Generalized Multiparameters Fractional Variational Calculus

This paper builds upon our recent paper on generalized fractional variational calculus (FVC). Here, we briefly review some of the fractional derivatives (FDs) that we considered in the past to develop FVC. We first introduce new one parameter generalized fractional derivatives (GFDs) which depend on two functions, and show that many of the one-parameter FDs considered in the past are special cases of the proposed GFDs. We develop several parts of FVC in terms of one parameter GFDs. We point out how many other parts could be developed using the properties of the one-parameter GFDs. Subsequently, we introduce two new two- and three-parameter GFDs. We introduce some of their properties, and discuss how they can be used to develop FVC. In addition, we indicate how these formulations could be used in various fields, and how the generalizations presented here can be further extended. 1. Introduction For over a century, many researchers have been in search for a fundamental law that can be used to describe the behavior of the nature. One law that comes very close to it is the universal law of extremum which states that the nature always behaves in a way such that some quantity is an extremum. A catenary takes a shape so that the total potential energy is minimum, light travels from a point to another so that the travel time is minimum, a particle in a flow takes a path of least resistance, and even in social settings, we behave so that our conflict with others within our conviction is minimum. Related laws, principles, and theories have been developed in almost every field of science, engineering, mathematics, biology, economics and social science. For example, applications of such laws, principles, and theories in continuum mechanics, classical and quantum mechanics, relativistic quantum mechanics, and electromagnetics could be found in [1–5] and many other textbooks, monographs, and papers. Opponent of the universal principle may argue that nature behaves in its own way, and the extremum principles are our creations where we design a functional that is extremum for the nature’s trajectory. Whatever may be the reality, the underlying theories have advanced our understanding of the nature tremendously. The field that deals with the mathematical theories of the extremum principles is known as the variational calculus. Excellent books have been written in this field, see for example [6, 7]. These books provide not only the foundations for theoretical work in the field, but they have also been a basis for many numerical techniques (see, [8, 9]). However, the