On a Leibnitz-type fractional derivative1 V. V. Kobelev2 Abstract. The new type of fractional derivative, referred to as α-derivative, is studied. The αderivative of fractional type obeys Leibnitz rule. Based on the definition of α-derivative the operations of analysis and differential geometry are studied. Lead paragraph. The new type of fractional derivative, referred to as α-derivative, is studied. The α-derivative of fractional type obeys Leibnitz rule. Based on the definition of α-derivative the operations of analysis and differential geometry are studied. The principal definition in the theory of fractional calculus is the classical Riemann-Liouville fractional derivative. The binomial Leibniz rule for derivatives is replaced by the infinite sum in terms of the Riemann-Liouville operator. The infinite summation complicates the foundations of geometry, based on the concept of fractional derivatives. That is, the binomial Leibniz rule, will be considered as a crucial and important property and is used as an axiom of the new theory of differentiation. We introduce a new definition of a derivative of fractional order, referred to as an α -derivative. We use the slightly adapted Leibniz's notation for the α -derivative to underline the fact, that the new fractional-type derivative must obey the binomial Leibnitz rule. The application of α-derivative to a polynomial with integer power series converts it to a polynomial with fractional power series. The application of α-derivative to a polynomial the polynomial with fractional power series converts it again to a polynomial with fractional power series. The ring of polynomials with fractional power series is known as Puiseux series. We demonstrate, that α-derivative of Puiseux series is Puiseux series, such that the polynomials with fractional power series are closed with respect to operation of αderivative. Starting with α-derivatives, we consider the α-differential equations. For the solution of linear α-differential equations with constant coefficients we use the method of power series. We introduce the co-joint equation as the ordinary differential equation with the same coefficients. Its general solution could be obtained by common methods. To evaluate the coefficients of the α-differential equations, the solution the co-joint equation should be expanded in power series. Then, the fractional power (Puiseux) series delivers the general solution of the α-differential equation. The α-analogues for trigonometric and exponential functions are studied. In the last Section we build up, following the standard modus operandi, the main objects of differential geometry, based on the concept of α-derivative.
1
Introduction
The concept of derivative is traditionally associated to an integer; given a function, we can derive it one, two, three times and so on. It can be have an interest to investigate the possibility to derive a real number of times a function. The classical theory of integrals and derivatives of non-integer order goes back to Leibniz, Liouville, Riemann, Grunwald, and Letnikov [1,2].
1
Submitted to “Chaos: An Interdisciplinary Journal of Nonlinear Science“, 2012.02.13
2
[email protected], Faculty IV: Science and Technology, University of Siegen, Siegen, D-57076, Germany
1
The simplest approach to a definition of fractional differentiation begins with the formula, which was mentioned in a letter from G.W. Leibnitz to G. F. A. l’Hopital (1695) [3]: Dα e az = aα e az . The order of derivative α could be arbitrary (integral, rational, irrational or complex). Based on this definition, J. Liouville wrote several memoirs on fractional derivatives. The alternative approach to fractional derivatives was proposed by L. E. Euler in 1731 and is based on the formula Γ( β + 1) α Dz z β = z β −α . Γ( β − α + 1) Worth while mentioned that the two above definitions are inconsistent. For the discussion on this subject and for the explanation the reader is addressed to the article [4]. One of the most frequently encountered tools in the theory of fractional calculus (that is, differentiation and integration of an arbitrary real or complex order) is furnished by the familiar differ-integral operator 1 z c ∈ R, Re(α ) < 0 (z − ς )−α −1 f (ς )dς ∫ Γ(− α ) c α { ( ) } D f z (1) = c z dm α −m { f (z )} m ∈ N := {1,2,3...}, m − 1 ≤ Re(α ) < m c Dz dz m provided that the integral exists. For c = 0 , α α (2) 0 Dz { f ( z )} = Dz { f ( z )} corresponds essentially to the classical Riemann-Liouville fractional derivative (or integral) of order α (or − α ). Moreover, when c → ∞ , Equation (1) may be identified with the definition of the familiar Weyl fractional derivative (or integral) of order order α (or − α ). An ordinary derivative corresponds α = 1 , such that (d dz ){ f (z )} = D1z { f (z )}. The binomial Leibniz rule for derivatives D1z { f ( z )g ( z )} = g ( z )D1z { f ( z )}+ f ( z )D1z {g ( z )} (3)
{ }
{ }
admits itself of the following extension in terms of the Riemann-Liouville operator Dzα :
∞ α Dzα { f ( z )g ( z )} = ∑ Dzα − n { f ( z )}Dzn {g ( z )} , n=0 n α Γ(α + 1) := α , k ∈ C, . k Γ(α − k + 1)Γ(k + 1) The infinite summation in the Eq. (4) complicates the foundations of geometry, based on the concept of fractional derivatives. That is, the binomial Leibniz rule, given by the Eq. (3) will be considered as a crucial and important property and is used as an axiom of the new theory of differentiation. The motivation is following. Let us start with a monomial function of the form z β . Then Γ(β + 1) λ −α (5) Dzα z β = z , Γ(β − α + 1) Re(α ) < 0, Re(β ) > −1 .
