Coadjoint Formalism: Nonorthogonal Basis Problems

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2016, Article ID 9718962, 9 pages http://dx.doi.org/10.1155/2016/9718962

Research Article Coadjoint Formalism: Nonorthogonal Basis Problems William Labecca, Osvaldo Guimarães, and José Roberto C. Piqueira Escola Politécnica da Universidade de São Paulo, Avenida Prof. Luciano Gualberto, Travessa 3, No. 158, 05508-900 São Paulo, SP, Brazil Correspondence should be addressed to José Roberto C. Piqueira; [email protected] 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.

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 𝑚 to 𝑚 + 1, 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 {𝜙𝑘 }, 𝑘 = 1, . . . , 𝑛 < ∞, are given by 𝜙𝑘 : R → C, with 𝜙𝑘 ∈ H, where H is the function space, equipped as a Hilbert space.

2

Mathematical Problems in Engineering

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 H. (ii) The finite dimension space with dimension 𝑛, generated by the series expansion with 𝑛 terms for the considered function, denoted by O𝑛 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]: | ⟩ ∼ [...] ,

(1)

⟨ | ∼ [⋅ ⋅ ⋅] . The kets can belong either to the function space, | ⟩∞ ∈ H or to the order space, | ⟩0𝑛 ∈ O𝑛 . 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: 𝑢𝑖 . 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 |𝑏𝑖 ⟩ ∈ V, a vector 𝑢 belonging to the order space, O𝑛 (V), can be described by a ket in two different ways: (i) Invariant representation: |𝑢⟩0𝑛 = ∑𝑛𝑖 𝑢𝑖 |𝑏𝑖 ⟩. 1

(ii) Coordinate representation: |𝑢⟩[𝐵] ∼ [

𝑢 .. . 𝑢𝑛

] , [𝐵]

with [𝐵] = {|𝑏𝑖 ⟩}. The spaces V where the spectral methods are generally applied are H, C and C𝑛 . Given a reference basis [𝜖] from V 𝑛 , for a basis composed of |𝑏𝑖 ⟩ ∈ V 𝑛 and described in terms of the basis [𝜖], where V can be either the Hilbert space H or any other complex vectorial space, two representations are possible: (i) |𝐵. ][𝜖] ∈ M𝑛 (V) : |𝐵. ][𝜖] ≃ [|𝑏1 ⟩[𝜖] ⋅ ⋅ ⋅ |𝑏𝑛 ⟩[𝜖] ]. |𝑏1 ⟩[𝜖] .. . |𝑏𝑛 ⟩[𝜖]

(ii) |𝐵. ⟩ ∈ V 𝑛 (V 𝑛 ) : |𝐵. ⟩ ≃ [

|𝑏1 ⟩ .. ] . . |𝑏𝑛 ⟩

Given a vector |𝑢⟩ with its components expressed in a generical basis |𝑒𝑖 ⟩, the following operations are defined: (i) Conjugation: |𝑢⟩∗ = |𝑢⟩, |𝑒𝑖 ⟩∗ = |𝑒𝑖 ⟩.

(ii) Transposition: |𝑢⟩󸀠 = ⟨𝑢|, |𝑒𝑖 ⟩󸀠 = ⟨𝑒𝑖 |. (iii) Adjunction: |𝑢⟩† = ⟨𝑢|, |𝑒𝑖 ⟩† = ⟨𝑒𝑖 |. (iv) Duality: |𝑢⟩⋆ = ⟨𝑢. |, |𝑒𝑖 ⟩⋆ = ⟨𝑒𝑖 |. 2.2. Inner Products. The inner products (IP), ⟨ | ⟩ : O𝑛 (V) × O𝑛 (V) → C, 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: ⟨𝑢 ∑𝑛𝑖=1 𝑢𝑖 V𝑖 .

|

V⟩[𝐵]

fl

|

V⟩[𝐵]

fl

(2) Dual product is as follows:

(ii) Covariant components: 𝑢𝑖 .

(ii) |𝐵. ⟩[𝜖] ∈ V 𝑛 (V 𝑛 ) : |𝐵. ⟩[𝜖] ≃ [

(i) |𝐵. ] ∈ M𝑛 (V) : |𝐵. ] ≃ [|𝑏1 ⟩ ⋅ ⋅ ⋅ |𝑏𝑛 ⟩],

],

with the inferior dot representing the covariant nature of the basis and M𝑛 represents the set of the 𝑛-order square matrices. In this kind of disposition, the bases are described by their coordinates in the reference basis [𝜖], but the reference basis can be omitted and the invariant representation is assumed. Consequently, there are two isomorphic representations, given by

(i) coordinate representation: ⟨𝑢. ∑𝑛𝑖=1

𝑖

𝑢𝑖 V ; (ii) invariant representation: ⟨𝑢. | V⟩0𝑛 fl ∑𝑛𝑖=1 𝑢𝑖 V𝑖 and ⟨𝑢 | V⟩0𝑛 fl ∑𝑛𝑖=1 𝑢𝑖 V𝑖 .

