Doctoral Thesis
DOI
10.11606/T.3.2009.tde-29062009-154349
Document
Author
Full name
José Osvaldo de Souza Guimarães
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2009
Supervisor
Committee
Netto, Marcio Lobo (President)
Campos Filho, Frederico Ferreira
Torres, Marcelo Augusto Santos
Title in Portuguese
Computação evolutiva na resolução de equações diferenciais ordinárias não lineares no espaço de Hilbert.
Keywords in Portuguese
Computação evolutiva
Equações diferenciais
Matrizes
Polinômios de Legendre
Problemas do valor inicial
Abstract in Portuguese
Title in English
Evolutive computation in the resolution of non-linear ordiinary diferential equations in the Hilbert space.
Keywords in English
Differential equation
Evolutive computation
Initial value problem
Legendres polynomials
Operational matrices of differentiation and integration
Abstract in English
This thesis shows a new method to get polynomial solutions to the initial value problems (IVP), with an error margin comparable to the consecrate numerical methods (NM), for both the function and its derivatives. The method works with differential equations (DEs) linear or not, beeing the developed tolls available until 4th order, whose can be expanded to higher orders. The solution is a polynomial high degree expression with coefficients expressed by the ratio between two integers. The method behaves efficiently even in some cases that NM cannot get started. The resolutions are gotten considering that, the solution space is a Hilbert space, equipped with a complete set basis of Legendre Polynomials. Due the method here developed, the errors majoratives for the function and its derivatives are found analytically by a matrix calculus, also derived in this thesis. Beside all analytical foundation, a software (SAM) was developed to automate the whole process, joining all the tasks involved in the search for solutions to the IVP. This thesis proposes, verifies and validates a new error criterion, which takes in account simultaneously the local and global errors. As sub-products of the results described before, also integrated to the SAM, the following achievements should be highlighted: (1) An objective criterion to analyze the quality of any NM, despite of the knowledge of its algorithm; (2) A tool for a polynomial approximation, of high precision, for functions whose square is integrable in a given limited domain, with an errors majorative; (3) A tool-kit for a generically transpose (linear or not) of the IVPs domain and form, taking into account its derivatives, until the 4th order; (4) The generic matrices for integration and differentiation for all the polynomial basis of the Hilbert space.