(4)
{ }
That is, the Riemann-Liouville derivative of the function z β is the product of C * (β , α ) and the monomial z β −α with an exponent β − α : (6)
{ }
Dzα z β = C * (β , α )z β −α ,
C * (β , α ) =
2
Γ(β + 1) . Γ(β − α + 1)
The Riemann-Liouville and Euler fractional derivatives of the function z β are equivalent. Recently new definitions of fractional derivatives were proposed. Definition of fractional derivatives as fractional powers of derivative operators is suggested [5]. 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 in the cited article to derive the definition of fractional derivative operator. The main idea is to examine the Leibniz property of the ordinary derivative and see where and how it is possible to generalize the concepts. In the articles [6, 7] the theory of probability of fractional order in which the exponential function is replaced by the Mittag-Leffler function is constructed. In this framework, some useful classical mathematical tools were generalized, so that they are more suitable in fractional calculus. Most of the definitions of fractional derivatives lack the Leibnitz property. Recently, Cottrill-Shepherd and Naber gave the definition of a fractional exterior derivative [8] and found that fractional-differential formal space generates new vector spaces of finite and infinite dimension, the definition of closed and exact forms are extended to the new fractional form spaces with closure and integrability condition worked out for a special case. Coordinate transformation rules are also computed. The field equations of general relativity are not easily fractionally generalized because of the covariance requirement on derivatives. The general relativity applications in [9, 10] did not modify the field equations or any of the usual GR tensors in any way; the fractional match simply provided a broader set of metric relations across a boundary. It was used to create a family of Israel layers parameterized by the non-integer order of the fractional derivative. There are, however, geometric objects used in general relativity, which can be fractionalized without altering the basic covariant structure of the theory. One of these is the Lie derivative, defined only with partial derivatives. The Lie derivatives take into account the difference between a tensor that is Taylor transported to a point and coordinate transformed at the same point. A fractional Lie derivative, valid in the thin shell limit, is developed in [11]. The nonlocal nature of the fractional derivative allows the inclusion of shell thickness in the stress energy description of zero thickness Israel layers.
2
Fractional derivative, that fulfils Leibnitz rule
Let us consider a new definition of a derivative of fractional order, referred to as an α derivative. We assume an α -derivative of the function z λ is the product of a coefficient C (β , α ) and the monomial z β −α with an exponent β − α : α β (7) z = C (β , α )z β −α . αz The α -derivative must satisfy the Leibnitz rule (3) by definition. We use the slightly adapted Leibniz's notation for the α -derivative to underline the fact, that the new fractional-type derivative obeys the Leibnitz rule (3). Parameter α will be referred to as a basis of derivative. The requirement (3) applies conditions on the function C (β , α ) .
{ }
At first, consider an application of α -derivative on the monomial z β . The Leibnitz rule (3) has to be valid for all β . That is, if the function is spitted into the product of two monomials
z β ≡ f (z )g ( z ), f ( z ) = z β −ε , g ( z ) = z ε . With an arbitrary ε , the Leibniz rule is applicable for the functions f ( z ) = z β −ε , g ( z ) = z ε . The application of (3) delivers immediately, that
3
{ }
(8)
α β α β −ε α ε z ≡ zε z + z β −ε z = αz αz αz z ε C (β − ε , α )z β −ε −α + z β −ε C (ε , α )z ε −α =
[C (β − ε ,α ) + C (ε ,α )] z β −α
Comparison of equations (7) and (8) delivers following functional equation (9) C (β − ε , α ) + C (ε , α ) = C (β , α ) . To guarantee the binomial Leibnitz rule (3), this functional equation must be satisfied for any β , ε , α . The functional equation (9) is the basic Cauchy's functional equation and its solution is [12] (10) C (β , α ) = A(α )β . Here A(α ) is the function of α only. That is, for the validity of Leibnitz rule the α -derivative must be of the form α β z = A(α )βz β −α . (11) αz Easily seeing, that C * (β , α ) is not of type (10) and that’s why the Riemann-Liouville operator
{ }
Dzα does not possess binomial Leibniz rule with an exception α = 1 [13]. The arbitrary function A(α ) could be chosen arbitrary and is assumed to A(α ) = 1 . With this assumption we define formally the α-derivative of a power function z β for any β as α β def (12) z = β z β −α . αz Particularly, α α α 0 z =α , z = 0 z −α ≡ 0 . αz αz The application of α-derivative to a polynomial with integer power series converts it to a polynomial with fractional power series. Fortunately, the application of α-derivative to a polynomial the polynomial with fractional power series converts it again to a polynomial with fractional power series. The ring of polynomials with fractional power series is known as Puiseux series [14]. As we show in the next section, α-derivative of Puiseux series is Puiseux series. This ring is algebraically closed. Actually we need only the closure of the polynomials with fractional power series with respect to operation of α-derivative.
