Mathematical Problems in Engineering

Volume 2016, Article ID 9718962, 9 pages

http://dx.doi.org/10.1155/2016/9718962

## 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.

#### Abstract

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.

#### 1. Introduction

Spectral numerical methods are increasingly used to solve differential equations, even when fractional derivatives appear. Concerning these methods, they frequently use orthogonal functions [1], in order to ease the calculations and to preserve the basis elements when dimensional expansions are necessary.

This is an important advantage when operational matrices are used, because if it is necessary to extend an orthogonal basis of functions from dimensions to , only the new element must be calculated with all the former elements preserved [2, 3]. If the basis functions are not orthogonal, this assumption is not true.

However, there are some situations that, even for numerical methods with operational matrices [4], solution of differential equations demands new basis of functions [5, 6], with some of them being nonorthogonal as, for instance, Bernstein’s polynomials, mainly in the cases where boundary or initial conditions must be considered [7–9].

Some iterative methods, based on Krylov formulation, for instance, present the use of such bases and are implemented by using the Gram-Schmidt process [8, 10, 11] that, in most cases, is difficult to be operationalized.

Trying to simplify this procedure, an alternative operational method is presented with the first ideas developed in [12] that is described by using tensorial language and called* coadjoint formalism* [13], allowing the direct use of nonorthogonal bases without any kind of previous conditioning process.

In a nutshell,* coadjoint formalism* adapts Dirac’s* bracket* notation [14] to spectral methods considering finite dimension complex vectorial spaces. The methodology is applied to nonorthogonal bases in finite dimension function spaces with a generalized tensorial approach [15–17] that simplifies the operational conditions.

In the next section, the theoretical fundamentals of the* coadjoint formalism* are presented, trying to connect it with the well-known Dirac* bracket*, followed by a section where two examples show how simple it is to apply the method developed here, with a conclusion section closing the work.

#### 2. Coadjoint Formalism: Theoretical Foundations

This section presents the concepts and definitions used to build the* coadjoint formalism*. The bases to be considered are finite sets of complex functions of a real variable; that is, the bases elements , , are given by , with , where is the function space, equipped as a Hilbert space.

##### 2.1. Actuation Spaces

Here, it is considered that any operation regarding series expansion of a function occurs in two distinct spaces:(i)A finite dimensional Hilbert space .(ii)The finite dimension space with dimension , generated by the series expansion with terms for the considered function, denoted by and called* order space*.

It is assumed that* kets* are represented by column vectors and vice versa. It is the same for* bras*, represented by line vectors, understood as covectors, that is, dual space elements. Consequently, an equivalence relation can be established [14]:

The* kets* can belong either to the function space, or to the order space, . The same is valid for* bras*.

Covariant and contravariant distinction are assumed [15], and the traditional notation of differential geometry and tensorial calculus is used:(i)Contravariant components: .(ii)Covariant components: .

Quantities described by* kets* are in the* original space*; quantities described by* bras* are in the* dual space*, in the traditional way of linear algebra [14].* Primitive space* is related to the space where the quantity is defined; that is, if a* bra* defines a quantity, the* dual space* assumes the* primitive space* condition of the quantity.

Given a covariant basis composed of , a vector belonging to the order space, , can be described by a* ket* in two different ways:(i)Invariant representation: .(ii)Coordinate representation: ,with . The spaces where the spectral methods are generally applied are , and .

Given a reference basis from , for a basis composed of and described in terms of the basis , where can be either the Hilbert space or any other complex vectorial space, two representations are possible:(i).(ii),with the inferior dot representing the covariant nature of the basis and represents the set of the -order square matrices. In this kind of disposition, the bases are described by their coordinates in the reference basis , but the reference basis can be omitted and the invariant representation is assumed.

Consequently, there are two isomorphic representations, given by(i),(ii)

Given a vector with its components expressed in a generical basis , the following operations are defined:(i)Conjugation: .(ii)Transposition: (iii)Adjunction: (iv)Duality:

##### 2.2. Inner Products

The inner products (IP), , are defined in several different ways, depending on the space and the representation, as it is shown below.

*Order Space*(1)Hermitian product is as follows:(i)coordinate representation: .(2)Dual product is as follows:(i)coordinate representation: ;(ii)invariant representation: and .

*Functions Space*. Considering the space of the functions with domain , the IP is defined as

*Hybrid Spaces*. The mixed inner product is defined as

##### 2.3. Duality Relations

It is possible to obtain the* dual basis * from an original basis by using the duality relations, given by

It is possible to express the same vector in several ways, considering the formalism described here, considering four distinct bases. Here, these representations are called* connected representations* and are shown in Table 1.