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Fractional Derivative as Fractional Power of Derivative

Definitions of fractional derivatives as fractional powers of derivative operators are suggested. The Taylor series and Fourier series are used to define fractional power of self-adjoint derivative operator. The Fourier integrals and Weyl quantization procedure are applied to derive the definition of fractional derivative operator. Fractional generalization of concept of stability is considered.

It is known that we can define a fractional power of operator [33,34,35,31,32,36,37]. The integer power of operator can be easily realized. Therefore, we can realize the fractional power as the integer power series. In this paper, we use equations that represent the fractional power as a series of integer powers series. This representation allows us to define the fractional power of operator as a series of integer powers of operator. As a result, we obtain the definition of fractional derivatives as a fractional power of derivative operator.
Note that the well-known Riemann-Liouville fractional derivative can be represented as a power series of derivatives of integer order [1]: where A n (x, a, α) = (−1) n−1 αΓ(n − α) Γ(1 − α)Γ(n + 1) (x − a) n−α Γ(n + 1 − α) for the functions that are analytical in the interval (a, b). In Sec. 2, we point out some well-known definitions of functions of bounded and unbounded operators. In Sec. 3, the fractional derivatives are defined as fractional powers of coordinates that considered as Taylor series. In Sec. 4, the fractional derivatives are considered as fractional powers of coordinates that described as Fourier series for the interval. In Sec. 5, the fractional derivatives are defined as fractional powers of coordinates by using the Fourier integrals. In Sec. 6, the fractional derivatives are defined as fractional powers of coordinates by using the Weyl quantization. In Sec. 7, using fractional derivatives, we define the stability with respect to fractional variations.

Function of bounded and unbounded operators
Let us point out some well-known definitions of functions of bounded and unbounded operators [31,32,33,34,35].

Power series
Let us consider a bounded linear operator A that is defined on the linear space E, and A ∈ L(E, E), where L(E, E) is a space of linear maps of E. Suppose the function f (x) is an analytical function of the variable x such that it can be represented as a power series Then, we can define The operator f (A) is a linear bounded operator A on space E. For example, the exponential function of operator is defined by

Cauchy's integral formula
The definition of operator function by power series can be generalized for wider class of functions. To realize this generalization, we use Cauchy's integral formula instead of power series. Cauchy's integral formula states that where the integral is a contour integral along the contour Γ enclosing the point z 0 . We can define the algebraic isomorphism between an operator algebra and some functions [34]. The function f (z) = z corresponds to the operator A.
then the resolvent operator exists. The function of linear bounded operator is defined by where Here Γ = ∂G ∈ σ(A), where σ(A) is a spectrum of operator A, and ρ(A) ⊂ G. For example, we can define the operator that is corresponded to the function exp(zt), and σ(A) ⊂ G.
As the second example, the operator E z that corresponds to the Heaviside function θ(z − z 0 ), where θ(z − z 0 ) = 0 for z 0 ≥ z, and θ(z − z 0 ) = 1 for z 0 < z is defined by and is called the spectral operator. The operator E z can be denoted by θ(zI − A).

Spectral representation of self-adjoint unbounded operator
It is known that spectral function E z exists for all self-adjoint operators A, and where Then, the function f (A) of self-adjoint operator A can be defined by the equation As operator A, we can consider the self-adjoint derivative −i∂/∂x. For the father information about this approach, we can use Refs. [31,32,35,45,46].

Definition of Taylor series
A one-dimensional Taylor series, which is an expansion of a real-valued function f (x) about a point x = a, is given by where and f (n) (a) is the nth derivative of f (x) evaluated at the point x = a.
Suppose the function f (x) has all derivatives in the interval |x−a| < a 0 , and the condition is satisfied, then the series converges to the function f (x) for all intervals |x−a| < a ′ , where a ′ < a 0 . This representation of the functions can be used to define fractional power of operator.

Taylor series for fractional power of coordinate
The Taylor series for fractional power of coordinate f (x) = x α about a point x = a > 0 is where f n = 1 n! (x α ) (n) (a) = B(α, n) n!a n−α , Using we rewrite Eq. (11) in the form where Equation (14) represents the fractional power of coordinate as a series of integer powers. This representation allows us to define the fractional power of operator as a series of integer powers of operator.
It is known that self-adjoint operators have the real eigenvalues. Using Eq. (14) with a > 0, we can define the fractional power of the self-adjoint operator A by For the operator or, in the equivalent form As a result, we get that fractional derivative is defined as a series of integer powers of self-adjoint derivative operator.

