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Generalized class of fractional inverse Lagrange expansion
Available online at www.sciencedirect.com
ScienceDirect Journal of Taibah University for Science 8 (2014) 175–185
Generalized class of fractional inverse Lagrange expansion F.A. Abd El-Salam a,b,∗ a
Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah, KSA Department of Astronomy, Faculty of Science, Cairo University, Cairo 12613, Egypt
b
Received 21 November 2012; received in revised form 16 August 2013; accepted 17 August 2013 Available online 2 October 2013
Abstract In this work the left and right Riemann–Liouville derivatives are introduced. A generalized operator based on left/right Riemann–Liouville derivatives is obtained. Basic definitions, lemmas and theorems in the fractional calculus which is related to our purpose are presented. The fractional form of Lagrange expansion theorem is obtained. © 2013 Taibah University. Production and hosting by Elsevier B.V. All rights reserved. Keywords: Fractional right/left Riemann–Liouville derivative; Generalized operator; Lagrange expansion; Delta-function
1. Introduction The development of the FC theory is due to the contributions of many mathematicians such as Euler, Liouville, Riemann, and Letnikov. Several definitions of a fractional derivative have been proposed. These definitions include Riemann–Liouville, Grunwald-Letnikov,Weyl, Caputo, Marchaud, and Riesz fractional derivatives, see [1–4]. Riemann–Liouville derivative is the most used generalization of the derivative. It is based on the direct generalization of Cauchy’s formula for calculating an n-fold or repeated integral. The right and the left Riemann–Liouville fractional derivatives, in brief, are denoted by RRLFD and LRLFD respectively, see [5]. In [6–8] Agrawal has studied Fractional variational problems using the Riemann–Liouville derivatives. He notes that even if the initial functional problems only deal with the left Riemann–Liouville derivative, the right Riemann–Liouville derivative appears naturally during the computations. In what follows, we construct an operator combining the left and right Riemann–Liouville (RL) derivative. We remind some results concerning functional spaces associated to the left and right RL derivative. In particular, we discuss the possibility to obtain a law of exponents. In 1770, Joseph Louis Lagrange (1736–1813) published his power series solution of the implicit equation. However, his solution used cumbersome series expansions of logarithms [9,10]. This expansion was generalized by Bürmann [11–13]. There is a straightforward derivation using complex analysis and contour integration; the complex formal power series version is clearly a consequence of knowing the formula for polynomials, so the theory of analytic functions may be applied. Actually, the machinery from analytic function theory enters only in a formal way in this proof. In ∗ Tel.: +966 543234803. Peer review under responsibility of Taibah University
1658-3655 © 2013 Taibah University. Production and hosting by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jtusci.2013.08.001
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1780, Pierre-Simon Laplace (1749–1827) published a simpler proof of the theorem, based on relations between partial derivatives with respect to the variable and the parameter, see [14,15]. Charles Hermite (1822–1901) presented the most straightforward proof of the theorem by using contour integration [16–18]. In mathematical analysis, this series expansion is known as Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, and gives the Taylor series expansion of the inverse function. Suppose z = f(w) where f is analytic function at a point a and f(a) = / 0. Then it is possible to invert or solve the equation for w such that w = g(z) on a neighborhood of f(a), where g is analytic at the point f(a). This is also called reversion of series. The series expansion of g is given by n ∞ (z − f (a))n d n−1 w−a lim w = g(z) = a + w→a n! dwn−1 f (w) − f (a) n=1
In this work we will apply the concepts of fractional calculus to obtain a fractional form of Lagrange expansion and some generalizations. Definition 1. By D, we denote the operator that maps a differentiable function onto its integer derivative, i.e. Df(x) = f , by Ja , we denote the integer integration operator that maps a function f, assumed to be (Riemann) integrable on the x compact interval [a,b], onto its primitive centered at a, i.e. Ja f (x) = a f (t)dt, ∀ a ≤ x ≤ b. Definition 2. By Dn and Jan , n ∈ N we denote the n-fold iterates of D and Ja , respectively. Note that Dn is the left inverse of Jan in a suitable space of functions. Lemma 1. Let f be Riemann integrable on [a,b]. Then, for a ≤ x ≤ b and n ∈ N, we have x 1 Jan f (x) = (x − t)n−1 f (t)dt, n ∈ N. (n − 1)! a
(1)
2. Left and right Riemann–Liouville derivatives and integrals We define the left and right Riemann–Liouville derivatives following [19,20]. In all the following definitions Γ (α) represents the Euler gamma function given by ∞ Γ (α) = xα−1 e−α dx = (α − 1)! 0
Definition 3. Let f be a function defined on (a, b) ⊂ R, and α ∈ R+ . Then the left and right Riemann–Liouville fractional integral of order α is on respective defined on Lebesgue space L1 [a, b] are given by x 1 −α (x − s)α−1 f (s)ds, a ≤ x ≤ b, α∈R (2) a Dx f (x) = Γ (α) a b 1 −α (3) (s − x)α−1 f (s)ds, a ≤ x ≤ b, α∈R x Db f (x) = Γ (α) x Definition 4. Let f be a function defined on (a, b) ⊂ R, and α ∈ R+ , the left and right Riemann–Liouville derivative α of order α defined on Lebesgue space L1 [a, b] and denoted by a D−α x and x Db respectively, are defined by n x 1 d α D f (x) = (x − s)n−α−1 f (s)ds a ≤ x ≤ b, α∈R (4) a x Γ (n − α) dx a n b (−1)n d α D f (x) = (x − s)n−α−1 f (s)ds, a ≤ x ≤ b, α∈R (5) x b Γ (n − α) dx x Left and right (RL) integrals satisfy some important properties like the semi-group property.
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n Remark 1. It is evident that a D−α x ≡ Ja , ∀α ∈ N, except for the fact that we have extended the domain from Riemann integrable functions to Lebesgue integrable functions (which will not lead to any problems in our development). Moreover, in the case α ≥ 1 it is obvious that the integral a D−α x f (x) exists for every x ∈ [a, b] because the integrand is the product of an integrable function f and the continuous function (x − • )α−1 . One important property of integer-order integral operators is preserved by this generalization. i.e. ∀ α, β > 0, f(x) ∈ L1 [a, b] −α −β a Dx (a Dx f (x))
−α −α−β = a D−β f (x) x (a Dx f (x)) = a Dx
(6)
The fractional differential and integral operators, namely differeintegral operator a D±α x can be obtained from the ±β generalization of the two operators x Db as, n n d d ±α ±α−n ±α ±α−n n , x Db = (−1) (7) a Dx = a Dx x Db dxn dxn where the number of additional differentiations n is equal to [α] + 1, where [α] is the whole part of α. The generalized ±β operator x Db satisfies ⎧ dα ⎪ , Fractional differentiation mapping ⎪ ⎪ ⎪ dxα ⎪ ⎪ ⎨ 1, Fractional unitary mapping ±α a Dx = x ⎪ ⎪ ⎪ ⎪ dt −α , Fractional integration mapping ⎪ ⎪ ⎩ a
If α = m, m ∈ N, and f ∈ Cm ([a, b]) we have dL±m f dR±m f ±m , D f = − . (8) x b dx±m dx±m This last relation which ensures the gluing of the left and right Riemann–Liouville (RL) differeintegral to the classical derivative and integral operator will be of fundamental importance. These fractional operators satisfy the semi-group property. If f(t) ∈ C0 with left and right-differeintegral at point t denoted by dL±m f/dt ±m and dR±m f/dt ±m , respectively. β β In what follows, we denote by αa I, Ib and αa Ib the functional spaces defined by α ±m a Dt f exists}, a I = {f ∈ C([a, b]), (9) β Ib = {f ∈ C([a, b]), t Dt f exists}, ±m a Dx f
=
and α β a Ib
β
= αa I ∩ Ib . β
(10) β
As is clear the set αa Ib is non-empty. We have AC([a, b]) ⊂ αa Ib , where AC([a, b]) is the set of absolutely continuous functions on the interval [a,b]. The operators of ordinary differentiation or integration of integer order satisfy a commutativity property and the law of exponents (the semi-group property) i.e. ±n ±m ±m ±n ±n±m dL,R dL,R dL,R dL,R dL,R f f f f f × = × = . (11) ±n ±m ±m ±n ±n±m dx dx dx dx dx These two properties in general fail to be satisfied by the left and right fractional RL differenitegral operator. These bad properties are responsible for several difficulties in the study of fractional differential equations.
