Parameter Uniform Numerical Method for Singularly Perturbed 2D Parabolic PDE with Shift in Space

Singularly perturbed 2D parabolic delay differential equations with the discontinuous source term and convection coefficient are taken into consideration in this paper. For the time derivative, we use the fractional implicit Euler method, followed by the fitted finite difference method with bilinear interpolation for locally one-dimensional problems. The proposed method is shown to be almost first-order convergent in the spatial direction and first-order convergent in the temporal direction. Theoretical results are illustrated with numerical examples

Solving a System of One-Dimensional Hyperbolic Delay Differential Equations Using the Method of Lines and Runge-Kutta Methods

In this paper, we consider a system of one-dimensional hyperbolic delay differential equations (HDDEs) and their corresponding initial conditions. HDDEs are a class of differential equations that involve a delay term, which represents the effect of past states on the present state. The delay term poses a challenge for the application of standard numerical methods, which usually require the evaluation of the differential equation at the current step. To overcome this challenge, various numerical methods and analytical techniques have been developed specifically for solving a system of first-order HDDEs. In this study, we investigate these challenges and present some analytical results, such as the maximum principle and stability conditions. Moreover, we examine the propagation of discontinuities in the solution, which provides a comprehensive framework for understanding its behavior. To solve this problem, we employ the method of lines, which is a technique that converts a partial differential equation into a system of ordinary differential equations (ODEs). We then use the Runge–Kutta method, which is a numerical scheme that solves ODEs with high accuracy and stability. We prove the stability and convergence of our method, and we show that the error of our solution is of the order O(Δt+h¯4), where Δt is the time step and h¯ is the average spatial step. We also conduct numerical experiments to validate and evaluate the performance of our method

Flexible PON Key technologies: digital advanced modulation formats and devices

Flexible PON are future paradigm in parallel with flexible and elastic optical networks are under research for core networks. In the same way as those backbone optical networks can be significantly improved by following software-defined network (SDN) techniques, it is described how SDN PONs can be implemented by highly spectral efficient digital modulation formats. A main challenge is the implementation by cost effective devices. We will show the progress in alternatives implementations and adequacy of diverse modulation formats to cost effective bandwidth limited optical sources and receivers.Postprint (published version

Flexible PON Key technologies: digital advanced modulation formats and devices

Flexible PON are future paradigm in parallel with flexible and elastic optical networks are under research for core networks. In the same way as those backbone optical networks can be significantly improved by following software-defined network (SDN) techniques, it is described how SDN PONs can be implemented by highly spectral efficient digital modulation formats. A main challenge is the implementation by cost effective devices. We will show the progress in alternatives implementations and adequacy of diverse modulation formats to cost effective bandwidth limited optical sources and receivers