scholarly journals Optimality conditions for variational problems involving distributed-order fractional derivatives with arbitrary kernels

2021 ◽  
Vol 6 (5) ◽  
pp. 5351-5369
Author(s):  
Fátima Cruz ◽  
◽  
Ricardo Almeida ◽  
Natália Martins
Keyword(s):  
Optimality Conditions ◽  
Fractional Derivatives ◽  
Variational Problems ◽  
Distributed Order

scholarly journals Variational Problems with Time Delay and Higher-Order Distributed-Order Fractional Derivatives with Arbitrary Kernels

Mathematics ◽  
2021 ◽  
Vol 9 (14) ◽  
pp. 1665
Author(s):  
Fátima Cruz ◽  
Ricardo Almeida ◽  
Natália Martins
Keyword(s):  
Time Delay ◽  
Fractional Derivatives ◽  
Variational Problems ◽  
Higher Order ◽  
Sufficient Optimality Conditions ◽  
Fractional Operator ◽  
Different Types ◽  
Generalized Fractional Derivatives ◽  
Necessary And Sufficient ◽  
Distributed Order

In this work, we study variational problems with time delay and higher-order distributed-order fractional derivatives dealing with a new fractional operator. This fractional derivative combines two known operators: distributed-order derivatives and derivatives with respect to another function. The main results of this paper are necessary and sufficient optimality conditions for different types of variational problems. Since we are dealing with generalized fractional derivatives, from this work, some well-known results can be obtained as particular cases.

scholarly journals New Variational Problems with an Action Depending on Generalized Fractional Derivatives, the Free Endpoint Conditions, and a Real Parameter

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 592
Author(s):  
Ricardo Almeida ◽  
Natália Martins
Keyword(s):  
Optimality Conditions ◽  
Fractional Derivatives ◽  
Lagrange Function ◽  
Sufficient Conditions ◽  
Necessary Optimality Conditions ◽  
Variational Problems ◽  
Final State ◽  
Fractional Variational Problems ◽  
Generalized Fractional Derivatives ◽  
Arbitrary Kernel

This work presents optimality conditions for several fractional variational problems where the Lagrange function depends on fractional order operators, the initial and final state values, and a free parameter. The fractional derivatives considered in this paper are the Riemann–Liouville and the Caputo derivatives with respect to an arbitrary kernel. The new variational problems studied here are generalizations of several types of variational problems, and therefore, our results generalize well-known results from the fractional calculus of variations. Namely, we prove conditions useful to determine the optimal orders of the fractional derivatives and necessary optimality conditions involving time delays and arbitrary real positive fractional orders. Sufficient conditions for such problems are also studied. Illustrative examples are provided.

scholarly journals Necessary optimality conditions of a reaction-diffusion SIR model with ABC fractional derivatives

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Moulay Rchid Sidi Ammi ◽  
Mostafa Tahiri ◽  
Delfim F. M. Torres
Keyword(s):  
Optimal Control ◽  
Fractional Derivative ◽  
Optimality Conditions ◽  
Fractional Derivatives ◽  
Necessary Optimality Conditions ◽  
Reaction Diffusion ◽  
Derivative Operator ◽  
Sir Epidemic ◽  
Non Local

<p style='text-indent:20px;'>The main aim of the present work is to study and analyze a reaction-diffusion fractional version of the SIR epidemic mathematical model by means of the non-local and non-singular ABC fractional derivative operator with complete memory effects. Existence and uniqueness of solution for the proposed fractional model is proved. Existence of an optimal control is also established. Then, necessary optimality conditions are derived. As a consequence, a characterization of the optimal control is given. Lastly, numerical results are given with the aim to show the effectiveness of the proposed control strategy, which provides significant results using the AB fractional derivative operator in the Caputo sense, comparing it with the classical integer one. The results show the importance of choosing very well the fractional characterization of the order of the operators.</p>

scholarly journals Optimality Conditions and Duality for Multiobjective Variational Problems with Generalized -B-Type-I Functions

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
C. Nahak ◽  
N. Behera
Keyword(s):  
Optimality Conditions ◽  
Variational Problems ◽  
Sufficient Optimality Conditions ◽  
Type I ◽  
Duality Results ◽  
Multiobjective Variational Problems ◽  
Generalized Type

We use -type-I and generalized -type-I functions to establish sufficient optimality conditions and duality results for multiobjective variational problems. Some of the related problems are also discussed.

scholarly journals Expansion formula for fractional derivatives in variational problems

2014 ◽  
Vol 409 (2) ◽  
pp. 911-924 ◽  
Author(s):  
Teodor M. Atanacković ◽  
Marko Janev ◽  
Sanja Konjik ◽  
Stevan Pilipović ◽  
Dušan Zorica
Keyword(s):  
Fractional Derivatives ◽  
Variational Problems ◽  
Expansion Formula

scholarly journals Variational problems with fractional derivatives: Invariance conditions and Nöther’s theorem

2009 ◽  
Vol 71 (5-6) ◽  
pp. 1504-1517 ◽  
Author(s):  
Teodor M. Atanacković ◽  
Sanja Konjik ◽  
Stevan Pilipović ◽  
Srboljub Simić
Keyword(s):  
Fractional Derivatives ◽  
Variational Problems

Generalized solutions of an equation with fractional derivatives

2009 ◽  
Vol 20 (2) ◽  
pp. 215-229
Author(s):  
B. STANKOVIC ◽  
T. M. ATANACKOVIC
Keyword(s):  
Mathematical Model ◽  
Fractional Derivatives ◽  
Variational Problems ◽  
Generalized Solutions ◽  
Singular Solutions ◽  
Space Of Distributions ◽  
Left And Right

We consider an equation with left and right fractional derivatives which appears as a mathematical model in the mechanics. The type of equations that we analyse appear, as a rule, in variational problems containing fractional derivatives. We look for solutions in a suitably defined sub-space of distributions which is sufficient to enclose different ‘singular’ solutions.

Sign in / Sign up

Export Citation Format

Share Document