- Email: [email protected]
On Fractional Derivatives with Exponential Kernel and their Discrete Versions
Vol. 80 (2017)
REPORTS ON MATHEMATICAL PHYSICS
No. 1
ON FRACTIONAL DERIVATIVES WITH EXPONENTIAL KERNEL AND THEIR DISCRETE VERSIONS T HABET A BDELJAWAD Department of Mathematics and Physical Sciences, Prince Sultan University, P. O. Box 66833, 11586 Riyadh, Saudi Arabia (e-mail: [email protected])
and D UMITRU BALEANU Department of Mathematics, C¸ankaya University, 06530 Ankara, Turkey and Institutes of Space Sciences, Magurele-Bucharest, Romania (e-mail: [email protected]) (Received June 15, 2016 — Revised November 11, 2016) In this paper we define the right fractional derivative and its corresponding right fractional integral with exponential kernel. We provide the integration by parts formula and we use the Q-operator to confirm our results. The related Euler–Lagrange equations are obtained and one example is reported. Moreover, we formulate and discuss the discrete counterparts of our results. Keywords: Caputo fractional difference, Q-operator, discrete exponential function, discrete nabla Laplace transform, convolution.
1.
Introduction
The techniques of fractional calculus are applied successfully in many branches of science and engineering [1–7]. The power law effects are described accurately within the fractional calculus approach. However, there are several complex phenomena which do not obey this law, thus we need to find alternatives, e.g. the nonlocal kernels without singularity in order to overcome this issue. On the other side the discrete version of any fractional derivative is one of the interesting topics nowadays [8–20]. Some applications of the discrete fractional Caputo derivative can be seen in [21, 22]. For two real numbers a < b, a ≡ b(mod1), we denote Na = {a, a + 1, . . . } and N = {b, b − 1, . . . }. Also, ∇g(t) = g(t) − g(t − 1). b The following is the action of the discrete Q-operator on fractional sums and differences [20, 23]: [11]
12
T. ABDELJAWAD and D. BALEANU
• (∇a−α Qf )(t) = Qb ∇ −α f (t), • (∇aα Qf )(t) = Qb ∇ α f (t), • (C ∇aα Qf )(t) = Qb C ∇ α f (t). In this article, we use the action of discrete Q-operator to analyze the proposed definitions of fractional differences with discrete exponential function kernels. We recall some facts about discrete transform within nabla. For more general theory of Laplace transform on time scales we refer to [24, 25]. The (nabla) exponential function on the time scale Z is written as t−t0 1 b e(λ, t − t0 ) = , (1) 1−λ while the (delta) exponential function becomes
e e(λ, t − t0 ) = (1 + λ)t−t0 .
(2)
In this paper, we follow the nabla discrete analysis.
DEFINITION 1 ([26]). The nabla discrete Laplace transform for a function f defined on N0 is ∞ X N f (z) = (1 − z)t−1 f (t). (3) t=1
For a function f defined on Na we have
Na f (z) =
∞ X
(1 − z)t−1 f (t).
(4)
t=a+1
DEFINITION 2 (see also [26]). Let s ∈ R, 0 < α < 1 and f, g : Na → R be functions. The nabla discrete convolution of f with g is defined by (f ∗ g)(t) =
t X
s=a+1
g(t − ρ(s))f (s).
(5)
PROPOSITION 1 (see also [27]). For any c ∈ R \ {. . . , −2, −1, 0}, s ∈ R and h, p defined on Nc we conclude that (Nc (h ∗ p))(z) = (Nc h)(z)(N p)(z). Proof : (Nc (f ∗ g))(z) = =
∞ X
(1 − z)t−1
t=c+1
t X c+1
f (s)g(t − ρ(s))
∞ X ∞ t X X (1 − z)t−1 f (s)g(t − ρ(s))
s=c+1 t=s
c+1
(6)
ON FRACTIONAL DERIVATIVES WITH EXPONENTIAL KERNEL AND. . .
13
∞ X ∞ X = (1 − z)r−1 (1 − z)s−1 f (s)g(r) s=c+1 r=1
= (Nc f )(z)(N g)(z), where r = t − ρ(s) was used.
For the delta Laplace convolution theory we refer to ([24], Section 3.10). LEMMA 1 ([26]). Let g be defined on N0 . Then (N ∇(g(t))(z) = z(N g)(z) − g(0).
(7)
The generalization of Lemma 1 is given below. LEMMA 2 ([26]). Let g be defined on Na . Then (Na ∇(g(t))(z) = z(Na g)(z) − (1 − z)a g(a).
(8)
We recall that the (delta or nabla) Laplace transform of the constant function 1 1 on any time scale is 1z and the Laplace transform of the exponential function is z−λ . 2.
Left and right fractional derivatives with exponential nonsingular kernel and their corresponding integral operators
Here we define the discrete fractional derivatives with nabla exponential kernels. From the nabla time scale calculus (see [24], p. 55), we recall that the nabla 1 t−a , exponential function on the Z (or Na ) time scale is given by b eλ (t, a) = 1−λ 1 t ∈ Na . According to [31], if f ∈ H (a, b), a < b, α ∈ [0, 1], then the new (left) Caputo fractional derivative is Z M(α) t ′ −α CFC α ( a D f )(t) = f (s) exp(λ(t − s))ds, λ= . (9) 1−α a 1−α The right new (Caputo) fractional derivative can be defined by Z M(α) b ′ −α CFC α f (s) exp(λ(s − t))ds, λ= , ( Db f )(t) = − 1−α t 1−α
(10)
λ=
−α , 1−α
(11)
λ=
−α . 1−α
(12)
the left new Riemann fractional derivative by Z t M(α) d CFR α ( a D f )(t) = f (s) exp(λ(t − s))ds, 1 − α dt a
and the right new Riemann fractional derivative by Z b M(α) d CFR α ( Db f )(t) = − f (s) exp(λ(s − t))ds, 1 − α dt t
It can be easily shown that (CFR a D α Qf )(t) = Q(CFR Dbα f )(t) and (CFC a D α Qf )(t) = Q(CFC Dbα f )(t).
14
T. ABDELJAWAD and D. BALEANU
Consider the equation (CFR a D α f )(t) = u(t). If we apply Laplace transform starting at a to both sides and use the convolution theorem, we conclude that F (s) =
α U (s) 1−α U (s) − , B(α) B(α) s
where F (s) = La {f (t)}(s) and U (s) = La {u(t)}(s). Then, apply the inverse Laplace to reach Z t 1−α α f (t) = u(t) + u(s)ds. B(α) B(α) a Therefore, we can define the corresponding left fractional integral of (CF a D α ) by Z t 1−α α CF α ( a I u)(t) = u(t) + u(s)ds. (13) B(α) B(α) a We define the right fractional integral (CF Ibα u)(t)
α 1−α u(t) + = B(α) B(α)
Z
b
u(s)ds,
(14)
t
so that (CF a I α Qu)(t) = Q(CF Ibα u)(t). Furthermore, by the action of the Q-operator we can easily show that (CF Ibα CF Dbα u)(t) = u(t). Conversely, consider the equation Z t 1−α α CF α ( a I u)(t) = u(t) + u(s)ds = f (t). B(α) B(α) a Apply the Laplace transform to see that 1−α α U (s) = F (s), B(α) sB(α) from which it follows that U (s) =
B(α) sF (s) . 1−α s−λ
(15)
The right-hand side of (15) is nothing more but the Laplace of (CFR a D α f )(t). Hence, (CF a I α CFR a D α f )(t) = f (t). Similarly or by the action of the Q-operator we can show that (CF Ibα CF Dbα f )(t) = f (t). REMARK 1 (The action of the Q-operator). The Q-operator acts regularly between left and right new fractional differences as follows: • (QCFR a D α f )(t) = (CFR Dbα Qf )(t), • (QCFC a D α f )(t) = (CFC Dbα Qf )(t).
