Analysis Of Initial Value Problems Research Articles

Verification and Reachability Analysis of Fractional-Order Differential Equations Using Interval Analysis

Interval approaches for the reachability analysis of initial value problems for sets of classical ordinary differential equations have been investigated and implemented by many researchers during the last decades. However, there exist numerous applications in computational science and engineering, where continuous-time system dynamics cannot be described adequately by integer-order differential equations. Especially in cases in which long-term memory effects are observed, fractional-order system representations are promising to describe the dynamics, on the one hand, with sufficient accuracy and, on the other hand, to limit the number of required state variables and parameters to a reasonable amount. Real-life applications for such fractional-order models can, among others, be found in the field of electrochemistry, where methods for impedance spectroscopy are typically used to identify fractional-order models for the charging/discharging behavior of batteries or for the dynamic relation between voltage and current in fuel cell systems if operated in a non-stationary state. This paper aims at presenting an iterative method for reachability analysis of fractional-order systems that is based on an interval arithmetic extension of Mittag-Leffler functions. An illustrating example, inspired by a low-order model of battery systems concludes this contribution.

Stabilization of spectral methods for the analysis of singular systems using piecewise constant basis functions

The analysis of initial value problems associated with singular linear dynamic systems using spectral methods is discussed. These methods, though extensively studied in the literature, can be very sensitive to numerical errors and to the presence of noisy input data. Moreover, using piecewise constant functions, like Walsh or Block-Pulse, spectral methods have been shown to be unstable with respect to incorrect initial conditions. In this paper regularization techniques are used to reduce arbitrarily the sensitivity of the spectral methods, and at the same time to stabilize them. Although the regularized spectral methods maintain the structure of the original algorithm, they can also be rewritten in recursive form, leading to the definition of a class of discrete time filters. Finally, some applicative numerical examples are discussed in order to show the effectiveness of the proposed algorithms.

On the use of stability regions in the numerical analysis of initial value problems

This paper deals with the stability analysis of one-step methods in the numerical solution of initial (-boundary) value problems for linear, ordinary, and partial differential equations. Restrictions on the stepsize are derived which guarantee the rate of error growth in these methods to be of moderate size. These restrictions are related to the stability region of the method and to numerical ranges of matrices stemming from the differential equation under consideration. The errors in the one-step methods are measured in arbitrary norms (not necessarily generated by an inner product). The theory is illustrated in the numerical solution of the heat equation and some other differential equations, where the error growth is measured in the maximum norm.

On the Use of Stability Regions in the Numerical Analysis of Initial Value Problems

This paper deals with the stability analysis of one-step methods in the numerical solution of initial (-boundary) value problems for linear, ordinary, and partial differential equations. Restrictions on the stepsize are derived which guarantee the rate of error growth in these methods to be of moderate size. These restrictions are related to the stability region of the method and to numerical ranges of matrices stemming from the differential equation under consideration. The errors in the one-step methods are measured in arbitrary norms (not necessarily generated by an inner product). The theory is illustrated in the numerical solution of the heat equation and some other differential equations, where the error growth is measured in the maximum norm.

Dynamic strain localization in elasto-(visco-)plastic solids, Part 1. General formulation and one-dimensional examples

Abstract Viscoplasticity is introduced as a procedure to regularize the elasto-plastic solid, especially for those situations in which the underlying inviscid material exhibits instabilities which preclude further analysis of initial-value problems. The procedure is general, and therefore has the advantage of allowing the regularization of any inviscid elastic-plastic material. Rate-dependency is shown to naturally introduce a length-scale into the dynamical initial-value problem. Furthermore, the width of the localized zones in which high strain gradients prevail and strain accumulations take place, is shown to be proportional to the characteristic length c η, which is the distance the elastic wave travels in the characteristic time η. Viscosity can thus be viewed either as a regularization parameter (computational point of view), or as a substructural/micromechanical parameter to be determined from observed shear-band widths (physical point of view). Finally, from a computational point of view, the proposed approach is shown to have striking advantages: (1) the wave speeds always remain real (even in the softening regime) and are set by the elastic moduli; (2) the elasto-(visco-)plastic constitutive equations are amenable to unconditionally stable integration; (3) the resulting well-posedness of the dynamical initial-value problem guarantees stable and convergent solutions with mesh refinements. The initial-value problems reported in this first part are essentially one-dimensional. They are used because they offer the simplest possible context to illustrate both the physical and computational significance of the proposed viscoplastic regularization procedure. The methods used in multi-dimensional analysis and examples will be reported in Part 2.

Sensitivity analysis of initial value problems with mixed odes and algebraic equations

An efficient method is described for sensitivity analysis of nonlinear initial value problems, which may include algebraic equations as well as ordinary differential equations. The linearity of the sensitivity equations is utilized to solve them directly via the local Jacobian of the state equations. The method is implemented with the implicit integrator DASSL and is demonstrated on a stiff industrial reaction model.