Examples of computation of fractional derivatives
Let us consider the fractional derivative (16) of constant c: Using we get Let us consider the fractional derivative of a power x m . From (17), Using for k ≤ m, and (x m ) (k) = 0 for k > m, we obtain 4 Fractional Derivatives by Fourier Series

Fourier series
Fourier series of a function f (x) ∈ L 2 [−l, l] is an expansion in terms of an infinite sum of sines and cosines. Since sines and cosines form a complete orthogonal system over [−l, l], the Fourier series is given by where and n is a positive integer number. Let us consider the Fourier series for f (x) = |x| α ∈ L 2 [−l; l] , where α is a positive fractional power. The Fourier series of this function for x ∈ [−l, +l] is where The cosine can be represented as a power series. Therefore, equation (26) allows us to present the fractional power as a series of integer powers series. Then, we can define fractional derivative on the interval [−l, l] as a fractional power of derivative.

Fractional derivative
The fractional power of derivative operator for the interval [−l; l], we can define by It is known that we can define exp(A) for the self-adjoint operator A = −id/dx by Using Equation (67) and (29) allows us to define the fractional power of operator as a series of integer powers of operator. Using (29), we rewrite (27) as or by the equivalent equation where Let us compute the coefficients a k by a k = 1 l α l 0 y α cos(πky)dy.

Complex Fourier series
The real-valued function f (x), which is defined on [-L/2,L/2], can be presented by where The operator function f (A) is defined by where f n are defined in (34), and Substitution of (36) into (35) gives Then the fractional power of self-adjoint derivative on [−L/2, +L/2] can be presented by where f (n, k) = (2πn) k k!L k+1 Equation (38) represents the fractional power as a series of integer powers. This representation defines the fractional power of derivative as a series with integer powers of derivatives.

Fractional derivative by Fourier integral
Let us consider a function f (x) with n variables x. Suppose A 1 , A 2 ,...,A n are n elements of commutative operator algebra A. For example, A is an algebra of operators in linear space. We denote byf the Fourier transform for f (x): where The operator function f (A) of elements A 1 , ..., A n is defined by Substitution of Eq. (40) into Eq. (42) gives In general, we must have the exact definition of these integrals and description of possible class of symbols and algebras of elements A 1 , .., A n . To define the fractional power of the operator A, we use (42) in the form This equation can be considered as a definition of fractional power of operator A. For f (x) = |x| α , where α = −1, −3, ..., we have [39]: Then Eq. (44) gives For the self-adjoint derivative operators As a result, we obtain the fractional derivative operator as a fractional power of self-adjoint derivative operator.

Fractional power of self-adjoint derivative operator
Let us consider the self-adjoint derivative operators It is easy to prove that f (p) = p α ∈ S ∞ (R n p ). Here, S ∞ is the space of symbols that are slowly growth on the infinity. This space is defined as where S k l (R n ) is a space of functions from the class C k (R n ) with the norm The space S ∞ (R n ) is defined by ∃r, ∀s ||f || r,s = sup The fractional powers of the operator (47) are elements of algebra L(S ∞ , S ∞ ) of all continuous linear maps of the space S ∞ (R n ). Then, the fractional derivative operator is a fractional power of self-adjoint derivative operator. The operator (48) acts on the arbitrary function u(x) ∈ C ∞ (R n ) by where F y→p u(y) = 1 2πi is the direct Fourier transform, and is the Fourier transform.
Proposition. The operator It is not hard to prove this proposition. Using Eq. (49), we get As a result, we obtain D α x e p (x) = p α e p (x).
Multiplying both sides of (54) on e −p (x), we get e −p (x)D α x e p (x) = p α that proves (53).

Quantization procedure for coordinate representation
Let us consider the quantum mechanics in coordinate representation. It is known that quantizationQ is a linear map of coordinate q and momentum p into self-adjoint operatorŝ Using linearity of quantization map [40,41], we get Q(aq + bp) = aq + bp.

Weyl quantization
Canonical quantization defines a map of real-valued functions into self-adjoint operators [40,41,42,43,44,45,46]. A classical observable is described by some real-valued function A(q, p) from a function space M. Quantization of this function leads to self-adjoint operator A(q,p) from some operator spaceM. Let us consider main points of the usual method of canonical quantization [40,42,45,46]. Suppose q k are canonical coordinates and p k are canonical momenta, where k = 1, ..., n. The basis of the space M of functions A(q, p) is defined by functions Quantization transforms coordinates q k and momenta p k to operatorsq k andp k . Weyl quantization of the basis functions (58) leads to the Weyl operatorŝ Operators (59) form a basis of the operator spaceM. Classical observable, which is characterized by the function A(q, p), can be represented in the form whereÃ i.e.,Ã(a, b) is the Fourier image of the function A(q, p). Quantum observableÂ(q,p), which corresponds to A(q, p), iŝ This formula can be considered as an operator expansion forÂ(q,p) in the operator basis (59). Substitution of (61) into (62) giveŝ The function A(q, p) is called the Weyl symbol of the operatorÂ(q,p). Canonical quantization defined by (63) is called the Weyl quantization. The Weyl operator (59) in formula (63) leads to the Weyl quantization. Another basis operator leads to different quantization scheme [42].

