Chapter 5
The Fractional Hyperboletry

5.1 The Fractional R₁-Hyperbolic Functions

In previous chapters, we developed the R-function as a fractional generalization of the exponential function. Because of its eigen-character, the R-function may be used as the basis of a generalized hyperboletry. This chapter presents that development. The classical hyperbolic functions are based on the exponential function (equations (1.20) and (1.21)). The R-function is now applied to create fractional generalizations of the hyperbolic functions. With the exception of Section 5.7 and Figures 5.4–5.11 and the related discussions, this chapter is adapted from Lorenzo and Hartley [76], with the permission of the ASME:

Figure 5.4 Effect of the a parameter on versus t-Time, for (a) q = 0.25, (b) q = 0.50, (c) q = 0.75, a = 0.2–2.0 in steps of 0.2, v = 0, and k = 0.

Figure 5.5 Effect of the a parameter on versus t-Time, for (a) q = 0.25, (b) q = 0.50, (c) q = 0.75, a = 0.2–2.0 in steps of 0.2, v = 0, and k = 0.

Figure 5.6 Effect of the a parameter on versus t-Time, for (a) q = 0.25, (b) q = 0.50, (c) q = 0.75, (d) q = 2.00, a = 0.2–2.0 in steps of 0.2, v = 0, and k = 0.

Figure 5.7 Effect of v on (a) , (b) , (c) versus t-Time, for v = q = 0.2–2.0 in steps of 0.2, a = 1, and k = 0.

Figure 5.8 Effect of v on versus t-Time, for v = q − 1, q = 1.0–2.0 in steps of 0.2, a = 1, and k = 0.

Figure 5.9 Effect of v on versus t-Time, for v = q − 1, q = 1.0–2.0 in steps of 0.2, a = 1, and k = 0.

Figure 5.10 Effect of v on versus t-Time, for v = q − 1, q = 1.0–2.0 in steps of 0.2, a = 1, and k = 0.

Figure 5.11 and versus t-Time, for k = 0 to 6.

The basis for the fractional hyperboletry is , with both a and t real. Therefore, expanding the R-function into a series of even and odd powers of a gives

5.1

5.2

By reference to the exponential and the traditional hyperbolic functions, the even and odd terms are defined as the -hyperbolic functions. Thus, the even part of , that is, even powers of a, will become the -function and the odd part of , that is, odd powers of a, will become the -function.

In these series, t is raised to a fractional power and is thus multivalued. We include this multivalued character of the fractional hyperbolic functions by considering the roots of t based on equation (3.122):

5.3

where the exponent of t is

5.4

and where and are assumed rational and irreducible, , and M/D is rational and in minimal form (i.e., common factors removed). Continuing from Ref. [76] with the permission of ASME:

Then, the multivalued, or indexed, definition for the is given by

5.5

where . The and are defined similarly:

5.6

with , and

5.7

also with . It is observed that the definitions given in equations (5.5)–(5.7) are indexed complex functions.

The principal -hyperbolic functions are obtained by taking k = 0, in equations (5.5)–(5.7), thus

5.8

and

5.9

and

5.10

where we have introduced a simplified notation when k = 0, for the principal functions. Also, the subscript “1” is only used to relate to the R₁ trigonometry in Chapter 6. In contrast to equations (5.5)–(5.7), these definitions (equations (5.8)–(5.10) are not indexed and are real functions. The principal functions are used as the basis of the identities and analyses of the following sections.

To demonstrate backward compatibility to the classical hyperbolic functions, we substitute , into the defining series (equations (5.5)–(5.7)) giving

5.11

5.12

and

5.13

Figures 5.1–5.3 show the principal (k = 0) -hyperbolic functions for various values of q. We observe the traditional and when , and k = 0. We also notice that . This is readily verified by appropriate substitution into the defining series (equation (5.6).

Figure 5.1 versus t-Time, for q = 0.2–2.0 in steps of 0.2, a = 1, v = 0, and k = 0.

Source: Lorenzo and Hartley 2005b [76]. Reproduced with permission of ASME.

Figure 5.2 versus t-Time, for q = 0.2–2.0 in steps of 0.2, a = 1, v = 0, and k = 0.

Source: Lorenzo and Hartley 2005b [76]. Reproduced with permission of ASME.

Figure 5.3 versus t-Time, for q = 0.2–3.0 in steps of 0.2, a = 1, v = 0, and k = 0.