ON FRACTIONAL DERIVATIVES WITH EXPONENTIAL KERNEL AND. . .
15
PROPOSITION 2 (The relation between Riemann- and Caputo-type fractional derivatives with exponential kernels). B(α) f (a)eλ(t−a) , 1−α B(α) • (CFC Dbα f )(t) = (CFR Dbα f )(t) − f (b)eλ(b−t) . 1−α Proof : From the relations • (CFC a D α f )(t) = (CFR a D α f )(t) −
B(α) sF (s) , 1−α s−λ
(16)
B(α) sF (s) B(α) 1 − f (a)e−as , 1−α s−λ 1−α s−λ
(17)
La {(CFR a D α f )(t)}(s) = and La {(CFC a D α f )(t)}(s) = we conclude that
1 B(α) f (a)e−as . (18) 1−α s−λ Applying the inverse Laplace to (18) and making use of the fact that La {f (t −a)}(s) = e−as L{f (t)}(s) we reach our conclusion in the first part. The second part can be proved by the first part and the action of the Q-operator. La {(CFC a D α f )(t)}(s) = La {(CFR a D α f )(t)}(s) −
The following result will be used as an important tool in our study. PROPOSITION 3. For 0 < α < 1, we conclude that AB
(
Similarly,
aI
α ABC
aD
α
f )(x) = f (x) − f (a)e
λ(x−a)
= f (x) − f (a).
α − f (a) 1−α
Z
x
eλ(x−s) ds a
(AB Ibα ABC Dbα f )(x) = f (x) − f (b).
(19)
3.
The integration by parts formula for fractional derivatives with exponential kernels Before we present an integration by parts for the new proposed fractional derivatives and integrals we introduce the following function spaces: For p ≥ 1 and α > 0, we define and
(CF a I α (Lp ) = {f : f = CF a I α ϕ,
ϕ ∈ Lp (a, b)}
(CF Ibα (Lp ) = {f : f = CF Ibα φ,
φ ∈ Lp (a, b)}. CFR
(20) (21) α
Above it was shown that the left fractional operator and its asaD sociated fractional integral CF a I α satisfy (CFR a D α CF a I α f )(t) = f (t) and that (CFR Dbα CF Ibα f )(t) = f (t). Also it was shown that (CF a I α CFR a D α f )(t) = f (t) and
16
T. ABDELJAWAD and D. BALEANU
(CF Ibα CFR Dbα f )(t) = f (t) and hence the function spaces (CF a I α (Lp ) and (CF Ibα (Lp ) are nonempty. THEOREM 1. Let α > 0, p ≥ 1, q ≥ 1, and 1 1 + ≤1+α p q (p 6 = 1 and q 6 = 1 in the case
1 p
+
1 q
= 1 + α). Then
• If ϕ(x) ∈ Lp (a, b) and ψ(x) ∈ Lq (a, b), then Z Z b 1−α b CF α ψ(x)ϕ(x)dx ϕ(x)( a I ψ)(x)dx = B(α) a a Z b Z b α ψ(x) ψ(t dt)dx + B(α) a x Z b = ψ(x)(CF Ibα ϕ(x)dx
(22)
a
and similarly, Z x Z Z b Z b α 1−α b CF α ψ(x)ϕ(x)dx + ψ(x) ϕ(t dt)dx ϕ(x)( Ib ψ)(x)dx = B(α) a B(α) a a a Z b = ψ(x)(CF a I α ϕ)(x)dx. (23) a
• If f (x) ∈
CF α Ib (Lp ) Z b a
and g(x) ∈ CF a I α (Lq ), then Z b CFR α f (x)( a D g)(x)dx = (CFR Dbα f )(x)g(x)dx. a
Proof : From the definition and the identity Z x Z Z b Z b ψ(x) ϕ(t dt)dx = ϕ(t) a
we have Z
a
b
a
a
t
b
ψ(x dx)dt
Z x α 1−α ψ(x) + ψ(t)dt dx ϕ(x) B(α) B(α) a a Z b 1−α = ϕ(x)ψ(x)dx B(α) a Z b Z b α + ψ(x) ϕ(t dt)dx B(α) a x Z b = ψ(x)(CF Ibα ϕ(x)dx.
ϕ(x)(CF a I α ψ)(x)dx =
Z
a
(24)
b
(25)
ON FRACTIONAL DERIVATIVES WITH EXPONENTIAL KERNEL AND. . .
17
The other case follows similarly by the definition of the right fractional integral in (14) and the identity (24). From definition and the first part we have Z b Z b CFR α f (x)( a D g)(x)dx = (CF Ibα φ)(x).(CFR a D α ◦ CF a I α ϕ)(x)dx a
a
= = =
Z
b
(CF Ibα φ)(x).ϕ(x)dx
a
Z
b
φ(x)(CF a I α ϕ)(x)dx
a
Z
a
b
(CFR Dbα f )(x)g(x)dx.
Before proving (a) by parts formula for Caputo fractional derivatives, we use the following notation: • The (left) exponential integral operator Z x (eλ,a + ϕ)(x) = eλ(t−a) ϕ(t)dt,
x > a.
(26)
x < b.
(27)
a
• The (right) exponential integral operator Z b (eλ,b− ϕ)(x) = eλ(b−t) ϕ(t)dt, x
THEOREM 2. One has Z b Z CFC α • ( a D f )(t)g(t) = a
•
Z
a
b
(CFC Dbα f )(t)g(t) =
Z
a
b
a
f (t)(CFR Dbα g)(t) +
b
f (t)(CFR a D α g)(t) −
B(α) f (t)e −α ,b− g)(t)|ba , 1−α 1−α
B(α) f (t)e −α ,a + g)(t)|ba . 1−α 1−α
Proof : The proof of the first part follows by Theorem 1 and the first part of Proposition 2, and the proof of the second part follows by Theorem 1 and the second part of Proposition 2. 4.