3
Puiseux series
Throughout the paper, K is an algebraically closed field of characteristic 0, K [x] is the corresponding polynomial ring, and K ( x ) is the field of rational functions. Consider a complex function F ( z ) that could be represented by convergent power series in the indeterminate z [15,16]. The Laurent series for the function about a point 0 is given by (13)
F (z ) =
∞
∑c z
k = −∞
k
∞
k
, F+ ( z ) = ∑ ck z k , F− ( z ) = k =0
−1
∑c z
k = −∞
k
k
ck ∈ K . For every k, the ck ’s are the coefficients of Laurent series for the function F . The
principal part F− ( z ) of a Laurent series is the series of terms with negative degree. There exits a k0 such that ck = 0, k ≤ k0 . The first derivative of F with respect to z is defined to be
4
∞ dF = ∑ ck kz k −1 dz k =1 The following properties of derivatives follow from the definition d 1. (F + G ) = dF + dG dz dz dz da d (aF ) dF = 0 , then =a . 2. If a is a constant dz dz dz n d (FG ) = G dF + F dG , dF = n F n−1 dF 3. dz dz dz dz dz The first part of 3° can be easily checked; the last part of 3° is obtained by repeated application of the first part. The formal power series can be added and multiplied just like polynomials, and they constitute a ring K [[x ]] . The quotient field K (( x )) of K [[x ]] is called the field of formal Laurent series. Our aim is to check the 1-3 for the function, represented by fractional power series. In accordance with the theory of analytic continuation of complex variable function all properties of a function analytic in some point are defined by its power expansion in this point [17]. The central practical problem of the theory is study of function properties immediately on series coefficients prescribed. At present there exist several methods of this problem solution based on assumption that we possess beforehand some information either about class of the function or about general tend of its power expansion terms behavior. We consider the union ∞
(( ))
K x = ∪ K x1/ k k =1
This becomes a field if we set
(x )
1 / rn n
x1/1 = x,
K x
= x1/ n ,
( )
x m / n = x1/ n
m
.
is called the field of fractional power series or the field of Puiseux series [18,19].
If f ∈ K x has the form
f =
∞
∑c x
k = k0
mi / ni
k
,
where c1 ≠ 0 and mi , ni ∈ N := {1,2,3...} , mi / ni < m j / n j for i < j ,
then the order of y is O( f ) = m n , where m = m1 , n = n1 and f ( x) = F x1/ n . Puiseux series allow for negative exponents of the indeterminate as long as these negative exponents are bounded (here by k0 ). Without loss of generality we will always assume that a maximal such r was chosen, even if that r is ∞. Let for any given ramification index of the Puiseux expansion n and complex z
( )
∞
(14)
f ( z ) = ∑ ck ( z − z 0 ) n k = −∞ ∞
k
= f + (z ) + f − (z ) ,
f + (z ) = ∑ ck ( z − z0 )n , k
k =0
5
f − (z ) =
−1
∑ c (z − z )
k = −∞
k
0
k n
,
ck = 0, k ≤ k0
be generalized Puiseux series converging in the disc δ r ( z0 ) defined by
δ r ( z0 ) = {z ∈ C : r > 0, z − z0 < r}.
The principal part f − ( z ) of generalized Puiseux series is the series of terms with negative degree. The regular part f + ( z ) of generalized Puiseux series is the series of terms with positive degree. Also note that it would be equivalent to begin with an analytic function defined on some small open set. We say that the vector g = ( z0 ,..., c−1 , c0 , c1 ,...) is a germ of f. The base g 0 of g is z0 , the stem of g is ( ..., c−1 , c0 , c1 ,... ) and the top g1 of g 0 is c0 . The top of g is the value of f at z0 , the bottom of g. Any vector
g = ( z0 ,..., c−1 , c0 , c1 ,...)
is a germ if it represents a power series of an analytic function around z0 with some radius of convergence r > 0 . Therefore, we can safely speak of the set of germs G . Let g and h be germs. If g 0 − h0 < r where r is the radius of convergence of g and if the power series that
g and h define identical functions on the intersection of the two domains, then we say that h is generated by (or compatible with) g , and we write g ≥ h . This compatibility condition is neither transitive, neither symmetric nor antisymmetric. If we extend the relation by transitivity, we obtain a symmetric relation, which is therefore also an equivalence relation on germs. This extension by transitivity is one definition of analytic continuation. We can define a topology on G . Let r > 0 , and let U r ( g ) = {h ∈ G : g ≥ h, g 0 − h0 < r }.
The sets U r ( g ) , for all r > 0 and g ∈ G define a basis of open sets for the topology on G . A connected component of G (i.e., an equivalence class) is called a sheaf. We also note that the map φg (h ) = h0 from U r ( g ) to C where r is the radius of convergence of g , is a chart. The set of such charts forms an atlas for G , hence G is a Riemann surface. G is referred to as the universal Puiseux function. Consider the universal Puiseux function f ( z ) in the disc δ r (0) .
