- Research Article
- Open Access
Fractional-Order Variational Calculus with Generalized Boundary Conditions
© Mohamed A. E. Herzallah and Dumitru Baleanu. 2011
- Received: 18 September 2010
- Accepted: 8 November 2010
- Published: 28 November 2010
This paper presents the necessary and sufficient optimality conditions for fractional variational problems involving the right and the left fractional integrals and fractional derivatives defined in the sense of Riemman-Liouville with a Lagrangian depending on the free end-points. To illustrate our approach, two examples are discussed in detail.
- Fractional Derivative
- Fractional Calculus
- Fractional Integral
- Transversality Condition
- Natural Boundary Condition
Fractional calculus is one of the generalizations of the classical calculus. Several fields of application of fractional differentiation and fractional integration are already well established, some others have just started. Many applications of fractional calculus can be found in turbulence and fluid dynamics, stochastic dynamical system, plasma physics and controlled thermonuclear fusion, nonlinear control theory, image processing, nonlinear biological systems, astrophysics, and so forth (see [1–11] and the references therein).
Real integer variational calculus plays a significant role in many areas of science, engineering, and applied mathematics. In recent years, there has been a growing interest in the area of fractional variational calculus and its applications which include classical and quantum mechanics, field theory, and optimal control (see [10, 12–20]).
In the papers cited above, the problems have been formulated mostly in terms of two types of fractional derivative, namely, Riemann-Liouville (RL) and Caputo derivatives.
The necessary optimality conditions for problems of the fractional calculus of variations with a Lagrangian that may also depend on the unspecified end-points , is proven in .
In  the two authors discussed the fractional variational problems with fractional integral and fractional derivative in the sense of Riemann-Liouville and the Caputo derivatives and give the fractional Euler-Lagrange equations with the natural boundary conditions.
Here we develop the theory of fractional variational calculus further by proving the necessary optimality conditions for more general problems of the fractional calculus of variations with a fractional integral and a Lagrangian that may also depend on the unspecified end-points or . The novelty is the dependence of the integrand on the free end-points , with replacing the ordinary integral by fractional integral in the functional.
The paper is organized as follows.
In Section 2, we present the principal definitions used in this paper. In Section 3, the necessary optimality conditions are proved for problems (1.1) and (1.2) by giving some special cases which prove the generalization of our problems. Sufficient conditions are shown in Section 4, and two examples are depicted in Section 5.
where is such that and
where is such that and .
3.1. Necessary Optimality Conditions for Problem (1.1)
These conditions are only necessary for an extremum. The question of sufficient conditions for the existence of an extremum is considered in the next section.
we get similar results as in .
3.2. Necessary Optimality Conditions for Problem (1.2)
In this section, we prove the sufficient conditions that ensure the existence of a minimum (maximum). Some conditions of convexity (concavity) are in order.
for all .
Let be jointly convex (concave) in . If satisfies conditions (3.5) (3.7), then is a global minimizer (maximizer) to problem (1.1).
Since satisfies conditions (3.5)–(3.7), thus we obtain which completes the proof.
Similar to proving the previous theorem, we can prove the following theorem.
Let be jointly convex (concave) in (y,z,u ). If satisfies conditions (3.16)–(3.18), then is a global minimizer (maximizer) to problem (1.2).
We will provide in this section two examples in order to illustrate our main results.
Note that it is difficult to solve the above fractional equations; for , a numerical method should be used, and where is a jointly convex then the obtained solution is a global minimizer to problem (5.1).
Using a numerical method, we get the solution which is a global minimizer to problem (5.3) where is a jointly convex.
The first author would like to thank Majmaah University in Saudi Arabia for financial support and for providing the necessary facilities.
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