Functions Space. Considering the space of the functions with domain [𝑎, 𝑏] ⊂ R, the IP ⟨ | ⟩ : H × H → C is defined as 𝑏

⟨𝑓 | 𝑔⟩∞ fl ∫ 𝑓 (𝑥) 𝑔 (𝑥) 𝑑𝑥. 𝑎

(2)

Hybrid Spaces. The mixed inner product ⟨ | ⟩ : O⋆𝑛 (C) × O𝑛 (H) → C is defined as 𝑛

󵄨 ⟨𝑐 | 𝜙⟩ fl ∑𝑐𝑟 󵄨󵄨󵄨𝜙𝑟 ⟩∞ . 𝑟

(3)

2.3. Duality Relations. It is possible to obtain the dual basis {|𝑒𝑘 ⟩} from an original basis {|𝑒𝑖 ⟩} by using the duality relations, given by ⟨𝑒𝑘 | 𝑒𝑖 ⟩ = 𝛿𝑘 𝑖 , ⟨𝑒𝑖 | 𝑒𝑘 ⟩ = 𝛿𝑖 𝑘 .

(4)

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. Considering that the covariant representation in the original space is considered to be natural, the dot below the


3 Table 1: Connected representations.

Basis

Notation and correspondence

Invariant representations

Coordinate representations

|𝐵. ] ≃ 𝐵

|𝑢⟩0𝑛 = ∑𝑢𝑖 |𝑒𝑖 ⟩

|𝑢⟩[𝐵]

[𝐵. | ≃ 𝐵†

⟨𝑢|0𝑛 = ∑ ⟨𝑒𝑖 |𝑢𝑖

⟨𝑢|[𝐵]

[𝐵|̇ ≃ 𝐵⋆

⟨𝑢. |0𝑛 = ∑ ⟨𝑒𝑖 |𝑢𝑖

⟨𝑢. |[𝐵]

|𝑢. ⟩0𝑛 = ∑𝑢𝑖 |𝑒𝑖 ⟩

|𝑢. ⟩[𝐵]

Original Adjoint Dual

𝑛

𝑖 𝑛

𝑖 𝑛

𝑖 𝑛

|𝐵]̇ ≃ (𝐵⋆ )†

Coadjoint

𝑖

with its 𝑟 eciprocal metric tensor given by

Table 2: Transformation relations between bases. Basis Original

Notation |𝐵. ]

𝛾𝑟𝑠 fl ⟨𝜙𝑟 | 𝜙𝑠 ⟩∞ .

Dual

[𝐵. | [𝐵|̇

Transformation relations |𝐵. ] ≃ 𝐵

Coadjoint

|𝐵]̇

[𝐵|̇ ≡ |𝐵. ]−1 |𝐵]̇ ≡ [𝐵|̇ † ≃ (𝐵−1 )†

Consequently, the IPs between two functions 𝑓, 𝑔 represented by series are given by

Adjoint

[𝐵. | ≡ |𝐵. ]

†

basis symbol can be omitted and the transformation relations between the bases are shown in Table 2. The covariant and contravariant components are obtained by applying (1) contravariant: 𝑢𝑘 = ⟨𝑒𝑘 | 𝑢⟩0𝑛 , (2) covariant: 𝑢𝑘 = ⟨𝑒𝑘 | 𝑢⟩0𝑛 .

(9)

𝑛

𝑛

𝑟,𝑠

𝑟,𝑠

󵄨 ̂ ∗𝑟 ̂ 𝑠 󵄨󵄨 ̂ ∗𝑟 𝑔 ̂𝑠 ⟨𝑓 | 𝑔⟩0𝑛 = ∑ ⟨𝜙𝑟 󵄨󵄨󵄨 𝑓 𝑔 󵄨󵄨𝜙𝑠 ⟩ = ∑𝛾𝑟𝑠 𝑓 𝑛

∗𝑟

̂ 𝑔 ̂𝑟, ≡ ∑𝑓 𝑟

⟨𝑓 | 𝑔⟩ = .

2.4. Series Expansions of Functions. Considering a basis {|𝜙𝑖 ⟩∞ } from O𝑛 (H) and elements 𝜙𝑖 : R → H, the infinite series expansion of a function 𝑓 is given by

0𝑛

𝑛

󵄨 ̂ ∗ ̂𝑠 ∑ ⟨𝜙𝑟 󵄨󵄨󵄨 𝑓 𝑟𝑔 𝑟,𝑠

󵄨󵄨 󵄨󵄨𝜙𝑠 ⟩ =

𝑛

(10)

𝑠 ̂ ∗𝑔 ∑𝛿𝑟 𝑠 𝑓 𝑟̂ 𝑟,𝑠

𝑛

𝑠 ̂ ∗𝑔 ≡ ∑𝑓 𝑠̂ . 𝑠

∞

󵄨󵄨 𝑟󵄨 󵄨󵄨𝑓⟩∞ = ∑𝑐 󵄨󵄨󵄨𝜙𝑟 ⟩∞ , 𝑟

(5)

𝑟

with 𝑐 being the expansion coefficients in that basis. Considering the mixed inner product definition, this expression can be modified as 󵄨󵄨 (6) 󵄨󵄨𝑓⟩∞ = ⟨𝑐 | 𝜙𝑟 ⟩ , O⋆𝑛 (C)

called coefficient covector. with the covector ⟨𝑐| ∈ In the subspace O𝑚 (H), projectors can be defined by

