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.

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 [

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 [

The subject of fractional variational calculus was initiated by Riewe [

Klimek [

Recently, the field of fractional variational calculus has indeed grown very rapidly. A citation search of [

In this paper, we provide further generalization of fractional variational calculus. Specifically we introduce Hadamard type, Erdélyi-Kober type fractional integral and fractional derivatives, and fractional integrals and derivatives of a function (say

At this point, we would like to emphasize that a comprehensive treatment and an excellent review of many generalized fractional operators proposed in the field could be found in [

In this paper, we will introduce several general multiparameter fractional integrals and derivatives, and show that many specific integrals and derivatives can be obtained from these general derivatives. For ease in the discussion to follow and to make this paper self-contained, we first introduce several symbols and notations, and provide some preliminaries. A large part of these symbols and definitions could be found in [

In this paper, we consider several fractional integrals and derivatives. We begin with the Riemann-Liouville and the Caputo fractional integrals and derivatives.

Many fractional derivatives are defined using the left/forward and the right/backward Riemann-Liouville fractional integrals (RLFIs). These integrals are defined as follows.

We now consider the

Using (

The left/forward and the right/backward Caputo fractional derivatives (CFDs) of order

Operators

The integral operators

The left/forward and the right/backward Hadamard fractional integrals (HFIs) of order

The left/forward and the right/backward Hadamard fractional derivatives (HFDs) of order

The HFDOs

We can develop integration by parts formula for Hadamard operators also. It can be demonstrated that the HFIOs

Similar to (

As stated earlier, one can take

Equations (

The left/forward and right/backward Erdélyi-Kober fractional integrals (EKFIs) are defined as [

As pointed out earlier, by setting

These integrals satisfy the following semigroup properties:

The Erdélyi-Kober fractional derivatives (EKFDs) of order

Equations (

It was pointed out above that the Erdélyi-Kober operators defined by (

These integrals satisfy the following semigroup properties

We define the modified EKFDs of order

For sufficiently good function

Equation (

In this section we define the left/forward and the right/backward fractional integrals and fractional derivatives of a function

We now define the left/forward and the right/backward weighted/scaled fractional integrals of a function with respect to another function as follows.

Like the GFIs, we define the left/forward and the right/backward weighted/scaled fractional derivatives of a function

Before we proceed further, we would like to note that the fractional integrals and derivatives of a function with respect to another function defined in [

We now list several properties of operators

References [

The generalized fractional integrals and derivatives defined in (

In this section, we present several fractional variational formulations in terms of one parameter generalized fractional derivatives. We shall consider formulations in terms of one variable and one fractional derivative term, specified and unspecified terminal conditions, one variable and multiple fractional derivative terms with different order of derivatives, multiple variables and multiple derivative terms but the same order of derivatives, geometric constraints, and parametric constrains to name a few. The approach presented here will also be applicable to multivariables and multiple fractional derivative terms with different order of derivatives, free end points, free end-point constraints, multidimensions, and many other formulations can also be considered. As a matter of fact almost all variational formulations can be recast in terms of generalized fractional derivatives, and these will be considered in the future. The derivations of almost all formulations follow the same pattern, and for this reason, a fractional variational formulation would be given in detail for a simple fractional variational problem only. For other fractional variational problems, the final Euler-Lagrange equation will be given but the details would be omitted.

In this subsection, we develop an Euler-Lagrange formulation for a simple fractional variational formulation. The functional considered in this case may contain the left and the right integrals and derivatives, and the derivatives could be of Type-1, Type-2, or both. In the simplest fractional variational problem considered here, we take only one fractional derivative term, namely the term

At this stage, two points can be made which are the same as those made in [

To derive the necessary conditions, we define

Let us assume that

As a quick modification of the above problem, assume that the functional

It should be pointed out that

In the above functional, we considered only the left fractional derivative. If the functional contains the right fractional derivative also, then the functional is written as

In the formulation above, we have considered Type-2 fractional derivatives in the functional. Using (

We now consider several variations of the above formulation. As a first variation, assume that the functional contains

Assume now that the functional is of the following type:

We now consider the following functional containing multiple functions,

Following the above approach, it can be shown that for this functional the Euler-Lagrange equation is