Fractional derivative by Weyl quantization map
Let us consider a quantization map of real-valued function f (p) = |p| α into self-adjoint operator. Quantization of this function leads to some self-adjoint operatorf (p) from the operator spaceM p , wherep k = −i∂/∂x k and k = 1, ..., n. Then The operators (65) form a basis of the operator spaceM p . Using Fourier transform, the function f (p) = |p| α can be presented by wheref i.e.,f(a) is the Fourier image of the function f (p). Quantum observablef (p), which corresponds to f (p), is defined by the formulâ This formula can be considered as an operator expansion forf (p) in the operator basis (59).
From (67) and (68), we obtain The function f (p) is the Weyl symbol of the operatorf (p). For f (p) = |p| α , where α is a positive real number, we obtain As a result, we have the definition of fractional derivatives on R n as a fractional power of self-adjoint derivative.

Fractional Stability
In this section, we use the fractional generalization of variations of variables. Fractional integrals and derivatives are used for stability problems [47,48,49,50,51]. In this paper, we consider the properties of dynamical systems with respect to fractional variations [25]. We formulate stability with respect to motion changes at fractional changes of variables. Some systems can be unstable "in sense of Lyapunov", and be stable with respect to fractional variations.

Fractional variation derivative
Let us consider dynamical system that is defined by the ordinary differential equations. Suppose that the motion of dynamical system is described by the equations Here y 1 , ..., y n be real variables that define the state of dynamical system. Let us consider the variation δy k of variables y k . The unperturbed motion is satisfied to zero value of variation δy k = 0. The variation describes that as function f (y) changes at changes of argument y. The first variation describes changes of function with respect to the first power of changes of y: δf (y) = D 1 y f (y)dy, where D 1 y f (y) = ∂f (y) ∂y .
The second variation describes changes of function with respect to the second power of changes of y: The variation δ n of integer order n is defined by the derivative of integer order D n y f (y) = ∂ n f /∂y n .
Let us define the variation of fractional order as a fractional exterior derivative of the function (zero-form) by the equation where D α y is a fractional derivative with respect to y. The fractional variation of order α describes the function f (y) changes with respect to fractional power of variable y changes. The variation of fractional order is defined by the derivative of fractional order.

Equations for fractional variations
Let us derive the equations for fractional variations δ α y k . We consider the fractional variation of equation (71) in the form: Using the definition of fractional variation (74), we have From Eq. (76), and the property of variation where y k = y k (t, a), we obtain Note that in the left hand side of Eq. (78), we have fractional variation of δ α y k , and in the right hand side -fractional power of variation (δy k ) α . Let us consider the fractional variation of the variable y k . Using Eq. (74), we get δ α y k = [D α y l y k ] (δy l ) α , k = 1, ..., n.
For the Riemann-Liouville fractional derivative, D α y l y k = 0 if k = l. Therefore for simplification of our transformations, we use the fractional derivative as a fractional power of derivative where δ kl is a Kronecker symbol. Substituting Eq. (80) into Eq. (79), we can express the fractional power of variations (δy k ) α through the fractional variation δ α y k : (δy k ) α = D α y k y k δ α y k .
Substitution of Eq. (81) into Eq. (78) gives d dt δ α y k = D α y l y l D α y l F k δ α y l .
Here we mean the sum on the repeated index l from 1 to n. Equation (82) is equations for fractional variations. Let us denote x k the fractional variations δ α y k : x k = δ α y k = D α y k y k (δy k ) α .
As a result, we obtain the differential equation for fractional variations where a kl (α) = D α y l y l D α y l F k .
Equation (86) is a linear differential equation. To define the stability with respect to fractional variations, we consider the characteristic equation with respect to λ. If the real part Re[λ k ] of all eigenvalues λ for the matrix A α are negative, then the unperturbed motion is asymptotically stable with respect to fractional variations. If the real part Re[λ k ] of one of the eigenvalues λ of the matrix A α is positive, then the unperturbed motion is unstable with respect to fractional variations. A system is said to be stable with respect to fractional variations if for every ǫ, there is a δ 0 such that: δ α y(t 0 ) < δ 0 => δ α y(α, t) < ǫ ∀t ∈ R + .
The dynamical system is said to be asymptotically stable with respect to fractional variations δ α y(t, α) if as t → ∞, δ α y(t, α) → 0.
The concept of stability with respect to fractional variations is wider than the usual Lyapunov or asymptotic stability. Fractional stability includes concept of "integer" stability as a special case α = 1. Some systems can be unstable with respect to first variation of states, and be stable with respect to fractional variation. Therefore fractional derivatives expand our possibility to research the properties of dynamical systems.