Discrete Approximations and Optimization of Evolution Inclusions

The paper studies discrete approximations of nonconvex valued evolution inclusions with the right-hand side satisfying Kamke condition which is more general than the Lipschitz one and more convenient than the variant of the one-sided Lipschitz condition used in Donchev et al. (J Differ Equ 243:301–328, 2007). We extend an interesting previous result of Mordukhovich to a large class of evolution systems appearing in the theory of parabolic partial differential equations. Examples of control systems governed by partial differential equations are provided.     

Structure–activity relationship models for rat carcinogenesis and assessing the role mutagens play in model predictivity

We previously demonstrated that fragment based cat-SAR carcinogenesis models consisting solely of mutagenic or non-mutagenic carcinogens varied greatly in terms of their predictive accuracy. This led us to investigate how well the rat cancer cat-SAR model predicted mutagens and non-mutagens in their learning set. Four rat cancer cat-SAR models were developed: Complete Rat, Transgender Rat, Male Rat and Female Rat, with leave-one-out (LOO) validation concordance values of 69%, 74%, 67% and 73%, respectively. The mutagenic carcinogens produced concordance values in the range 69–76% compared with only 47–53% for non-mutagenic carcinogens. As a surrogate for mutagenicity, comparisons between single site and multiple site carcinogen SAR models were analysed. The LOO concordance values for models consisting of 1-site, 2-site and 4+-site carcinogens were 66%, 71% and 79%, respectively. As expected, the proportion of mutagens to non-mutagens also increased, rising from 54% for 1-site to 80% for 4+-site carcinogens. This study demonstrates that mutagenic chemicals, in both SAR learning sets and test sets, are influential in assessing model accuracy. This suggests that SAR models for carcinogens may require a two-step process in which mutagenicity is first determined before carcinogenicity can be accurately predicted.     

Bifurcation analysis and chaos control in a discrete‐time plant quality and larch budmoth interaction model with Ricker equation

We investigate the dynamics of two‐dimensional discrete‐time model of leaf quality and larch budmoth interaction with Ricker equation. More precisely, the qualitative behavior of larch budmoth model is discussed in which the effect of food source upon the moth population is through intrinsic growth rate. We find the parametric conditions for local asymptotic stability of the unique positive fixed point. It is also proved that under certain parametric conditions, the system undergoes period‐doubling bifurcation with the help of center manifold theory. The parametric conditions for existence and direction of Neimark‐Sacker bifurcation at positive fixed point is investigated with the help of standard mathematical techniques of bifurcation theory. The chaos control in the system is discussed through implementation of hybrid control methodology. Finally, numerical simulations are provided to illustrate theoretical results. These results of numerical simulations demonstrate chaotic long‐term behavior over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the system.     

Finite-difference analysis of natural convection flow of a viscous fluid in a porous channel with constant heat sources

The fully developed natural convection flow of a viscous fluid in a porous channel is modeled and studied numerically. The walls are kept at constant temperatures. The effects of various dimensionless parameters emerging in the model are studied graphically. It has been noted that the velocity and temperature both depend on the heat source and the free convection parameters.     

Numerical Approximations of Impulsive Delay Differential Equations

The article deals with numerical approximations of impulsive delay differential equations with a non-fixed time of impulses. The right-hand side of the approximation is assumed to be Lipschitz with respect to the norm of the measurable functions, which allows us to estimate the distance between functions with different times of jumps. Illustrative examples are provided.     

Neimark–Sacker bifurcation and chaos control in discrete-time predator–prey model with parasites

In this paper, we discuss the qualitative behavior of a four-dimensional discrete-time predator–prey model with parasites. We investigate existence and uniqueness of positive steady state and find parametric conditions for local asymptotic stability of positive equilibrium point of given system. It is also proved that the system undergoes Neimark–Sacker bifurcation (NSB) at positive equilibrium point with the help of an explicit criterion for NSB. The system shows chaotic dynamics at increasing values of bifurcation parameter. Chaos control is also discussed through implementation of hybrid control strategy, which is based on feedback control methodology and parameter perturbation. Finally, numerical simulations are conducted to illustrate theoretical results.     

A high order kinetic flux-vector splitting method for the reduced five-equation model of compressible two-fluid flows

We present a high order kinetic flux-vector splitting (KFVS) scheme for the numerical solution of a conservative interface-capturing five-equation model of compressible two-fluid flows. This model was initially introduced by Wackers and Koren (2004) [21]. The flow equations are the bulk equations, combined with mass and energy equations for one of the two fluids. The latter equation contains a source term in order to account for the energy exchange. We numerically investigate both one- and two-dimensional flow models. The proposed numerical scheme is based on the direct splitting of macroscopic flux functions of the system of equations. In two space dimensions the scheme is derived in a usual dimensionally split manner. The second order accuracy of the scheme is achieved by using MUSCL-type initial reconstruction and Runge–Kutta time stepping method. For validation, the results of our scheme are compared with those from the high resolution central scheme of Nessyahu and Tadmor [14]. The accuracy, efficiency and simplicity of the KFVS scheme demonstrate its potential for modeling two-phase flows.     

Kinetic schemes for the relativistic gas dynamics

Summary. A kinetic solution for the relativistic Euler equations is presented. This solution describes the flow of a perfect gas in terms of the particle density n, the spatial part of the four-velocity u and the inverse temperature . In this paper we present a general framework for the kinetic scheme of relativistic Euler equations which covers the whole range from the non-relativistic limit to the ultra-relativistic limit. The main components of the kinetic scheme are described now. (i) There are periods of free flight of duration _M, where the gas particles move according to the free kinetic transport equation. (ii) At the maximization times t_n=n_M, the beginning of each of these free-flight periods, the gas particles are in local equilibrium, which is described by Jüttners relativistic generalization of the classical Maxwellian phase density. (iii) At each new maximization time t_n>0 we evaluate the so called continuity conditions, which guarantee that the kinetic scheme satisfies the conservation laws and the entropy inequality. These continuity conditions determine the new initial data at t_n. iv If in addition adiabatic boundary conditions are prescribed, we can incorporate a natural reflection method into the kinetic scheme in order to solve the initial and boundary value problem. In the limit _M0 we obtain the weak solutions of Eulers equations including arbitrary shock interactions. We also present a numerical shock reflection test which confirms the validity of our kinetic approach. Mathematics Subject Classification (1991):65M99, 76Y05This work is supported by the project Long-time behaviour of nonlinear hyperbolic systems of conservation laws and their numerical approximation, contract # DFG WA 633/7-2.     

A general Struble's technique for solving an nth order weakly non-linear differential system with damping

A general formula (based on the method of variation of parameters) has been presented for determining an approximate solution of an nth order n=2,3,… weakly non-linear differential system with several damping effects. The general solution covers the under-damped, undamped and over-damped cases. The formulation as well as determination of the solution is simple. The method is illustrated by several examples.