𝑛

𝑟,𝑠

(7)

󵄨 󵄨 P†𝑚 fl ∑ 󵄨󵄨󵄨𝜙𝑟 ⟩ ⟨𝜙𝑟 󵄨󵄨󵄨 , 𝑟

with the second one called adjoint projector. The eigenprojectors in the O𝑛 (H) space are defined as the projectors in the 𝑛-dimensional proper space; that is, P = P𝑛 and P† = P†𝑛 . In a way analogous to that followed in differential geometry, fundamental metric tensor of a basis, {|𝜙𝑖 ⟩∞ } from O𝑛 (H), is defined as 𝛾𝑟𝑠 fl ⟨𝜙𝑟 | 𝜙𝑠 ⟩∞ ,

(11)

⋆ 󵄨 󵄨 ← → 𝛾 ≡ PP† fl ∑ 󵄨󵄨󵄨𝜙𝑟 ⟩ 𝛾𝑟𝑠 ⟨𝜙𝑠 󵄨󵄨󵄨 .

󵄨 󵄨 P𝑚 fl ∑ 󵄨󵄨󵄨𝜙𝑟 ⟩ ⟨𝜙𝑟 󵄨󵄨󵄨 , 𝑚

𝑛

󵄨 󵄨 ← → 𝛾 ≡ P† P fl ∑ 󵄨󵄨󵄨𝜙𝑟 ⟩ 𝛾𝑟𝑠 ⟨𝜙𝑠 󵄨󵄨󵄨 ; 𝑟,𝑠

𝑚 𝑟

̂ and 𝑔 ̂ , respectively, representing the 𝑓 and 𝑔 expanwith 𝑓 sion coefficients, in the basis {|𝜙⟩}. The metric tensorial fundamental operator and its reciprocal are defined as

(8)

2.5. Covariant-Contravariant Transformation Relations. Considering the covariant and contravariant matrix representations and defining 𝐵 = {|𝜙𝑘 ⟩}, the transformation relations for bases and functions can be written as 𝑛

󵄨 𝑟 󵄨󵄨 󵄨󵄨𝜙𝑘 ⟩ = ∑ 󵄨󵄨󵄨𝜙 ⟩ 𝛾𝑟𝑘 , 𝑟

󵄨 |𝐵] = 󵄨󵄨󵄨󵄨𝐵]̇ 𝛾, 󵄨 ̇ ; |𝐵⟩ = 𝛾󸀠 󵄨󵄨󵄨󵄨𝐵⟩

(12)

4

Mathematical Problems in Engineering 𝑛 󵄨󵄨 𝑖 󵄨󵄨𝜙 ⟩ = ∑ 󵄨󵄨󵄨󵄨𝜙𝑟 ⟩ 𝛾𝑟𝑖 , 󵄨 𝑟

it is possible to write

󵄨󵄨 ̇ 󵄨󵄨𝐵] = |𝐵] 𝛾⋆ , 󵄨 󵄨󵄨 ̇ ⋆ 󵄨󵄨󵄨𝐵⟩ = 𝛾̃ |𝐵⟩ ;

𝑏

󸀠 󵄨 ⟨𝑐| = ∫ ⟨𝜙 (𝜉)󵄨󵄨󵄨 𝑓 (𝜉) 𝑑𝜉 [𝛾⋆ ] .

(13)

𝑎

As a consequence, the expanded function becomes

𝑛

𝑓𝑟 = ∑𝛾𝑟𝑘 𝑓𝑘 ,

𝑏

𝑘

󵄨 󵄨󵄨 ⋆ 󸀠󵄨 󵄨󵄨𝑓𝑛 ⟩∞ = ∫ ⟨𝜙 (𝜉)󵄨󵄨󵄨 𝑓 (𝜉) 𝑑𝜉 [𝛾 ] 󵄨󵄨󵄨𝜙⟩∞ . 𝑎

(14)

󵄨󵄨 󵄨󵄨𝑓⟩ = 𝛾 󵄨󵄨󵄨𝑓⟩ ; 󵄨󵄨 . 󵄨 |𝐵] 󵄨 |𝐵] 𝑛 𝑟

(15)

󵄨 󵄨󵄨 ⋆󵄨 󵄨󵄨𝑓⟩|𝐵] = 𝛾 󵄨󵄨󵄨󵄨𝑓⟩ . 󵄨 . |𝐵]

In this section, two examples are developed showing the expansion of functions using nonorthogonal bases, one using the canonical basis and the other using a basis of nonorthogonal complex functions. 3.1. Canonical Basis. Here, the more common case of nonorthogonality is developed, considering the canonical basis of functions defined in the real interval, defined by the polynomials:

Therefore, 𝑛

∑𝛾𝑖𝑟 𝛾𝑟𝑘 fl 𝛿𝑖 𝑘 , 𝑟

(16)

𝜖𝑚 (𝑥) = 𝑥𝑚−1 , 𝑚 = 1, 2, . . . , 𝑛.

𝑖

𝑟𝑖

∑𝛾𝑘𝑟 𝛾 fl 𝛿𝑘 , 𝑟

𝛾⋆ = 𝛾−1 .