2.1. Fractional functional spaces Following [21] we introduce the left and right fractional differenitegral as well as convenient functional spaces on which we have the semi-group property. In [21] several useful functional spaces are introduced. Let I ⊂ R be an
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open interval (which may be unbounded). We denote by C0∞ (I) the set of all functions f ∈ C∞ (I) that vanish outside a compact subset K of I. Definition 5. [Left fractional differenitegral space] Let α > 0. Define the semi-norm (A nonzero vector may have a zero length) |f |JLα (R) = ||a D±α x f ||J 2 (R) L
and norm 1/2
||f ||JLα (R) = (||f ||2J 2 (R) + |f |2J 2 (R) ) L
L
and let JLα (R) denote the closure of C0∞ (R) with respect to || · ||JLα (R) . Definition 6. [Right fractional differenitegral space] Let α > 0. Define the semi-norm |f |JLα (R) = ||x D±α b f ||J 2 (R) L
1/2
||f ||JLα (R) = (||f ||2J 2 (R) + |f |2J 2 (R) ) L
L
and let JLα (R) denote the closure of C0∞ (R) with respect to || · ||JLα (R) . Similarly, we can define the right fractional derivative space. We now assume that I is a bounded open subinterval of R. We restrict the fractional differeintegral spaces to I. Define α (I), J α (I) as the closure of C ∞ (I) under their respective norms. These spaces have very interesting the spaces JL,0 R,0 0 ±α properties with respect to a D±α x and x Db . In particular, we have the following semi-group property: β semi-Group: For f ∈ JL,0 (I), 0 < α < β we have ±γ a Dx f
±γ∓α = a D±α f x a Dx
(12)
β
and similarly for x ∈ JR,0 (I), ±γ x Db f
±γ∓α
= x D±α b x Db
(13)
f
see [22] Lemma 2.9 there for a proof. α (I) and J α (I) have been characterized when α > 0. We denote by W s,α (I) the The fractional derivative spaces JL,0 R,0 0 fractional Sobolev space. α (I), J α (I) and W s,α (I) spaces are equal, see [22] Theorem 2.13 for a proof. In fact, when Let α > 0. Then the JL,0 R,0 0 α (I), J α (I) and Iα (I) spaces have equivalent semi-norms and αn − 1/2, n ∈ N we have a stronger result as the JL,0 0 R,0 norms. 2.2. Generalized operator As we want to deal with dynamical systems exhibiting the arrow of time, we need to consider the operator a D±α x ±β and x Db , in order to keep track of the past and future of the dynamics. The fact that we consider αβ is only here for convenience. This can be used to take into account a different quantity of information from the past and the future. ±β
α,β Definition 7. Let a D±α of the form x and x Db be given. We look for an operator D ±β
Dα,β = M(a D±α x , x Db ) where M : R2 → C is a mapping which does not depend on (α, β), satisfying the following general principles:
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d ±m f
L,R • E1: Gluing property: If f(x) ∈ C1 then when α = β = m, m ∈ N, Dm,m f (x) = dx ±m . • E2: M is a R-linear mapping. • E3: Reconstruction: The mapping M is invertible. E1 is fundamental in the embedding framework. It follows that all or constructions can be seen as a continuous two-parameters deformation of the corresponding classical one. This can be of importance dealing with the fractional quantization problem in classical mechanics. In other words E1 is not the usual condition underlying the stochastic or quantum embedding theories. In general, we have an injective mapping ι from the set of differentiable functions C1 in a bigger functional space ε such that the operator D that we define on ε reduces to the classical derivative on ι(C1 ) meaning that for f ∈ C1 , D(ι(f )) = D(ι(f )) where f (x) = df/dx. As a consequence, we have a true embedding in this case, meaning that the embed theory already contains the classical one via the mapping ι. Here, the classical theory is not contained in the embedded theory but can be recovered by a continuous two-parameters deformation. E2 does not have a particular meaning. This is only the simplest dependence of the operator D with respect to a D±α x ±β and x Db . E3 is important. It means that the data of D±α,β on a given function f at point x allow us to recover the left and right RL derivatives of f at x, so information about x in a neighborhood of f(x). Operators satisfying E1, E2, and E3 are of the form
±β
±β
±α D±α,±β = [pa D±α x + (p − 1)x Db ] + iq[a Dx + x Db ]
(14)
where p, q ∈ R and p − q = 1, q = / 0. By E2, we denote M(η, ξ) = pη + qξ + i(rη + sξ), with p, q, r, s ∈ R and r − s ∈ R. By E1, we must have with ξ = − η corresponding to the operator a choice of operators ±df/dx. We then already have an operator of the form (14). The reconstruction assumption only imposes that q = / 0 in (14). A more rigid form for these operators is obtained imposing a symmetry condition. • E4: Let f ∈ C0 be a real valued function possessing left and right classical derivatives at point x, denoted by d+ f/dx and d− f/dx, respectively. E4: must be seen as the non-differentiable pendant of E1. Indeed, E1 can be rephrased as follows: if f ∈ C0 is such that d+ f and d− f exist and satisfy d+ f = d− f then D1,1 f = d + f . E4 is then equivalent to the commutativity of the following diagram, where R2 is seen as the R-vector space associated to C: C
τ
→
η − iη →
C η + iη
(15)
where τ : C → C is defined by τ(z) = iz, z ∈ C and we have used the fact that M(η, − η) = 0 following E1. The unique operator satisfying E1, E2, E3 and E4 is given by D±α,±β =
1 i ±β ±β ±α [a D±α x − x D b ] + [a D x + x D b ] 2 2
(16)
By (iv), we must have p + (p − 1) = 0 and 2q = 1, so that p = q = 1/2. 3. The fractional operator of order (α, β) For all a, b ∈ R, a < b, the fractional operator of order (α, β), α > 0, β > 0, denoted by Dα,β μ , is defined by 1 iμ ±β ±β [a D±α [a D±α μ∈C (17) x − x Db ] + x + x Db ], 2 2 when α = β = ±1, we obtain D1,1 D−1,−1 = , respectively. μ = d/dx, μ The free parameter μ can be used to reduce the operator Dα,β μ to some special cases of importance. Let us denotes by f(x) a given real valued function. D±α,±β = μ