4
Properties of α-derivative
We consider in this Article the α-derivative for a basis α = m n, 0 < m < n , m, n ∈ N := {1,2,3...} . The basis of derivative is considered to be a rational number. The α-derivative of a Puiseux function of the order O( f ) = 1 / n is again a Puiseux function of the order (1 − m ) n . The derivative with α = 1 n is of principal importance. In this case the α-derivative of a Puiseux function k ∞ ∞ k f + = ∑ ck z n = ∑ ck z β with β ≡ β (k ) = n k =0 k =0 6
is Puiseux function of the same order: k k ∞ ∞ ∞ α α β ∞ β −α n f + ( z ) = ∑ ck z ≡ ∑ α β ck z = ∑ α β ck −1 z = ∑ cˆk z n , αz α z k =0 k =0 k =1 k =0
cˆk = α β ck −1 .
−1 −1 −2 −1 α α f − ( z ) = ∑ ck z β ≡ ∑ α β ck z β −α = ∑ α β ck −1 z n = ∑ cˆk z n , cˆ−1 = 0 . αz αz k = −∞ k = −∞ k = −∞ k = −∞ The α-derivative with α = m n of a generalized Puiseux function k
∞
f =
∑c z
k = −∞
k n
k
=
∞
∑c zβ
k = −∞
β ≡ β (k ) =
with
k
k
k n
is once again the generalized Puiseux function: k k ∞ ∞ ∞ ∞ α α β β −α n (15) f ( z ) = ∑ ck z ≡ ∑ α β ck z = ∑ α β ck −1 z = ∑ cˆk z n αz α z k = −∞ k = −∞ k = −∞ k = −∞ From the linearity of α-derivative follows α ( f + g ) = αf + αg , 1°. αz αz αz α 2°. (cf (z )) = c α f (z ) with a constant c . αz αz Using the common properties of power series one get the Leibnitz rule for Puiseux functions
α ( fg ) = g αf + f αg . αz αz αz For proof of Leibnitz rule we use the principle of induction. Firstly, for an arbitrary ε we have using (8) and (9): def α β −ε ε α β z z =(β − ε ) z β −ε −α z ε + ε z β −α z 0 = β z β −α = z . αz αz Secondly, assume that the Leibnitz rule is valid for arbitrary generalized Puiseux functions 3°.
(
f =
)
∞
∑ ek z β k , g =
k = −∞
∞
∑ d zβ
l = −∞
l
l
:
α ( fg ) = g αf + f αg . αz αz αz ~ Using this property, we demonstrate the Leibnitz rule for Puiseux functions f , g~ , such that ~ f = f + e z ε , g~ = g + d z δ with arbitrary e, d , ε , δ . ~ α-derivatives of the functions f , g~ respectively are α ~ α α α f = f + e zε ≡ f + e(ε − α )z ε −α , αz αz αz αz α α ~ α α δ α g= g+d z ≡ g + d (δ − α )z δ −α . α z αz αz αz αz ~ Immediate calculation of α-derivative of the function fg~ = f + e z ε g + d z δ delivers ∞ ∞ α ~~ α fg = fg + e ∑ (ε + β l )d l z ε + β l −α + d ∑ (δ + β k )ek z δ + β kl −α + ed (ε + δ )z δ +ε −α = αz αz l = −∞ k = −∞ αf αg α ε α α α ~ ~ α ~ =g +f + g ez + f d zδ + e z ε d z δ ≡ g~ f +f g. αz αz αz αz αz αz αz
(
(
)
7
)(
)
Applying the principle of induction we prove the validity of Leibnitz rule for any pair of ~ Puiseux functions f , g~ . The validity of Leibniz rule is essentially important for differential geometry, based on the concept of α-derivative. r α ∂F αg k ( z ) 4°. F ( g1 ( z ),..., g r ( z )) = ∑ , where αz αz k =1 ∂g k ∂F ∂F ( g1 ,..., g r ) = ∂g k ∂g k is the ordinary partial derivative of function F ( x1 ,.., xk ) with respect to xk (Chain rule). The expression for second α-derivative of a power function reads α α β α2 zβ (16) = β (β − α )z β − 2α . z ≡ 2 α z αz αz Analogously, the expression for the l -fold α-derivative is l −1 αl β β − lα (17) z = x (β − iα ) = x β −lα (− α )l Γ(l − β / α ) . ∏ l αz Γ(− β / α ) i =0 Substitution β = k / n , α = m / n delivers its equivalent form
( )
l −1 αl k / n k − im m Γ(l − k / m ) n z = x =x n − . ∏ l αz n n Γ(− k / m ) i =0 Parameter l is an order of the differentiation. The multiple differentiation possess the semigroup property with respect to order l . From Eq.(17) follows α m β −lα l −1 α m α l β α m β −lα l −1 ( ) ( ) z = x − k = − k β α β α ∏ mx = ∏ m αz m αz l α α z z k = 0 k = 0
( )
k − ml
k − ml
l
( )
αl +m β z . αz l + m k =0 j =0 k =0 Thus, the essence of parameters α and n strongly are different. The α−derivative of order 1 obeys Leibnitz rule. The application of this identity for all terms of a Puiseux function proves the associativity of differentiation with respect to its order for Puiseux functions: k k ∞ α m αl α m αl ∞ α m α l n α l +m n f = c z = c z = l +m ( f ) ∑ k k∑ k m l αz m αz l αz m αz l k = k 0 αz = k 0 αz αz Operation of taken a higher derivative of Puiseux function for a given fixed basis α is a semigroup [20]. This semigroup has the property that α m αl αl α m f= f α z m αz l αz l α z m (Schwarz rule). The semigroup is a commutative or Abelian semigroup. On the contrary, the classical Riemann-Liouville fractional derivative forms a semigroup with respect to parameter α, but lucks a Leibnitz property. The indefinite α-integral of the function is defined as l −1
m −1
= ∏ (β − kα )∏ (β − lα − jα )x β −nα −mα =
(18)
∫
∞
f (z )αz = ∑ ck ∫ z β αz k =0
with 8
l + m −1
∏ (β − kα )x β −lα −mα
=
( )
def
(19)
β ∫ z αz =
z β +α β +α
β +α ≠ 0 .
