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Derivatives and Integrals: Matrix Order Operators As an Extension of the Fractional Calculus Cleiton Bittencourt Da Porciúncula

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Derivatives and Integrals: Matrix Order Operators As an Extension of the Fractional Calculus Cleiton Bittencourt Da Porciúncula Derivatives and integrals: matrix order operators as an extension of the fractional calculus Cleiton Bittencourt da Porciúncula UERGS, State University of Rio Grande do Sul, Brazil Independência, Av., Santa Cruz do Sul Campus, ZIP Code: 96816-501 Corresponding author: Phone/Fax Number: +55 51 3715-6926 Email: [email protected] Abstract. A natural consequence of the fractional calculus is its extension to a matrix order of differentiation and integration. A matrix-order derivative definition and a matrix-order integration arise from the generalization of the gamma function applied to the fractional differintegration definition. This work focuses on some results applied to the Riemann- Liouville version of the fractional calculus extended to its matrix-order concept. This extension also may apply to other versions of fractional calculus. Some examples of possible ordinary and partial matrix-order differential equations and related exact solutions are shown. Keywords: matrix-order differintegration, fractional calculus, matrix gamma function, differential operator 1. Introduction There are several good works and reviews about fractional derivatives and integrals. In a general fashion, we may consider a general fractional derivative operator as: 푑훼 푑푝 푥 퐷훼푓(푥) = 푓(푥) = ∫ 퐾 (푥, 푡, 훼)휙 (푓(푡), 푓푛(푡))푑푡 (1.1) 푎 푑푥훼 푑푥푝 푎 퐷 퐷 and a fractional integral operator as: 푥 퐽훼푓(푥) = 퐷−훼푓(푥) = 퐾 (푥, 푡, 훼)휙 (푓(푡), 푓푛(푡))푑푡 푎 ∫푎 퐽 퐽 (1.2) with 훼 ∈ ℂ, 푡, 푥, 푎 ∈ ℜ and 푝 ∈ ℕ. In eqs. (1.1) and (1.2), the kernels 퐾퐷 and 퐾푓 vary according to the version of the fraction differintegration. Functions 휙퐷 and 휙퐽 also may assume any form as a function of 푓(푡) and the 푛-derivatives of 푓(푡), 푓푛(푡). Table 1 shows different forms of these operators under different kernels and shapes of integrable functions, with some corresponding references. 훼-derivative 훼-integral 푛 훼 1 푑 푥 (푛−훼−1) −훼 1 푥 (훼−1) 퐷 푓(푥) = ∫ (푥 − 푡) 푓(푡)푑푡 푐퐷푥 푓(푥) = ∫ (푥 − 푡) 푓(푡)푑푡 푐 푥 Γ(푛−훼) 푑푥푛 푐 Γ(훼) 푐 a 푥 > 푐, 푛 > 훼 + 1 푥 > 푐 훼 −∞퐷푥 푓(푥) = 1 푑푛 푥 1 푥 = ∫ (푥 − 푡)(푛−훼−1)푓(푡)푑푡 퐷 −훼푓(푥) = ∫ (푥 − 푡)(훼−1)푓(푡)푑푡 Γ(푛−훼) 푑푥푛 −∞ −∞ 푥 Γ(훼) −∞ b 푛 > 훼 + 1 1 푑 푝 푥 ( ) ∫ (푥 − 푡)−(훼−푛+1)푓(푡)푑푡 Γ(푛−훼) 푑푥 푎 푥 > 푎; 푛 − 1 < 훼 ≤ 푛 1 푥 퐷 −훼푓(푥) = ∫ (푥 − 푡)(훼−1)푓(푡)푑푡 0 푥 Γ(훼) 0 c 1 푑 푝 푏 (− ) ∫ (푥 − 푡)−(훼−푛+1)푓(푡)푑푡 ; Γ(푛−훼) 푑푥 푥 푥 < 푏 ; 푛 − 1 < 훼 ≤ 푛 1 2휋 퐷훼푓(푥) = ∫ 푓(푥 − 푡)휓 ′(푡)푑푡 ; 2휋 0 1−훼 0 < 훼 < 1 −훼 1 ∞ (훼−1) ∞ 퐷 푓(푥) = ∫ (푥 − 푡) 푓(푡)푑푡 exp(푘푡) 0 푥 Γ(훼) 푥 d 