A Review of Operational Matrices and Spectral Techniques for Fractional Calculus

Recently, operational matrices were adapted for solving several kinds of fractional differential equations (FDEs). The use of numerical techniques in conjunction with operational matrices of some orthogonal polynomials, for the solution of FDEs on finite and infinite intervals, produced highly accurate solutions for such equations. This article discusses spectral techniques based on operational matrices of fractional derivatives and integrals for solving several kinds of linear and nonlinear FDEs. More precisely, we present the operational matrices of fractional derivatives and integrals, for several polynomials on bounded domains, such as the Legendre, Chebyshev, Jacobi and Bernstein polynomials, and we use them with different spectral techniques for solving the aforementioned equations on bounded domains. The operational matrices of fractional derivatives and integrals are also presented for orthogonal Laguerre and modified generalized Laguerre polynomials, and their use with numerical techniques for solving FDEs on a semi-infinite interval is discussed. Several examples are presented to illustrate the numerical and theoretical properties of various spectral techniques for solving FDEs on finite and semi-infinite intervals.

An efficient numerical scheme for solving multi-dimensional fractional optimal control problems with a quadratic performance index

The shifted Legendre orthogonal polynomials are used for the numerical solution of a new formulation for the multi-dimensional fractional optimal control problem (M-DFOCP) with a quadratic performance index. The fractional derivatives are described in the Caputo sense. The Lagrange multiplier method for the constrained extremum and the operational matrix of fractional integrals are used together with the help of the properties of the shifted Legendre orthonormal polynomials. The method reduces the M-DFOCP to a simpler problem that consists of solving a system of algebraic equations. For confirming the efficiency and accuracy of the proposed scheme, some test problems are implemented with their approximate solutions.

Determination of ochratoxin A in bread: evaluation of microwave-assisted extraction using an orthogonal composite design coupled with response surface methodology

An analytical method using microwave-assisted extraction (MAE) and liquid chromatography (LC) with fluorescence detection (FD) for the determination of ochratoxin A (OTA) in bread samples is described. A 24 orthogonal composite design coupled with response surface methodology was used to study the influence of MAE parameters (extraction time, temperature, solvent volume, and stirring speed) in order to maximize OTA recovery. The optimized MAE conditions were the following: 25 mL of acetonitrile, 10 min of extraction, at 80 °C, and maximum stirring speed. Validation of the overall methodology was performed by spiking assays at five levels (0.1–3.00 ng/g). The quantification limit was 0.005 ng/g. The established method was then applied to 64 bread samples (wheat, maize, and wheat/maize bread) collected in Oporto region (Northern Portugal). OTAwas detected in 84 % of the samples with a maximum value of 2.87 ng/g below the European maximum limit established for OTA in cereal products of 3 ng/g.

Root-locus practical sketching rules for fractional-order system

For integer-order systems, there are well-known practical rules for RL sketching. Nevertheless, these rules cannot be directly applied to fractional-order (FO) systems. Besides, the existing literature on this topic is scarce and exclusively focused on commensurate systems, usually expressed as the ratio of two noninteger polynomials. The practical rules derived for those do not apply to other symbolic expressions, namely, to transfer functions expressed as the ratio of FO zeros and poles. However, this is an important case as it is an extension of the classical integer-order problem usually addressed by control engineers. Extending the RL practical sketching rules to such FO systems will contribute to decrease the lack of intuition about the corresponding system dynamics. This paper generalises several RL practical sketching rules to transfer functions specified as the ratio of FO zeros and poles. The subject is presented in a didactic perspective, being the rules applied to several examples.

Overcoming the objectification of western classical music with the concept of music as a process

Dissertation to obtain the degree of Master in Music - Artistic Interpretation