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.

Les amours de feu Mr Tristan et autres pièces tres-curieuses ([Reprod.])

Collection : French books 1601-1700 ; 218.2

Recueil de généalogies.

Contient : De Créqui ; Comtes de Hainaut ; D'Ailly ; D'Anglure ; D'Angennes ; D'Enghien ; Adhémar de Grignan ; Bracque ; De Champagne ; De Clérembault ; De Chabanes ; De Daillon ; De Damas ; De Clermont ; Gouffier ; Comtes d'Eu ; Comtes de Ponthieu ; De Saint-Gelais ; Chastaignier ; De Ghistelle ; Lettre de Trincant à A. Duchesne ; Note sur la famille L'Hermite de Solier ; Extraits des mémoires de Jean de Nostradamus, concernant la maison des Baux ; Maison de Joyeuse ; De Montbron ; De Maillé ; De Sainte-Maure ; De Talleyrand-Périgord ; De Pons ; De Prie ; De Blanchefort ; De Roye ; Turpin ; De Bueil ; De Villeneuve ; D'Aunoy ; De Trie et Dammartin ; De l'Isle-Adam ; De Villiers ; De Beaumont ; Villiers de l'Isle-Adam ; D'Offémont ; Extraits de l'inventaire des titres de Dammartin ; Extrait de l'inventaire des titres de Chantilly ; Extraits des titres de la baronnie de Préaux ; Famille de Chauvigny ; De Grandpré ; De Mello ; De Noyers ; De Nesle ; De Trainel