Fractional Euler–Lagrange equations
THEOREM 3. Let 0 < α ≤ 1 be noninteger, c, d ∈ R, c < d. Assume that the functional J : C 2 [c, d] → R of the form Z d J (f ) = L(t, f (t), CFC c D α f (t))dt c
18
T. ABDELJAWAD and D. BALEANU
admits a local extremum in S = {z ∈ C 2 [c, d] : z(c) = C, z(d) = D} at some f ∈ S, where L : [c, d] × R × R → R. Then, [L1 (s) + ABR Ddα L2 (s)] = 0, for all s ∈ [0, d],
where L1 (s) =
∂L (s) ∂f
and
L2 (s) =
(28)
∂L (s). α 0D f
∂ CFC
Proof : We suppose that J posseses a local maximum in S at f . As a result, there exists an ǫ > 0 and J (fb) − J (f ) ≤ 0 for all fb ∈ S and kfb − f k = supt∈Na ∩b N |fb(t) − f (t)| < ǫ. For any fb ∈ S there is an η ∈ H = {z ∈ C 2 [c, d], z(c) = z(d) = 0} such that fb = f + ǫη. Thus, the ǫ-Taylor’s theorem implies the following L(t, f, fb) = L(t, f + ǫη, CFC c D α f + ǫ CFC c D α η) = L(t, f, CFC c D α f ) + ǫ[ηL1 + CFC c D α ηL2 ] + O(ǫ 2 ).
Then J (fb) − J (f ) =
Z
=ǫ
d
c
Z
c
L(t, fb(t), CFC c D α fb(t)) −
Z
d
L(t, f (t), CFC c D α f (t))
0
d
[η(t)L1 (t) + (CFC c D α η)(t)L2 (t)] + O(ǫ 2 ).
(29)
Rd Let the quantity δJ (η, y) = c [η(t)L1 (t) + (CFC c D α η)(t)L2 (t)]dt denote the first variation of J . If η ∈ H , then −η ∈ H , and δJ (η, y) = −δJ (−η, y), respectively. For ǫ small, the sign of J (fb) − J (f ) is given by the first variation sign, unless δJ (η, y) = 0 for all η ∈ H . To make η free, we use the integration by parts formula in Theorem 2, to reach Z d B(α) (e −α − L2 )(t)|dc = 0, δJ (η, y) = η(s)[L1 (s) + CFR Ddα L2 (s)] + η(t) 1 − α 1−α ,b 0
for all η ∈ H , therefore the result comes by the fundamental lemma of calculus of variation (FLCV).
We call (e −α ,c− L2 )(t)|c0 = 0 the natural boundary condition. 1−α When the Lagrangian depends on the right Caputo fractional derivative, we conclude the following. THEOREM 4. Let 0 < α ≤ 1 be noninteger, c, d ∈ R, c < d. Assume that the functional J : C 2 [c, d] → R of the form Z d J (f ) = L(t, f (t), CFC Ddα f (t))dt c
ON FRACTIONAL DERIVATIVES WITH EXPONENTIAL KERNEL AND. . .
19
has a local extremum in S = {y ∈ C 2 [c, d] : y(c) = C, y(d) = D} at some f ∈ S, where L : [c, d] × R × R → R. Then [L1 (s) + CFR c D α L2 (s)] = 0, for all s ∈ [c, d], where L1 (s) =
∂L (s) ∂f
and
L2 (s) =
∂L ∂ CFC Ddα f
(30)
(s).
Proof : This proof is similar to the one of Theorem 3 by use the second integration by parts in Proposition 2 to obtain (e −α ,0+ L2 )(t)|dc = 0. 1−α
EXAMPLE 1. In order to exemplify our results we study an example of physical interest under Theorem 3. We consider the following Z b 1 CFC α 2 J (z) = ( a D z(t)) − V (z(t)) , 2 a where 0 < α < 1 and with z(a), z(b) are assigned or with (e1α,1, −α ,b− CFC 0 D α z(t))(t)|ba = 0. 1−α
Then, by using Theorem 3 the Euler–Lagrange equation becomes (CFR Dbα oCFC a D α z)(s) −
dV (s) = 0 for all s ∈ [a, b]. dz
Clearly, if we let α → 1, then the Euler–Lagrange equation for the above functional is reduced to z′′ = 0 when the potential functions is zero. Its solution is then linear, which agrees with the classical case. For equations in the classical fractional case with composition of the left and right fractional derivatives with the action of the Q-operator we refer to [30].
5.
Discrete versions of fractional derivatives with exponential kernels
Using the time scale notation, the nabla discrete exponential kernel can be expressed as t−ρ(s) 1 b eλ (t, ρ(s)) = = (1 − α)t−ρ(s) 1−λ
and
where λ =
−α . 1−α
e eλ (t, σ (s)) = (1 + λ)t−σ (s) =
1 − 2α 1−α
t−σ (s)
,
Hence, we can propose the following discrete versions.
20
T. ABDELJAWAD and D. BALEANU
DEFINITION 3. For α ∈ (0, 1) and f defined on Na , or b N in the right case, we define: • The left (nabla) new Caputo fractional difference by (CFC a ∇ α f )(t) =
t B(α) X (∇s f )(s)(1 − α)t−ρ(s) 1 − α s=a+1
= B(α)
t X
(∇s f )(s)(1 − α)t−s .
(31)
s=a+1
• The right (nabla) new Caputo fractional difference by (CFC ∇bα f )(t) =
b−1 B(α) X (−1s f )(s)(1 − α)s−ρ(t) 1 − α s=t
= B(α)
b−1 X (−1s f )(s)(1 − α)s−t .
(32)
s=t
• The left (nabla) new Riemann fractional difference by (CFR a ∇ α f )(t) =
t X B(α) f (s)(1 − α)t−ρ(s) ∇t 1 − α s=a+1
= B(α)∇t
t X
s=a+1
f (s)(1 − α)t−s .
(33)
• The right (nabla) new Riemann fractional difference by (CFR ∇bα f )(t) =
b−1 X B(α) (−1t ) f (s)(1 − α)s−ρ(t) 1−α s=t
= B(α)(−1t )
b−1 X s=t
f (s)(1 − α)s−t .
(34)
REMARK 2. In the limiting case α → 0 and α → 1, we remark the following relations (a) and (b) and
(CFC a ∇ α f )(t) → f (t) − f (a) (CFC a ∇ α f )(t) → ∇f (t)) (CFC ∇bα f )(t) → f (t) − f (b) (CFC ∇bα f )(t) → −1f (t)
as as as as
α → 0, α → 1. α → 0, α → 1.
ON FRACTIONAL DERIVATIVES WITH EXPONENTIAL KERNEL AND. . .
(c) and (d) and
(CFR a ∇ α f )(t) → f (t) (CFR a ∇ α f )(t) → ∇f (t) (CFR ∇bα f )(t) → f (t) (CFR ∇bα f )(t) → −1f (t)
as as as as
21
α → 0, α → 1. α → 0, α → 1.
