## Abstract

A new and simplified method for the solution of linear constant coefficient fractional differential equations of any commensurate order is presented. The solutions are based on the *R*-function and on specialized Laplace transform pairs derived from the principal fractional meta-trigonometric functions. The new method simplifies the solution of such fractional differential equations and presents the solutions in the form of real functions as opposed to fractional complex exponential functions, and thus is directly applicable to real-world physics.

## 1. Background

### (a) Fractional differential equations

The primary objective of the fractional calculus is the formation and solution of a fractional integral and, more importantly, fractional differential equations. There have been many important contributions in this regard going back to the work of Bush [1], Davis [2] and others. More modern contributions include the works of Oldham & Spanier [3], Miller & Ross [4], Samko *et al.* [5], West *et al.* [6], Oustaloup [7], Magin [8] and the important book dedicated to the subject by Podlubny [9]. These studies and others present various powerful approaches to the solution of fractional differential equations.

The objective of this paper is to solve linear constant coefficient fractional differential equations of commensurate order. These equations are of the form
1.1
where the *c*_{i} are constants. Solutions are obtained using the fractional meta-trigonometric functions and the real fractional exponential function in the form of the *R*-function. This parallels the solution of ordinary linear differential equations, with constant coefficients, using real exponential functions together with the classical sine and cosine functions. For ordinary differential equations, solutions to high-order equations are expressed as sums of first- and second-order subsystems. Similarly, solutions for fractional differential equations of the form of equation (1.1) will be determined as the sum of subsystem elements of the *q*th and 2*q*th order. The solutions will be developed using the Laplace transforms of the required functions. Fractional differential equations of the form of equation (1.1) have been solved in [9] using infinite summations of generalized Mittag–Leffler functions. The forms to be presented in this paper will be in terms of the considerably simpler fractional trigonometric functions. Previous application of the fractional meta-trigonometric functions to this problem [10] was based on a simplification of the functions that allowed non-principal functions to be used. Here, the functions are simplified by restriction to principal functions.

Linear fractional differential equations of the form
1.2
have been referred to as the fundamental fractional differential equation [11]. The solutions of this equation may be expressed in terms of *R*-functions with real arguments, defined later in equation (2.1). The more general form
1.3
will be shown to have solutions that can be represented as fractional trigonometric functions. These solutions are based on *R*-functions with complex arguments. The fractional trigonometries [12–15] define a wide variety of oscillatory fractional functions that are the solutions or components of solutions to such linear fractional differential equations.

## 2. Fractional meta-trigonometry

The fractional meta-trigonometry [12,13] is a generalization of the *R*_{1} and *R*_{2} fractional trigonometries [14,15] which are in turn generalizations of the classical trigonometry. The fractional meta-trigonometries are, in fact, an infinite set of fractional trigonometries and hyperboletries that are based on the *R*-function [16] with fully complex arguments, namely
2.1
and are based on the definition
2.2
The meta-trigonometric functions based on *complexity*, that is, the real and the imaginary parts of equation (2.1) are defined for *t*>0 as
2.3
and
2.4
where *k*=0,1,2,…,(*D*−1),*t*>0, and *D*=*D*_{q}*D*_{v}*D*_{α}*D*_{β} is the product of the denominators of the rational arguments *q*=*N*_{q}/*D*_{q}, *v*=*N*_{v}/*D*_{v}, *α*=*N*_{α}/*D*_{α} and *β*=*N*_{β}/*D*_{β} with repeated multipliers being removed. These functions are simplified by considering only the principal, *k*=0, functions, thereby eliminating the index *k* in the function arguments. Thus, we have
2.5
and
2.6
where *σ*=(*α*+*βq*)(*π*/2+2*πk*) and *λ*=*β*[*q*−1−*v*](*π*/2+2*πk*).

Figures 1 and 2 show typical fractional sine and cosine functions plotted versus time (*t*) for particular values of the arguments as given.

The corresponding Laplace transforms for these functions [13] are given by
2.7
and
2.8
where *σ*=(*α*+*βq*)(*π*/2),*λ*=*β*[*q*−1−*v*](*π*/2). Previously in [10], the function transforms were simplified by taking *σ* and *λ* directly, as the function arguments in equations (2.7) and (2.8).

