Sensitivity Analysis of Tech 1 - A Systems Dynamics Model for Technological Shift

This paper deals with the sensitivity analysis of TECH1 -- a system dynamics model, which describes the technological shift from an old technology to a new one, within a specific scenario. However, its goal is not to describe the model, which was done by Robinson (1979), in this case the paper's goal is threefold: 1. To show with mathematical tools which factors are important for an invention to become an innovation, by interpreting in an economic sense the results of the performed analysis. 2. To make it possible for a broader range of people to understand system dynamics models -- especially TECH1 and consequently to improve them. 3. To show what kind of mathematical analysis is useful for a class of economic models represented by differential equations. Although TECH1 has not yet been applied to the real world, the author hopes that this paper will help to produce a better understanding of the innovation process in the real world, as well as of system dynamics models and their limits

Numerical Solution of Parabolic Problems with Non-Smooth Solutions

This paper deals with the convergence of stable and consistent one-step approximations for linear parabolic initial-boundary-value problems with non-smooth solutions. The proofs given may be extended to semilinear parabolic problems using H.B. Keller's stability concept. Finally an extension to Lax's convergence theorem is given

An Optimal Control Model for the Diffusion of Innovation

The diffusion of technology has been modeled using different modeling techniques, and these models should fulfill different needs. For example, there are spatial models, models describing the technological side or the consumer side, and so on. On the other hand there are many models which just describe the process but few really give advice for necessary regulations or outside influence. Therefore, I have set up a model called DIFFOPT, which should describe the production aspect and the societal aspect of the diffusion process, and should also give explicit advice for investment and price policy of innovative technologies. This paper deals with the development of DIFFOPT which has the structure of an optimal control model, and with the mathematical description. Moreover the descriptive model which is basic for the optimal control model is tested computationally

Low Momentum Classical Mechanics with Effective Quantum Potentials

A recently introduced effective quantum potential theory is studied in a low momentum region of phase space. This low momentum approximation is used to show that the new effective quantum potential induces a space-dependent mass and a smoothed potential both of them constructed from the classical potential. The exact solution of the approximated theory in one spatial dimension is found. The concept of effective transmission and reflection coefficients for effective quantum potentials is proposed and discussed in comparison with an analogous quantum statistical mixture problem. The results are applied to the case of a square barrier.Comment: 4 figure

A nonlinear equation for ionic diffusion in a strong binary electrolyte

The problem of the one dimensional electro-diffusion of ions in a strong binary electrolyte is considered. In such a system the solute dissociates completely into two species of ions with unlike charges. The mathematical description consists of a diffusion equation for each species augmented by transport due to a self consistent electrostatic field determined by the Poisson equation. This mathematical framework also describes other important problems in physics such as electron and hole diffusion across semi-conductor junctions and the diffusion of ions in plasmas. If concentrations do not vary appreciably over distances of the order of the Debye length, the Poisson equation can be replaced by the condition of local charge neutrality first introduced by Planck. It can then be shown that both species diffuse at the same rate with a common diffusivity that is intermediate between that of the slow and fast species (ambipolar diffusion). Here we derive a more general theory by exploiting the ratio of Debye length to a characteristic length scale as a small asymptotic parameter. It is shown that the concentration of either species may be described by a nonlinear integro-differential equation which replaces the classical linear equation for ambipolar diffusion but reduces to it in the appropriate limit. Through numerical integration of the full set of equations it is shown that this nonlinear equation provides a better approximation to the exact solution than the linear equation it replaces.Comment: 4 pages, 1 figur

On the Long Time Behavior of the Quantum Fokker-Planck equation

We analyze the long time behavior of transport equations for a class of dissipative quantum systems with Fokker-planck type scattering operator, subject to confining potentials of harmonic oscillator type. We establish the conditions under which there exists a thermal equilibrium state and prove exponential decay towards it, using (classical) entropy-methods. Additionally, we give precise dispersion estimates in the cases were no equilibrium state exists

On a one-dimensional steady-state hydrodynamic model for semiconductors

AbstractWe present a hydrodynamic model for semiconductors, where the energy equation is replaced by a pressure-density relationship. We prove existence of smooth solutions and a uniqueness result in the subsonic case, which is characterized by a smallness assumption on the current flowing through the device