REMARK 3 (The action of the discrete Q-operator). The Q-operator acts regularly between left and right new fractional differences as follows: (a) (QCFR a ∇ α f )(t) = (CF R ∇bα Qf )(t), (b) (QCFC a ∇ α f )(t) = (CF C ∇bα Qf )(t). Consider the equation (CFR a ∇ α f )(t) = u(t). If we apply the discrete nabla Laplace transform starting at a to both sides and use the convolution theorem, we conclude that λ 1−α α U (s) 1−α [U (s) − U (s)] = U (s) − , F (s) = B(α) s B(α) B(α) s where F (s) = (Na {f (t)}(s) and U (s) = (Na {u(t)}(s). Then, apply the inverse Laplace to get t 1−α α X f (t) = u(t) + u(s)ds. B(α) B(α) s=a+1 Therefore, we can define the corresponding left fractional integral of (CFR a D α ) by (CF a ∇ −α u)(t) =
t 1−α α X u(t) + u(s)ds. B(α) B(α) s=a+1
(35)
We define the right fractional integral (CF ∇b−α u)(t) =
b−1 α X 1−α u(t) + u(s)ds, B(α) B(α) s=t
(36)
so that (CF a ∇ −α Qu)(t) = Q(CF ∇b−α u)(t). Furthermore, by the action of the Qoperator we can easily show that (CF ∇b−α CF ∇bα u)(t) = u(t). Conversely, consider the equation (CF a ∇ −α u)(t) =
t 1−α α X u(s) = f (t). u(t) + B(α) B(α) a+1
22
T. ABDELJAWAD and D. BALEANU
Apply the nabla discrete Laplace transform to see that α 1−α U (s) = F (s). B(α) sB(α) It follows that U (s) =
B(α) sF (s) . 1−α s−λ
(37)
The right-hand side of (37) is nothing more but the Laplace of (CF a ∇ α f )(t). Hence, (CF a ∇ −α CF a ∇ α f )(t) = f (t).
Similarly or by the action of the Q-operator, we can show that (CF ∇b−α CF ∇bα f )(t) = f (t). PROPOSITION 4. One has (i) (CFC a ∇ α f )(t) = (CFR a ∇ α f )(t) − B(α) f (a)(1 − α)(t−a) . 1−α (ii) (CFC ∇bα f )(t) = (CFR ∇bα f )(t) − B(α) f (b)(1 − α)(b−t) . 1−α Proof : From the relations B(α) sF (s) , 1−α s−λ
(38)
B(α) 1 B(α) sF (s) − f (a)(1 − s)a , 1−α s−λ 1−α s−λ
(39)
Na {(CFR a ∇ α f )(t)}(s) = and
Na {(CFC a ∇ α f )(t)}(s) = we conclude that
Na {(CFC a D α f )(t)}(s) = Na {(CFR a D α f )(t)}(s) −
B(α) 1 f (a)(1 − s)a . 1−α s−λ
(40)
Applying the inverse Laplace to (40) and making use of the fact that Na {f (t−a)}(s) = (1 − s)a N {f (t)}(s) we reach our conclusion in the first part. The second part can be proved by the first part and the action of the Q-operator. 6. Integration by parts formula for fractional differences with discrete exponential kernels Above it was shown that the left fractional operator CFR a ∇ α and its associate fractional integral CF a ∇ −α satisfy (CFR a ∇ α CFa ∇ −α f )(t) = f (t) and that (CFR Dbα CF ∇b−α f )(t) = f (t) . Also it was shown that (CF a ∇ −α CFR a ∇ α f )(t) = f (t) and (CF ∇b−α CFR ∇bα f )(t) = f (t).
ON FRACTIONAL DERIVATIVES WITH EXPONENTIAL KERNEL AND. . .
23
THEOREM 5. Assume thaat 0 < α < 1 and the used functions are defined on Na ∩ b N, a ≡ b(mod1). Then (i) for functions ϕ(t) and ψ(t) we have b−1 X
CF
ϕ(t)(
t=a+1
a∇
−α
b−1 b−1 b−1 X 1−α X α X ϕ(s) ψ)(t) = ψ(t)ϕ(t) + ψ(t) B(α) t=a+1 B(α) t=a+1 s=t
=
b−1 X
t=a+1
ψ(t)CF ∇b−α ϕ(t),
(41)
and similarly b−1 X
t=a+1
ϕ(t)(CF ∇b−α ψ)(t) = =
b−1 b−1 t X α X 1−α X ψ(t)ϕ(t) + ψ(t)( ϕ(s)) B(α) t=a+1 B(α) t=a+1 s=a+1 b−1 X
t=a+1
ψ(t)(CF a ∇ −α ϕ)(t),
(42)
(ii) for functions f (t) and g(t) defined on Na we have b−1 X
t=a+1
f (t)(CFR a ∇ α g)(t) =
b−1 X
(CFR ∇bα f )(t)g(t).
t=a+1
Proof : (i) From the definition and the identity b−1 X a+1
ψ(t)(
t X
s=a+1
ϕ(s)) =
b−1 X a+1
b−1 X ϕ(s)( ψ(t)),
(43)
t=s
the first part follows. The other case follows similarly by the definition of the right fractional difference in (36) and the identity (43). (ii) From the statement before Theorem 5 and the first part we have b−1 X
t=a+1
f (t)(CFR a ∇ α g)(t) = = =
b−1 X
(CF ∇b−α CFR ∇bα f )(t)(CFR a ∇ α g)(t)
b−1 X
(CFR ∇bα f )(t)(CF a ∇ −α CFR a ∇ α g)(t)
b−1 X
(CFR ∇bα f )(t)g(t).
t=a+1
t=a+1
t=a+1
Before proving the by part formula for Caputo fractional derivatives, we use the following notation:
24
T. ABDELJAWAD and D. BALEANU
(a) the (left) discrete exponential integral operator (t−a) t X 1 (Eλ,a + ϕ)(t) = ϕ(s), 1−λ s=a+1 (b) the (right) discrete exponential integral operator (b−t) b−1 X 1 ϕ(t), (Eλ,b− ϕ)(x) = t=s 1 − λ
t ∈ Na ,
(44)
t ∈ b N.
(45)
THEOREM 6. One has b−1 b−1 X X B(α) ρ α (i) (CFC a−1 ∇ α f )(t)g(t) = f (t)(CFR ∇b−1 g)(t) + f (t)E −α ,b− g)(t)|ba , 1−α 1 − α t=a t=a (ii)
b X
α (CFC ∇b+1 f )(t)g(t)
t=a+1
=
b X
t=a+1
f (t)(CFR a+1 ∇ α g)(t) −
B(α) σ f (t)E −α ,a + g)(t)|ba . 1−α 1−α
Proof : The proof of the first part follows by Theorem 5 and the first part of Proposition 4. The proof of the second part can be established by Theorem 5 and the second part of Proposition 4. 7. New discrete fractional Euler–Lagrange equations THEOREM 7. Let 0 < α < 1 be noninteger, c, d ∈ R, c < d, c ≡ d(mod1). Assume that d−1 X J (f ) = L(t, f ρ (t), CFC c−1 ∇ α f (t)) t=c
possesses a local extremum in S = {z : (Nc−1 ∩ d−1 N) → R : z(c − 1) = C, z(d − 1) = D} at some f ∈ S, where L : (Nc−1 ∩ d−1 N) × R × R → R. Then where
α [L1 (s) + CFR ∇c−1 L2 (s)] = 0, for all s ∈ (Nc−1 ∩ d−1 N),
L1 (s) =
∂L (s) ∂f ρ
and
L2 (s) =
∂L ∂ CFC
c−1 ∇
αf
(46)
(s).