𝑛

⋆󵄨 󵄨 󵄨󵄨 → 𝛾 󵄨󵄨󵄨𝑓⟩∞ = ∑𝑐𝑟 󵄨󵄨󵄨𝜙𝑟 ⟩∞ , 󵄨󵄨𝑓𝑛 ⟩∞ = ← 𝑟

𝑐𝑟 = ⟨𝜙𝑟 | 𝑓⟩∞ ≡ ∑𝛾𝑟𝑠 ⟨𝜙𝑠 | 𝑓⟩∞ . Under these conditions, the coefficient covector ⟨𝑐| ∈ O⋆𝑛 (C) can be expressed as ⟨𝑐| ≃ [𝑐1 ⋅ ⋅ ⋅ 𝑐𝑛 ] .

(19)

Then, the expansion is given by the mixed IP: |𝑓𝑛 ⟩∞ = ] being the vector associated with the

second representation chosen basis. Equation (18) can be expressed by using matrices, giving a useful computational expression, in order to find the coefficient covector. Considering the coefficient covector

𝑠

𝑎

󵄨 𝑐𝑟 = ∑ ∫ ⟨𝜙𝑠 (𝜉)󵄨󵄨󵄨 𝑓 (𝜉) 𝑑𝜉 𝛾𝑠𝑟 , 󸀠

𝑟+𝑠−2

−1

=

󵄨1 𝑥𝑟+𝑠−1 󵄨󵄨󵄨 󵄨󵄨 𝑑𝑥 = 𝑟 + 𝑠 − 1 󵄨󵄨󵄨−1

𝑟+𝑠−1

(24)

1 − (−1) . 𝑟+𝑠−1

Therefore, the analytical expression for the metric tensor elements in the domain 𝐷1 can be calculated, resulting in 𝛾𝑟𝑠 =

2par (𝑟 + 𝑠) , 1 ≤ 𝑟, 𝑠 ≤ 𝑛, 𝑟+𝑠−1

(25)

with

𝑠

𝑏

𝛾𝑟𝑠 [𝜖] = ⟨𝜖𝑟 | 𝜖𝑠 ⟩∞ fl ∫ 𝑥

(18)

𝑛

𝑛

1

(17)

2.6. Finite Expansions. The 𝑛-order finite expansion of a function 𝑓 can be expressed by the action of the reciprocal tensorial metric operator over the function; that is,

𝜙1 .. . 𝜙𝑛

(23)

The metric tensor can be calculated as

meaning that the reciprocal tensor matrix is the inverse of the metrical tensor matrix. Calling the first one 𝛾⋆ :

⟨𝑐 | 𝜙⟩, with |𝜙⟩ = [

(22)

3. Application Examples

𝑓𝑖 = ∑𝛾𝑖𝑟 𝑓𝑟 ,

𝑛

(21)

(20)

{1, par (𝑘) fl 1 − mod (𝑘, 2) = { 0, {

if 𝑘 is even; if 𝑘 is odd.

(26)

Following the calculations for 𝑛 = 5, the metric matrix is given by 2 [ [ [0 [ [ [2 𝛾 [𝜖] = [ [3 [ [ [0 [ [2 [5

0 2 3 0 2 5 0

2 3 0 2 5 0 2 7

0 2 5 0 2 7 0

2 5] ] 0] ] ] 2] ], 7] ] ] 0] ] 2] 9]

(27)


5

allowing, by using expression (17), the calculation of the matrix representing its reciprocal tensor: 525 945 225 0 − 0 [ 128 64 128 ] [ ] 105 75 [ 0 0 − 0 ] [ ] [ ] 8 8 [ ] 525 2205 4725 −1 [ ]. 𝛾 [𝜖] = [− 0 0 − 32 64 ] [ 64 ] [ ] 105 175 [ 0 − 0 0 ] [ ] 8 8 [ 945 4725 11025 ] 0 − 0 [ 128 64 128 ]

changing order implies a whole recalculation of the basis elements. In order to illustrate the ideas in a particular case, the expansion of the function 𝑔(𝑥) = 2 cos 1 − 2𝑥2 cos 𝑥 is considered. The development for 𝑛 = 5 gives 5

(28)

𝑟=1𝑠=1

5

1 [𝑥] [ ] [ ] 1 𝑇 [ 2] 󵄨󵄨 ⋆ 󵄨󵄨𝑔5 ⟩ = ∫ [1 𝜉 𝜉2 𝜉3 𝜉4 ] 𝑔 (𝜉) 𝑑𝜉 [𝛾 ] [𝑥 ] , [ ] −1 [𝑥3 ] [ ]

2 4 225 (525𝑥 ) (945𝑥 ) − + ; 𝜖 (𝑥) = 128 64 128 3 (75𝑥) (105𝑥 ) − ; 8 8

2 4 525 (2205𝑥 ) (4725𝑥 ) + − ; 𝜖 (𝑥) = − 64 32 64

𝜖4 (𝑥) = − 𝜖5 (𝑥) =

(31)

4

[𝑥 ] or (29)

1 [𝑥] [ ] [ ] 󵄨󵄨 1 2 3 4 5 [ 2] 󵄨󵄨𝑔5 ⟩ = ⟨𝑐 | 𝜖⟩ = [𝑐 𝑐 𝑐 𝑐 𝑐 ] [𝑥 ] . [ ] [𝑥3 ] [ ]