The principal meta-trigonometric functions based on *parity* [13], that is, the real and the imaginary parts of the even and odd terms of equation (2.1) are defined for *k*=0, and *t*>0 as
2.9
2.10
2.11
2.12
Many properties, identities and graphics associated with these meta-functions are given in [12,13].

The corresponding Laplace transforms for the parity functions are given by
2.13
2.14
2.15
2.16
where *σ*=(*α*+*βq*)(*π*/2), *λ*=*β*[*q*−1−*v*](*π*/2).

## 3. Fractional differential equations

Our interest in this paper is the solution of linear fractional differential equations of commensurate order, *nq*, and with constant coefficients *c*_{0},*c*_{1},*c*_{3},…,
3.1
using the fractional meta-trigonometric functions and the *R*-function. The notation has been used to indicate the initialized fractional derivative of the *q*th order. Then where is the uninitialized derivative and *ψ*(*t*) is the time-varying initialization function. The net effect of the initialization terms is to add a time-varying term to the right-hand side of equation (3.1) which contributes the initialization response to the solution. Including such terms will distract from the objectives of this paper and, for simplicity of presentation, only the forced response will be considered. Thus, our objective will be the solution of uninitialized fractional differential equations of the form
3.2
Sections 3*a* and 3*c*–*h* derive a set of Laplace transform pairs useful for the solution of fractional differential equations of the form of equation (3.2). These transforms are simplifications of the general transforms for the *Sin*_{q,v}(*a*,*α*,*β*,*t*), *Cos*_{q,v}(*a*,*α*,*β*,*t*) and Flut_{q,v}(*a*,*α*,*β*,*t*) functions given in equations (2.7), (2.8) and (2.15), respectively.

### (a) Fundamental fractional differential equation

The fractional differential equation
3.3
is known as the fundamental fractional differential equation [11]. Assuming quiescent conditions and taking the Laplace transform yields
3.4
For *f*(*t*)=*δ*(*t*) a unit impulse function, the solution may be written in terms of the *F*-function [11] or the *R*-function. Its Laplace transform is given as [16]
3.5
Then
3.6

### (b) Fundamental fractional differential equation of the second kind

Here, we consider the solution to fractional differential equations of the form 3.7 The Laplace transform associated with this equation then is given by 3.8 Solutions are possible using either the fractional cosine or fractional sine functions.

### (c) Fractional cosine function

The Laplace transform of the fractional cosine function is given as
3.9
where *σ*=(*α*+*βq*)(*π*/2), *λ*=*β*[*q*−1−*v*](*π*/2). For *f*(*t*)=*δ*(*t*) a unit impulse function we take *v*=*u*, *a*^{2}=*c*_{0}, , , and . From these conditions the following requirements are determined:
thus
3.10
and the superscript asterisk is introduced to indicate the values that apply only to the formulation of this subsection. For simplicity of results take *m*=1, then
3.11
Now, we also require
Thus,
3.12
But we must also have that
3.13
Solving for *α*
3.14
Now, we can write
3.15
These results give the useful transform
3.16
where *α**, *β** and *K** are given by equations (3.14), (3.11) and (3.15), respectively, and the superscript asterisks are introduced to indicate the values that apply only to the formulation of this subsection. Thus, the solution for the forced response of (3.7) when *f*(*t*) is a unit impulse function is given byQ2
3.17
When *f*(*t*) is not the unit impulse function, the convolution theorem may be applied to evaluate the solution.

### (d) Fractional sine function

Equation (3.7) may also be solved using the fractional sine function; this requires specialization of the transform pair
3.18
where *σ*=(*α*+*βq*)(*π*/2), *λ*=*β*[*q*−1−*v*](*π*/2). The process for use of the sine function proceeds in much the same manner as that for the cosine. For *f*(*t*)=*δ*(*t*), a unit impulse function, we take *v*=*u*, *a*^{2}=*c*_{0}, , and . From these conditions, the following requirements are determined:
3.19
and
3.20
thus
3.21
Again for simplicity of results take *m*=0⇒*λ**=0, Q3and this requires
3.22
From the condition
3.23
But again, we must also have that
3.24
Solving for *α*
3.25
and
3.26
The condition now becomes
3.27
These results give the *Sin*_{q,v}(*a*,*α*,*β*,*t*)-based transform as
3.28
where *α** is given by equation (3.26), and again the superscript asterisks are introduced to indicate the values that apply only to the formulation of this subsection. Thus, the forced solution of equation (3.7) for *f*(*t*) a unit impulse function is given by
3.29
a particularly simple result.