Proof : We suppose that J reaches a local maximum in S at f . As a result, there exists an ǫ > 0 fulfilling J (fb) − J (f ) ≤ 0 for all fb ∈ S such that kfb − f k = supt∈Nc ∩d N |fb(t) − f (t)| < ǫ. For any fb ∈ S there is an η ∈ H = {y : (Nc−1 ∩ d−1 N) → R, z(a − 1) = z(b − 1) = 0} such that fb = f + ǫη. The ǫ-Taylor’s theorem and the assumption lead us to the conclusion that δJ (η, z) =
d−1 X [ηρ (t)L1 (t) + (CF C c−1 ∇ α η)(t)L2 (t)]dt = 0 t=c
ON FRACTIONAL DERIVATIVES WITH EXPONENTIAL KERNEL AND. . .
25
for all η ∈ H . Thus, to make η free, we use Theorem 6 to obtain δJ (η, f ) =
d−1 X s=c
α ηρ (s)[L1 (s) + CFR ∇d−1 L2 (s)] + ηρ (t)
B(α) (E −α − L2 )(t)|dc = 0, 1 − α 1−α ,d
for all η ∈ H , and hence the result comes as a result of the discrete FLCV.
(E −α ,d − L2 )(t)|dc 1−α
Here = 0 denotes the natural boundary condition. For the case when the Lagrangian contains the discrete right Caputo fractional derivative, we obtain the theorem. THEOREM 8. Let 0 < α ≤ 1 be noninteger, c, d ∈ R, c < d, c ≡ d(mod1). Suppose J , that d X α J (f ) = L(t, f σ (t), CFC ∇d+1 f (t)), c+1
admits a local extremum in S = {z : (Nc+1 ∩ d+1 N) → R : z(c +1) = A, z(d +1) = B} at some f ∈ S, where L : (Nc+1 ∩ d+1 N) × R × R → R. Then where
[L1 (s) + CFR c+1 ∇ α L2 (s)] = 0, for all s ∈ (Nc+1 ∩ d+1 N), L1 (s) =
∂L (s) ∂f σ
and
L2 (s) =
∂L α f ∂ CF C ∇d+1
(47)
(s).
Proof : The proof looks similar to the one of Theorem 7 by using Theorem 6 to obtain the natural boundary condition of the form (E −α ,c+ L2 )(t)|dc = 0. 1−α
EXAMPLE 2. Below we provide a discrete example of physical interest under Theorem 7, namely d−1 X 1 J (z) = [ (CFC c−1 ∇ α z(t))2 − V (zρ (t))], t=c 2
where 0 < α < 1 and with z(c − 1), z(d − 1) given or with the natural boundary condition (E −α ,d − CFC c−1 ∇ α z(t))(t)|dc = 0. 1−α
Thus, the related Euler–Lagrange equation becomes dV α oCFC c−1 ∇ α z)(s) − (CFR ∇d−1 (s) = 0 for all s ∈ (Nc−1 ∩ d−1 N). dz To see the difference with the classical discrete case within nabla we suggest the reader [29]. 8.
Conclusions The Caputo–Fabrizio derivative was already applied successfully since it was recently suggested. Several applications of it were implemented in some areas
26
T. ABDELJAWAD and D. BALEANU
where the nonlocality appears in real world phenomena. However, the properties of this derivative should be deeply explored and applied to describe the real world phenomena. In this manuscript we elaborated the discrete version of the Caputo–Fabrizio derivative. By proving the integration by parts we open the gate for application in discrete variational principles and their applications in control theory. The reported discrete fractional Euler–Lagrange will play a crucial role in constructing the related Lagrangian and Hamiltonian dynamics. REFERENCES [1] I. Podlubny: Fractional Differential Equations, Academic Press: San Diego, CA 1999. [2] G. Samko, A. A. Kilbas and S. Marichev: Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon 1993. [3] A. A. Kilbas, M. H. Srivastava and J. J. Trujillo: Theory and Application of Fractional Differential Equations, North Holland Mathematics Studies 204, 2006. [4] R. L. Magin: Fractional Calculus in Bioengineering, Begell House Publishers 2006. [5] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo: Fractional Calculus Models and Numerical Methods (Series on Complexity, Nonlinearity and Chaos), World Scientific, Singapore 2012. [6] Y. Zhou: Basic Theory of Fractional Differential Equations, World Scientific, Singapore 2014. [7] Y. Zhou: Fractional Evolution Equations and Inclusions: Analysis and Control, Academic Press 2016. [8] T. Abdeljawad: On Riemann and Caputo fractional differences, Comput. Math. Appl. 62 (2011), 1602–1611. [9] F. M. Atıcı and P. W. Eloe: A Transform method in discrete fractional calculus, Int. J. Differ. Equ. 2 (2007), 165–176. [10] F. M. Atıcı and P. W. Eloe: Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc. 137 (2009), 981–989. [11] F. M. Atıcı and P. W. Eloe: Discrete fractional calculus with the nabla operator, Electr. J. Qualit. Theor. Differ. Equ. 1–12 (2009). [12] F. M. Atıcı and S. S¸eng¨ul: Modelling with fractional difference equations, J. Math. Anal. Appl. 369 (2010), 1–9. [13] K. S. Miller and B. Ross: Fractional difference calculus, Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, Japan, (1989) 139–152. [14] T. Abdeljawad and D. Baleanu: Fractional differences and integration by parts, J. Comput. Anal. Appl. 13 (2011), 574–582. [15] N. R. O. Bastos, R. A. C. Ferreira and D. F. M. Torres: Discrete-time fractional variational problems, Signal Proc. 91 (2011), 513–524. [16] H. L. Gray and N. F. Zhang: On a new definition of the fractional difference, Math. Comput. 50 (1988) , 513–529. [17] G. A. Anastassiou: Principles of delta fractional calculus on time scales and inequalities, Math. Comput. Model. 52 (2010), 556–566. [18] G. A. Anastassiou: Nabla discrete calculus and nabla inequalities, Math. Comput. Model., 51 (2010), 562–571. [19] T. Abdeljawad and F. Atici: On the definitions of nabla fractional differences, Abstr. Appl. Anal. 2012 (2012), Article ID 406757, 13 pages, doi:10.1155/2012/406757. [20] T. Abdeljawad: On Delta and Nabla Caputo fractional differences and dual identities, Discr. Dyn. Nat. Soc. 2013 (2013), Article ID 406910, 12 pages. [21] G. C. Wu, D. Baleanu, S. D. Zeng and Z. G. Deng: Discrete fractional diffusion equation, Nonlinear Dyn. 80 (2015), 281–286. [22] G. C. Wu, D. Baleanu and S. D. Zeng: Several fractional differences and their applications to discrete maps, J. Appl. Nonlinear Dyn. 4 (2015), 339–348.
ON FRACTIONAL DERIVATIVES WITH EXPONENTIAL KERNEL AND. . .
27
[23] T. Abdeljawad: Dual identities in fractional difference calculus within Riemann, Adv. Differ. Equ. 2013 (2013), 36. [24] M. Bohner and A. Peterson: Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston-BaselBerlin 2003. [25] Bohner M. and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston 2003. [26] T. Abdeljawad, F. Jarad and D. Baleanu: A semigroup-like property for discrete Mittag-Leffler functions, Adv. Differ. Equ. 2012 (2012), 72. [27] T. Abdeljawad and D. Baleanu: Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Differ. Equ. 2016 (2016), 232. [28] M. Caputo and M. Fabrizio: A new definition of fractional derivative without singular kernel, Progr. Frac. Differ. Appl. 1 (2015), 73–85. [29] T. Abdeljawad: Nabla Euler–Lagrange equations in discrete fractional variational calculus within Riemann and Caputo, Int. J. Math. Comput.22 (2014), 144-153. [30] T. Abdeljawad (Maraaba), D. Baleanu and F. Jarad: Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, J. Math. Phys. 49, (2008). [31] M. Caputo and M. Fabrizio: A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 1 (2) (2015), 73-85.
