Xu and agrawal deduced the eulerlagrange equation of the fractional variational problem involving a modified hilfer fractional derivative. Almeida investigated optimality conditions for fractional variational problems with a lagrangian depending on the rieszcaputo derivative 5 and the caputo. Computing eigenelements of boundary value problems with. Baleanu, integration by parts and its application of a new nonlocal fractional derivative with mittagleffler kernel, j. The main result of this section is stated in the following theorem. Variational methods for nonlocal fractional problems. Vibrations and diffusion processes and fractional calculus with applications in mechanics.
Variational calculus with fractional and classical derivatives. Eulerlagrange equations t m atanackovi, s konjik and s pilipovi fractional variational principles in action dumitru baleanugeneralized hamiltons principle with fractional derivatives t m atanackovi, s konjik, lj oparnica et al. New variational problems with an action depending on. Generalized eulerlagrange equations for fractional. Jan 15, 2011 we generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized. If at least one fractional term exists in the formulation, it is called a fractional variational problem. The solutions of our models equations are calculated in the form of convergent series with easily. This is a generalization of the classical theory, where derivatives and integrals can only appear in integer orders. Research article a note on fractional order derivatives and. Pdf higher order fractional variational optimal control.
Finally, part iii contains a selection of recent results on critical fractional equations. Necessary and sufficient conditions for the fractional calculus of variations with. In 9 we obtained necessary conditions that every local. Fractional problems have attracted many scholars attention as the actual problems can be much better simulated by using the fractional derivatives than by using traditional integral derivatives 16. Variational principles in the frame of certain generalized. A close relationship between these problems and multiobjective fractional symmetric dual problems is also. In the case when fractional derivatives model memory effects the lower bound a in the definition of derivative should not b. Adiabatic invariants for disturbed fractional hamiltonian. Agrawal o p 2006 fractional variational calculus and the transversality conditions j. Calculus of variations, rieszcaputo fractional derivative, isoperimetric problem. The fractional variational problems under our consideration are formulated in terms of c d, the eulerlagrange equations for the basic and isoperimetric problems, as well as transversality conditions, are proved. Conference paper fractional variational problems depending on. The purpose of this paper is to solve fractional calculus of variational herglotz problem depending on an atanganabaleanu fractional derivative.
Sufficient and necessary conditions of the first and second. Using a relation between the hadamard fractional operator and a sum involving integerorder derivatives, we rewrite the fractional problem into a classical optimal control problem. Several linear fractional differential equations are analytically solved as examples and the methodology is demonstrated. Chen 22 proposed a symmetric dual problem for a class of multiobjective fractional variational problems. Fractional derivatives 2 can be regarded as a generalization of usual fractional derivatives. We obtain necessary optimality conditions for variational problems with a lagrangian depending on a caputo fractional derivative, a fractional and an inde. Similar kind of fractional derivatives were used earlier in 23,24. The papers of mond and hanson 9, 10, bector 1, mond, chandra and husain 12, mond and husain 11, smart and mond. Part ii is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. The left and right riemannliouville fractional derivatives of order. Pdf variational problems with fractional derivatives. Symmetry free fulltext new variational problems with. This chapter presents some results for variational problems in which the lagrangian density involves derivatives of real fractional order, as well as a generalization of the classical hamilton prin. Then in section 3, we explicitly construct the strong solution representation via fractional order integral operators.
Variational problems with fractional derivatives fractional. Then the method is successfully extended to fractional differential equations. In this article, we study generalized fractional derivatives that contain kernels depending on a function on the space of absolute continuous functions. Computational methods in the fractional calculus of variations. Therefore, it is expected that the fractional derivative operation involves some sort of boundary. These fractional derivatives lead to nonsymmetric boundary value problems, which are investigated from a variational point of view. The concept of symmetric duality for multiobjective fractional problems has been extended to the class of multiobjective variational problems. The proposed method is directly extended to study the fractional order roseauhyman and fractional order inhomogeneous partial differential equations without any transformation to convert the given problem into integer order. Variational iteration method for solving system of fractional. In 16 the fractional variational problems with fractional integral and frac tional derivative in the sense of riemannliouville and the caputo derivatives were. Fractional herglotz variational principles with generalized caputo.
Also, for the standard case when these two bounds coincide, we derive a new form of eulerlagrange equations. Fractional variational herglotz problems of variable order are considered and two cases. Farhadinia 9, fard 10 and soolaki 11 established the necessary optimality conditions for fuzzy fractional variational problems by using the generalized hukuhara differentiability concept. Fractional herglotz variational problems with atangana. The readers interested in the discrete fractional calculus of variations are. We generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with. The variational problem for the riemannliouville case is coercive on the space h 2 0 0. Fractional derivatives have recently played a significant role in many areas of sciences, engineering, fluid mechanics, biology, physics and economies. Optimality conditions for fractional variational problems. Optimality conditions for fractional variational problems with caputo. Fractional eulerlagrange differential equations via caputo. In that paper the author formulated the problem in several cases such as for.