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±α 1. For μ = − i, we have Dα,β μ = a Dx then dealing with an operator using the future state denoted by Px (f ) of the underlying function, i.e. Px (f ) = {f (s), s ∈ (a, x]}. ±β 2. For μ = i, we obtain Dα,β μ = −x Db then dealing with an operator using the past state denoted by Pt (x) of the underlying function, i.e. Px (f ) = {f (s), s ∈ (x, b}. ±β
As a consequence, our operator can be used to deal with problems using, a D±α x , x Db , or both operators in a unified way, only particularizing the value of μ at the end to recover the desired framework. 3.1. Product rules The classical product rule for Riemann–Liouville derivatives is for all α > 0 b b ±α f (x)x D±α a Dx f (x)g(x)dx = b g(x)dx a
(18)
a
as long as f(a) = f(b) or g(a) = g(b). ±β This formula gives a strong connection between a D±α x and x Db via a generalized integration by part. This relation ±β is responsible for the emergence of x Db in problems of fractional calculus of variations only dealing with a D±α x . This result also justifies our approach to the construction of a fractional operator which put on the same level as the left and right RL derivatives. As a consequence, we obtain the following formula for our fractional operator: β For all f, g ∈ αa Fb , we have b b α,β α,β D f (x)g(x)dx = − f (x)x D−μ g(x)dx (19) a μ a
a
provided that f(a) = f(b) or g(a) = g(b). We have b b f (x)g(x)dx = − f (x)[(−) + iμ(+)](g(x))dx a
(20)
a
Exchanging the role of α and β in [(−) + iμ(+)], we obtain the operator [(−) + iμ(+)] which can be written as α,β
−[(−) − iμ(+)] = −D−μ
(21)
This concludes the proof. Here again, we see that it is convenient to keep the parameter μ free. In what follows we shall generalize some lemmas and theorems proved by Abd El-Salam [23]. m m−α,m−β . Lemma 2. Let α, β ∈ R+ and let m ∈ N such that m > α, m > β. Then Dα,β μ = Da D μ
Proof. Since m > α, m > β yields m≥ α , m≥ β. Thus 1 iμ m−β m−β Dm Dm−α,m−β = Dm [a Dm−α − x Db ] + Dm [a Dm−α + x Db ] x x μ 2 2 1 β−β m−β α−α = [Dα Dm−α a Dm−α − Dβ Dm−β x Db ] x Db a Dx x 2 iμ β−β m−β α−α + [Dα Dm−α a Dm−α + Dβ Dm−β x Db ] a Dx x Db x 2 1 = [Dα+m−α+α−m+α−α − Dβ+m−β+β−m+β−β ] 2 iμ + [Dα+m−α+α−m+α−α + Dβ+m−β+β−m+β−β ] 2 =
1 α iμ [D − Dβ ] + [Dα + Dβ ] = Dα,β μ 2 2
(22)
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−α,−β Theorem 1. Let (α, β≥0) ∈ R+ . Then, for every f(x) ∈ L1 [a, b], we have Dα,β f (x) = f (x). μ Dμ
Proof. For α = β = 0 both operators are the identities. For α, β > 0, let m≥ α , m≥ β, then −α,−β Dα,β f (x) μ Dμ
= Dam Dα−m,β−m D−α,−β f (x) = Dam D−(m−α),−(m−β) D−α,−β f (x) μ μ μ μ = Dam D−(m−α+α),−(m−β+β) f (x) = D−(m−m−α+α),−(m−m−β+β) f (x) = f (x) μ μ
(23)
Corollary 1. Let f be analytic in (a − h, a + h) for some h > 0, and let (α, β≥0) ∈ R+ . Then we have I-1), & D-1) ∀ a ≤ x < a + h/2 and we have I-2), & D-2) ∀ a ≤ x < a + h I-1)
1 iμ −β −β μ∈C [a D−α [a D−α x − x Db ]f (x) + x + x Db ]f (x), 2 2 1 iμ −β −β = [a D−α [a D−α x f (x) − x Db f (x)] + x f (x) + x Db f (x)] 2 2
D−α,−β f (x) = μ
D−α,−β f (x) μ
∞
∞ 1 (−1)m (x − μ)k+α k (−1)m (x − μ)k+β k = Dμ f (x) − Dμ f (x) 2 k!(α + k)Γ (α) k!(β + k)Γ (β) k=0 k=0
∞ ∞ iμ (−1)m (x − μ)k+α k (−1)m (x − μ)k+β k + Dμ f (x) + D f (x) 2 k!(α + k)Γ (α) k!(β + k)Γ (β) μ k=0
∞
∞ k+β (x − μ)k+α (x − μ) 1 I-2) D−α,−β f (x) = Dk f (x) − Dk f (x) μ 2 Γ (k + 1 + α) μ Γ (k + 1 + β) μ k=0 k=0 ∞
∞ k+α iμ (x − μ) (x − μ)k+β k k + D f (x) + D f (x) 2 Γ (k + 1 + α) μ Γ (k + 1 + β) μ k=0
D-1)
Dα,β μ f (x)
D-2)
∞ ∞ β (x − μ)k−β k 1 α (x − μ)k−α k = D f (x) − D f (x) k (k + 1 − α) μ k (k + 1 − β) μ 2 k=0 k=0 ∞
∞ β (x − μ)k−β k iμ α (x − μ)k−α k D f (x) + D f (x) + k (k + 1 − α) μ k (k + 1 − β) μ 2
(26)
k=0
∞ ∞ (x − μ)k−β k 1 (x − μ)k−α k = D f (x) − D f (x) 2 (k + 1 − α) μ (k + 1 − β) μ k=0 k=0
∞ ∞ (x − μ)k−β k iμ (x − μ)k−α k D f (x) + D f (x) + 2 (k + 1 − α) μ (k + 1 − β) μ k=0
(25)
k=0
k=0
Dα,β μ f (x)
(24)
k=0
(27)
k=0
the binomial coefficients for α, β ∈ R and k ∈ N is defined as α α(α − 1)(α − 2). . .(α − k + 1) α! = = k k! k!(α − k)! Proof. For the first two statements I-1), I-2), we use the definition of the Riemann–Liouville integral operator D−α a , and expand f(t) into a power series about x. Since x ∈ [a, a + h/2), the power series converges in the entire interval
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of integration. And exchange summation and integration. Then we use the explicit representation for the fractional integral of the power function k D−α a (x − a) =
Γ (k + 1) (x − a)k+α (α + k + 1)
I-1) follows immediately. For the second statement, we proceed in a similar way, but we now expand the power series at a and not at x. This allows us again to conclude the convergence of the series in the required interval. The analyticity of D−α a follows immediately from the second statement. To prove D-1) use is made of Dαμ = Dα D−α−α μ and
k!Γ (α)(α + k)
−α k
= (−1)k Γ (k + 1 + α)
This allows us to rewrite the statement I-1) as ∞ − α − α (x − a)k−α−α α−α Dμ f (x) = Dk f (x) k Γ (k + 1 − α − α) μ k=0
Differentiating α times with respect to x, we find α D−α−α f (x) Dμ μ
= Dαμ f (x) =
∞
⎛ ⎝
α − α k
k=0
⎞ ⎠
1 k f ](x) Dα [(• − μ)k+α−α Dμ Γ (k + 1 + α − α) μ
The classical version of Leibniz’s formula yields α ∞ α − α α 1 α−j • α k+j Dμ Dμ f (x) = [( − μ)k+α−α ](x)Dμ f j k Γ (k + 1 + α − α) j=0
k=0
which yields Dαμ f (x)
α ∞ α − α α [(x − a)k+j−α ] k+j = f (x)Dμ j k Γ (k + 1 + j − α) k=0
j=0
i By definition, = 0 if i ∈ N and i < j. Thus we may replace the upper limit in the inner sum by ∞ without j changing the expression. The substitution j = l − k gives ∞ ∞ α α − α [(x − μ)l−α ] l Dαμ f (x) = (x)Dμ f l − k Γ (l + 1 − α) k l=0 k=0
∞ Σ∞ = Σ∞ Σl Using the fact that Σl=0 k=0 l=0 k=0 ∞ l α α − α [(x − μ)l−α ] α l Dμ f (x) = (x)Dμ f l − k Γ (l + 1 − α) k l=0 k=0
and the explicit calculation yields l α α α − α = l−k k l k=0
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The last three summations can be obtained similarly, thus D − 1) follows directly.