REPORTS ON MATHEMATICAL PHYSICS
No. 1
ON FRACTIONAL DERIVATIVES WITH EXPONENTIAL KERNEL AND THEIR DISCRETE VERSIONS T HABET A BDELJAWAD Department of Mathematics and Physical Sciences, Prince Sultan University, P. O. Box 66833, 11586 Riyadh, Saudi Arabia (e-mail: [email protected])
and D UMITRU BALEANU Department of Mathematics, C¸ankaya University, 06530 Ankara, Turkey and Institutes of Space Sciences, Magurele-Bucharest, Romania (e-mail: [email protected]) (Received June 15, 2016 — Revised November 11, 2016) In this paper we define the right fractional derivative and its corresponding right fractional integral with exponential kernel. We provide the integration by parts formula and we use the Q-operator to confirm our results. The related Euler–Lagrange equations are obtained and one example is reported. Moreover, we formulate and discuss the discrete counterparts of our results. Keywords: Caputo fractional difference, Q-operator, discrete exponential function, discrete nabla Laplace transform, convolution.
1.
Introduction
The techniques of fractional calculus are applied successfully in many branches of science and engineering [1–7]. The power law effects are described accurately within the fractional calculus approach. However, there are several complex phenomena which do not obey this law, thus we need to find alternatives, e.g. the nonlocal kernels without singularity in order to overcome this issue. On the other side the discrete version of any fractional derivative is one of the interesting topics nowadays [8–20]. Some applications of the discrete fractional Caputo derivative can be seen in [21, 22]. For two real numbers a < b, a ≡ b(mod1), we denote Na = {a, a + 1, . . . } and N = {b, b − 1, . . . }. Also, ∇g(t) = g(t) − g(t − 1). b The following is the action of the discrete Q-operator on fractional sums and differences [20, 23]: [11]
12
T. ABDELJAWAD and D. BALEANU
• (∇a−α Qf )(t) = Qb ∇ −α f (t), • (∇aα Qf )(t) = Qb ∇ α f (t), • (C ∇aα Qf )(t) = Qb C ∇ α f (t). In this article, we use the action of discrete Q-operator to analyze the proposed definitions of fractional differences with discrete exponential function kernels. We recall some facts about discrete transform within nabla. For more general theory of Laplace transform on time scales we refer to [24, 25]. The (nabla) exponential function on the time scale Z is written as t−t0 1 b e(λ, t − t0 ) = , (1) 1−λ while the (delta) exponential function becomes
e e(λ, t − t0 ) = (1 + λ)t−t0 .
(2)
In this paper, we follow the nabla discrete analysis.
DEFINITION 1 ([26]). The nabla discrete Laplace transform for a function f defined on N0 is ∞ X N f (z) = (1 − z)t−1 f (t). (3) t=1
For a function f defined on Na we have
Na f (z) =
∞ X
(1 − z)t−1 f (t).
(4)
t=a+1
DEFINITION 2 (see also [26]). Let s ∈ R, 0 < α < 1 and f, g : Na → R be functions. The nabla discrete convolution of f with g is defined by (f ∗ g)(t) =
t X
s=a+1
g(t − ρ(s))f (s).
(5)
PROPOSITION 1 (see also [27]). For any c ∈ R \ {. . . , −2, −1, 0}, s ∈ R and h, p defined on Nc we conclude that (Nc (h ∗ p))(z) = (Nc h)(z)(N p)(z). Proof : (Nc (f ∗ g))(z) = =
∞ X
(1 − z)t−1
t=c+1
t X c+1
f (s)g(t − ρ(s))
∞ X ∞ t X X (1 − z)t−1 f (s)g(t − ρ(s))
s=c+1 t=s
c+1
(6)
ON FRACTIONAL DERIVATIVES WITH EXPONENTIAL KERNEL AND. . .
13
∞ X ∞ X = (1 − z)r−1 (1 − z)s−1 f (s)g(r) s=c+1 r=1
= (Nc f )(z)(N g)(z), where r = t − ρ(s) was used.
For the delta Laplace convolution theory we refer to ([24], Section 3.10). LEMMA 1 ([26]). Let g be defined on N0 . Then (N ∇(g(t))(z) = z(N g)(z) − g(0).
(7)
The generalization of Lemma 1 is given below. LEMMA 2 ([26]). Let g be defined on Na . Then (Na ∇(g(t))(z) = z(Na g)(z) − (1 − z)a g(a).
(8)
We recall that the (delta or nabla) Laplace transform of the constant function 1 1 on any time scale is 1z and the Laplace transform of the exponential function is z−λ . 2.
Left and right fractional derivatives with exponential nonsingular kernel and their corresponding integral operators
Here we define the discrete fractional derivatives with nabla exponential kernels. From the nabla time scale calculus (see [24], p. 55), we recall that the nabla 1 t−a , exponential function on the Z (or Na ) time scale is given by b eλ (t, a) = 1−λ 1 t ∈ Na . According to [31], if f ∈ H (a, b), a < b, α ∈ [0, 1], then the new (left) Caputo fractional derivative is Z M(α) t ′ −α CFC α ( a D f )(t) = f (s) exp(λ(t − s))ds, λ= . (9) 1−α a 1−α The right new (Caputo) fractional derivative can be defined by Z M(α) b ′ −α CFC α f (s) exp(λ(s − t))ds, λ= , ( Db f )(t) = − 1−α t 1−α
(10)
λ=
−α , 1−α
(11)
λ=
−α . 1−α
(12)
the left new Riemann fractional derivative by Z t M(α) d CFR α ( a D f )(t) = f (s) exp(λ(t − s))ds, 1 − α dt a
and the right new Riemann fractional derivative by Z b M(α) d CFR α ( Db f )(t) = − f (s) exp(λ(s − t))ds, 1 − α dt t
It can be easily shown that (CFR a D α Qf )(t) = Q(CFR Dbα f )(t) and (CFC a D α Qf )(t) = Q(CFC Dbα f )(t).
