11,469 research outputs found
Fractional variational problems with the Riesz-Caputo derivative
No full textIn this paper we investigate optimality conditions for fractional variational problems, with a Lagrangian depending on the Riesz-Caputo derivative. First we prove a generalized Euler-Lagrange equation for the case when the interval of integration of the functional is different from the interval of the fractional derivative. Next we consider integral dynamic constraints on the problem, for several different cases. Finally, we determine optimality conditions for functionals depending not only on the admissible functions, but on time also, and we present a necessary condition for a pair function-time to be an optimal solution to the problem. © 2011 Elsevier Ltd. All rights reserved.FCTCIDM
Fractional variational problems depending on indefinite integrals
No full textWe obtain necessary optimality conditions for variational problems with a Lagrangian depending on a Caputo fractional derivative, a fractional and an indefinite integral. Main results give fractional Euler-Lagrange type equations and natural boundary conditions, which provide a generalization of the previous results found in the literature. Isoperimetric problems, problems with holonomic constraints and depending on higher-order Caputo derivatives, as well as fractional Lagrange problems, are considered. © 2011 Elsevier Ltd. All rights reserved
Numerical approach to the Caputo derivative of the unknown function
No full textIf a function can be explicitly expressed, then one can easily compute its Caputo derivative by the known methods. If a function cannot be explicitly expressed but it satisfies a differential equation, how to seek Caputo derivative of such a function has not yet been investigated. In this paper, we propose a numerical algorithm for computing the Caputo derivative of a function defined by a classical (integer-order) differential equation. By the properties of Caputo derivative derived in this paper, we can change the original typical differential system into an equivalent Caputo-type differential system. Numerical examples are given to support the derived numerical method
Local density of Caputo-stationary functions in the space of smooth functions
No full textWe consider the Caputo fractional derivative and say that a function is Caputo-stationary if its Caputo derivative is zero. We then prove that any Ck( [ 0,1 ]) function can be approximated in [0,1] by a function that is Caputo-stationary in [0,1], with initial point a< 0. Otherwise said, Caputo-stationary functions are dense in Ckloc(R)
An ABC of Citizenship, vo. 1
No full textE' il primo di due fascicoli monografici che raccoglie i lavori e le intuizioni nate anche in sinergia con alcune associazioni del territorio pugliese: creare degli Abbecedari della cittadinanza, legati a grappoli di eventi, capaci di coinvolgere territori, scuole e in generale agenzie educative. L'esperienza ha a sua volta sollecitato docenti e ricercatori universitari italiani e stranieri, che hanno inviato i loro contributi. La molteplicità di materiale e l'interesse del percorso teorico-pratico ci ha appunto convinto a dividere quanto giunto in redazione in due fascicoli, che raccolgono saggi legati a 21 sezioni, una per ogni lettera dell'alfabeto di questo ideale abbecedario comunitario: dalla A di amicizia alla Z di Zero povertà. Con contributi anche di tipo internazionale: G. Taylor, Università di Pittsburg, USA; R. Savage, Los Angeles; C. Waktin, Monash – Australia; T.D. Moratalla – Madrid; P. Mena Malet, Santiago del Cile; G. Malkassian, Parigi; Patrik Friedlund, Univ di Lund, Svezia
High Order Approximation to Caputo Derivative On Graded Mesh and Time-Fractional Diffusion Equation for Non-smooth Solutions
No full textIn this paper, a high-order approximation to Caputo-type time-fractional diffusion equations involving an initial-time singularity of the solution is proposed. At first, we employ a numerical algorithm based on the Lagrange polynomial interpolation to approximate the Caputo derivative on the non-uniform mesh. Then truncation error rate and the optimal grading constant of the approximation on a graded mesh are obtained as min{4 - a, ra} and 4-a/a , respectively, where α ∈ (0, 1) is the order of fractional derivative and r ≥ 1 is the mesh grading parameter. Using this new approximation, a difference scheme for the Caputo-type time-fractional diffusion equation on the graded temporal mesh is formulated. The scheme proves to be uniquely solvable for general r. Then we derive the unconditional stability of the scheme on uniform mesh. The convergence of the scheme, in particular for r = 1, is analyzed for non-smooth solutions and concluded for smooth solutions. Finally, the accuracy of the scheme is verified by analyzing the error through a few numerical examples.The first author acknowledges the support provided by the Council of Scientific and Industrial Research, India, under grant number
09/086(1483)/2020-EMR-I. The fourth author acknowledges the support provided by the SERB, a statutory body of DST,
India, under the award SERB–POWER fellowship (grant
number SPF/2021/000103)
Approximation of the Riesz–Caputo derivative by cubic splines
No full textDifferential problems with the Riesz derivative in space are widely used to model anomalous diffusion. Although the Riesz–Caputo derivative is more suitable for modeling real phenomena, there are few examples in literature where numerical methods are used to solve such differential problems. In this paper, we propose to approximate the Riesz–Caputo derivative of a given function with a cubic spline. As far as we are aware, this is the first time that cubic splines have been used in the context of the Riesz–Caputo derivative. To show the effectiveness of the proposed numerical method, we present numerical tests in which we compare the analytical solution of several boundary differential problems which have the Riesz–Caputo derivative in space with the numerical solution we obtain by a spline collocation method. The numerical results show that the proposed method is efficient and accurate
α-Tolylsulfinylation of ketones via their trimethylsilyl enol ethers. One step synthesis of β-ketosulfoxides R. Caputo; C. Ferreri; L. Longobardo; G. Palumbo; S. Pedatella
No full textKetones are reported to be conveniently converted into their α-tolylsulfinylated derivatives. This new procedure is based on the reaction of their corresponding trimethylsilyl enol ethers with p-toluenesulfinyl p-tolylsulfone in the presence of tris(dimethylamino)sulfur trimethylsilyldifluoride (TAS-F). Considering that β-ketosulfoxides are key intermediates for the preparation of α,β-unsaturated ketones, this procedure turns out to be of rather broad synthetic relevance
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