183
4. Generalized fractional form of Lagrange’s expansion We can use the standard form of Lagrange’s expansion for one implicitly defined independent variable and the Definition 4 to obtain the fractional form of Lagrange’s Expansion as follows: Let z be a function of (ζ, ε) and in terms of another function μ such that z = z(ζ, ε) ≡ ζ + εμ(z)
(28)
Then for any function f ∞ n n−1 ε d df (ζ) n μ (ζ) f (z) = f (ζ) + n! dζ n−1 dζ
(29)
n=1
for small ε. If f is the identity z=ζ+
∞ n n−1 ε d [μn (ζ)] n! dζ n−1
(30)
n=1
Then ∞ n n−1 ε d df (ζ) n μ (ζ) f (z) = f (ζ) + n! dζ n−1 dζ
(31)
n=1
This classical result can be obtained using the following integral f (z) = δ(εμ(τ) − τ + ζ)f (τ)(1 − εμ (τ))dτ Now, we are going to introduce a fractional form of the Lagrange inversion formula. Rewrite the integral (15) in the fractional form as f (z)
= z D−α,−β δ(εμ(z) − z + ζ)f (z)(1 − εz Dα,β a a μ(z)) z (1 + i) (z − τ)α−1 δ(εμ(τ) − τ + ζ)f (τ)(1 − ετ Dαa μ(τ))dτ = 2Γ (α) a (1 − i) a + (τ − z)β−1 δ(εμ(τ) − τ + ζ)f (τ)(1 − εa Dβτ μ(τ))dτ 2Γ (β) z
Writing the delta-function as an integral we have: (1 + i) z π f (z) = (z − τ)α−1 e(iξ[εμ(τ)−τ+ζ]) f (τ)(1 − ετ Dαa μ(τ))dξdτ 2πΓ (α) a −π a π (1 − i) + (τ − z)β−1 e(iξ[εμ(τ)−τ+ζ]) f (τ)(1 − εa Dβτ μ(τ))dξdτ 2πΓ (β) z −π ∞
f (z) =
(1 + i) 2πΓ (α)
z π
n=0 a ∞
+
(1 − i) 2πΓ (β)
n=0 z
−π
(iξεμ(τ))n eik(ζ−τ) (z − τ)α−1 f (τ)(1 − ετ Dαa μ(τ))dξdτ n!
a π −π
(iξεμ(τ))n eik(ζ−τ) (τ − z)β−1 f (τ)(1 − εa Dβτ μ(τ))dξdτ n!
(32)
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F.A. Abd El-Salam / Journal of Taibah University for Science 8 (2014) 175–185 ∞
f (z) =
(1 + i) α n (z D a ) 2πΓ (α)
n=0
−π
a
∞
+
z π
(1 − i) n (a Dβz ) 2πΓ (β) n=0
z
(εμ(τ))n eiξ(ζ−τ) (z − τ)α−1 f (τ)(1 − ετ Dαa μ(τ))dξdτ n!
a π −π
(εμ(τ))n eiξ(ζ−τ) (τ − z)β−1 f (τ)(1 − εa Dβτ μ(τ))dξdτ n!
The integral over ξ then gives δ(ζ − τ) = 0, ∀ ζ − τ = 0 and we have: ∞
f (z) =
(εμ(ζ))n (1 + i) (ζ Dαa )n f (ζ)(1 − εζ Dαa μ(ζ)) Γ (α) n! n=0
∞
+
n (1 − i) β n (εμ(ζ)) β f (ζ)(1 − εa Dζ μ(ζ)) (a D ζ ) Γ (β) n! n=0
∞
f (z) =
(1 + i) (ζ Dαa )n [εn (μ(ζ))n f (ζ) − εn+1 f (ζ)(μ(ζ))n (ζ Dαa μ(ζ))] n!Γ (α) n=0
∞
+
(1 − i) β n β (a Dζ ) [εn (μ(ζ))n f (ζ) − εn+1 f (ζ)(μ(ζ))n (a Dζ μ(ζ))] n!Γ (β) n=0
∞
f (z) =
(1 + i) (ζ Dαa )n [εn (μ(ζ))n f (ζ) − εn+1 f (ζ)Γ (1 + α)(μ(ζ))n (μ(ζ)1−α )] n!Γ (α) n=0
∞
+
(1 − i) β n (a Dζ ) [εn (μ(ζ))n f (ζ) − εn+1 f (ζ)Γ (1 + β)(μ(ζ))n (μ(ζ)1−β )] n!Γ (β) n=0
∞
f (z) =
(1 + i) (ζ Dαa )n [εn (μ(ζ))n f (ζ) − εn+1 f (ζ)Γ (1 + α)(μ(ζ)n+1−α )] n!Γ (α) n=0
∞
+
(1 + i) (a Dαζ )n [εn (μ(ζ))n f (ζ) − εn+1 f (ζ)Γ (1 + β)(μ(ζ)n+1−β )] n!Γ (β) n=0
n ∞ (1 + i) εn+1 Γ (1 + α) α n ε n α n+1 α n+1 (ζ D a ) (μ(ζ)) f (ζ) − × {ζ Da [f (ζ)(μ(ζ)) ] − ζ Da f (ζ)(μ(ζ)) } f (z) = Γ (α) n! Γ (n + 1 + α) n=0
+
n ∞ εn+1 Γ (1 + β) (1 − i) β n ε β β (a D ζ ) (μ(ζ))n f (ζ) − × {a Dζ [f (ζ)(μ(ζ))n+1 ] − a Dζ f (ζ)(μ(ζ))n+1 } Γ (β) n! Γ (n + 1 + β) n=0
On extraction, the first term out of summation, set n + 1 = k ⇒ n = k − 1 and re-arranging the terms then gives the result: ∞
f (z) =
α(εk ) (1 + i) f (ζ) + [(μ(ζ))k ζ Dαa f (ζ)] (ζ Dαa )k−1 Γ (α) Γ (k + α) k=0
∞
k (1 − i) β k−1 β(ε ) β (a D ζ ) f (ζ) + [(μ(ζ))k a Dζ f (ζ)] + Γ (β) Γ (k + β) k=0
(33)
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185
5. Conclusion The author introduced the left and right Riemann–Liouville derivatives. He obtained a generalized operator based on left/right Riemann–Liouville derivatives. He proved some new basic definitions, lemmas and theorems in the fractional calculus which is related to the purpose of the presented work. A generalized fractional form of Lagrange expansion theorem is obtained. In a forthcoming work a more generalized class of the fractional Lagrange expansion theorem will be derived. Acknowledgments The author is deeply indebted and thankful to the dean of the scientific research and his helpful and dedicated team of employees at the Taibah university, Al-Madinah Al-Munawwarah, K.S.A. This research work was supported by a grant No. (1600/1433). Also I wish to express my deep gratitude and very indebtedness for the referees for all very fruitful discussions, constructive comments, remarks and suggestions on the manuscript. The author wishes also to record his indebtedness and thankfulness to Drs. M. R. El-Zawy and A. El-Abed for their extensive revision of the manuscript. Also the authors wishes to thank deeply the reviewers for their valuable and fruitful comments as well as for their powerful reading and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
[10] [11]
[12]
[13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
K.S. Miller, B. Ross, An Introduction to the Fractional Integrals and Derivatives – Theory and Applications, John Wiley and Sons, 1993. F. Riewe, Nonconservative Lagrangian and Hamiltonian mechanics, Phys. Rev. E 53 (1996) 1890–1899. F. Riewe, Mechanics with fractional derivatives, Phys. Rev. E 55 (1997) 3581–3592. D. Baleanu, New applications of fractional variational principles, Rep. Math. Phys. 61 (2) (2008). K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974. Om.P. Agrawal, Formulation of Euler–Lagrange equations for fractional variational problems, J. Math. Anal. Appl. 272 (2002) 368–379. Om.P. Agrawal, J. Phys. A: Math. Gen. 39 (2006) 10375. Om. P. Agrawal, J. Phys. A: Math. Theor. 40 (2007) 5469. J.L. Lagrange, Nouvelle méthode pour résoudre les équations littérales par le moyen des séries Mémoires de l’Académie Royale des Sciences et Belles-Lettres de Berlin, vol. 24, 1770, pp. 251–326 (available on-line at: http://gdz.sub.uni-goettingen.de/ no cache/dms/load/img/?IDDOC=41070). M. Abramowitz, I.A. Stegun (Eds.), 3. 6. 6. Lagrange’s Expansion”. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1972, p. 14. H.H. Bürmann, “Essai de calcul fonctionnaire aux constantes ad-libitum,” submitted in 1796 to the Institut National de France. For a summary of this article, see: C.F. Hindenburg (Ed.), Versuch einer vereinfachten Analysis; ein Auszug eines Auszuges von Herrn Bürmann [Attempt at a simplified analysis; an extract of an abridgement by Mr. Bürmann]. Archiv der reinen und angewandten Mathematik [Archive of pure and applied mathematics], vol. 2, Schäferischen Buchhandlung, Leipzig, Germany, 1798, pp. 495–499. Bürmann, Hans Heinrich, “Formules du développement, de retour et d’integration,” submitted to the Institut National de France. Bürmann’s manuscript survives in the archives of the École Nationale des Ponts et Chaussées [National School of Bridges and Roads] in Paris. (See ms. 1715.). J.-L. Lagrange, A.-M. Legendre, A report on Bürmann’s theorem appeared Rapport sur deux mémoires d’analyse du professeur Burmann, “Mémoires de l’Institut National des Sciences et Arts: Sciences Mathématiques et Physiques, vol. 2, 1799, pp. 13–17. P.S. de Laplace, Mémoire sur l’usage du calcul aux différences partielles dans la théories des suites Mémoires de l’Académie Royale des Sciences de Paris, vol., 1777, pp. 99–122. P.S. de Laplace, Oeuvres [Paris], vol. 9, 1843, pp. 313–335. E. Goursat, A Course in Mathematical Analysis (E.R. Hedrick and O. Dunkel, Trans.), vol. I, Dover, New York, N.Y., 1959, pp. 404–405. C. Hermite, Sur quelques développements en série de fonctions de plusieurs variables Comptes Rendus de l’Académie des Sciences des Paris, vol. 60, 1865, pp. 1–26. C. Hermite, Oeuvres [Paris], vol. 2, 1908, pp. 319–346. S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives – Theory and Applications, Gordon and Breach, Longhorne, PA, 1993. I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering V198, Academic Press, San Diego, 1999. V.J. Erwin, J.P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Meth. P. D. E. 22 (2006) 558–576. P.M. Morse, H. Feshbach, Methods of Theoretical Physics, Part I, McGraw-Hill, New York, 1953. F.A. Abd El-Salam, N-Dimensional Fractional Lagrange’s Inversion Theorem, Abstract and Applied Analysis, Hindawi Publishing Corporation, 2013, http://dx.doi.org/10.1155/2013/310679, Article ID 310679, 11 pages.