Agrawal 2, 3, 4 studied the fractional eulerlagrange equations for general fractional variational problems fvp involving riemannliouville, caputo and riesz fractional derivatives. Apr 02, 2021 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. In this paper, the necessary and sufficient conditions of optimality for variational problems with caputo partial fractional derivative are established. The a th derivative of a function f x at a point x is a local property only when a is an integer. We generalize the laplace transform in order to be applicable for the generalized fractional integrals and derivatives and apply this transform to solve some ordinary differential equations in the frame of the fractional derivatives under. The continuous variational formulations aredeveloped insection4, andtheir wellposedness is also established. Variational approach to solutions for a class of fractional. Conference paper fractional variational problems depending. Part i begins the book with some basic facts about fractional sobolev spaces. New iterative approach for the solutions of fractional. Existence of minimizers for fractional variational. Symmetry free fulltext new variational problems with an. Wave propagation, impact and variational principles contain various applications of fractional calculus to the fields of classical mechanics. In this paper, we study the necessary and sufficient optimality conditions for problems of the fractional calculus of variations with a lagrange.
R of class cn and z 2r thatminimize or maximizethe functional. Aug 01, 2011 riewe, was the first to propose eulerlagrange equations for the variational problems with fractional derivatives. Necessary conditions for normal efficient solutions to a class of multiobjective fractional variational problems mfp with nonlinear equality and inequality constraints are established using a parametric approach to relate efficient solutions of a fractional problem and a non fractional problem. Duality, fractional variational problem,quasiinvexity. The left conformable fractional derivative of order 0 fractional integrodifferential equations fides arise in the mathematical modelling of physical phenomena and play an important role in various branches of science and engineering.
Since the introduction of fractional calculus of variations by riewe 1, fractional. To solve the above problems, the variableorder vo fractional di usion equation models have been suggested for use. Variational problems involving a caputotype fractional. A numerical method for solving fractional variational. Two different types of fractional derivatives are considered. For advantages of applying them to fractional variational problems see 8,12. Fractional variational problems depending on fractional. Sep 01, 2009 now we will consider the fractional variational problem in the case a and in the case b.
Fractional variational problems with the rieszcaputo derivative. Variational iteration method for solving system of. The fractional derivatives are described in the caputo sense, because it allows traditional initial and boundary conditions to be included in the formulation of the problem. This paper is concerned with the solution of fides by the variational iteration. Eulerlagrange equation for fractional variational problems with multiple integrals were. For the origin of the calculus of variations with fractional operators we should look back to 19961997, when riewe used noninteger order derivatives to better describe nonconservative systems in mechanics 22, 23. Main results give fractional eulerlagrange type equations and natural boundary conditions, which provide a generalization of previous results found in the literature.
Eulerlagrange equations for discrete variational problems 1. The aim of this paper is to study certain problems of calculus of variations that are dependent upon a lagrange function on a caputotype fractional derivative. Existence of minimizers for fractional variational problems. Agrawal also presented eulerlagrange equations for some kinds of variational problems with fractional derivatives in the riemannliouville sense. Introduction for the first results on the necessity of the optimal solutions of the variational problems we cite the valentines paper17. Variational methods for a fractional dirichlet problem involving jumarie s derivative rafa bkamocki faculty of mathematics and computer science, university of lodz, banacha, lodz, poland correspondence should be addressed to rafa kamocki. Pdf we generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not. Apr 02, 2021 with this, we obtain what is known as a fractional variational problem with arbitrary higher order fractional derivatives.
Examples include riemannliouville, hadamard, grunwaldletnikov, ca. These examples suggest that the eulerlagrange equation for the fractional variational problem 4 with a a and with the boundary condition specified at t a, should be reformulated as followsl. Research article variational methods for a fractional. The variational fractional order is a continuous function of two variables, a. Since 17th century, when fractional calculus was born, several kinds of fractional derivatives have been proposed. Fractional variational iteration method for fractional. Symmetric duality for multiobjective fractional variational. The fractional derivatives considered in this paper are the riemannliouville and the caputo derivatives with respect to an arbitrary kernel.
Variational iteration method for fractional calculus a. We generalize the fractional variational problem by allowing the possibility that the lower bound in the fractional derivative does not coincide with the lower bound of the integral that is minimized. Nov 18, 2020 the fractional derivatives are considered in the caputo sense whose order belongs to the closed interval 0, 1. Variational problems with partial fractional derivative. Fractional calculus with applications in mechanics wiley. Fractional calculus series on complexity, nonlinearity. Fractional variational calculus deals with problems of minimizing functionals that involves some fractional derivatives andor fractional integrals. A discretization method to solve fractional variational.
The books fractional calculus with applications in mechanics. Fractional variational calculus deals with problems of. Research article a note on fractional order derivatives. The study of fractional variational problems also has a long history. Throughout the book, we call the problems in the calculus of variations and optimal control variational problems. Let l u be a fractional variational problem which is being solved in the case a or b. Feb 19, 2008 variational problems with fractional derivatives. Jan 21, 20 a novel modification of the variational iteration method vim is proposed by means of the laplace transform. We provide a fast and simple method to solve fractional variational problems with dependence on hadamard fractional derivatives.
Published 19 february 2008 2008 iop publishing ltd journal of physics a. Introduction to the fractional calculus of variations. Baleanu, fractional differences and integration by parts, j. With hes variational iteration method, it is possible to obtain exact or better approximate solutions of differential equations. We use approximations for fractional derivatives in the lagrangian and obtain the. In this paper, the necessary and sucient conditions of optimality for variational problems with caputo partial fractional derivativeareestablished.
Weak, strong and converse duality theorems are proved under generalized invexity assumptions. Since the new atanganabaleanu fractional derivative is nonsingular and nonlocal, the eulerlagrange equations are proposed for the problems of herglotz. New iterative approach for the solutions of fractional order. Since then, numerous works on the fractional variational calculus have been written. Variational order fractional derivatives via fractional di erence.
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