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Mathematical Problems in Engineering
Volume 2016, Article ID 9718962, 9 pages
Research Article

Coadjoint Formalism: Nonorthogonal Basis Problems

Escola Politécnica da Universidade de São Paulo, Avenida Prof. Luciano Gualberto, Travessa 3, No. 158, 05508-900 São Paulo, SP, Brazil

Received 12 March 2016; Accepted 28 June 2016

Academic Editor: Gisele Mophou

Copyright © 2016 William Labecca et al. 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.


Using nonorthogonal bases in spectral methods demands considerable effort, because applying the Gram-Schmidt process is a fundamental condition for calculations. However, operational matrices numerical methods are being used in an increasing way and extensions for nonorthogonal bases appear, requiring simplified procedures. Here, extending previous work, an efficient tensorial method is presented, in order to simplify the calculations related to the use of nonorthogonal bases in spectral numerical problems. The method is called coadjoint formalism and is based on bracket Dirac’s formulation of quantum mechanics. Some examples are presented, showing how simple it is to use the method.