Browsing by Author "Almeida, Ricardo"
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- Caputo derivatives of fractional variable order: numerical approximationsPublication . Tavares, Dina; Almeida, Ricardo; Torres, Delfim F. M.We present a new numerical tool to solve partial differential equations involving Caputo derivatives of fractional variable order. Three Caputo-type fractional operators are considered, and for each one of them an approximation formula is obtained in terms of standard (integer-order) derivatives only. Estimations for the error of the approximations are also provided. We then compare the numerical approximation of some test function with its exact fractional derivative. We end with an exemplification of how the presented methods can be used to solve partial fractional differential equations of variable order.
- Combined Fractional Variational Problems of Variable Order and Some Computational AspectsPublication . Tavares, Dina; Almeida, Ricardo; Torres, Delfim F. M.We study two generalizations of fractional variational problems by considering higher-order derivatives and a state time delay. We prove a higher-order integration by parts formula involving a Caputo fractional derivative of variable order and we establish several necessary optimality conditions for functionals containing a combined Caputo derivative of variable fractional order. Because the endpoint is considered to be free, we also deduce associated transversality conditions. In the end, we consider functionals with a time delay and deduce corresponding optimality conditions. Some examples are given to illustrate the new results. Computational aspects are discussed using the open source software package Chebfun.
- Constrained fractional variational problems of variable orderPublication . Tavares, Dina; Almeida, Ricardo; Torres, Delfim F. M.Isoperimetric problems consist in minimizing or maximizing a cost functional subject to an integral constraint. In this work, we present two fractional isoperimetric problems where the Lagrangian depends on a combined Caputo derivative of variable fractional order and we present a new variational problem subject to a holonomic constraint. We establish necessary optimality conditions in order to determine the minimizers of the fractional problems. The terminal point in the cost integral, as well the terminal state, are considered to be free, and we obtain corresponding natural boundary conditions.
- Fractional Herglotz variational problems of variable orderPublication . Tavares, Dina; Almeida, Ricardo; Torres, Delfim F. M.We study fractional variational problems of Herglotz type of variable order. Necessary optimality conditions, described by fractional differential equations depending on a combined Caputo fractional derivative of variable order, are proved. Two different cases are considered: the fundamental problem, with one independent variable, and the general case, with several independent variables. We end with some illustrative examples of the results of the paper.
- Optimality Conditions for Fractional Variational Problems with Dependence on a Combined Caputo Derivative of Variable OrderPublication . Tavares, Dina; Almeida, Ricardo; Torres, Delfim F. M.We establish necessary optimality conditions for variational problems with a Lagrangian depending on a combined Caputo derivative of variable fractional order. The endpoint of the integral is free, and thus transversality conditions are proved. Several particular cases are considered illustrating the new results.