3 (105𝑥) (175𝑥 ) + ; 8 8

2 4 945 (4725𝑥 ) (11025𝑥 ) − + . 128 64 128

(32)

4

[𝑥 ]

It is important to notice that, if the basis is orthogonal, to obtain its reciprocal is simply scale changing. Considering a nonorthogonal basis, the procedure is described here and

In these expressions, the coefficient covector is ⟨𝑐| = ∫−1 [1 𝜉 𝜉2 𝜉3 𝜉4 ] 𝑔(𝜉)𝑑𝜉[𝛾⋆ ]𝑇 , and, according to (21), can be calculated resulting in 1

1.7578 0 −8.2031 0 7.3828 [ ] [ 0 9.3750 0 −13.125 0 ] [ ] [ 0 68.906 0 −73.828] ⟨𝑐| = [1.2047 0 0.1881 0 0.0683] [−8.2031 ], [ ] [ 0 −13.125 0 21.875 0 ] [ ] 7.3828 0 −73.828 0 86.133 [ ]

generating the expansion for the function 𝑔5 (𝑥) = 1.0789 − 1.9639𝑥2 + 0.89037𝑥4 .

(30)

According to (22), the matrix expression is

1

3

5

2par (𝑟 + 𝑠) 𝑟−1 1 𝑠−1 = ∑∑ 𝑥 ∫ 𝜉 𝑔 (𝜉) 𝑑𝜉. −1 𝑟=1𝑠=1 𝑟 + 𝑠 − 1

By using the methodology developed in the former section and applying the transformation relation (13), the reciprocal basis is composed of polynomials that, for 𝑛 = 5, are listed below:

𝜖2 (𝑥) =

5

⋆󵄨 󵄨 󵄨󵄨 → 𝛾 󵄨󵄨󵄨𝑔⟩∞ = ∑ ∑ 󵄨󵄨󵄨𝜖𝑟 ⟩ 𝛾𝑟𝑠 ⟨𝜖𝑠 | 𝑔⟩∞ 󵄨󵄨𝑔5 ⟩∞ = ←

(33)

Defining the basis (34)

Figure 1(a) shows the almost perfect superposition of the function 𝑔(𝑥) and its expansion 𝑔5 (𝑥). Figure 1(b) shows the local error for the expansion. 3.2. Expansion in a Complex Nonorthogonal Basis. Here, considering a complex basis {|𝑄𝑖 ⟩∞ }, composed of elements 𝑄𝑖 : R → H, 𝑖 = 1, . . . , 𝑛, the real function 𝑓(𝑥) = exp(−2𝜋𝑥2 ) is expanded, showing efficient and concise results.

𝑄𝑟 (𝑥) = 𝑟𝑥𝑟−1 + 𝑖 (1 − 𝛿1𝑟 ) 𝑥𝑟−2 , 𝑟 = 1, . . . , 𝑛, the metric tensor is expressed by 1

𝛾𝑟𝑠 [𝑄] = ⟨𝑄𝑟 | 𝑄𝑠 ⟩∞ = ∫ 𝑄𝑟∗ (𝑥) 𝑄𝑠 (𝑥) 𝑑𝑥 −1

1

= ∫ [𝑟𝑥𝑟−1 − 𝑖 (1 − 𝛿1𝑟 ) 𝑥𝑟−2 ] −1

⋅ [𝑠𝑥𝑠−1 + 𝑖 (1 − 𝛿1𝑠 ) 𝑥𝑠−2 ] 𝑑𝑥 󳨐⇒

(35)

6


1.4

×10−3 3

1.2

2 1 0

0.8

R 5 (x)

g(x), g5 (x)

1

0.6

−1 −2 −3

0.4

−4 0.2

−5

0 −1 −0.8 −0.6 −0.4 −0.2 0 x

0.2 0.4 0.6 0.8

1

(a) Superposition of the graphics of 𝑔(𝑥) and 𝑔5 (𝑥)

−6 −1

−0.5

0 x

0.5

1

(b) Local error of the expansion 𝑔5 (𝑥) of 𝑔(𝑥)

Figure 1: Approximation of 𝑔(𝑥).

𝛾𝑟𝑠 [𝑄] = 2par (𝑟 + 𝑠) [ + ⋅

where 𝜇𝑘 = ∑5𝑟=1 𝛾𝑘𝑟 ⟨𝑄𝑟 | 𝑓⟩∞ or

𝑟𝑠 𝑟+𝑠−1

1 [ 2𝑥 + 𝑖 ] [ ] [ ] ] 󵄨󵄨 2 1 2 3 4 5 [ 󵄨󵄨𝑓5 ⟩ = [𝜇 𝜇 𝜇 𝜇 𝜇 ] [ 3𝑥 + 𝑖𝑥 ] ≡ ⟨𝜇 | 𝑄⟩ , (40) [ ] [4𝑥3 + 𝑖𝑥2 ] [ ]

(1 − 𝛿1𝑟 ) (1 − 𝛿1𝑠 ) ] + 2𝑖 mod (𝑟 + 𝑠, 2) 𝑟+𝑠−3

[𝑟 (1 − 𝛿1𝑠 ) − 𝑠 (1 − 𝛿1𝑟 )] . 𝑟+𝑠−2

4 3 [5𝑥 + 𝑖𝑥 ]