for
Particularly, (20)
zα
∫ αz =
α
.
Immediately follows, that α αf ( z ) f ( z )αz = f ( z ) , αz = f ( z ) (21) ∫ ∫ αz αz The partial derivative of the Puiseux series ∞
f ( z1 ,..., z N ) =
∑
k1 = k 01
∞
...
∑c
k N = k0 N
z
k1 ...k N 1
k1 / n
...z N
.
kN / n
with respect to z j may then be defined in a usual way ∞ ∞ α k /n α k /n k /n f ( z1 ,..., z N ) = ∑ ... ∑ ck1 ...k N z1 1 ... z j j ...z N N . αz j k1 = k 01 k N = k 0 N αz j
5 5.1
α-differential equations α-differential equations and conjoint ordinary differential equations
In this section we study some equations, which involve α-derivatives. Consider a linear αdifferential equation with constant coefficients M αl (22) a f ( z ) = h( z ) , α = 1 / n , ∑ l αz l l =0 ∞
h( z ) = ∑ hk z
k n
k =0
For the solution of α-differential equations we use the method of power series ∞
(23)
k
f ( z ) = ∑ ck z n . k =0
For the solution we account the auxiliary linear ordinary differential equation M dl ~ (24) al l F ( x ) = H ( x ) , ∑ dx l =0 ∞
H ( x ) = ∑ hk x k . k =0
The ordinary differential equation (24) will be referred to as a co-joint equation to the equation (22). The solution of the co-joint equation (24) with the corresponding boundary conditions is obtained using standard methods [21, Chapter IV]. This solution is assumed a-priory known and is represented in the form ∞
(25)
F = ∑ ck x k . k =0
Because the solution of co-joint ordinary differential equation is known, the coefficients ck of its Taylor expansion are also known. Substitution of (25) in (22) delivers k − ml l −1 k ∞ M k − im n − hk z n = 0 . ∑ al ck z ∑ ∏ n k = 0 l =0 i=0 9
Equating the terms with the equal exponents we get the equation for coefficients ck : l −1 k − im a c ∑ l k ∏ n − hk −ml = 0 . l =0 i =0 Consider the case m = 1, α = 1 / n . The last equation reduces to the following indicial equation M
l −1 l −1 a c α (k − i ) − hk −l = 0 ∑ ∏ l k l =0 i =0 The same procedure being applied to the equation (26) results in the series l −1 ∞ M ~ k −l a c x (k − i ) − hk x k = 0 ∑ ∏ ∑ l k k =0 l =0 i =0 and corresponding indicial equation for coefficients ck M
(26)
~ l −1 ∑ al ck ∏ (k − i ) − hk −l = 0 . l =0 i=0 Comparison of expressions (26) and (27) demonstrates, then if the coefficients of the co-joint ordinary differential equation are ~ (28) a~l = alα l −1 , bl = blα l −1 M
(27)
the formal power series (23) and (25) contain the same coefficients ck . This observation allows an immediate method for solution of α-differential equations with coefficients al , bl . The co-joint equation is an ordinary differential equation. Its general solution F ( x ) of Eq. (24) could be obtained by common methods. To evaluate the coefficients ck ,
the solution F ( x ) should be expanded in power series. Then, the fractional power series with coefficients ck delivers the general solution of the α-differential equation (22). 5.2
The α-differential equations of the first order
The α-exponent is defined as (29)
(z
Eα ( z ) = ∑
α
α)
zα ≡ exp k = 0 Γ (k + 1) α E1 ( z ) = exp( z ) , Eα (0 ) = 1 for 0 < α < 1 . ∞
def
k
,
This definition is motivated by the fact, that the function Eα ( z ) satisfies the α-differential equation α Eα ( z ) = Eα ( z ) with Eα (0 ) = 1 . αz The term-to-term differentiation of series (29) proves this solution of α-differential equation: ∞ α 1 α αk ∞ αk Eα ( z ) = ∑ z =∑ zα (k −1) = k k ( ) ( ) αz Γ k + 1 α z Γ k + 1 α α k =0 k =1 ∞
=∑
1
z α (k −1) = k −1
∞
1
∑ Γ(n + 1)α
zαn ≡ Eα ( z )
n = k −1 Γ(k )α n=0 Worth while mentioned, that Eα (z ) does not possess the semigroup property, k =1
n
Eα ( z1 + z2 ) ≠ Eα ( z1 )Eα ( z2 ) .