We now consider fractional variational formulation for constrained systems. First, we consider an isoperimetric problem defined as follows: find the curve

Next, we consider a system subjected to holonomic constraints. For simplicity, we consider the functional defined by (

The constraints can also be given in the form of fractional differential equations which govern the dynamics of the system. Such problems arise in optimal controls. For simplicity, we consider one state variable

An approach to the above problem is to redefine the functional as

The fractional variational principle discussed above allows us to develop fractional Hamilton principles. However, depending on the form and the boundary conditions considered, the resulting fractional differential equations would be different, and accordingly the Hamilton equation would also be different. Here we discuss one such principle. Thus, one form of the fractional Hamilton principle can be stated as follows: the fractional Hamilton’s principle states that the path traced by a system of particles which are described by

To demonstrate an application of the formulations presented above, consider the following problem:

For this case, function

We now consider 4 different cases of this example.

As a first case, take

As a second case, take

Next, consider

We now consider

At this stage we would like to emphasize the following two points. First, we have considered only a few type of

The fractional variational formulations developed here can be extended in many directions. In this regard, we emphasize that the additional remarks made in [

First, we have considered one dimensional domain only. The formulation above could be easily extended to multi dimensional domains. For example, by replacing

Second, the formulations developed here could be extended to symmetric and antisymmetric fractional derivatives. Depending on whether Type-1 or Type-2 fractional derivatives are selected, two types of symmetric and two types of antisymmetric fractional derivatives can be defined. We call them Type-3 and Type-4 fractional derivatives, and they are defined as follows.

The above approach could also be applied to functionals defined in terms of sequential generalized fractional derivatives. To demonstrate this, consider the following functional:

Third, we define the Hilfer type two parameter generalized fractional derivatives as follows.

One of the most important keys to developing a fractional variational formulation or for that matter any variational formulation is to develop an integration by parts formula for the associated differential operators. Once that is done, many of the standard techniques from variational calculus can be used to develop the desired formulations. The fractional integration by parts formula for the generalized two parameter fractional derivatives can be developed using (

Fourth, we now define a generalized three-parameter fractional derivative as follows

In [

Fifth, the fractional integrals and the derivatives are defined here using kernels of type

It can also be demonstrated that operators

It can be demonstrated that operators

Sixth, several different kernel functions have been considered to develop generalized fractional operators and generalized fractional calculus (see [

As a first example, consider the integral equation

As a second example, consider the integral equation

The above observation will have several consequences: (a) it will allow us to write many integral equations in terms of generalized fractional integral and differential operators, and use the properties of these operators to find the solution of the integral equations using the properties of the generalized fractional operators in elegant way. (b) It will initiate a new class of generalized differential equations, and blur the distinction between differential and integral equations. (c) It will allow us to write many of the equations, physical, and social laws, and so forth, in the field of science, engineering, economics, and bioengineering in terms generalized fractional operators, and thus broaden the area where fractional calculus could be applied. (d) It will also impact the history of fractional calculus. In the history of fractional calculus, Abel is attributed for solving a practical fractional calculus problem. Recently, many integral equations have been recast in terms of fractional integrals and derivative operators. The new operators and their properties proposed above will allow one to write equations and physical and social laws in terms of generalized fractional operators. Thus, while the researchers were proposing these equations and physical and social laws, they were indeed, indirectly, proposing applications of generalized fractional calculus; and when they solved the associated integral equations, they were indeed developing analytical tools for fractional calculus.

Seventh, we have largely focused here on the discrete systems. However, it could be easily extended to field/distributed-order-dynamic-systems (see [

Finally, note that we developed a generalized variational calculus in [

In this paper, we first introduced some one-parameter FDs, and listed some of their properties useful in developing FVC. We then introduced new one-parameter GFDs, developed their properties, and used them to develop several parts of FVC. These parts include fractional variational formulations for functionals containing one function and multi functions, specified and unspecified terminal conditions, multiorder of FDs, holonomic, parametric, and dynamic constraints. These parts also include formulations for fractional Lagrangian and Hamiltonians and fractional optimal controls. Subsequently, we introduced two- and three-parameters GFDs and developed some Euler-Lagrange type necessary conditions, and pointed out how other multiparameter fractional derivatives could be developed. We also discussed many areas where the formulations developed here could be applied.