Many phenomena and systems in science and engineering may be accurately modeled by systems of IVP (initial value problem) ODEs (ordinary differential equations). As two simple examples, at rather different scales, we cite the dynamics of an automobile suspension and the motion of the planets in our solar system. In this nutshell we describe the essential preparations and ingredients for numeric...

page 1 Linear Multistep Methods Note: The authoritative reference for the material on convergence is the book by Peter Henrici, Discrete Variable Methods in Ordinary Differential Equations , Wiley, 1962. The best reference on absolute stability is the book by Jack Lambert, Numerical Methods for Ordinary Differential Systems, Wiley, 1991. We consider the Initial Value Problem (IVP) y′ = f(x, y),...

in this paper, we introduce the new class of implicit l-stable generalized hybrid methods forthe numerical solution of first order initial value problems. we generalize the hybrid methodswith utilize ynv directly in the right hand side of classical hybrid methods. the numericalexperimentation showed that our method is considerably more efficient compared to wellknown methods used for the numer...

In this paper, we introduce a new technique to find the approximate solution of a nonlinear initial value problem (IVP). By introducing an artificial zero cost function and a linear functional, the problem is modified into one consisting of the minimization of a positive linear functional over a set of Radon measures. Then we obtain an optimal measure which is approximated by a finite combinati...

In this paper we prove the arcwise connectedness of the solution set for the initial value problems (IVP for short), for a class of nonconvex and nonclosed functional hyperbolic differential inclusions of fractional order.

We present some open problems pertaining to the approximation theory involved in the solution of the Nonlinear Schrödinger (NLS) equation. For this important equation, any Initial Value Problem (IVP) can be theoretically solved by the Inverse Scattering Transform (IST) technique whose main steps involve the solution of Volterra equations with structured kernels on unbounded domains, the solutio...