Source: Lorenzo and Hartley 2005b [76]. Reproduced with permission of ASME.

The effect of the parameter a on the , , and , for values of q = 0.25, 0.50, and 0.75, is shown in Figures 5.4–5.6, respectively. It is seen by comparison with the a = 1 cases shown in Figures 5.1–5.3 that the parameter does not act as a time scaling parameter. The effect of the parameter v on the -fractional hyperbolic functions is presented in Figure 5.7. This is later shown to be the same as a fractional differentiation of order v.

In later chapters, it will be seen that for q > 1, special behavior is observed when v = q-1. Figures 5.8–5.10 examine the effect on the fractional hyperbolic functions under these conditions. The most notable changes here occur with overlapping of the -functions with t < 2.

The behavior of the multivalued functions is shown in Figure 5.11 for with various k values. The order q = 6/7 was chosen for the strongly symmetric behavior observed in the imaginary part of the -function. It is important to observe here that there is only one real -function, that is, k = 0. This behavior does not exist for all q. For example, with q = 7/8, not shown, there are two real functions, they are k = 0 and k = 4. The remainder of this section develops some fundamental relationships between the fractional hyperbolic functions and the R-functions. Continuing from Ref. [76] with the permission of the ASME:

It is directly observed that the -function is an even function (with respect to a), that is, for t > 0

5.14

Also, the -function is odd, that is,

5.15

From the definition of the fractional hyperbolic functions based on equation (5.2) with (5.5) and (5.6), it is clear that

5.16

Furthermore, by a similar expansion of based on (5.2), we have

5.17

Adding and subtracting equations (5.16) and (5.17) gives

5.18

and

5.19

These equations ((5.18) and (5.19) exactly parallel equations (1.20) and (1.21) and may equally well be taken as the definitions for the principal fractional -hyperbolic functions.

Following from these relations and definition (5.10), we have

5.20

The right-hand sides of equations (5.5) and (5.6) are recognized as -functions, that is,

5.21

5.22

Combining the results of equations (5.18) and (5.21), we have the identity

5.23

Similarly, combining the results of equations (5.19) and (5.22), we have

5.24

which also introduces two new R-function relationships. Taking the ratio of equations (5.23) and (5.24) and combining with the results of equation (5.7) yields

5.25

Now, adding equations (5.23) and (5.24) yields

5.26

while subtraction yields

5.27

Subtracting the square of equation (5.19) from the square of equation (5.18) yields the following:

5.28

or using the center terms of equations (5.23) and (5.24)

5.29

Taking , equation (5.29) is seen that is a generalization of the well-known equation

5.30

Furthermore, adding the square of equation (5.18) to the square of equation (5.19) yields

5.31

Again, using the center terms of equations (5.23) and (5.24) with (5.31) yields

5.32

Taking q = 1, v = 0 shows that this is a generalization of the duplication formula:

5.33

5.2 R₁-Hyperbolic Function Relationship

An interrelationship between the R₁-hyperbolic functions is readily determined. Using the definition (5.6), consider

5.34

Thus, recognizing the form of equation (5.6), we have

5.35

Because of the nature of this relation may also be considered to be a fractional calculus operation when k = 0. Continuing from Ref. [76] with the permission of ASME:

5.3 Fractional Calculus Operations on the R₁-Hyperbolic Functions

The -order differintegral of , is determined as follows:

5.36

Oldham and Spanier [104] p. 67 provide the following differintegration relation:

5.37

which is valid for all q. Then, fractionally differintegrating term-by-term (see Section 3.16) yields

5.38

Thus,

5.39

Applying equation (5.35), this can be written as

5.40

If , we have

5.41

By similar logic,

5.42

and

5.43

If , we have

5.44

5.4 Laplace Transforms of the R₁-Hyperbolic Functions

The Laplace transforms for the principal -hyperbolic functions are easily determined from the Laplace transforms of the related R-functions (equation (3.22)). Then, the transform of equation (5.23) is

5.45

Then, using equation (3.22), we have

5.46

In a similar manner, the transform of equation (5.24) is

5.47

Again, using equation (3.22)

5.48

5.5 Complexity-Based Hyperbolic Functions

In Section 5.1, fractional hyperbolic functions that parallel the familiar classical hyperbolic functions were derived based on the parity-based series (equation (5.2). In this section, we shall consider complexity-based functions starting from equation (5.1). Using equation (5.3) (5.1) is written as

5.49

Now, the real and imaginary parts define the new functions:

5.50

and

5.51

The related principal functions, k = 0, are

5.52

and

5.53

For the case with q = 1 and v = 0, we have

5.54

Because the principal functions return to the R-function and zero, further properties are not pursued.

