We consider numerical solutions of a class of nonlinear (nonstandard) Volterra integral equations. We first prove the existence and uniqueness of the solution of the Volterra integral equation in the context of the space of continuous funtions over a closed interval. We then use one point collocation methods and quadrature methods with a uniform mesh to construct solutions of the nonlinear VIE....

The Taylor series method is one of the earliest analytic-numeric algorithms for approximate solution of initial value problems for ordinary differential equations. The main idea of the rehabilitation of this algorithms is based on the approximate calculation of higher derivatives using well-known finite-difference technique for the partial differential equations. The approximate solution is giv...

We study a boundary integral equation method for solving Laplace s equation u with nonlinear boundary conditions This nonlinear boundary value problem is reformulated as a nonlinear boundary in tegral equation with u on the boundary as the solution being sought The integral equation is solved numerically by using the collocation method with piecewise quadratic functions used as approximations t...

In this work, the issue of estimation reachable sets in continuous bimodal piecewise affine systems is studied. A new method proposed, framework ellipsoidal bounding, using quadratic Lyapunov functions. Although can be seen as a special class hybrid systems, reachability methods developed for might inappropriately complex dynamics. This work goes direction exploiting dynamical structure system ...

Keywords: Volterra integro-differential equations Multistep collocation Superconvergence Stability a b s t r a c t Multistep collocation methods for Volterra integro-differential equations are derived and analyzed. They increase the order of convergence of classical one-step collocation methods, at the same computational cost. The numerical stability analysis is carried out and classes of A 0-s...

Space-time nite element methods for dynamic Signorini problems are discussed in this article. The discretization scheme is based on a mixed space-time formulation of the continuous problem, where the Lagrange multipliers represent the contact stress. To construct the trial space for the displacement and the velocity, we use piecewise polynomial and globally continuous basis functions in space a...

Parry showed that every continuous transitive piecewise monotonic map τ of the interval is conjugate (by an order preserving homeomorphism) to a uniformly piecewise linear map T (i.e., one with slopes ±s). In the current article, it is shown that the map T is unique. This is proven by showing that any order-preserving conjugacy between two continuous transitive uniformly piecewise linear maps i...

In recent years, new and more effective procedures for applying collocation have been published. This article is devoted to present a revision of this subject and complement its developments. From the general theory two broad approaches are derived, which yield the direct and the indirect TH-collocation methods. The former approach had not been published before, and it is a dual of the indirect...

We develop a new approach to the computation of the Hausdorff dimension of the invariant set of an iterated function system or IFS. In the one dimensional case that we consider here, our methods require only C3 regularity of the maps in the IFS. The key idea, which has been known in varying degrees of generality for many years, is to associate to the IFS a parametrized family of positive, linea...

In Brokate-Sprekels 1996, it is shown that hysteresis operators acting on scalar-valued, continuous, piecewise monotone input functions can be represented by functionals acting on alternating strings. In a number of recent papers, this representation result is extended to hysteresis operators dealing with input functions in a general topological vector space. The input functions have to be cont...