ScienceDirect Journal of Taibah University for Science 8 (2014) 175–185
Generalized class of fractional inverse Lagrange expansion F.A. Abd El-Salam a,b,∗ a
Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah, KSA Department of Astronomy, Faculty of Science, Cairo University, Cairo 12613, Egypt
b
Received 21 November 2012; received in revised form 16 August 2013; accepted 17 August 2013 Available online 2 October 2013
Abstract In this work the left and right Riemann–Liouville derivatives are introduced. A generalized operator based on left/right Riemann–Liouville derivatives is obtained. Basic definitions, lemmas and theorems in the fractional calculus which is related to our purpose are presented. The fractional form of Lagrange expansion theorem is obtained. © 2013 Taibah University. Production and hosting by Elsevier B.V. All rights reserved. Keywords: Fractional right/left Riemann–Liouville derivative; Generalized operator; Lagrange expansion; Delta-function
1. Introduction The development of the FC theory is due to the contributions of many mathematicians such as Euler, Liouville, Riemann, and Letnikov. Several definitions of a fractional derivative have been proposed. These definitions include Riemann–Liouville, Grunwald-Letnikov,Weyl, Caputo, Marchaud, and Riesz fractional derivatives, see [1–4]. Riemann–Liouville derivative is the most used generalization of the derivative. It is based on the direct generalization of Cauchy’s formula for calculating an n-fold or repeated integral. The right and the left Riemann–Liouville fractional derivatives, in brief, are denoted by RRLFD and LRLFD respectively, see [5]. In [6–8] Agrawal has studied Fractional variational problems using the Riemann–Liouville derivatives. He notes that even if the initial functional problems only deal with the left Riemann–Liouville derivative, the right Riemann–Liouville derivative appears naturally during the computations. In what follows, we construct an operator combining the left and right Riemann–Liouville (RL) derivative. We remind some results concerning functional spaces associated to the left and right RL derivative. In particular, we discuss the possibility to obtain a law of exponents. In 1770, Joseph Louis Lagrange (1736–1813) published his power series solution of the implicit equation. However, his solution used cumbersome series expansions of logarithms [9,10]. This expansion was generalized by Bürmann [11–13]. There is a straightforward derivation using complex analysis and contour integration; the complex formal power series version is clearly a consequence of knowing the formula for polynomials, so the theory of analytic functions may be applied. Actually, the machinery from analytic function theory enters only in a formal way in this proof. In ∗ Tel.: +966 543234803. Peer review under responsibility of Taibah University
1658-3655 © 2013 Taibah University. Production and hosting by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jtusci.2013.08.001
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1780, Pierre-Simon Laplace (1749–1827) published a simpler proof of the theorem, based on relations between partial derivatives with respect to the variable and the parameter, see [14,15]. Charles Hermite (1822–1901) presented the most straightforward proof of the theorem by using contour integration [16–18]. In mathematical analysis, this series expansion is known as Lagrange inversion theorem, also known as the Lagrange–Bürmann formula, and gives the Taylor series expansion of the inverse function. Suppose z = f(w) where f is analytic function at a point a and f(a) = / 0. Then it is possible to invert or solve the equation for w such that w = g(z) on a neighborhood of f(a), where g is analytic at the point f(a). This is also called reversion of series. The series expansion of g is given by n ∞ (z − f (a))n d n−1 w−a lim w = g(z) = a + w→a n! dwn−1 f (w) − f (a) n=1
In this work we will apply the concepts of fractional calculus to obtain a fractional form of Lagrange expansion and some generalizations. Definition 1. By D, we denote the operator that maps a differentiable function onto its integer derivative, i.e. Df(x) = f , by Ja , we denote the integer integration operator that maps a function f, assumed to be (Riemann) integrable on the x compact interval [a,b], onto its primitive centered at a, i.e. Ja f (x) = a f (t)dt, ∀ a ≤ x ≤ b. Definition 2. By Dn and Jan , n ∈ N we denote the n-fold iterates of D and Ja , respectively. Note that Dn is the left inverse of Jan in a suitable space of functions. Lemma 1. Let f be Riemann integrable on [a,b]. Then, for a ≤ x ≤ b and n ∈ N, we have x 1 Jan f (x) = (x − t)n−1 f (t)dt, n ∈ N. (n − 1)! a
(1)
2. Left and right Riemann–Liouville derivatives and integrals We define the left and right Riemann–Liouville derivatives following [19,20]. In all the following definitions Γ (α) represents the Euler gamma function given by ∞ Γ (α) = xα−1 e−α dx = (α − 1)! 0
Definition 3. Let f be a function defined on (a, b) ⊂ R, and α ∈ R+ . Then the left and right Riemann–Liouville fractional integral of order α is on respective defined on Lebesgue space L1 [a, b] are given by x 1 −α (x − s)α−1 f (s)ds, a ≤ x ≤ b, α∈R (2) a Dx f (x) = Γ (α) a b 1 −α (3) (s − x)α−1 f (s)ds, a ≤ x ≤ b, α∈R x Db f (x) = Γ (α) x Definition 4. Let f be a function defined on (a, b) ⊂ R, and α ∈ R+ , the left and right Riemann–Liouville derivative α of order α defined on Lebesgue space L1 [a, b] and denoted by a D−α x and x Db respectively, are defined by n x 1 d α D f (x) = (x − s)n−α−1 f (s)ds a ≤ x ≤ b, α∈R (4) a x Γ (n − α) dx a n b (−1)n d α D f (x) = (x − s)n−α−1 f (s)ds, a ≤ x ≤ b, α∈R (5) x b Γ (n − α) dx x Left and right (RL) integrals satisfy some important properties like the semi-group property.