(36) For 𝑛 = 5 [ 2 [ [ [ [ −2𝑖 [ [ [ 𝛾 [𝑄] = [ [ 2 [ [ [ 2𝑖 [− [ 3 [ [ 2 [

2𝑖 3 14 2𝑖 58 − 3 3 15 2𝑖 64 2𝑖 − 3 15 5 58 2𝑖 174 15 5 35 6𝑖 164 2𝑖 5 35 7 2𝑖

2 ] ] 6𝑖 ] ] − ] 5] ] 164 ] ], 35 ] ] ] 2𝑖 ] − ] 7] ] 368 ]

2

with 󵄨 ⟨𝜇󵄨󵄨󵄨 1

= ∫ [𝑄1∗ (𝜉) 𝑄2∗ (𝜉) 𝑄3∗ (𝜉) 𝑄4∗ (𝜉) 𝑄5∗ (𝜉)] 𝑓 (𝜉) 𝑑𝜉 (41)

(37)

Considering 𝛾⋆ [𝑄] formerly calculated, the expansion becomes

63 ]

𝑓5 (𝑥) = (1.38287 − 8.50196 × 10−16 𝑖) 𝑄1 (𝑥)

⋆

𝛾 [𝑄] −6.48244𝑖 −7.7305

4.08105 ] 2.60449𝑖 ] ] 4.87307𝑖 8.6524 −1.75798𝑖 −5.20898 ] ]. ] −2.51961 1.75798𝑖 1.58252 −0.861328𝑖] ] −2.60449𝑖 −5.20898 0.861328𝑖 3.44531 ] 5.05372

𝑇

⋅ [𝛾 [𝑄]] .

and, for the reciprocal metrics, 9.66896 [ [ 6.48244𝑖 [ [ = [ −7.7305 [ [−2.88875𝑖 [ [ 4.08105

−1

⋆

+ (8.46164 × 10−16 + 0.502949𝑖) Q2 (𝑥)

2.88875𝑖

−4.87307𝑖 −2.51961

(38)

Expanding the given function for 𝑛 = 5, 5

5

𝑟=1

𝑟=1

⋆󵄨 󵄨󵄨 󵄨 󵄨 → 𝛾 󵄨󵄨󵄨𝑓⟩∞ = ∑𝛾𝑘𝑟 𝜇𝑟 󵄨󵄨󵄨𝑄𝑘 ⟩∞ = ∑𝜇𝑘 󵄨󵄨󵄨𝑄𝑘 ⟩∞ , (39) 󵄨󵄨𝑓5 ⟩∞ = ←

− (1.0059 − 2.95875 × 10−16 𝑖) 𝑄3 (𝑥)

(42)

− (6.04446 × 10−16 + 0.110628𝑖) 𝑄4 (𝑥) + (0.442512 − 9.03615 × 10−16 𝑖) 𝑄5 (𝑥) . Figure 2 shows the superposition of the function 𝑓(𝑥) and its expansion 𝑓5 (𝑥). It can be noticed that the five-term approximation is not good.


7

1.2 1

f(x), f5 (x)

0.8 0.6 0.4 0.2 0 −0.2 −1

−0.5

0 x

0.5

1

Figure 2: Superposition of the graphics of 𝑓(𝑥) and 𝑓5 (𝑥).

From the explicit expression (26), the metrical matrix for 𝑛 = 10 can be determined: 2 [ [ [ [ −2𝑖 [ [ [ 2 [ [ [ 2𝑖 [− [ [ 3 [ [ 2 [ 𝛾 [𝑄] = [ [ 2𝑖 [− [ 5 [ [ [ 2 [ [ [ 2𝑖 [− [ 7 [ [ [ 2 [ [ 2𝑖 − [ 9

2𝑖 14 3 2𝑖 3 58 15 6𝑖 5 134 35 10𝑖 7 242 63 14𝑖 9 382 99

2 2𝑖 3 64 15 2𝑖 5 164 35 6𝑖 7 104 21 10𝑖 9 508 99 14𝑖 11

−

2𝑖 2𝑖 2 2 3 5 58 6𝑖 134 10𝑖 − − 15 5 35 7 2𝑖 164 6𝑖 104 − − 5 35 7 21 174 2𝑖 118 2𝑖 − − 35 7 21 3 2𝑖 368 2𝑖 652 − 7 63 9 99 118 2𝑖 670 2𝑖 − 21 9 99 11 2𝑖 652 2𝑖 1104 3 99 11 143 598 6𝑖 1082 2𝑖 99 11 143 13 10𝑖 1016 6𝑖 556 11 143 13 65 906 10𝑖 106 2𝑖 143 13 13 5

2𝑖 7 242 63 10𝑖 − 9 598 99 6𝑖 − 11 1082 143 2𝑖 − 13 1694 195 2𝑖 15 2434 255

2 14𝑖 9 508 99 10𝑖 − 11 1016 143 6𝑖 − 13 556 65 2𝑖 − 15 2464 255 2𝑖 17 −

2𝑖 9 ] 382 ] ] ] 99 ] 14𝑖 ] ] − ] 11 ] 906 ] ] ] 143 ] 10𝑖 ] ] − 13 ] ], 106 ] ] 13 ] ] 2𝑖 ] − ] 5 ] ] 2434 ] ] 255 ] ] 2𝑖 ] − ] 17 ] 3438 ]

