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International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 21, Pages 1105-1120
http://dx.doi.org/10.1155/S0161171204108028

Travelling wave solutions to some PDEs of mathematical physics

1Department of Physics, Faculty of Basic Sciences, University of Mazandaran, P.O. Box 47415-453, Babolsar, Iran
2Department of Mathematics, Faculty of Basic Sciences, University of Mazandaran, P.O. Box 47415-453, Babolsar, Iran

Received 6 August 2001

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Nonlinear operations such as multiplication of distributions are not allowed in the classical theory of distributions. As a result, some ambiguities arise when we want to solve nonlinear partial differential equations such as differential equations of elasticity and multifluid flows, or some new cosmological models such as signature changing space-times. Colombeau's new theory of generalized functions can be used to remove these ambiguities. In this paper, we first consider a simplified model of elasticity and multifluid flows in the framework of Colombeau's theory and show how one can handle such problems, investigate their jump conditions, and resolve their ambiguities. Then we consider as a new proposal the case of cosmological models with signature change and use Colombeau's theory to solve Einstein equation for the beginning of the Universe.