In this work the Adomian decomposition method (ADM) is used to solve the non-linear equation that represents the generalized model of Black-Scholes, that is to say that considers the volatility as a nonconstant function. The efficiency of this method is illustrated by investigating the convergence results for this type of models. The numerical results show the reliability and accuracy of the ADM.

The Stochastic Partial Differential Equations are part of a set of non-linear partial differential equations (PDE), which by their random behavior are difficult to solve analytically and numerically; One of them has been known as the Black-Scholes PDE since 1973, which determines the valuation of goods and/or assets called financial options. The development of the present work is to find numerical approximations to the solution of Black-Scholes PDE by the Method of Lines (MOL). The previous was achieved by means of a methodology based on an analytical study of the classic solution of the BlackScholes PDE. Finally, numerical methods and algorithms were applied to the Black Scholes PDE.

This document describes the extended linearization method for system control. The purpose of which is to determine a non-linear control signal in such a way that linearization of said signal around an operating point results in the approximate linear controller. When the approximate linear controller is obtained, a non-linear control signal is inferred from it, which allows the system to be controlled around the family of points associated with the initially established operating point. The importance of this type of control technique allows to analyze non-linear systems around the operating points with a wider margin than what could be done traditionally. its implementation on a converter is of great importance since electrical systems every day incorporate more elements of power electronics.

This document shows an analysis of the stability of a Buck type power converter, which, being a switched electronic circuit, has complex dynamics. The analysis consists of determining the influence on its stability of the variations of each of its parameters and determining the points where these changes occur, known as the bifurcation point. This analysis allows investigating these types of problems from the bifurcation theory allowing to understand the stability and operation of the converters.

Taking as a starting point the exact feedback linearization to transform a non-linear system into a linear one through coordinate transformation and state-space feedback based on its non-linear model, we have presented in this document the modelling, analysis and non-linear control of a boost-type power converter, with the objective of comparing the results with a proportional, integral and derivative controller. This technique was then applied on a power converter from continuous current to continuous current and the dynamics were analyzed zero to verify the validity of the controller, which was simulated in the Simulink-Matlab software. Finally, the simulation results showed that the system with the exact linearization controller is not only free of excess, but also improves the stability in the output voltage.