<p style='text-indent:20px;'>For differential equations with state-dependent delays the associated initial value problem is well-posed, differentiable solution operators, on submanifolds of space <inline-formula><tex-math id="M1">\begin{document}$ C^1_n=C^1([-r,0],\mathbb{R}^n) $\end{document}</tex-math></inline-formula>, under mild smoothness assumptions. We study...

Suppose you are given a simple first order smooth ordinary differential equation with a given initial condition. If you cannot write down its solution explicitly, you might find a discrete approximate solution by using the Euler or Runga-Kutta algorithm, just to have some initial idea how the smooth solution behaves. In this case, your interest in the approximate solution is only as a stepping ...

hybrid system is a dynamic system that exhibits both continuous and discrete dynamic behavior‎. ‎the hybrid differential equations have a wide range of applications in science and engineering‎. ‎the hybrid systems are devoted to modeling‎, ‎design‎, ‎and validation of interactive systems of computer programs and continuous systems‎. ‎hybrid fuzzy differential equations (hfdes) is considered by ...

In this note we propose a numerical method to approximate the solution of a Backward Stochastic Differential Equations with Jumps (BSDEJ). This method is based on the construction of a discrete BSDEJ driven by a complete system of three orthogonal discrete time-space martingales, the first a random walk converging to a Brownian motion; the second, another random walk, independent of the first o...

This paper studies the discrete-time sampled-data approximation of the input-constrained finite-horizon linear quadratic regulator problem. Explicit estimates for the error in the performance index rendered by the solution of a discrete-time approximation are provided. This result can be used to evaluate the level of sub-optimality of the sampled-data solution corresponding to a given discretis...

The aim of this paper is to study the high order difference scheme for the solution of a fractional partial differential equation (PDE) in the electroanalytical chemistry. The space fractional derivative is described in the Riemann-Liouville sense. In the proposed scheme we discretize the space derivative with a fourth-order compact scheme and use the Grunwald- Letnikov discretization of the Ri...

In this paper, system planning network is formulated with mixed-integer programming. Two meta-heuristic procedures are considered for this problem. The cost function of this problem consists of the capital investment cost in discrete form, the cost of transmission losses and the power generation costs. The DC load flow equations for the network are embedded in the constraints of the mathematica...

We present a fourth order numerical solution method for the singular Neumann boundary problem of Poisson equations. Such problems arise in the solution process of incompressible Navier–Stokes equations and in the time-harmonic wave propagation in the frequence space with the zero wavenumber. The equation is first discretized with a fourth order modified Collatz difference scheme, producing a si...

In this paper, a positive definite semi-discrete mixed finite element method was presented for two-dimensional parabolic equations. In the new positive definite systems, the gradient equation and flux equations were separated from their scalar unknown equations.  Also, the existence and uniqueness of the semi-discrete mixed finite element solutions were proven. Error estimates were also obtaine...