# American Institute of Mathematical Sciences

March  2022, 15(3): 519-534. doi: 10.3934/dcdss.2021149

## Optimality conditions involving the Mittag–Leffler tempered fractional derivative

 1 Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810–193 Aveiro, Portugal 2 Center for Computational and Stochastic Mathematics, Instituto Superior Técnico and Department of Mathematics, University of Trás-os-Montes e Alto Douro, UTAD, 5000-801, Vila Real, Portugal

* Corresponding author: Ricardo Almeida

Received  November 2019 Revised  February 2021 Published  March 2022 Early access  November 2021

Fund Project: R. Almeida is supported by Portuguese funds through the CIDMA - Center for Research and Development in Mathematics and Applications, and the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciência e a Tecnologia), within project UIDB/04106/2020. M. L. Morgado acknowledges the financial support of the Portuguese FCT - Fundação para a Ciência e a Tecnologia, through projects UIDB/04621/2020 and UIDP/04621/2020

In this work we study problems of the calculus of the variations, where the differential operator is a generalization of the tempered fractional derivative. Different types of necessary conditions required to determine the optimal curves are proved. Problems with additional constraints are also studied. A numerical method is presented, based on discretization of the variational problem. Through several examples, we show the efficiency of the method.

Citation: Ricardo Almeida, M. Luísa Morgado. Optimality conditions involving the Mittag–Leffler tempered fractional derivative. Discrete & Continuous Dynamical Systems - S, 2022, 15 (3) : 519-534. doi: 10.3934/dcdss.2021149
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##### References:
Maximum of the committed absolute error in the approximation of the solution
 $N$ 5 10 20 40 $E$ 8.81 $\times 10^{-3}$ 3.61 $\times 10^{-3}$ 1.40 $\times 10^{-3}$ 5.24 $\times 10^{-4}$
 $N$ 5 10 20 40 $E$ 8.81 $\times 10^{-3}$ 3.61 $\times 10^{-3}$ 1.40 $\times 10^{-3}$ 5.24 $\times 10^{-4}$
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