14
T. ABDELJAWAD and D. BALEANU
Consider the equation (CFR a D α f )(t) = u(t). If we apply Laplace transform starting at a to both sides and use the convolution theorem, we conclude that F (s) =
α U (s) 1−α U (s) − , B(α) B(α) s
where F (s) = La {f (t)}(s) and U (s) = La {u(t)}(s). Then, apply the inverse Laplace to reach Z t 1−α α f (t) = u(t) + u(s)ds. B(α) B(α) a Therefore, we can define the corresponding left fractional integral of (CF a D α ) by Z t 1−α α CF α ( a I u)(t) = u(t) + u(s)ds. (13) B(α) B(α) a We define the right fractional integral (CF Ibα u)(t)
α 1−α u(t) + = B(α) B(α)
Z
b
u(s)ds,
(14)
t
so that (CF a I α Qu)(t) = Q(CF Ibα u)(t). Furthermore, by the action of the Q-operator we can easily show that (CF Ibα CF Dbα u)(t) = u(t). Conversely, consider the equation Z t 1−α α CF α ( a I u)(t) = u(t) + u(s)ds = f (t). B(α) B(α) a Apply the Laplace transform to see that 1−α α U (s) = F (s), B(α) sB(α) from which it follows that U (s) =
B(α) sF (s) . 1−α s−λ
(15)
The right-hand side of (15) is nothing more but the Laplace of (CFR a D α f )(t). Hence, (CF a I α CFR a D α f )(t) = f (t). Similarly or by the action of the Q-operator we can show that (CF Ibα CF Dbα f )(t) = f (t). REMARK 1 (The action of the Q-operator). The Q-operator acts regularly between left and right new fractional differences as follows: • (QCFR a D α f )(t) = (CFR Dbα Qf )(t), • (QCFC a D α f )(t) = (CFC Dbα Qf )(t).
ON FRACTIONAL DERIVATIVES WITH EXPONENTIAL KERNEL AND. . .
15
PROPOSITION 2 (The relation between Riemann- and Caputo-type fractional derivatives with exponential kernels). B(α) f (a)eλ(t−a) , 1−α B(α) • (CFC Dbα f )(t) = (CFR Dbα f )(t) − f (b)eλ(b−t) . 1−α Proof : From the relations • (CFC a D α f )(t) = (CFR a D α f )(t) −
B(α) sF (s) , 1−α s−λ
(16)
B(α) sF (s) B(α) 1 − f (a)e−as , 1−α s−λ 1−α s−λ
(17)
La {(CFR a D α f )(t)}(s) = and La {(CFC a D α f )(t)}(s) = we conclude that
1 B(α) f (a)e−as . (18) 1−α s−λ Applying the inverse Laplace to (18) and making use of the fact that La {f (t −a)}(s) = e−as L{f (t)}(s) we reach our conclusion in the first part. The second part can be proved by the first part and the action of the Q-operator. La {(CFC a D α f )(t)}(s) = La {(CFR a D α f )(t)}(s) −
The following result will be used as an important tool in our study. PROPOSITION 3. For 0 < α < 1, we conclude that AB
(
Similarly,
aI
α ABC
aD
α
f )(x) = f (x) − f (a)e
λ(x−a)
= f (x) − f (a).
α − f (a) 1−α
Z
x
eλ(x−s) ds a
(AB Ibα ABC Dbα f )(x) = f (x) − f (b).
(19)
3.
The integration by parts formula for fractional derivatives with exponential kernels Before we present an integration by parts for the new proposed fractional derivatives and integrals we introduce the following function spaces: For p ≥ 1 and α > 0, we define and
(CF a I α (Lp ) = {f : f = CF a I α ϕ,
ϕ ∈ Lp (a, b)}
(CF Ibα (Lp ) = {f : f = CF Ibα φ,
φ ∈ Lp (a, b)}. CFR
(20) (21) α
Above it was shown that the left fractional operator and its asaD sociated fractional integral CF a I α satisfy (CFR a D α CF a I α f )(t) = f (t) and that (CFR Dbα CF Ibα f )(t) = f (t). Also it was shown that (CF a I α CFR a D α f )(t) = f (t) and
16
T. ABDELJAWAD and D. BALEANU
(CF Ibα CFR Dbα f )(t) = f (t) and hence the function spaces (CF a I α (Lp ) and (CF Ibα (Lp ) are nonempty. THEOREM 1. Let α > 0, p ≥ 1, q ≥ 1, and 1 1 + ≤1+α p q (p 6 = 1 and q 6 = 1 in the case
1 p
+
1 q
= 1 + α). Then
• If ϕ(x) ∈ Lp (a, b) and ψ(x) ∈ Lq (a, b), then Z Z b 1−α b CF α ψ(x)ϕ(x)dx ϕ(x)( a I ψ)(x)dx = B(α) a a Z b Z b α ψ(x) ψ(t dt)dx + B(α) a x Z b = ψ(x)(CF Ibα ϕ(x)dx
(22)
a
and similarly, Z x Z Z b Z b α 1−α b CF α ψ(x)ϕ(x)dx + ψ(x) ϕ(t dt)dx ϕ(x)( Ib ψ)(x)dx = B(α) a B(α) a a a Z b = ψ(x)(CF a I α ϕ)(x)dx. (23) a
• If f (x) ∈
CF α Ib (Lp ) Z b a
and g(x) ∈ CF a I α (Lq ), then Z b CFR α f (x)( a D g)(x)dx = (CFR Dbα f )(x)g(x)dx. a
Proof : From the definition and the identity Z x Z Z b Z b ψ(x) ϕ(t dt)dx = ϕ(t) a
we have Z
a
b
a
a
t
b
ψ(x dx)dt
Z x α 1−α ψ(x) + ψ(t)dt dx ϕ(x) B(α) B(α) a a Z b 1−α = ϕ(x)ψ(x)dx B(α) a Z b Z b α + ψ(x) ϕ(t dt)dx B(α) a x Z b = ψ(x)(CF Ibα ϕ(x)dx.
ϕ(x)(CF a I α ψ)(x)dx =
Z
a
(24)
b
(25)
ON FRACTIONAL DERIVATIVES WITH EXPONENTIAL KERNEL AND. . .
17
The other case follows similarly by the definition of the right fractional integral in (14) and the identity (24). From definition and the first part we have Z b Z b CFR α f (x)( a D g)(x)dx = (CF Ibα φ)(x).(CFR a D α ◦ CF a I α ϕ)(x)dx a
a
= = =
Z
b
(CF Ibα φ)(x).ϕ(x)dx
a
Z
b
φ(x)(CF a I α ϕ)(x)dx
a
Z
a
b
(CFR Dbα f )(x)g(x)dx.
Before proving (a) by parts formula for Caputo fractional derivatives, we use the following notation: • The (left) exponential integral operator Z x (eλ,a + ϕ)(x) = eλ(t−a) ϕ(t)dt,
x > a.
(26)
x < b.
(27)
a
• The (right) exponential integral operator Z b (eλ,b− ϕ)(x) = eλ(b−t) ϕ(t)dt, x
THEOREM 2. One has Z b Z CFC α • ( a D f )(t)g(t) = a
•
Z
a
b
(CFC Dbα f )(t)g(t) =
Z
a
b
a
f (t)(CFR Dbα g)(t) +
b
f (t)(CFR a D α g)(t) −
B(α) f (t)e −α ,b− g)(t)|ba , 1−α 1−α
B(α) f (t)e −α ,a + g)(t)|ba . 1−α 1−α
Proof : The proof of the first part follows by Theorem 1 and the first part of Proposition 2, and the proof of the second part follows by Theorem 1 and the second part of Proposition 2. 4.