### (e) Fractional sine function: higher-order numerator dynamics

A Laplace transform with a higher order of the numerator Laplace variable, *s*, is obtained as follows. Using the fractional sine function we have
3.30
where *σ*=(*α*+*βq*)(*π*/2), *λ*=*β*[*q*−1−*v*](*π*/2). For *f*(*t*)=*δ*(*t*) a unit impulse function, we take *v*=*u*, *a*^{2}=*c*_{0}, , and . From these conditions, the following requirements are determined:
3.31
from ,Q4
3.32
Then from
3.33
thus
3.34
Now from *λ*=*β*[*q*−1−*v*](*π*/2)
3.35
and from *σ*=(*α*+*βq*)(*π*/2)
3.36
Note that when *u*=0⇒*α**=−*β**. We may now determine *K**:
3.37
Thus, we have the transform pair
3.38
where *α** and *β** are given by equations (3.36) and (3.35), respectively.

### (f) Fractional cosine function: higher-order numerator dynamics

The results for the cosine function are 3.39 where 3.40 3.41 3.42

### (g) Parity functions—flutter function

The parity functions may also be used to solve the fundamental fractional differential equation of the second kind. Here only the principal Flut function is considered. Its Laplace transform is given as
3.43
where *σ*=(*α*+*βq*)(*π*/2),*λ*=*β*[*q*−1−*v*](*π*/2). Our objective is to match the transfer function of the form of equation (3.8). Now let *q*=*w*/2 and *v*=*u*. This gives
3.44
where *σ*=(*α*+*βw*/2)(*π*/2), *λ*=*β*[(*w*/2)−1−*u*](*π*/2).

To obtain the form of equation (3.8), let
3.45
Now from
3.46
from which
3.47
Now
3.48
For simplicity of results take *m*=0, thus
3.49
and
3.50
From the definition of *λ*, that is, *λ*=*β*[(*w*/2)−1−*u*](*π*/2),
3.51
From the definition of *σ*, namely *σ*=(*α*+*βw*/2)(*π*/2),
3.52
The above results substituted into yield
3.53
3.54
3.55
Thus, the useful transform pair based on the fractional meta-trigonometric parity function is given as
3.56
where *α**, *β** and *K** are given by equations (3.52), (3.51) and (3.55), respectively. Similar results may be derived for the Cofl_{q,v}(*a*,*α*,*β*,*t*), Vib_{q,v}(*a*,*α*,*β*,*t*) and *Covib*_{q,v}(*a*,*α*,*β*,*t*) functions.

### (h) Transform pairs: elementary transfer functions of the second kind

Sections 3*c*–*g* have presented simplified Laplace transforms generally of the form
3.57
based on the general Laplace transforms (2.7), (2.8) and (2.15). These simplified transforms are given by equations (3.16), (3.28), (3.38), (3.39) and (3.56). Importantly, the constraints applicable to these equations differ. Also, the user is allowed the choice of the fractional trigonometric function in some circumstances. Forms similar to equation (3.57) are known as the ‘elementary transfer functions of the second kind’ [17]. These forms are defined by the Laplace transform
where *ω*_{0} is a fractional generalization of natural frequency and *ζ* has been named the pseudo-damping factor [17]. Here, we consider the form
3.58
Both the fractional sine and cosine functions may be used to achieve such transforms. Here, the sine function, equation (2.7), basis is examined. In equation (2.7), we take *v*=*v*, and .

From
3.59
3.60
3.61
Since , from the constraint on (2.7), then
3.62
and
3.63
Furthermore,
3.64
Thus, we have
3.65
where *α** is given by equation (3.63).