The function Eα ( z / λ ) with an arbitrary κ satisfies equation 10
(30) 5.3
λα
α z z Eα = Eα with Eα (0 ) = 1 . αz λ λ
The α-differential equations of the second order
The following equality holds for z > 0 : (31) Eα (i z ) = Cα ( z ) + i Sα ( z ) , where i is the imaginary unit. The functions α zα απ z απ Cα ( z ) = exp cos cos sin 2 α 2 α
(
)
α ∞ απ z α k , = ∑ cos k =0 2 Γ(k + 1)
(
k
)
α α zα απ z α απ z απ ∞ Sα ( z ) = exp cos k sin sin = ∑ sin 2 α 2 k =1 2 Γ(k + 1) α could be considered as the α-trigonometric functions: C1 ( z ) = cos( z ) , S1 ( z ) = sin ( z ) . The functions Cα ( z ) and Sα ( z ) appear in the solutions of α-differential equation the second order. To show this, consider the following equation α α α f ( z ) + 2d f (z ) + ω 2 f (z ) = 0 . (32) α z αz αz The substitution of (23) in (32) delivers the indicial equation α 2ck + 2 (k + 1)(k + 2) + 2dαck +1 (k + 1) + ω 2ck = 0 with the solution (− 1)k (d + p )k ( p − d ) + ( p − d )k (d + p ) c + ( p − d )k − (− 1)k (d + p )k c ck = 0 1 2 pΓ(k + 1)α k 2 pΓ(k + 1)α k −1 k
q = − d + p, p = d 2 −ω2 The general solution of Eq. (32) is in form α-trigonometric functions f ( z ) = c0 f 0 ( z ) + c1 f1 ( z ) , (33) with zα d zα ω 2 − d 2 zα d z α ω 2 − d 2 d sin cos f0 (z ) = exp − + exp − α ω α ω2 − d 2 ω zα d zα ω 2 − d 2 α sin f1 ( z ) = exp − ω α ω2 − d 2 and the following properties lim f 0 ( z ) = 0 , lim f 1 ( z ) = 0 , lim f 0 ( z ) = −1 , lim f1 ( z ) = 0 . z →0
z →0
z →∞
z →∞
The α-derivative of f ( z ) is α α α (34) f ( z ) = c0 f 0 ( z ) + c1 f1 ( z ) = c0 g 0 ( z ) + c1 g1 (z ) ≡ g ( z ) αz αz αz with z α d z α ω 2 − d 2 ω2 sin g 0 (z ) = exp − , ω α ω2 − d 2
11
−1 ,
2 2 zα ω 2 − d 2 α zα d α − d sin z ω − d ω 2 − d 2 cos g1 ( z ) = exp − 2 2 α α ω ω − d and lim g 0 ( z ) = 0 , lim g1 ( z ) = α , lim g 0 ( z ) = 0 , lim g1 ( z ) = 0 . z →0
z →0
z →∞
z →∞
The following boundary value problem for the equation (32) is correctly stated: α f (0) = A , f (z ) = B. αz z →∞ For this problem is c0 = − A, c1 = B / α . Surprisingly, two boundary conditions of α-differential equation the second order for 0 < α < 1 must be separately assigned at the points (0, ∞) . The ordinary differential equation the second order requires two boundary conditions at the point z = 0 . As we can see, some properties of α-differential equations are familiar to those of ordinary differential equations.
6 6.1
α-differential geometry Manifold
In this Section we build up, following the standard modus operandi, the main objects of differential geometry, based on the concept of α-derivative. Let RN be the set of all N -tuples of real numbers ( z1 ,..., z N ) . A set M of points is a topological manifold if each point P in it has an open neighborhood U homeomorphic to some open set V in RN . In other words, there is a bi-continuous bijection (1-1 onto map)22
φ :U →V by P ֏ φ (P ) = ( z1 (P ),..., z N (P ))
for all P in M . The N numbers z1 (P ),..., z N (P ) are called the coordinates of P . N is the dimension of M . Thus, the topology of M is the same as RN locally. The pair (U , φ ) is called a chart, or a local coordinate system. An atlas on M is a set {(U a , φa )} of charts so that the domains {U a } covers M . Thus, every P is in some U a . An atlas of class C k requires the maps φb φa −1 : φa U a ∩U b → φb U a ∩U b .