휓훼(푡) = ∑ (±푘)훼 푘=−∞,푘≠0 1 푥 퐷훼 푓(푥) = ∫ (푥 − 푡)−(훼−푛+1)푓푛(푡)푑푡 ; 푎+ Γ(푛−훼) 푎 푥 > 푎; 푛 − 1 < 훼 ≤ 푛 1 푥 −훼 ( ) ( )(훼−1) ( ) (−1)푛 푏 푎퐷푥 푓 푥 = ∫푎 푥 − 푡 푓 푡 푑푡 e 퐷훼 푓(푥) = ∫ (푥 − 푡)−(훼−푛+1)푓푛(푡)푑푡 ; Γ(훼) 푏− Γ(푛−훼) 푥 푥 < 푏 ; 푛 − 1 < 훼 ≤ 푛 Table 1 – Different versions of the fractional differintegral operators (푅푒(훼 > 0): a) Riemann: [1] Miller and Ross, 1993; b) Liouville: [1]Miller and Ross, 1993, [2] Oliveira e Machado, 2014; c) Riemann-Liouville, Abel equation: [1]Miller and Ross, 1993, [3]Trujillo et al, 2006, [3] [4]Samko, Kilbas and Marichev; d) Weyl: [5]Rudolf Hilfer, 1993, [6]Fausto Ferrari; e) Caputo: [2] Trujillo et al, 2006, [7] Caputo, 1966, [8] Caputo, 1967, [9]Kumar et al, 2019. Starting from the Rieman-Liouville version by assuming 푛 = 1, 푝 = 1, with the lower integration limit 푎 = 0, the fractional derivative becomes: 푑훼 1 푑 푥 퐷훼푓(푥) = 푓(푥) = ∫ (푥 − 푡)−훼푓(푡)푑푡 (1.3) 푑푥훼 Γ(1−훼) 푑푥 0 The corresponding fractional integral is: 1 푥 퐷−훼푓(푥) = 퐽훼푓(푥) = ∫ (푥 − 푡)(훼−1)푓(푡)푑푡 (1.4) Γ(훼) 0 In this work, the following developments base on eqs. (1.3), (1.4). A natural extension of definitions (1.1)-(1.4) is possible if thinking in the sense of a differentiation/integration order as a squared matrix as follows: 푎11 푎12 … 푎1푛 푑푀 푎21 푎22 … 푎2푛 푓(푥), with 푀 = [ ] 푑푥푀 ⋮ ⋮ ⋮ ⋮ 푎푛1 푎푛2 … 푎푛푛 푀 is a matrix 푛 × 푛 with, 푛 ∈ ℤ and 푎푖푗, 1<푖<푛,1<푗<푛 elements; 푎 ∈ ℂ, and 푓(푥) a continuous scalar or matrix real or complex function function (푥 ∈ ℂ). The “matrix- order differentiation” may look improbable at a first glance; however, it is possible to define it also in terms of the gamma function by simply following the known definitions of fractional derivative and integral in eqs. (1.3), (1.4). The differintegration operators expressed as a matrix order (in terms of a generic kernel) become: 푑푀 푑푝 푥 퐷푀퐹(푥) = 퐹(푥) = ∫ 퐾 (푥, 푡, 푀)휙 (퐹(푡), 퐹푛(푡))푑푡 (1.5) 푎 푑푥푀 푑푥푝 푎 퐷 퐷 푥 퐽푀푓(푥) = 퐷−푀푓(푥) = 퐾 (푥, 푡, 푀)휙 (퐹(푡), 퐹푛(푡))푑푡 푎 ∫푎 퐽 퐽 (1.6) Expressing the matrix-order differintegration operators as a Rieman-Liouville according to eqs. (1.3) - (1.4), one obtains: 푑푀 푑 푥 퐷푀푓(푥) = 푓(푥) = [Γ−1(퐼 − 푀)] ∫ (푥 − 푡)−푀퐹(푡)푑푡 (1.7) 푑푥푀 푑푥 0 푥 퐷−푀푓(푥) = 퐽푀푓(푥) = [Γ−1(푀)] (푥 − 푡)(푀−퐼)퐹(푡)푑푡 ∫0 (1.8) where 퐼 is the identity matrix and 푀 ∈ ℂ푛×푛. In eqs. (1.5) - (1-8), the function 푓 is now expressed with an uppercase 퐹 to account the possibility that the function 퐹 may be a matrix of functions (since that dim 퐹 = dim 푀), that is, the number of rows and columns in 퐹 is equal to those in 푀, both with the same dimension), and not only a single function 푓. 