1. In a recent series of articles [9, 10, 11, 12]*), I have developed parts of the general theory of linear and non-linear difference-differential equations. The significance and usefulness of the general results are somewhat obscured by the comparative lack of Information about specific examples, such s is readily available in the special cases of differential and difference equations. For this reason I discuss here the particular nonlinear equation (1.1) y'(x) = —*y(x — i ) { i + y(*)} (*> 0).

A non-linear difference-differential equation.

The general linear difference-differential equation takes the form where x is a real variable, ν( x ) and A μν ( x ) are known functions and

Linear difference-differential equations

An approach using the Lambert W function for the analytical solution, free and forced, to systems of delay differential equations with a single delay has been developed by Asl And Ulsoy (2003) and Yi and Ulso (2006). The solution is expressed in the form of an infinite series of modes written in terms of the matrix Lambert W function. In this paper, we utilize the analytical solution to present a solution in the Laplace domain, present validation examples, and emphasize the analogy of the solution method to systems of ordinary differential equations

Solution of Systems of Linear Delay Differential Equations via Laplace Transformation

Stability Criteria and the Real Roots of a Transcendental Equation

Several nonlinear Internet congestion control schemes have been proposed in the previous years based on primal and dual decompositions of convex yet centralized optimization problems that describe the bandwidth allocation problem. This design methodology uses (sub)gradient methods for the design of node/link dynamics and guarantees nominal functionality for arbitrary system sizes. In this technical note we analyze the robustness of such nonlinear schemes to the presence of multiple, non-commensurate (heterogeneous) communication delays.

Delay Robustness of Nonlinear Internet Congestion Control Schemes

The non-linear difference-differential equation

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The non-linear difference-differential equation