(43)

323 ]

with the reciprocal matrix given by 𝛾⋆ [𝑄] 194.628 [ 182.705𝑖 [ [−245.619 [ [ 449.976𝑖 [ [ 431.287 [ =[ [ 837.612𝑖 [ [−402.514 [ [ 768.318𝑖 [ [ 142.611 [ 264.765𝑖

182.705𝑖 173.81 227.829𝑖 −441.019 395.46𝑖 831.682 366.933𝑖 −766.695 129.629𝑖 264.765

−245.619 227.829𝑖 338.212 529.699𝑖 −651.753 967.53𝑖 641.618 887.652𝑖 −234.589 307.523𝑖

449.976𝑖 −441.019 529.699𝑖 1306.76 825.923𝑖 −2699.58 706.803𝑖 2604.53 236.383𝑖 −922.568

431.287 395.46𝑖 −651.753 825.923𝑖 1425.36 1406.66𝑖 −1518.75 1258.88𝑖 583.739 434.151𝑖

837.612𝑖 831.682 967.53𝑖 −2699.58 1406.66𝑖 5953.4 1114.41𝑖 −5973.08 348.073𝑖 2170.75

−402.514 366.933𝑖 641.618 706.803𝑖 −1518.75 1114.41𝑖 1703.66 958.733𝑖 −677.455 325.651𝑖

768.318𝑖 −766.695 887.652𝑖 2604.53 1258.88𝑖 −5973.08 958.733𝑖 6154.17 286.437𝑖 −2279.55

142.611 129.629𝑖 −234.589 236.383𝑖 583.739 348.073𝑖 −677.455 286.437𝑖 275.799 95.1856𝑖

264.765𝑖 264.765 ] ] 307.523𝑖 ] ] −922.568] ] 434.151𝑖 ] ] ]. 2170.75 ] ] 325.651𝑖 ] ] −2279.55] ] 95.1856𝑖 ] 856.67 ]

(44)

8

Mathematical Problems in Engineering 1.2

0.015

1

0.01 0.005

f(x), f10 (x)

0.8

0 R10 (x)

0.6 0.4

−0.005 −0.01 −0.015

0.2

−0.02

0

−0.025

−0.2 −1

−0.5

0 x

0.5

1

(a) Superposition of the graphics of 𝑓(𝑥) and 𝑓10 (𝑥)

−0.03 −1

−0.5

0 x

0.5

1

(b) Local error of the expansion 𝑓10 (𝑥) of 𝑓(𝑥)

Figure 3: Approximation of 𝑓(𝑥).

where 𝛾⋆ [𝑄] is given by the new matrix 𝑄 or, analogously, calculating the coefficients by the equation

Consequently, the ten-term expansion is given by 10

⋆󵄨 󵄨󵄨 󵄨 → 𝛾 󵄨󵄨󵄨𝑓⟩∞ = ∑𝛾𝑘𝑟 𝜇𝑟 󵄨󵄨󵄨𝑄𝑘 ⟩∞ 󵄨󵄨𝑓10 ⟩∞ = ← 𝑟=1

𝑛

(45)

10

󵄨 = ∑𝜇𝑘 󵄨󵄨󵄨𝑄𝑘 ⟩∞ ,

(48)

+ (1.82597 × 10−13 + 1.04832𝑖) 𝑄2 (𝑥)

󵄨󵄨 󵄨󵄨𝑓10 ⟩ 1 ] [ [ 2𝑥 + 𝑖 ] ] [ [ 3𝑥2 + 𝑖𝑥 ] ] [ ] [ [ 4𝑥3 + 𝑖𝑥2 ] ] [ ] [ [ 5𝑥4 + 𝑖𝑥3 ] ] [ ] = [𝜇1 𝜇2 𝜇3 𝜇4 𝜇5 𝜇6 𝜇7 𝜇8 𝜇9 𝜇10 ] [ [ 6𝑥5 + 𝑖𝑥4 ] (46) ] [ ] [ [ 7𝑥6 + 𝑖𝑥5 ] ] [ ] [ [ 8𝑥7 + 𝑖𝑥6 ] ] [ ] [ [ 9𝑥8 + 𝑖𝑥7 ] ] [ 9 8 [10𝑥 + 𝑖𝑥 ]

− (2.09663 − 2.15423 × 10−13 𝑖) 𝑄3 (𝑥) − (5.49712 × 10−13 + 0.661302𝑖) 𝑄4 (𝑥) + (2.64521 − 3.28239 × 10−13 𝑖) 𝑄5 (𝑥) + (1.1432 × 10−12 + 0.320826𝑖) 𝑄6 (𝑥)

(49)

− (1.92496 − 2.75674 × 10−13 𝑖) 𝑄7 (𝑥) − (1.10567 × 10−12 + 0.071728𝑖) 𝑄8 (𝑥) + (0.573824 − 9.09373 × 10−14 𝑖) 𝑄9 (𝑥) + (3.92054 − 3.2685𝑖) × 10−13 𝑄10 (𝑥) .