The fractional cosine function may also be used to obtain transform pairs related to equation (3.58). For this case, we have the result 3.66 where 3.67 and 3.68

## 4. Fractional differential equations of higher order

The results of §3*a*–*h* may now be used to solve linear constant-coefficient fractional differential equations of commensurate order, i.e. equations of the form
4.1
Again for clarity of presentation, we take *f*(*t*)=*δ*(*t*) a unit impulse function and restrict attention to the forced response only, avoiding the initialization response. Taking the Laplace transform of equation (4.1) (without initialization) gives
4.2
The general path to solution is to factor the denominator polynomial in *s*^{q}, and use the method of partial fractions to reduce the problem to multiple solutions of fractional transfer functions of the first and second kinds. These solutions then are those presented in the above sections. The process is best illustrated by example. Here, a fractional version from ordinary differential equations [18] is studied.

Consider the forced response for
4.3
The Laplace transform is applied to yield
4.4
The roots of the denominator are found to be −1, 2±3i. Application of partial fractions yields
4.5
Solving for *A*, *B* and *C*, we have
4.6
or
4.7
Inverse transforming term-by-term, using equation (3.5) for the first term we have
4.8
Using equation (3.38) with *v*=0, *c*_{0}=13 and *c*_{1}=−4 to inverse transform the second term,
4.9
4.10
4.11
4.12
Note that in [9] the *q*>*v* constraint was violated when this term was inverse transformed.

The third term of equation (4.7) is evaluated using equation (3.28) with *v*=0, *c*_{0}=13 and *c*_{1}=−4:
4.13
4.14
4.15
This yields
4.16
for the forced response of equation (4.3). Care must be taken when comparing this result with that of [9] since the meta-trigonometric functions have different arguments for the two cases, specifically, *Sin*_{q,v}(*a*,*σ*,*λ*,*t*) for [9] and *Sin*_{q,v}(*a*,*α*,*β*,*t*) for this paper. The parameters for both cases are given following equations (2.3) and (2.4).

### (a) Comparison with contemporary methods

Contemporary methods to solve equation (4.3) may be found in [4,9]. Here, we show the structure of the solution using the fractional Green’s function as presented by Podlubny [9], §5.5. Then, for the four-term fractional differential equation with constant coefficients of the form
4.17
the fractional Green’s function is derived as
4.18
where
4.19
and where *E*_{λ,μ}(*y*) is the two-parameter Mittag–Leffler function, that is,
4.20
This result, equation (4.18), is simplified to the commensurate case by letting *a*=1, *γ*=3*q*, *β*=2*q*, *α*=*q*. This provides the somewhat simpler form
4.21
For the application of this result to equation (4.3), we now take *b*=−3, *c*=9 and *d*=13:
4.22
Clearly, the application of the fractional trigonometric functions to the solution of equations of the form of equation (4.1) is considerably simpler than the application of the fractional Green’s function. It is important to note, however, that Green’s function approach is not limited to commensurate-order fractional differential equations and is therefore a more general solution.

## 5. Discussion

This paper has presented a method for the solution of linear constant coefficient fractional differential equations of commensurate order with unrepeated poles. It is based on the Laplace transforms of the fractional meta-trigonometric functions and the *R*-function. Current methods with equations of this type require considerably more difficult approaches and/or complex coefficients.

An important benefit of the fractional trigonometric approach is that the solutions are in the form of real functions as opposed to fractional exponential functions with complex arguments. The method parallels the method used for the solution of ordinary differential equations. It is expected that the new fractional trigonometric forms will provide important insights and connections to physical processes and their descriptions.

Multiple Laplace transform pairs for the solution of fundamental fractional differential equations of the second kind have been presented. All of the new transform pairs presented are based on the reduced argument principal, *k*=0, fractional meta-trigonometric functions. Future effort should be directed towards analysis of constant coefficient fractional differential equations of commensurate order with repeated poles.

## Acknowledgements

The authors gratefully acknowledge the support of the NASA Glenn Research Center.

## Footnotes

One contribution of 14 to a Theme Issue ‘Fractional calculus and its applications’.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.