Considered herein are a number of variants of the classical Boussinesq system and their higher-order generalizations. Such equations were first derived by Boussinesq to describe the two-way propagation of small-amplitude, long wavelength, gravity waves on the surface of water in a canal. These systems arise also when modeling the propagation of long-crested waves on large lakes or the ocean and...

General linear methods were derived approximately thirty years ago as a unifying approach for the study of consistency, stability and convergence of the Runge-Kutta and the linear multistep methods. Their discovery opened the possibility of obtaining essentially new methods which were neither Runge-Kutta nor linear multistep methods nor slight variations of these methods. It was hoped that gene...

The solution of a linear time-invariant differential equation can be obtained as the output of a so-called canonical signal processing filter with the right hand side of the differential equation as input. Such a filter is build up with integrators, adders, multipliers and so on. One distinguishes in the literature between a series, a cascade and a parallel realization of the filter.The coeffic...

A historical development of the theory of linear partial differential equation is reviewed with comments. A recent development in the theory of linear partial differential equations is discussed.

Given an arbitrary initial value for the differential-algebraic equation _ ( ) + ( ) = ( ), an initial value can be selected from among all consistent initial values by means of the Laplace transform. We show that this choice is the only one that fulfills some simple, physically reasonable assumptions on linear systems’ behavior. Our derivation is elementary compared to previous justifications ...

Using inequalities for a certain function appearing in the half-linear version of Picone’s identity, we show that oscillatory properties of the half-linear second-order differential equation (r(t)Φ(x′))′ + c(t)Φ(x) = 0, Φ(x) = |x|p−2x, p > 1, can be investigated via oscillatory properties of a certain associated second-order linear differential equation. This linear equation plays the role of a...

For a linear ordinary differential equation the Lie algebra of its infinitesimal Lie symmetries is compared with its differential Galois group. For this purpose an algebraic formulation of Lie symmetries is developed. It turns out that there is no direct relation between the two above objects. In connection with this a new algorithm for computing the Lie symmetries of a linear ordinary differen...

Abstract. The aim of this paper is to show that any second order nonoscillatory linear differential equation can be converted into an oscillating system by applying a “sufficiently large”nonlinear perturbation. This can be achieved through a detailed analysis of possible nonoscillatory solutions of the perturbed differential equation which may exist when the perturbation is “sufficiently small”...