The weak and strong ergodic properties of the positive solutions of nonautonomous linear ordinary differential equations will be considered. We will show that if the coefficient matrix function is bounded, essentially nonnegative and uniformly irreducible, then the normalized positive solutions are asymptotically equivalent to the Perron vectors of the strongly positive transition matrix at infinity (weak ergodicity). If, in addition, the coefficient matrix function is uniformly continuous, then the convergence of the normalized positive solutions to the same strongly positive limiting vector (strong ergodicity) is equivalent to the convergence of the Perron vectors of the coefficient matrices.
This is a joint work with Professor Christian Pötzsche (Alpen-Adria University Klagenfurt, Austria).
The work to be presented concerns numerical solution of nonlinear two-point boundary value problems. We show that the quasi-linearization method (Newton method on operator level) can be used as a basis to derive (i) the FDM with Newton method and (ii) the shooting by Newton method. The same relation holds for the Picard and the constant-slope methods. Based on these results, we propose (i) a replacement of the FDMs for nonlinear problems (the relaxation methods) by respective successive application of the linear shooting method and (ii) a shooting by Picard method (shooting-projection method). We discuss the advantages of the proposed approaches and present examples.
Our world is not ideal, in reality the processes are dissipative. The framework of non-equilibrium thermodynamics offers lot of possibilities to derive models, constitutive equations that describe the
behavior of a dissipative system. The level of modeling is arbitrary, depends on our choice which is
reflected by these models. However, they should be remain as simple as possible to be applicable for practical problems in question.
First, the non-equilibrium thermodynamical background and the structure of equations are discussed. Here mostly the parabolic - hyperbolic properties of the resulted partial differential equations (PDE) are emphasized. The role of boundary conditions and their effect on solutions are also presented through different examples. Such example is related to a particular experimental arrangement called heat pulse (or laser flash) experiment that used to detect different dissipative wave propagation phenomena.
In this presentation the way from generating the equations to their solutions for experiments is presented. It covers analytical solution of a PDE for time dependent boundary condition and a particular numerical method that allows us to eliminate certain boundary conditions and related to the specific structure resulted by non-equilibrium thermodynamics.