5.6 Fractional Hyperbolic Differential Equations

With the availability of the Laplace transforms, various fractional differential equations may be identified that are solvable using -hyperbolic functions. Based on the Laplace transforms, we may easily determine associated fractional differential equations for which they are solutions. For clarity of presentation, we are only concerned with uninitialized fractional differential equations (to include initialization effects, see, e.g., Ref. [71]). Continuing from Ref. [76] with the permission of ASME:

Consider the uninitialized fractional differential equation

5.55

taking the Laplace transform gives

5.56

Based on equation (5.46), this has a solution of the form

5.57

when , that is, a unit impulse function. For an arbitrary , the solution is given by convolution

5.58

Assuming composition, equation (5.55) may also be written as a fractional integro-differential equation, namely

5.59

Taking in equation (5.46) infers the uninitialized fractional differential equation

5.60

when , this has a solution of the form

5.61

In a similar manner, equation (5.48) indicates that the uninitialized fractional differential equation

5.62

has a solution of the form

5.63

when , that is, an impulse function. Again for an arbitrary , the solution is given by convolution

5.64

Equation (5.62) may also be written as a fractional integro-differential equation, namely

5.65

Linear combinations of fractional hyperbolic functions may also be used to infer fractional differential forms. For example, from the linear combination

5.66

or using the relationships (5.22) and (5.23)

5.67

we may infer the transformed equation

5.68

From this, it is observed that the noncommensurate order fractional differential equation

5.69

where

5.70

will have a solution of the form given by equation (5.66) for .

5.7 Example

Consider the generation of an expanding viscous foam in a cylindrical tank. The foam volume in the tank (Figure 5.12) is described by

5.71

where is the volumetric rate of foam entering the tank and is the volumetric rate of foam leaving the tank through a long baffled pipeline.

Figure 5.12 Expanding foam reactor.

The volumetric foam generation rate, , which, for short timescales, is proportional to the volume of foam in the tank and is given by

5.72

The foam volume in the cylindrical tank then is directly proportional to the height of foam in the tank, that is,

The tank is drained by a baffled diffusive pipeline that we consider to be of semi-infinite length in terms of the time–distance scaling of the problem. The foaming process is photosensitively ended as it enters the pipeline. The volumetric flow into the pipe is proportional to the semiderivative of the pressure (height of foam) at the pipeline inlet. Thus,

5.73

or

where is the initialization function, Lorenzo and Hartley [78] representing the flow history of the long pipeline. The reaction process is started at t = 0 by a unit impulse of the foam volume rate, . Taking the Laplace transform of the aforementioned equations and solving for L{v(t)} gives

5.74

To illustrate the application of the fractional hyperbolic functions, we take the radius of the tank to be small relative to the constants k and k₁ thus ignoring the first term in the denominator. Then, with , the volume of foam in the tank can be approximated as

5.75

Note that from equation (5.48)

5.76

Inverse transforming equation (5.75) and applying the convolution theorem give the solution for the volume of foam in the tank as a function of time to be

5.77

5.8 Discussions

Based on the R-function with real parameters a and t, namely , we have defined the fractional hyperbolic functions. These functions are backward compatible with the classical hyperbolic functions containing them as subfunctions when q = 1 and v = 0. Along with the Laplace transforms for these functions, various properties and identities have been developed. In particular, the fractional differintegrals of the functions have been derived.

The fractional hyperbolic functions are applicable to the solutions of certain fractional integro-differential equations. Because of the growing nature of these functions with time (see Figures 5.1–5.10), they will generally be associated with unstable physical phenomena. This is demonstrated in the reactor example provided.

In Chapter 6, a similar development based on the R-function is used to develop the R₁-trigonometry; the first of several fractional trigonometries. We will find in Chapter 9 that the R₁-hyperbolic functions and the fractional -trigonometric functions are both contained as special cases of the fractional meta-trigonometry. In other words, the fractional meta-trigonometry will be seen to unify the hyperboletry and the trigonometries.