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177
n Remark 1. It is evident that a D−α x ≡ Ja , ∀α ∈ N, except for the fact that we have extended the domain from Riemann integrable functions to Lebesgue integrable functions (which will not lead to any problems in our development). Moreover, in the case α ≥ 1 it is obvious that the integral a D−α x f (x) exists for every x ∈ [a, b] because the integrand is the product of an integrable function f and the continuous function (x − • )α−1 . One important property of integer-order integral operators is preserved by this generalization. i.e. ∀ α, β > 0, f(x) ∈ L1 [a, b] −α −β a Dx (a Dx f (x))
−α −α−β = a D−β f (x) x (a Dx f (x)) = a Dx
(6)
The fractional differential and integral operators, namely differeintegral operator a D±α x can be obtained from the ±β generalization of the two operators x Db as, n n d d ±α ±α−n ±α ±α−n n , x Db = (−1) (7) a Dx = a Dx x Db dxn dxn where the number of additional differentiations n is equal to [α] + 1, where [α] is the whole part of α. The generalized ±β operator x Db satisfies ⎧ dα ⎪ , Fractional differentiation mapping ⎪ ⎪ ⎪ dxα ⎪ ⎪ ⎨ 1, Fractional unitary mapping ±α a Dx = x ⎪ ⎪ ⎪ ⎪ dt −α , Fractional integration mapping ⎪ ⎪ ⎩ a
If α = m, m ∈ N, and f ∈ Cm ([a, b]) we have dL±m f dR±m f ±m , D f = − . (8) x b dx±m dx±m This last relation which ensures the gluing of the left and right Riemann–Liouville (RL) differeintegral to the classical derivative and integral operator will be of fundamental importance. These fractional operators satisfy the semi-group property. If f(t) ∈ C0 with left and right-differeintegral at point t denoted by dL±m f/dt ±m and dR±m f/dt ±m , respectively. β β In what follows, we denote by αa I, Ib and αa Ib the functional spaces defined by α ±m a Dt f exists}, a I = {f ∈ C([a, b]), (9) β Ib = {f ∈ C([a, b]), t Dt f exists}, ±m a Dx f
=
and α β a Ib
β
= αa I ∩ Ib . β
(10) β
As is clear the set αa Ib is non-empty. We have AC([a, b]) ⊂ αa Ib , where AC([a, b]) is the set of absolutely continuous functions on the interval [a,b]. The operators of ordinary differentiation or integration of integer order satisfy a commutativity property and the law of exponents (the semi-group property) i.e. ±n ±m ±m ±n ±n±m dL,R dL,R dL,R dL,R dL,R f f f f f × = × = . (11) ±n ±m ±m ±n ±n±m dx dx dx dx dx These two properties in general fail to be satisfied by the left and right fractional RL differenitegral operator. These bad properties are responsible for several difficulties in the study of fractional differential equations.
2.1. Fractional functional spaces Following [21] we introduce the left and right fractional differenitegral as well as convenient functional spaces on which we have the semi-group property. In [21] several useful functional spaces are introduced. Let I ⊂ R be an
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open interval (which may be unbounded). We denote by C0∞ (I) the set of all functions f ∈ C∞ (I) that vanish outside a compact subset K of I. Definition 5. [Left fractional differenitegral space] Let α > 0. Define the semi-norm (A nonzero vector may have a zero length) |f |JLα (R) = ||a D±α x f ||J 2 (R) L
and norm 1/2
||f ||JLα (R) = (||f ||2J 2 (R) + |f |2J 2 (R) ) L
L
and let JLα (R) denote the closure of C0∞ (R) with respect to || · ||JLα (R) . Definition 6. [Right fractional differenitegral space] Let α > 0. Define the semi-norm |f |JLα (R) = ||x D±α b f ||J 2 (R) L
1/2
||f ||JLα (R) = (||f ||2J 2 (R) + |f |2J 2 (R) ) L
L
and let JLα (R) denote the closure of C0∞ (R) with respect to || · ||JLα (R) . Similarly, we can define the right fractional derivative space. We now assume that I is a bounded open subinterval of R. We restrict the fractional differeintegral spaces to I. Define α (I), J α (I) as the closure of C ∞ (I) under their respective norms. These spaces have very interesting the spaces JL,0 R,0 0 ±α properties with respect to a D±α x and x Db . In particular, we have the following semi-group property: β semi-Group: For f ∈ JL,0 (I), 0 < α < β we have ±γ a Dx f
±γ∓α = a D±α f x a Dx
(12)
β
and similarly for x ∈ JR,0 (I), ±γ x Db f
±γ∓α
= x D±α b x Db
(13)
f
see [22] Lemma 2.9 there for a proof. α (I) and J α (I) have been characterized when α > 0. We denote by W s,α (I) the The fractional derivative spaces JL,0 R,0 0 fractional Sobolev space. α (I), J α (I) and W s,α (I) spaces are equal, see [22] Theorem 2.13 for a proof. In fact, when Let α > 0. Then the JL,0 R,0 0 α (I), J α (I) and Iα (I) spaces have equivalent semi-norms and αn − 1/2, n ∈ N we have a stronger result as the JL,0 0 R,0 norms. 2.2. Generalized operator As we want to deal with dynamical systems exhibiting the arrow of time, we need to consider the operator a D±α x ±β and x Db , in order to keep track of the past and future of the dynamics. The fact that we consider αβ is only here for convenience. This can be used to take into account a different quantity of information from the past and the future. ±β
α,β Definition 7. Let a D±α of the form x and x Db be given. We look for an operator D ±β
Dα,β = M(a D±α x , x Db ) where M : R2 → C is a mapping which does not depend on (α, β), satisfying the following general principles:
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d ±m f
L,R • E1: Gluing property: If f(x) ∈ C1 then when α = β = m, m ∈ N, Dm,m f (x) = dx ±m . • E2: M is a R-linear mapping. • E3: Reconstruction: The mapping M is invertible. E1 is fundamental in the embedding framework. It follows that all or constructions can be seen as a continuous two-parameters deformation of the corresponding classical one. This can be of importance dealing with the fractional quantization problem in classical mechanics. In other words E1 is not the usual condition underlying the stochastic or quantum embedding theories. In general, we have an injective mapping ι from the set of differentiable functions C1 in a bigger functional space ε such that the operator D that we define on ε reduces to the classical derivative on ι(C1 ) meaning that for f ∈ C1 , D(ι(f )) = D(ι(f )) where f (x) = df/dx. As a consequence, we have a true embedding in this case, meaning that the embed theory already contains the classical one via the mapping ι. Here, the classical theory is not contained in the embedded theory but can be recovered by a continuous two-parameters deformation. E2 does not have a particular meaning. This is only the simplest dependence of the operator D with respect to a D±α x ±β and x Db . E3 is important. It means that the data of D±α,β on a given function f at point x allow us to recover the left and right RL derivatives of f at x, so information about x in a neighborhood of f(x). Operators satisfying E1, E2, and E3 are of the form
±β
±β
±α D±α,±β = [pa D±α x + (p − 1)x Db ] + iq[a Dx + x Db ]
(14)
where p, q ∈ R and p − q = 1, q = / 0. By E2, we denote M(η, ξ) = pη + qξ + i(rη + sξ), with p, q, r, s ∈ R and r − s ∈ R. By E1, we must have with ξ = − η corresponding to the operator a choice of operators ±df/dx. We then already have an operator of the form (14). The reconstruction assumption only imposes that q = / 0 in (14). A more rigid form for these operators is obtained imposing a symmetry condition. • E4: Let f ∈ C0 be a real valued function possessing left and right classical derivatives at point x, denoted by d+ f/dx and d− f/dx, respectively. E4: must be seen as the non-differentiable pendant of E1. Indeed, E1 can be rephrased as follows: if f ∈ C0 is such that d+ f and d− f exist and satisfy d+ f = d− f then D1,1 f = d + f . E4 is then equivalent to the commutativity of the following diagram, where R2 is seen as the R-vector space associated to C: C
