Advanced Methods In The Fractional Calculus Of Variations Springerbriefs In

Are you ready to take your knowledge of calculus to the next level? In this article, we will explore advanced methods in the fractional calculus of variations. By delving into the fascinating world of fractional derivatives and integrals, we can uncover powerful techniques that allow us to solve complex problems within various fields of science and engineering.

The Basics of Fractional Calculus

Before diving into advanced methods, let's quickly recap the basics of fractional calculus. Traditional calculus deals with integer-order derivatives and integrals, which are suitable for modeling many physical phenomena. However, fractional calculus extends these concepts to non-integer orders, introducing a new level of flexibility and precision in mathematical modeling.

By using fractional derivatives and integrals, we can describe systems with memory, anomalous diffusion, and other non-local effects. This makes fractional calculus a powerful tool in areas such as control theory, physics, finance, and biology.

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

by Robert N. Cahn(2015th Edition, Kindle Edition)

4 out of 5

Language	:	English
File size	:	5389 KB
Screen Reader	:	Supported
Print length	:	147 pages
Paperback	:	328 pages
Item Weight	:	1.47 pounds
Dimensions	:	6.22 x 0.98 x 9.2 inches
Hardcover	:	288 pages

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Fractional Variational Calculus

Variational calculus aims to find the function that minimizes or maximizes a given functional. In the fractional variant of variational calculus, we seek to optimize functionals containing fractional derivatives.

The fractional calculus of variations has its roots in the Euler-Lagrange equation, which provides necessary conditions for a functional to be extremal. By generalizing this equation to include fractional derivatives, we can develop advanced techniques for solving optimization problems involving non-local effects.

Applications

The applications of advanced methods in the fractional calculus of variations are vast and varied. Here are just a few examples:

1. Physics

Fractional variational principles have been successfully applied to describe the dynamics of anomalous diffusion, viscoelastic materials, and fractal media. These methods allow for a more accurate representation of real-world systems, leading to improved predictions and understanding.

2. Control Theory

Fractional optimal control problems arise in various engineering fields, such as aerospace, robotics, and electrical engineering. By incorporating fractional derivatives in the cost function, we can account for memory effects and achieve better performance in controlling complex systems.

3. Finance

The fractional calculus of variations has found applications in finance for modeling and optimizing investment strategies. By considering fractional dynamics, we can capture long-term correlations and adapt our models to better fit real-world financial data.

Advanced Techniques

Now that we understand the potential applications, let's explore some advanced methods used in the fractional calculus of variations:

1. Fractional Euler-Lagrange Equations

Generalizing the classical Euler-Lagrange equation to fractional derivatives allows us to find extremal functions for fractional functionals. Various methods, such as Caputo and Grünwald-Letnikov, are used to solve these equations and obtain optimal solutions.

2. Fractional Variational Principles

By formulating variational principles involving fractional derivatives, we can develop new mathematical frameworks for solving optimization problems. These principles provide a theoretical foundation for a wide range of applications and allow for the development of numerical algorithms.

3. Fractional Optimal Control

Optimal control problems involving fractional derivatives require specialized techniques to solve. These methods often include fractional dynamic programming, fractional Pontryagin's minimum principle, and fractional maximum principle. By utilizing these advanced tools, engineers can design control strategies that outperform traditional approaches.

The fractional calculus of variations opens up new possibilities for solving complex problems in various scientific and engineering disciplines. By incorporating fractional derivatives and integrals, advanced methods allow us to describe non-local effects and capture intricate system dynamics.

Whether you're a researcher, student, or simply an enthusiast of mathematics, exploring the advanced methods in the fractional calculus of variations will undoubtedly broaden your understanding of this fascinating field. So, why not embark on this exciting journey and discover the power of fractional calculus in solving real-world challenges?

https://www.springer.com/gp/book/9783319176640

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

by Robert N. Cahn(2015th Edition, Kindle Edition)

4 out of 5

Language	:	English
File size	:	5389 KB
Screen Reader	:	Supported
Print length	:	147 pages
Paperback	:	328 pages
Item Weight	:	1.47 pounds
Dimensions	:	6.22 x 0.98 x 9.2 inches
Hardcover	:	288 pages

FREE

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.