Submit Manuscript | http://medcraveonline.com Abbreviations: 1D: One-Dimensional; 2D: Two-Dimensional; 3D: Three-Dimensional; DGTD: Discontinuous Galerkin TimeDomain; FEs: Finite Elements; TM: Transverse Magnetic; TE: Transverse Electric; ODE: Ordinary Differential Equations; LSERK: Low-Storage Explicit Runge-Kutta; CFL: Courant-Friedrichs-Levy; TD: Time-Domain; PEC: Perfect Electric Conductor;...

In this paper, we show that there are Initial Value Problems defined with polynomial ordinary differential equations that can simulate universal Turing machines in the presence of bounded noise. The polynomial ODE defining the IVP is explicitly obtained and the simulation is performed in real time.

Submit Manuscript | http://medcraveonline.com Abbreviations: 1D: One-Dimensional; 2D: Two-Dimensional; 3D: Three-Dimensional; DGTD: Discontinuous Galerkin TimeDomain; FEs: Finite Elements; TM: Transverse Magnetic; TE: Transverse Electric; ODE: Ordinary Differential Equations; LSERK: Low-Storage Explicit Runge-Kutta; CFL: Courant-Friedrichs-Levy; TD: Time-Domain; PEC: Perfect Electric Conductor;...

Submit Manuscript | http://medcraveonline.com Abbreviations: 1D: One-Dimensional; 2D: Two-Dimensional; 3D: Three-Dimensional; DGTD: Discontinuous Galerkin TimeDomain; FEs: Finite Elements; TM: Transverse Magnetic; TE: Transverse Electric; ODE: Ordinary Differential Equations; LSERK: Low-Storage Explicit Runge-Kutta; CFL: Courant-Friedrichs-Levy; TD: Time-Domain; PEC: Perfect Electric Conductor;...

Submit Manuscript | http://medcraveonline.com Abbreviations: 1D: One-Dimensional; 2D: Two-Dimensional; 3D: Three-Dimensional; DGTD: Discontinuous Galerkin TimeDomain; FEs: Finite Elements; TM: Transverse Magnetic; TE: Transverse Electric; ODE: Ordinary Differential Equations; LSERK: Low-Storage Explicit Runge-Kutta; CFL: Courant-Friedrichs-Levy; TD: Time-Domain; PEC: Perfect Electric Conductor;...

This paper proposes Runge-Kutta neural networks (RKNNs) for identification of unknown dynamical systems described by ordinary differential equations (i.e., ordinary differential equation or ODE systems) with high accuracy. These networks are constructed according to the Runge-Kutta approximation method. The main attraction of the RKNNs is that they precisely estimate the changing rates of syste...

We consider inital-value problems governed by a system of autonomous ordinary differential equations (ODEs). When the required integration stepsize of the ODE system is very very small in comparison to the time domain of interest. Then the initial-value problem is said to be stiff.

Continuous approximations that are ordinary differential equations (ODEs) or stochastic differential equations (SDEs) are often used to study the properties of discrete stochastic processes. We show that different ways of taking the continuous limit of the same model may result in either an ODE or a SDE and study the manner in which each approximates the discrete model. We compare the asymptoti...

We develop a geometric approach to definable sets in differentially closed fields, with emphasis on the question of orthogonality to a given strongly minimal set. Equivalently, within a family of ordinary differential equations, we consider those equations that can be transformed, by differential-algebraic transformations, so as to yield solutions of a given fixed first-order ODE X. We show tha...

Modeling dynamical systems with ordinary differential equations implies a mechanistic view of the process underlying the dynamics. However in many cases, this knowledge is not available. To overcome this issue, we introduce a general framework for nonparametric ODE models using penalized regression in Reproducing Kernel Hilbert Spaces (RKHS) based on operator-valued kernels. Moreover, we extend...