τ
→
η − iη →
C η + iη
(15)
where τ : C → C is defined by τ(z) = iz, z ∈ C and we have used the fact that M(η, − η) = 0 following E1. The unique operator satisfying E1, E2, E3 and E4 is given by D±α,±β =
1 i ±β ±β ±α [a D±α x − x D b ] + [a D x + x D b ] 2 2
(16)
By (iv), we must have p + (p − 1) = 0 and 2q = 1, so that p = q = 1/2. 3. The fractional operator of order (α, β) For all a, b ∈ R, a < b, the fractional operator of order (α, β), α > 0, β > 0, denoted by Dα,β μ , is defined by 1 iμ ±β ±β [a D±α [a D±α μ∈C (17) x − x Db ] + x + x Db ], 2 2 when α = β = ±1, we obtain D1,1 D−1,−1 = , respectively. μ = d/dx, μ The free parameter μ can be used to reduce the operator Dα,β μ to some special cases of importance. Let us denotes by f(x) a given real valued function. D±α,±β = μ
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±α 1. For μ = − i, we have Dα,β μ = a Dx then dealing with an operator using the future state denoted by Px (f ) of the underlying function, i.e. Px (f ) = {f (s), s ∈ (a, x]}. ±β 2. For μ = i, we obtain Dα,β μ = −x Db then dealing with an operator using the past state denoted by Pt (x) of the underlying function, i.e. Px (f ) = {f (s), s ∈ (x, b}. ±β
As a consequence, our operator can be used to deal with problems using, a D±α x , x Db , or both operators in a unified way, only particularizing the value of μ at the end to recover the desired framework. 3.1. Product rules The classical product rule for Riemann–Liouville derivatives is for all α > 0 b b ±α f (x)x D±α a Dx f (x)g(x)dx = b g(x)dx a
(18)
a
as long as f(a) = f(b) or g(a) = g(b). ±β This formula gives a strong connection between a D±α x and x Db via a generalized integration by part. This relation ±β is responsible for the emergence of x Db in problems of fractional calculus of variations only dealing with a D±α x . This result also justifies our approach to the construction of a fractional operator which put on the same level as the left and right RL derivatives. As a consequence, we obtain the following formula for our fractional operator: β For all f, g ∈ αa Fb , we have b b α,β α,β D f (x)g(x)dx = − f (x)x D−μ g(x)dx (19) a μ a
a
provided that f(a) = f(b) or g(a) = g(b). We have b b f (x)g(x)dx = − f (x)[(−) + iμ(+)](g(x))dx a
(20)
a
Exchanging the role of α and β in [(−) + iμ(+)], we obtain the operator [(−) + iμ(+)] which can be written as α,β
−[(−) − iμ(+)] = −D−μ
(21)
This concludes the proof. Here again, we see that it is convenient to keep the parameter μ free. In what follows we shall generalize some lemmas and theorems proved by Abd El-Salam [23]. m m−α,m−β . Lemma 2. Let α, β ∈ R+ and let m ∈ N such that m > α, m > β. Then Dα,β μ = Da D μ
Proof. Since m > α, m > β yields m≥ α , m≥ β. Thus 1 iμ m−β m−β Dm Dm−α,m−β = Dm [a Dm−α − x Db ] + Dm [a Dm−α + x Db ] x x μ 2 2 1 β−β m−β α−α = [Dα Dm−α a Dm−α − Dβ Dm−β x Db ] x Db a Dx x 2 iμ β−β m−β α−α + [Dα Dm−α a Dm−α + Dβ Dm−β x Db ] a Dx x Db x 2 1 = [Dα+m−α+α−m+α−α − Dβ+m−β+β−m+β−β ] 2 iμ + [Dα+m−α+α−m+α−α + Dβ+m−β+β−m+β−β ] 2 =
1 α iμ [D − Dβ ] + [Dα + Dβ ] = Dα,β μ 2 2
(22)
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181
−α,−β Theorem 1. Let (α, β≥0) ∈ R+ . Then, for every f(x) ∈ L1 [a, b], we have Dα,β f (x) = f (x). μ Dμ
Proof. For α = β = 0 both operators are the identities. For α, β > 0, let m≥ α , m≥ β, then −α,−β Dα,β f (x) μ Dμ
= Dam Dα−m,β−m D−α,−β f (x) = Dam D−(m−α),−(m−β) D−α,−β f (x) μ μ μ μ = Dam D−(m−α+α),−(m−β+β) f (x) = D−(m−m−α+α),−(m−m−β+β) f (x) = f (x) μ μ
(23)
Corollary 1. Let f be analytic in (a − h, a + h) for some h > 0, and let (α, β≥0) ∈ R+ . Then we have I-1), & D-1) ∀ a ≤ x < a + h/2 and we have I-2), & D-2) ∀ a ≤ x < a + h I-1)
1 iμ −β −β μ∈C [a D−α [a D−α x − x Db ]f (x) + x + x Db ]f (x), 2 2 1 iμ −β −β = [a D−α [a D−α x f (x) − x Db f (x)] + x f (x) + x Db f (x)] 2 2
D−α,−β f (x) = μ
D−α,−β f (x) μ
∞
∞ 1 (−1)m (x − μ)k+α k (−1)m (x − μ)k+β k = Dμ f (x) − Dμ f (x) 2 k!(α + k)Γ (α) k!(β + k)Γ (β) k=0 k=0
∞ ∞ iμ (−1)m (x − μ)k+α k (−1)m (x − μ)k+β k + Dμ f (x) + D f (x) 2 k!(α + k)Γ (α) k!(β + k)Γ (β) μ k=0
∞
∞ k+β (x − μ)k+α (x − μ) 1 I-2) D−α,−β f (x) = Dk f (x) − Dk f (x) μ 2 Γ (k + 1 + α) μ Γ (k + 1 + β) μ k=0 k=0 ∞
∞ k+α iμ (x − μ) (x − μ)k+β k k + D f (x) + D f (x) 2 Γ (k + 1 + α) μ Γ (k + 1 + β) μ k=0
D-1)
Dα,β μ f (x)
D-2)
∞ ∞ β (x − μ)k−β k 1 α (x − μ)k−α k = D f (x) − D f (x) k (k + 1 − α) μ k (k + 1 − β) μ 2 k=0 k=0 ∞
∞ β (x − μ)k−β k iμ α (x − μ)k−α k D f (x) + D f (x) + k (k + 1 − α) μ k (k + 1 − β) μ 2
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k=0
∞ ∞ (x − μ)k−β k 1 (x − μ)k−α k = D f (x) − D f (x) 2 (k + 1 − α) μ (k + 1 − β) μ k=0 k=0
∞ ∞ (x − μ)k−β k iμ (x − μ)k−α k D f (x) + D f (x) + 2 (k + 1 − α) μ (k + 1 − β) μ k=0
(25)
k=0
k=0
Dα,β μ f (x)
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k=0
(27)
k=0
the binomial coefficients for α, β ∈ R and k ∈ N is defined as α α(α − 1)(α − 2). . .(α − k + 1) α! = = k k! k!(α − k)! Proof. For the first two statements I-1), I-2), we use the definition of the Riemann–Liouville integral operator D−α a , and expand f(t) into a power series about x. Since x ∈ [a, a + h/2), the power series converges in the entire interval
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of integration. And exchange summation and integration. Then we use the explicit representation for the fractional integral of the power function k D−α a (x − a) =
Γ (k + 1) (x − a)k+α (α + k + 1)
I-1) follows immediately. For the second statement, we proceed in a similar way, but we now expand the power series at a and not at x. This allows us again to conclude the convergence of the series in the required interval. The analyticity of D−α a follows immediately from the second statement. To prove D-1) use is made of Dαμ = Dα D−α−α μ and
k!Γ (α)(α + k)
−α k
= (−1)k Γ (k + 1 + α)
This allows us to rewrite the statement I-1) as ∞ − α − α (x − a)k−α−α α−α Dμ f (x) = Dk f (x) k Γ (k + 1 − α − α) μ k=0
Differentiating α times with respect to x, we find α D−α−α f (x) Dμ μ
= Dαμ f (x) =
∞
⎛ ⎝
α − α k
k=0
⎞ ⎠
1 k f ](x) Dα [(• − μ)k+α−α Dμ Γ (k + 1 + α − α) μ
The classical version of Leibniz’s formula yields α ∞ α − α α 1 α−j • α k+j Dμ Dμ f (x) = [( − μ)k+α−α ](x)Dμ f j k Γ (k + 1 + α − α) j=0
k=0
which yields Dαμ f (x)
α ∞ α − α α [(x − a)k+j−α ] k+j = f (x)Dμ j k Γ (k + 1 + j − α) k=0
j=0
i By definition, = 0 if i ∈ N and i < j. Thus we may replace the upper limit in the inner sum by ∞ without j changing the expression. The substitution j = l − k gives ∞ ∞ α α − α [(x − μ)l−α ] l Dαμ f (x) = (x)Dμ f l − k Γ (l + 1 − α) k l=0 k=0
∞ Σ∞ = Σ∞ Σl Using the fact that Σl=0 k=0 l=0 k=0 ∞ l α α − α [(x − μ)l−α ] α l Dμ f (x) = (x)Dμ f l − k Γ (l + 1 − α) k l=0 k=0
and the explicit calculation yields l α α α − α = l−k k l k=0
F.A. Abd El-Salam / Journal of Taibah University for Science 8 (2014) 175–185
The last three summations can be obtained similarly, thus D − 1) follows directly.
