Fractional variational calculus in terms of Riesz fractional derivatives
Fractional variational calculus in terms of Riesz fractional derivatives
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DOI：
10.1088/1751-8113/40/24/003
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发表时间：
2007-06
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影响因子：
--
通讯作者：
O. Agrawal
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文献类型：
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作者：
O. Agrawal
本文对传统的变分法在含Riesz分数阶导数系统中的应用进行了推广。具体来说，我们提出了广义欧拉-拉格朗日方程和横截性条件的分数阶变分问题（FVP）定义的RFDs。我们考虑两个问题，一个简单的有限边值问题和一个拉格朗日有限边值问题。第一个问题的结果被扩展到包含多个分数阶导数，函数和参数的问题，以及未指定的边界条件。对于第二个问题，我们提出了拉格朗日型乘数规则。对于这两个问题，我们开发的Euler-Lagrange型的必要条件，必须满足给定的功能是极值。问题被认为是证明配方的应用。具体介绍了分数阶动量、分数阶哈密顿量、分数阶汉密尔顿运动方程、分数阶场论和分数阶最优控制。给出的公式和所得方程类似于Agrawal（2002 J. Math. Anal. Appl.272368，2006 J.Phys.A：Math.Gen.3910375）和出现在经典变分法领域中的那些。这些公式是简单的，可以推广到其他问题的分数阶变分法领域。
This paper presents extensions of traditional calculus of variations for systems containing Riesz fractional derivatives (RFDs). Specifically, we present generalized Euler–Lagrange equations and the transversality conditions for fractional variational problems (FVPs) defined in terms of RFDs. We consider two problems, a simple FVP and an FVP of Lagrange. Results of the first problem are extended to problems containing multiple fractional derivatives, functions and parameters, and to unspecified boundary conditions. For the second problem, we present Lagrange-type multiplier rules. For both problems, we develop the Euler–Lagrange-type necessary conditions which must be satisfied for the given functional to be extremum. Problems are considered to demonstrate applications of the formulations. Explicitly, we introduce fractional momenta, fractional Hamiltonian, fractional Hamilton equations of motion, fractional field theory and fractional optimal control. The formulations presented and the resulting equations are similar to the formulations for FVPs given in Agrawal (2002 J. Math. Anal. Appl. 272 368, 2006 J. Phys. A: Math. Gen. 39 10375) and to those that appear in the field of classical calculus of variations. These formulations are simple and can be extended to other problems in the field of fractional calculus of variations.