Problems with different time scales

Difference Methods for Problems With Different Time Scales

We consider the use of difference methods for weakly nonlinear systems of ordinary differential equations with rapidly oscillating solutions and develop a general approach which depends only on the smoothness of the coefficients and the nonlinearities. In particular, one is led to a strategy which is suitable for the detection and resolution of turning points in such systems. A computational ex...

Hyperbolic Systems with Different Time Scales

Time domain simulation of power systems with different time scales

The time evolution of power systems is modeled by a system of differential and algebraic equations. The variables involved in the system may exhibit different time scales. In standard numerical time integration methods the most active variables impose the time step for the whole system. We present a strategy, which allows the use of different, local time steps over the variables. The partitioni...

Isoperimetric problems on time scales with nabla derivatives

We prove a necessary optimality condition for isoperimetric problems under nabla-differentiable curves. As a consequence, the recent results of [M.R. Caputo, A unified view of ostensibly disparate isoperimetric variational problems, Appl. Math. Lett. (2008), doi:10.1016/j.aml.2008.04.004], that put together seemingly dissimilar optimal control problems in economics and physics, are extended to ...

Periodic Boundary Value Problems on Time Scales

We extend the results concerning periodic boundary value problems from the continuous calculus to time scales. First we use the Schauder fixed point theorem and the concept of lower and upper solutions to prove the existence of the solutions and then we investigate a monotone iterative method which could generate some of them. Since this method does not work on each time scale, a condition cont...