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.

Database evolution on an orthogonal persistent programming system: A semi-transparent approach

In this paper the problem of the evolution of an object-oriented database in the context of orthogonal persistent programming systems is addressed. We have observed two characteristics in that type of systems that offer particular conditions to implement the evolution in a semi-transparent fashion. That transparency can further be enhanced with the obliviousness provided by the Aspect-Oriented Programming techniques. Was conceived a meta-model and developed a prototype to test the feasibility of our approach. The system allows programs, written to a schema, access semi-transparently to data in other versions of the schema.

Orthogonal persistence in Java supported by aspect- oriented programming and reflection

The persistence concern implemented as an aspect has been studied since the appearance of the Aspect-Oriented paradigm. Frequently, persistence is given as an example that can be aspectized, but until today no real world solution has applied that paradigm. Such solution should be able to enhance the programmer productivity and make the application less prone to errors. To test the viability of that concept, in a previous study we developed a prototype that implements Orthogonal Persistence as an aspect. This first version of the prototype was already fully functional with all Java types including arrays. In this work the results of our new research to overcome some limitations that we have identified on the data type abstraction and transparency in the prototype are presented. One of our goals was to avoid the Java standard idiom for genericity, based on casts, type tests and subtyping. Moreover, we also find the need to introduce some dynamic data type abilities. We consider that the Reflection is the solution to those issues. To achieve that, we have extended our prototype with a new static weaver that preprocesses the application source code in order to introduce changes to the normal behavior of the Java compiler with a new generated reflective code.

An aspect-oriented framework for orthogonal persistence

The life cycle of software applications in general is very short and with extreme volatile requirements. Within these conditions programmers need development tools and techniques with an extreme level of productivity. We consider the code reuse as the most prominent approach to solve that problem. Our proposal uses the advantages provided by the Aspect-Oriented Programming in order to build a reusable framework capable to turn both programmer and application oblivious as far as data persistence is concerned, thus avoiding the need to write any line of code about that concern. Besides the benefits to productivity, the software quality increases. This paper describes the actual state of the art, identifying the main challenge to build a complete and reusable framework for Orthogonal Persistence in concurrent environments with support for transactions. The present work also includes a successfully developed prototype of that framework, capable of freeing the programmer of implementing any read or write data operations. This prototype is supported by an object oriented database and, in the future, will also use a relational database and have support for transactions.