183
4. Generalized fractional form of Lagrange’s expansion We can use the standard form of Lagrange’s expansion for one implicitly defined independent variable and the Definition 4 to obtain the fractional form of Lagrange’s Expansion as follows: Let z be a function of (ζ, ε) and in terms of another function μ such that z = z(ζ, ε) ≡ ζ + εμ(z)
(28)
Then for any function f ∞ n n−1 ε d df (ζ) n μ (ζ) f (z) = f (ζ) + n! dζ n−1 dζ
(29)
n=1
for small ε. If f is the identity z=ζ+
∞ n n−1 ε d [μn (ζ)] n! dζ n−1
(30)
n=1
Then ∞ n n−1 ε d df (ζ) n μ (ζ) f (z) = f (ζ) + n! dζ n−1 dζ
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n=1
This classical result can be obtained using the following integral f (z) = δ(εμ(τ) − τ + ζ)f (τ)(1 − εμ (τ))dτ Now, we are going to introduce a fractional form of the Lagrange inversion formula. Rewrite the integral (15) in the fractional form as f (z)
= z D−α,−β δ(εμ(z) − z + ζ)f (z)(1 − εz Dα,β a a μ(z)) z (1 + i) (z − τ)α−1 δ(εμ(τ) − τ + ζ)f (τ)(1 − ετ Dαa μ(τ))dτ = 2Γ (α) a (1 − i) a + (τ − z)β−1 δ(εμ(τ) − τ + ζ)f (τ)(1 − εa Dβτ μ(τ))dτ 2Γ (β) z
Writing the delta-function as an integral we have: (1 + i) z π f (z) = (z − τ)α−1 e(iξ[εμ(τ)−τ+ζ]) f (τ)(1 − ετ Dαa μ(τ))dξdτ 2πΓ (α) a −π a π (1 − i) + (τ − z)β−1 e(iξ[εμ(τ)−τ+ζ]) f (τ)(1 − εa Dβτ μ(τ))dξdτ 2πΓ (β) z −π ∞
f (z) =
(1 + i) 2πΓ (α)
z π
n=0 a ∞
+
(1 − i) 2πΓ (β)
n=0 z
−π
(iξεμ(τ))n eik(ζ−τ) (z − τ)α−1 f (τ)(1 − ετ Dαa μ(τ))dξdτ n!
a π −π
(iξεμ(τ))n eik(ζ−τ) (τ − z)β−1 f (τ)(1 − εa Dβτ μ(τ))dξdτ n!
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184
F.A. Abd El-Salam / Journal of Taibah University for Science 8 (2014) 175–185 ∞
f (z) =
(1 + i) α n (z D a ) 2πΓ (α)
n=0
−π
a
∞
+
z π
(1 − i) n (a Dβz ) 2πΓ (β) n=0
z
(εμ(τ))n eiξ(ζ−τ) (z − τ)α−1 f (τ)(1 − ετ Dαa μ(τ))dξdτ n!
a π −π
(εμ(τ))n eiξ(ζ−τ) (τ − z)β−1 f (τ)(1 − εa Dβτ μ(τ))dξdτ n!
The integral over ξ then gives δ(ζ − τ) = 0, ∀ ζ − τ = 0 and we have: ∞
f (z) =
(εμ(ζ))n (1 + i) (ζ Dαa )n f (ζ)(1 − εζ Dαa μ(ζ)) Γ (α) n! n=0
∞
+
n (1 − i) β n (εμ(ζ)) β f (ζ)(1 − εa Dζ μ(ζ)) (a D ζ ) Γ (β) n! n=0
∞
f (z) =
(1 + i) (ζ Dαa )n [εn (μ(ζ))n f (ζ) − εn+1 f (ζ)(μ(ζ))n (ζ Dαa μ(ζ))] n!Γ (α) n=0
∞
+
(1 − i) β n β (a Dζ ) [εn (μ(ζ))n f (ζ) − εn+1 f (ζ)(μ(ζ))n (a Dζ μ(ζ))] n!Γ (β) n=0
∞
f (z) =
(1 + i) (ζ Dαa )n [εn (μ(ζ))n f (ζ) − εn+1 f (ζ)Γ (1 + α)(μ(ζ))n (μ(ζ)1−α )] n!Γ (α) n=0
∞
+
(1 − i) β n (a Dζ ) [εn (μ(ζ))n f (ζ) − εn+1 f (ζ)Γ (1 + β)(μ(ζ))n (μ(ζ)1−β )] n!Γ (β) n=0
∞
f (z) =
(1 + i) (ζ Dαa )n [εn (μ(ζ))n f (ζ) − εn+1 f (ζ)Γ (1 + α)(μ(ζ)n+1−α )] n!Γ (α) n=0
∞
+
(1 + i) (a Dαζ )n [εn (μ(ζ))n f (ζ) − εn+1 f (ζ)Γ (1 + β)(μ(ζ)n+1−β )] n!Γ (β) n=0
n ∞ (1 + i) εn+1 Γ (1 + α) α n ε n α n+1 α n+1 (ζ D a ) (μ(ζ)) f (ζ) − × {ζ Da [f (ζ)(μ(ζ)) ] − ζ Da f (ζ)(μ(ζ)) } f (z) = Γ (α) n! Γ (n + 1 + α) n=0
+
n ∞ εn+1 Γ (1 + β) (1 − i) β n ε β β (a D ζ ) (μ(ζ))n f (ζ) − × {a Dζ [f (ζ)(μ(ζ))n+1 ] − a Dζ f (ζ)(μ(ζ))n+1 } Γ (β) n! Γ (n + 1 + β) n=0
On extraction, the first term out of summation, set n + 1 = k ⇒ n = k − 1 and re-arranging the terms then gives the result: ∞
f (z) =
α(εk ) (1 + i) f (ζ) + [(μ(ζ))k ζ Dαa f (ζ)] (ζ Dαa )k−1 Γ (α) Γ (k + α) k=0
∞
k (1 − i) β k−1 β(ε ) β (a D ζ ) f (ζ) + [(μ(ζ))k a Dζ f (ζ)] + Γ (β) Γ (k + β) k=0
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F.A. Abd El-Salam / Journal of Taibah University for Science 8 (2014) 175–185
185
5. Conclusion The author introduced the left and right Riemann–Liouville derivatives. He obtained a generalized operator based on left/right Riemann–Liouville derivatives. He proved some new basic definitions, lemmas and theorems in the fractional calculus which is related to the purpose of the presented work. A generalized fractional form of Lagrange expansion theorem is obtained. In a forthcoming work a more generalized class of the fractional Lagrange expansion theorem will be derived. Acknowledgments The author is deeply indebted and thankful to the dean of the scientific research and his helpful and dedicated team of employees at the Taibah university, Al-Madinah Al-Munawwarah, K.S.A. This research work was supported by a grant No. (1600/1433). Also I wish to express my deep gratitude and very indebtedness for the referees for all very fruitful discussions, constructive comments, remarks and suggestions on the manuscript. The author wishes also to record his indebtedness and thankfulness to Drs. M. R. El-Zawy and A. El-Abed for their extensive revision of the manuscript. Also the authors wishes to thank deeply the reviewers for their valuable and fruitful comments as well as for their powerful reading and suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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