the rate of convergence of the Euler/Monte Carlo approximations, when pricing European, Asian and digital type options. The aim of the present work is to express the approximation error in terms of the time step size, denoted by h, used for the Euler scheme. We recover an already known result, obtained by other authors using PDE approximations, for European options. Namely we show that for a Lipschitz coefficient of the driving equations for the asset price and Lipschitz payoffs, we obtain an error of the order of . Moreover, using Malliavin Calculus techniques, we show that with a regular coefficient we may attain an error of the order of h for regular payoffs and of the order of for non Lipschitz payoffs. Finally we show some numerical simulations supporting our theoretical results.

In this paper, a fractional-order model which describes the human immunodeficiency type-1 virus (HIV-1) infection is presented. Numerical solutions are obtained using a generalized Euler method (GEM) to handle the fractional derivatives. The fractional derivatives are described in the Caputo sense. We show that the model established in this paper possesses non-negative solutions. Comparisons between the results of the fractional-order model, the results of the integer model and the measured real data obtained from 10 patients during primary HIV-1 infection are presented. These comparisons show that the results of the fractional-order model give predictions to the plasma virus load of the patients better than those of the integer model.

Stochastic spatial models are becoming a popular tool for understand the ecological and evolution of ecosystem problems. We consider the predator prey interactions in term of stochastic representation of this Lotka-Volterra model and explore the use of stochastic processes to extinction behavior of the interacting populations. Here, we present simulation of stochastic processes of continuous time Lotka-Volterra model. Euler method has been used to solve the predator prey system. The trajectory spiral graph has been plotted based on obtained solution to show the population cycle of predator as a function of time.