Advanced Methods in the Fractional Calculus of Variations (SpringerBriefs in Applied Sciences and Technology)

This brief presents a general unifying perspective on the fractional calculus. It brings together results of several recent approaches in generalizing the least action principle and the Euler–Lagrange equations to include fractional derivatives.

The dependence of Lagrangians on generalized fractional operators as well as on classical derivatives is considered along with still more general problems in which integer-order integrals are replaced by fractional integrals. General theorems are obtained for several types of variational problems for which recent results developed in the literature can be obtained as special cases. In particular, the authors offer necessary optimality conditions of Euler–Lagrange type for the fundamental and isoperimetric problems, transversality conditions, and Noether symmetry theorems. The existence of solutions is demonstrated under Tonelli type conditions. The results are used to prove the existence of eigenvalues and corresponding orthogonal eigenfunctions of fractional Sturm–Liouville problems.

Advanced Methods in the Fractional Calculus of Variations is a self-contained text which will be useful for graduate students wishing to learn about fractional-order systems. The detailed explanations will interest researchers with backgrounds in applied mathematics, control and optimization as well as in certain areas of physics and engineering.

本文提出了一個關於分數微積分的整體統一觀點。它將幾種最近的方法結果結合起來，將最小作用原理和歐拉-拉格朗日方程推廣到包括分數導數。

本文考慮了拉格朗日量對廣義分數算子和傳統導數的依賴性，以及在整數階積分被分數積分取代的更一般問題。對於幾種變分問題，我們得到了一般定理，其中最近在文獻中發展的結果可以作為特殊情況得到。特別是，作者提供了歐拉-拉格朗日類型的必要最佳性條件，用於基本問題和等周問題、橫截條件和諾特對稱定理。在Tonelli類型條件下證明了解的存在性。利用這些結果證明了分數Sturm-Liouville問題的特徵值和相應的正交特徵函數的存在性。

《分數變分計算的高級方法》是一本獨立的教材，對於希望了解分數階系統的研究生非常有用。詳細的解釋將吸引應用數學、控制和優化以及某些物理和工程領域的研究人員。