(
)
(
)
to be maps of class C k . −1 Note that φb φa is a map between open sets of RN . In fact, it represents a coordinate transformation for points in the overlap region U a ∩U b of two coordinate systems given by φa and φb . A manifold with an atlas of class C k is said to be a C k manifold. Those with k > 1 are called differentiable manifolds. For convenience, we shall deal only with C ∞ manifolds. 6.2
Curve
A curve is a differentiable mapping C from an open set of R into M , i.e., C : R → M with λ ֏ P(λ ) = x i (λ ), i = 1,..., N .
{
}
where λ is the parameter of the curve. α − differentiability here means that x i (λ ) are α − differentiable functions of λ . 12
6.3
Functions
A function f on M is an assignment of a real number f (P ) to each point P in M . This is denoted by f :M → R with P ֏ f (P ) If a region U ⊂ M is mapped α − differentially onto some region of Rn with
{
} {}
( )
P ֏ x i , i = 1,.., N = x i , we can write f (P ) = f x i so that f is a function on RN . If f is
α − differentiable in RN , we say f is α − differentiable in M . 6.4
Vectors as Tangents to Curves
( )
Consider a curve C (λ ) described by x i = xi (λ ) in M . Let f x i be a function on M . Consider the points on the curve. Function f can be taken as a function of
λ through g (λ ) = f [x i (λ )] . Thus, N αg df =∑ αλ i =1 dxi N αx α =∑ i αλ i =1 αλ
αxi , αλ ∂ ∂xi
From the definition, it follows that the functions x i = x i (λ ) , which represent the curve C in
{ }
other coordinates x i , are determined by the functions x i = xi (λ ) :
(35)
(
)
x = x x (λ ),..., x N (λ ) . i
i
i
(
1
1
Here the x x ,..., x
N
) are the transformation functions from {x } to {x }. i
i
The α − tangential vector t i (α ) to C at a point P(λ ) is determined by its components with
{}
respect to x i . If the point corresponds to the value λ0 of the parameter λ we have
αx i t (α ) = at λ = λ0 , i = 1,..., N . αλ In differential geometry a very important question is how a quantity transforms if the coordinates are changed. j N N αx i ∂x i αx j ∂x i i i αx i t (α ) = =∑ =∑Xj , Xj = j at λ = λ0 αλ j =1 ∂x j αλ αλ ∂x j =1 Hence, the tangential vector is an example of a quantity with the following properties: 1. It is always connected to a particular point P(λ0 ) of M . i
{}
(
)
2. In the coordinates x i around P(λ0 ) , it is represented by n components t 1 ,..., t N . 3. These components transform as follows: N
t i (α ) = ∑ X ij t j (α ) . j =1
Such a quantity is a vector. A α − vector is defined as the α − tangent to some curve in the αxi manifold. Now are the components of a vector α − tangent to the curve C . Thus, if we αλ ∂ α treat e i = , the vector can be identified as the α − tangent vector to the curve C(λ ) ∂xi αλ at point P(λ ) . 13
It is straightforward to verify that the set of all α − tangent vectors at a point P forms a vector space called the α − tangent space to M at P and denoted by TP ,α . Closure under addition and scalar multiplication is obviously proved. Note that the vector space defined above consists only of tangents at the same point in M . We define a α − vector field as a rule for assigning a vector at each point of M . Given a coordinate system x i for a neighborhood U of M , we call ∂ ∂ x i the coordinate basis of TP ,α for all points in U. One important characteristics of a coordinate basis is that its members commute, ∂ ∂ ∂ ∂ ∂ ∂ ∂x i , ∂x j = ∂x i ∂x j − ∂x j ∂x i = 0 . Thus N N ∂ V = ∑ V i i = ∑ V ′i e i ∂x i =1 i =1 i where V ′ is the component of V along e i .
{}
6.5
{
}
Vector Fields and Integral Curves
A α − vector field is a rule that selects a vector from the α − tangent vector space at each point of M . Consider a α − vector field V i (P ) for P ∈ M . Given a coordinate system x i , we have V i (P ) = v i x i .
{}
( )
The tangent vector to a curve x i (λ ) is given by αxi = vi xi . αλ It is just a set of α − differential equations, assuming that a solution exists in some neighborhood around any given point. Hence, given a vector field v i x i , a solution, called an integral curve, o is a curve whose α − tangent is everywhere equal to the vector field. By judicial choice of initial conditions, one can find a set of integral curves that fills up M . Such a set of curves is called a α − congruence.