퐹 may also be a column vector of functions, provided that the number of rows of 퐹 has the same number of columns of 푀 or (푥 − 푡)−푀. This extension of the fractional calculus to a matrix order calculus is not a new idea. Naber develops this theme in their works [10], [11]. Phillips [12], [13] uses the concept of matrix order differentiation and integration in the development of the econometric theory and advanced probability distributions. However, there are no references, for example, about developments of exact or numeric matrix-order differential equations, other related problems of such objects, and possible applications. Before continuing the analysis of this matrix order calculus, Table 2 summarizes the nomenclature and symbols used along with this text. Notation / Symbol Used for: The uppercase letter indicates a square 푀 matrix with 푛 × 푛 elements. 퐼 Identity matrix with 푛 × 푛 elements. 푓(푥), 푓(푡) A single function depending on 푥 or 푡 A matrix of functions 푓푖푗(푥) or 푓푖푗(푡) in 푓11 … 푓1푛 the form: 퐹 = [ ⋮ ⋱ ⋮ ] 푓푛1 … 푓푛푛 or a column vector function in the form: 퐹(푥), 퐹(푡) 푓1 퐹 = [ ⋮ ] where 푛 = 푚, i.e., the number 푓푚 of columns of 푀 is equal to the number of rows of 퐹. 푑푀 퐷푀, The M-derivative of a specified function. 푑푥푀 휕푀 The partial M-derivative of a specified 휕푀, , 휕푀 휕푥푀 푥 function concerning some variable 푥. 퐷−푀, 퐽푀, ∫ 푑푥푀 The M-integral of a specified function The matrix gamma function of some Γ(퐴) square matrix A. The matrix beta function of some square Β(퐴) matrix A. The inverse of the matrix gamma Γ−1(퐴) function of some square matrix A. The inverse of the matrix beta function Β−1(퐴) of some square matrix A. Equal to [(퐾(푥, 푡))퐴]−1 or 퐾−퐴(푥, 푡) 푛푣 [(퐾(푥, 푡)퐴] Table 2 – Notation used in this article. In eqs. (1.7) and (1.8), the matrix gamma functions are: +∞ Γ(퐼 − 푀) = 푡−푀 exp(−푡) 푑푡 ∫0 (1.9) +∞ Γ(푀) = 푡(푀−퐼) exp(−푡) 푑푡 ∫0 (1.10) Eqs. (1.9) – (1.10) are the matrix gamma function, and its properties are broadly covered in the literature, as well as several properties and applications derived from those definitions, especially applied to multivariate statistics [14], [15], [16], [17]. The term 푡−푀 in eq. (1.9) is: 푡−푀 = exp (ln 푡−푀) = exp(−푀 ln(푡)) (1.11) The exponential of a matrix is a well-known result based on the decomposition of eigenvalues and eigenvectors of a diagonalizable matrix [19]: exp(퐴푡′) = 푉 exp(Λ푡′) 푉−1 (1.12) with 푡′ = ln 푡. For a 푛 × 푛 matrix, one obtains: ′ exp(휆1푡 ) ⋯ 0 ′ exp(Λ푡 ) = [ ⋮ ⋱ ⋮ ] (1.13) ′ 0 ⋯ exp(휆푛푡 ) The corresponding eigenvector matrix is: 푣11 ⋯ 푣1푛 푉 = [푣1 ⋯ 푣푛] = [ ⋮ ⋱ ⋮ ] (1.14) 푣푛1 ⋯ 푣푛푛 The vectors 풗ퟏ up to 풗풏 are the associated eigenvectors with the eigenvalues 휆1 up to 휆푛. These eigenvalues are computed as: det(휆퐼 − 퐴) = 0 (1.15) The eigenvalues 휆푖 associated with each eigenvector 푣푖 is determined via the well- known relation 퐴푣푖 = 휆푖푣푖, being for eq. (1.11), 퐴 = −푀 , and changing (1.15) to: det(휆퐼 + 푀) = 0 (1.16) When applying the decomposition of matrices to eq.
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