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Oscillation results for higher order nonlinear neutral differential equations with positive and negative coefficients

Panigrahi S, Basu R.

Journal of Applied Analysis

2015

Vol. 21, nr 2

109--121

In this paper, the authors investigated oscillatory and asymptotic behavior of solutions of a class of nonlinear higher order neutral differential equations with positive and negative coefficients. The results in this paper generalize the results of Tripathy, Panigrahi and Basu [5]. We establish new conditions which guarantees that every solution either oscillatory or converges to zero. Moreover, using the Banach Fixed Point Theorem sufficient conditions are obtained for the existence of bounded positive solutions. Examples are considered to illustrate the main results.

Implicit difference methods for Hamilton-Jacobi differential functional equations

Kępczyńska A.

Demonstratio Mathematica

2007

Vol. 40, nr 1

125-150

Classical solutions of the local Cauchy problem on the Haar pyramid are approximated in the paper by solutions of suitable quasilinear systems of difference functional equations. The numerical methods are difference schemes which are implicit with respect to time variable. A complete convergence analysis for the methods is given and it is shown that the new methods are considerable better than the explicit schemes. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type. Numerical examples are given.

On the extension of Lie group analysis to functional differential equations

Oberlack M., Wacławczyk M.

Archives of Mechanics

2006

Vol. 58, nr 6

597-618

In the present paper the classical point symmetry analysis is extended from partial differential to functional differential equations. In order to perform the group analysis and deal with the functional derivatives, we extend the quantities such as infinitesimal transformations, prolongations and invariant solutions. For the sake of example, the procedure is applied to the functional formulation of the Burgers equation. The method can further lead to important applications in continuum mechanics.