( )
( )
6.6
Non-Coordinate Basis
Arbitrary vector field need not commute. For α α V= ,W = αλ αµ we have N N ∂W j ∂V j ∂ α α α α ∂ i i [[V, W ]] = − =∑ V − W = U j j. ∑ i i i αλ αµ αµ αλ i , j =1 ∂x ∂x ∂x ∂x j =1 Here N ∂W j ∂V j i . U j = ∑ V i − W ∂x i ∂x i i
14
6.7
α − Affine Connection of a Manifold
The definition of α − affine connection is the following. Let M be a N -manifold. For physical or geometric reasons a class of curves may be selected in a way that the coordinate representation x i (λ ) of any such curve satisfies the following differential equations: (36)
α αx i N N i αx j α x k + ∑∑ γ jk = 0. αλ αλ j =1 k =1 αλ αλ
Any such solution of this system defines a curve of this class. Here, the functions γ ijk are assumed to be Puiseux functions of x i , and we demand γ ijk ( x ) = γ kji ( x ), ∀x, i, j , k . The curves then define an affine connection on M . The γ ijk are called the components of the
α − affine connection and the curves the α − autoparallels of the α − affine connection, and the manifold is called α − affine connected. The system (36) consists of N coupled, ordinary, non-linear differential equations of order 2α . They are all solved for 2α derivatives; thus for every point P ∈ M with coordinates x i i and each vector t(α ) we get exactly one α − autoparallel in this point. Moreover the differential equation (36) is invariant with respect to affine transformations of the parameter λ. The parameterization of the α − autoparallel is therefore, given up to an α − affine transformation. An index-carrying quantity (IQ) is defined in [23] as a multidimensional table of numbers, the so-called components of the IQ, which are labeled by indices. We want to apply the above rules and conventions to calculations with IQs in order to derive the transformation law for the components of the α − affine connection. Let us calculate the derivatives N αx i αx j ∂x i = ∑ X ij , X ij = j αλ j =1 αλ ∂x N α αx i α α x j N N i α x j αx k = ∑ X ij + ∑∑ X , αλ αλ j =1 αλ αλ j =1 k =1 jk αλ αλ
∂ 2 xi . ∂x j ∂x k Insert these expressions in (36) we obtain α αx j αx j αx k + X ijk + γ lmi X lj X km X ij =0 αλ αλ αλ αλ and finally α αx r αx j αx k + X ir X ijk + γ lmi X ir X lj X km = 0. αλ αλ α λ αλ This equation delivers desired transformation law for the components of the α − affine connection γ rjk = X ir X ijk + γ lmi X ir X lj X km .
X ijk =
(
(
)
)
The transformation law for the components of the α − affine connection is the same as the transformation law for the components of the common affine connection.
7
α − Torsion and α − Riemann tensor
With the above formulas for the derivatives of basis vectors, we can find the derivatives of arbitrary tensors. If U = α αλ , then 15
(
(
)
)
∇ (Uα ) V = U i ∇ e(αi ) V j e j = U i ∇ (eαi )V j e j + U iV j ∇ e(αi )e j .
The first term in this expression is the α − derivative αV j = U i ∇ e(αi )V j . αλ With the expression ∇ e(αi )e j = γ kji e k we get
(
(37)
∇ (Uα ) V =
)
αV j e j + U iV j γ kji e k . αλ (α )
The two quantities [U, V ] and ∇ (Uα ) V − ∇ (Vα ) U are both vector fields and both antisymmetric in U and V . The α − Torsion is introduced by the expression (38)
[[
]]
∇ e(αj )ei − ∇ e(αi )e j − ei , e j ≡ T jik e k . (α )
The operator R is defined by (39)
[[∇
(α ) U
]]
, ∇ (Vα ) − ∇ (α )
[[U, V ]]
(α )
≡ R (U, V ) .
For an arbitrary function f we have (α )
(α )
R (U, V ) fW = f R (U, V )W ,
(α )
(α )
(α )
(α )
R ( fU, V ) fW = f R (U, V )W ,
R (U, fV ) fW = f R (U, V )W . Because of these properties, (39) actually defines a tensor, which will be referred to as the α − Riemann tensor. Since the left-hand side operates on a vector to give a new vector, (39) (α ) 1 shows that R (U, V ) is a tensor for given vectors U, V . With U, V also regarded as vari 1
1 able arguments, the α − Riemann tensor becomes a tensor. 3 (α )
The components of the α − Riemann tensor R ijkl , are defined by (40)
[[∇
(α ) i
]]
(α )
l , ∇ (jα ) e k − ∇ [([αei),e j ]]e k ≡ Rkij el . (α ) i jkl
In an α − coordinate basis the components of the α − Riemann tensor R , are (41)
8
(α ) l kij
R =
αγ kjl αxi
−
αγ kil l l + γ kjmγ mi − γ kimγ mj . αx j
Conclusion
In the Article the new fractional derivative, that fulfils the Leibnitz rule, is introduced. The application of α-derivative to a polynomial the polynomial with fractional power series converts it again to a polynomial with fractional power series. α-derivative of Puiseux series is Puiseux series. The fractional differential equations are solved by the formal power expansion method. The objects of the fractional differential geometry are formally introduced via the common methods of Riemannian geometry.
16
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