≡ ⟨𝜇 | 𝑄⟩ ,

Figure 3(a) shows the almost perfect superposition of the function 𝑓(𝑥) and the real part of its expansion 𝑓10 (𝑥). Figure 3(b) shows the local rest for the expansion.

with 󵄨 ⟨𝜇󵄨󵄨󵄨 −1

𝑟=1 −1

𝑓10 (𝑥) = (2.03777 − 1.8553 × 10−13 𝑖) 𝑄1 (𝑥)

𝑘𝑟 where 𝜇𝑘 = ∑10 𝑟=1 𝛾 ⟨𝑄𝑟 | 𝑓⟩∞ or, in matrix form,

=∫

󸀠

Consequently, the new approximation is given by

𝑟=1

1

1

𝜇𝑘 = ∑ ∫ Q⋆𝑟 (𝜉) 𝑓 (𝜉) 𝑑𝜉 (𝛾𝑟𝑘 ) .

4. Conclusion [𝑄1∗

𝑄2∗

𝑄3∗ 𝑇

⋅ 𝑓 (𝜉) 𝑑𝜉 [𝛾⋆ [𝑄]] ,

𝑄4∗

𝑄5∗

𝑄6∗

𝑄7∗

𝑄8∗

𝑄9∗

∗ 𝑄10 ] (𝜉)

(47)

Considering that using nonorthogonal bases is being increased in spectral numerical methods requiring GramSchmidt procedures that are generally difficult from the

Mathematical Problems in Engineering operational point of view, this paper presents a simpler method for expanding functions, based on bracket formalism and called coadjoint method. The mathematical ideas of the coadjoint method were presented and the examples have shown its practicability. It can be added that coadjoint method is an efficient and concise tool for nonorthogonal bases with low computational costs. As it is not necessary to have orthogonal bases, more general function can be used in the numerical methods, increasing the quality of the whole process.

9

[13]

[14] [15] [16]

Competing Interests The authors declare that there are no competing interests regarding the publication of this paper.

References [1] E. H. Doha, “On the construction of recurrence relations for the expansion and connection coefficients in series of Jacobi polynomials,” Journal of Physics A: Mathematical and General, vol. 37, no. 3, pp. 657–675, 2004. [2] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, U.S. Department of Commerce, National Bureau of Standards, Washington, DC, USA, 10th edition, 1972. [3] G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists: A Comprehensive Guide, Academic Press, New York, NY, USA, 2011. [4] O. Guimarães, J. R. C. Piqueira, and M. L. Netto, “Direct computation of operational matrices for polynomial bases,” Mathematical Problems in Engineering, vol. 2010, Article ID 139198, 12 pages, 2010. [5] E. H. Doha and A. H. Bhrawy, “Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials,” Applied Numerical Mathematics, vol. 58, no. 8, pp. 1224–1244, 2008. [6] E. H. Doha, A. H. Bhrawy, and S. S. Ezz-Eldien, “A new Jacobi operational matrix: an application for solving fractional differential equations,” Applied Mathematical Modelling, vol. 36, no. 10, pp. 4931–4943, 2012. [7] D. Baleanu, M. Alipour, and H. Jafari, “The Bernstein operational matrices for solving the fractional quadratic Riccati differential equations with the Riemann-Liouville derivative,” Abstract and Applied Analysis, vol. 2013, Article ID 461970, 7 pages, 2013. [8] E. H. Doha, A. H. Bhrawy, and M. A. Saker, “Integrals of Bernstien polynomials: an application for the solution of high even-order diferential equations,” Mathematical and Computer Modelling, vol. 53, pp. 1820–1823, 2011. [9] A. K. Singh, V. K. Singh, and O. P. Singh, “The Bernstein operational matrix of integration,” Applied Mathematical Sciences, vol. 49, pp. 2427–2436, 2009. [10] P. N. Paraskevopoulos, P. Sklavounos, and G. Georgiou, “The operational matrix of integration for Bessel functions,” Journal of the Franklin Institute, vol. 327, no. 2, pp. 329–341, 1990. [11] P. N. Paraskevopoulos, P. D. Sparis, and S. G. Mouroutsos, “The Fourier series operational matrix of integration,” International Journal of Systems Science, vol. 16, no. 2, pp. 171–176, 1985. [12] W. Labecca, O. Guimarães, and J. R. Piqueira, “Dirac’s formalism combined with complex Fourier operational matrices to

[17]

solve initial and boundary value problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 8, pp. 2614–2623, 2014. W. Labecca, Matrizes operacionais e formalismo coadjunto em equaço˜ es diferenciais fracionais [Ph.D. thesis], Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil, 2015. V. Vedral, Introduction to Quantum Information Science, Oxford University Press, Oxford, UK, 2006. A. I. Borisenko and I. E. Tarapov, Vector and Tensor Analysis with Applications, Dover, New York, NY, USA, 1968. D. Lovelock and H. Rund, Tensors, Differential Forms and Variational Principles, John Wiley & Sons, New York, NY, USA, 1975. E. Kreyszig, Introduction to Differential Geometry and Riemannian Geometry, University of Toronto Press, 1975.

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