A new method for solving the 1D Poisson equation is presented using the finite difference method. This method is based on the exact formulation of the inverse of the tridiagonal matrix associated with the Laplacian. This is the first time that the inverse of this remarkable matrix is determined directly and exactly. Thus, solving 1D Poisson equation becomes very accurate and extremely fast. This method is a very important tool for physics and engineering where the Poisson equation appears very often in the description of certain phenomena.

An innovative, extremely fast and accurate method is presented for Neumann-Dirichlet and Dirichlet-Neumann boundary problems for the Poisson equation, and the diffusion and wave equation in quasi-stationary regime; using the finite difference method, in one dimensional case. Two novels matrices are determined allowing a direct and exact formulation of the solution of the Poisson equation. Verification is also done considering an interesting potential problem and the sensibility is determined. This new method has an algorithm complexity of () O N , its truncation error goes like () 2 O h , and it is more precise and faster than the Thomas algorithm.

A new and innovative method for solving the 1D Poisson Equation is presented, using the finite differences method, with Robin Boundary conditions. The exact formula of the inverse of the discretization matrix is determined. This is the first time that this famous matrix is inverted explicitly, without using the right hand side. Thus, the solution is determined in a direct, very accurate (O(h 2 )), and very fast (O(N)) manner. This new approach treats all cases of boundary conditions: Dirichlet, Neumann, and mixed. Therefore, it can serve as a reference for solving the Poisson equation in one dimension.

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Prior to hardware implementation, simulation is an important step in the study of systems such as Direct Sequence Code Division Multiple Access (DS-CDMA). A useful technique is presented, allowing to model and simulate Linear Feedback Shift Register (LFSR) for CDMA. It uses the Scilab package and its modeling tool for dynamical systems Xcos. PN-Generators are designed for the quadrature-phase modulation and the Gold Code Generator for Global Positioning System (GPS). This study gives a great flexibility in the conception of LFSR and the analysis of Maximum Length Sequences (MLS) used by spread spectrum systems. Interesting results have been obtained, which allow the verification of generated sequences and their exploitation by signal processing tools.