This thesis is devoted to use Lie group analysis to obtain all invariant solutions by constructing optimal system of one-dimensional subalgebras of the Lie algebra L5 for a nonlinear wave equation. I will show how the given symmetries ( Eq.2) are admitted by using partial differential equation (Eq.1), In addition to obtain the commutator table by using the same given symmetries. Subsequently, I calculate the transformations of the generators with the Lie algebra L5, which provide the 5-parameter group of linear transformations for the operators. Finally, I construct the invariant solutions for each member of the optimal system.

The main purpose of this thesis is to use modern goal-oriented adaptive methods of Lie group analysis to construct the optimal sys- tem of Black-Scholes equation. We will show in this thesis how to obtain all invariant solutions by constructing what has now become so popular, optimal system of sub-algebras, the main Lie algebra admit- ted by the Black-Scholes equation. First, we obtain the commutator table of already calculated symmetries of the Black-Scholes equation. We then followed with the calculations of transformation of the gen- erators with the Lie algebra L6 which provides one-parameter group of linear transformations for the operators. Here we make use of the method of Lie equations to solve the partial di®erential equations. Next, we consider the construction of optimal systems of the Black- Scholes equation where the method requires a simpli¯cation of a vector to a general form to each of the transformations of the generators. Further, we construct the invariant solutions for each of the op- timal system. This study is motivated by the analysis of Lie groups which is being taken to another level by ALGA here in Blekinge In- stitute Technology, Sweden. We give a practical and in-depth steps and explanation of how to construct the commutator table, the calcu- lation of the transformation of the generators and the construction of the optimal system as well as their invariant solutions. Keywords: Black-Scholes Equation, commutators, commutator table, Lie equa- tions, invariant solution, optimal system, generators, Airy equation, structure constant, / It was an accolade for us to work with Professor Nail.H. Ibrgimov. +46762600953

Information system, ERP, IS selection, analysis, risk.

Dans ce travail de thèse, on s'intéresse é l'emploi de techniques de traitement de signal de systèmes de communication MIMO (Multiple Input Multiple Output) pour des applications aux réseaux de capteurs sans fil. Les contraintes énergétiques de cette classe de réseau font appel à des topologies particulières et le réseau peut être perçu comme étant un ensemble de grappes de nœuds capteurs. Ceci ouvre la porte à des techniques avancées de communication de type MIMO. Dans un premier temps, les différents aspects caractérisant les réseaux de capteurs sans fil sont introduits. Puis, les efforts engagés pour optimiser la conservation de l'énergie dans ces réseaux sont résumés. Les concepts de base de systèmes MIMOs sont abordés dans le deuxième chapitre et l'exploration par voie numérique de différentes pistes de la technologie MIMO sont exposées. Nous nous intéressons à des techniques de diversité de polarisation dans le cadre de milieux de communication riches en diffuseurs. Par la suite, des méthodes de type beamforming sont proposées pour la localisation dans les réseaux de capteurs sans fil. Le nouvel algorithme de localisation est présenté et les performances sont évaluées. Nous identifions la configuration pour la communication inter-grappes qui permet pour les meilleurs compromis entre énergie et efficacité spectrale dans les réseaux de capteurs sans fil. Finalement, nous envisageons la technique de sélection de nœuds capteurs afin de réduire la consommation de l'énergie dans le réseau de capteur sans fil. / The aim of this work is to study from a signal processing point of view the use of MIMO (Multiple Input Multiple Output) communication systems for algorithms dedicated to wireless sensor networks. We investigate energy-constrained wireless sensor networks and we focus on cluster topology of the network. This topology permits for the use of MIMO communication system model. First, we review different aspects that characterize the wireless sensor network. Then, we introduce the existing strategies for energy conservation in the network. The basic concepts of MIMO systems are presented in the second chapter and numerical results are provided for evaluating the performances of MIMO techniques. Of particular interest, polarization diversity over rich scattering environment is studied. Thereafter, beamforming approach is proposed for the development of an original localization algorithm in wireless sensor network. The novel algorithm is described and performances are evaluated by simulation. We determine the optimal system configuration between a pair of clusters that permits for the highest capacity to energy ratio in the fourth chapter. The final chapter is devoted to sensor nodes selection in wireless sensor network. The aim of using such technique is to make energy conservation in the network.