Fractional Euler–Lagrange equations
THEOREM 3. Let 0 < α ≤ 1 be noninteger, c, d ∈ R, c < d. Assume that the functional J : C 2 [c, d] → R of the form Z d J (f ) = L(t, f (t), CFC c D α f (t))dt c
18
T. ABDELJAWAD and D. BALEANU
admits a local extremum in S = {z ∈ C 2 [c, d] : z(c) = C, z(d) = D} at some f ∈ S, where L : [c, d] × R × R → R. Then, [L1 (s) + ABR Ddα L2 (s)] = 0, for all s ∈ [0, d],
where L1 (s) =
∂L (s) ∂f
and
L2 (s) =
(28)
∂L (s). α 0D f
∂ CFC
Proof : We suppose that J posseses a local maximum in S at f . As a result, there exists an ǫ > 0 and J (fb) − J (f ) ≤ 0 for all fb ∈ S and kfb − f k = supt∈Na ∩b N |fb(t) − f (t)| < ǫ. For any fb ∈ S there is an η ∈ H = {z ∈ C 2 [c, d], z(c) = z(d) = 0} such that fb = f + ǫη. Thus, the ǫ-Taylor’s theorem implies the following L(t, f, fb) = L(t, f + ǫη, CFC c D α f + ǫ CFC c D α η) = L(t, f, CFC c D α f ) + ǫ[ηL1 + CFC c D α ηL2 ] + O(ǫ 2 ).
Then J (fb) − J (f ) =
Z
=ǫ
d
c
Z
c
L(t, fb(t), CFC c D α fb(t)) −
Z
d
L(t, f (t), CFC c D α f (t))
0
d
[η(t)L1 (t) + (CFC c D α η)(t)L2 (t)] + O(ǫ 2 ).
(29)
Rd Let the quantity δJ (η, y) = c [η(t)L1 (t) + (CFC c D α η)(t)L2 (t)]dt denote the first variation of J . If η ∈ H , then −η ∈ H , and δJ (η, y) = −δJ (−η, y), respectively. For ǫ small, the sign of J (fb) − J (f ) is given by the first variation sign, unless δJ (η, y) = 0 for all η ∈ H . To make η free, we use the integration by parts formula in Theorem 2, to reach Z d B(α) (e −α − L2 )(t)|dc = 0, δJ (η, y) = η(s)[L1 (s) + CFR Ddα L2 (s)] + η(t) 1 − α 1−α ,b 0
for all η ∈ H , therefore the result comes by the fundamental lemma of calculus of variation (FLCV).
We call (e −α ,c− L2 )(t)|c0 = 0 the natural boundary condition. 1−α When the Lagrangian depends on the right Caputo fractional derivative, we conclude the following. THEOREM 4. Let 0 < α ≤ 1 be noninteger, c, d ∈ R, c < d. Assume that the functional J : C 2 [c, d] → R of the form Z d J (f ) = L(t, f (t), CFC Ddα f (t))dt c
ON FRACTIONAL DERIVATIVES WITH EXPONENTIAL KERNEL AND. . .
19
has a local extremum in S = {y ∈ C 2 [c, d] : y(c) = C, y(d) = D} at some f ∈ S, where L : [c, d] × R × R → R. Then [L1 (s) + CFR c D α L2 (s)] = 0, for all s ∈ [c, d], where L1 (s) =
∂L (s) ∂f
and
L2 (s) =
∂L ∂ CFC Ddα f
(30)
(s).
Proof : This proof is similar to the one of Theorem 3 by use the second integration by parts in Proposition 2 to obtain (e −α ,0+ L2 )(t)|dc = 0. 1−α
EXAMPLE 1. In order to exemplify our results we study an example of physical interest under Theorem 3. We consider the following Z b 1 CFC α 2 J (z) = ( a D z(t)) − V (z(t)) , 2 a where 0 < α < 1 and with z(a), z(b) are assigned or with (e1α,1, −α ,b− CFC 0 D α z(t))(t)|ba = 0. 1−α
Then, by using Theorem 3 the Euler–Lagrange equation becomes (CFR Dbα oCFC a D α z)(s) −
dV (s) = 0 for all s ∈ [a, b]. dz
Clearly, if we let α → 1, then the Euler–Lagrange equation for the above functional is reduced to z′′ = 0 when the potential functions is zero. Its solution is then linear, which agrees with the classical case. For equations in the classical fractional case with composition of the left and right fractional derivatives with the action of the Q-operator we refer to [30].
5.
Discrete versions of fractional derivatives with exponential kernels
Using the time scale notation, the nabla discrete exponential kernel can be expressed as t−ρ(s) 1 b eλ (t, ρ(s)) = = (1 − α)t−ρ(s) 1−λ
and
where λ =
−α . 1−α
e eλ (t, σ (s)) = (1 + λ)t−σ (s) =
1 − 2α 1−α
t−σ (s)
,
Hence, we can propose the following discrete versions.
20
T. ABDELJAWAD and D. BALEANU
DEFINITION 3. For α ∈ (0, 1) and f defined on Na , or b N in the right case, we define: • The left (nabla) new Caputo fractional difference by (CFC a ∇ α f )(t) =
t B(α) X (∇s f )(s)(1 − α)t−ρ(s) 1 − α s=a+1
= B(α)
t X
(∇s f )(s)(1 − α)t−s .
(31)
s=a+1
• The right (nabla) new Caputo fractional difference by (CFC ∇bα f )(t) =
b−1 B(α) X (−1s f )(s)(1 − α)s−ρ(t) 1 − α s=t
= B(α)
b−1 X (−1s f )(s)(1 − α)s−t .
(32)
s=t
• The left (nabla) new Riemann fractional difference by (CFR a ∇ α f )(t) =
t X B(α) f (s)(1 − α)t−ρ(s) ∇t 1 − α s=a+1
= B(α)∇t
t X
s=a+1
f (s)(1 − α)t−s .
(33)
• The right (nabla) new Riemann fractional difference by (CFR ∇bα f )(t) =
b−1 X B(α) (−1t ) f (s)(1 − α)s−ρ(t) 1−α s=t
= B(α)(−1t )
b−1 X s=t
f (s)(1 − α)s−t .
(34)
REMARK 2. In the limiting case α → 0 and α → 1, we remark the following relations (a) and (b) and
(CFC a ∇ α f )(t) → f (t) − f (a) (CFC a ∇ α f )(t) → ∇f (t)) (CFC ∇bα f )(t) → f (t) − f (b) (CFC ∇bα f )(t) → −1f (t)
as as as as
α → 0, α → 1. α → 0, α → 1.
ON FRACTIONAL DERIVATIVES WITH EXPONENTIAL KERNEL AND. . .
(c) and (d) and
(CFR a ∇ α f )(t) → f (t) (CFR a ∇ α f )(t) → ∇f (t) (CFR ∇bα f )(t) → f (t) (CFR ∇bα f )(t) → −1f (t)
as as as as
21
α → 0, α → 1. α → 0, α → 1.
REMARK 3 (The action of the discrete Q-operator). The Q-operator acts regularly between left and right new fractional differences as follows: (a) (QCFR a ∇ α f )(t) = (CF R ∇bα Qf )(t), (b) (QCFC a ∇ α f )(t) = (CF C ∇bα Qf )(t). Consider the equation (CFR a ∇ α f )(t) = u(t). If we apply the discrete nabla Laplace transform starting at a to both sides and use the convolution theorem, we conclude that λ 1−α α U (s) 1−α [U (s) − U (s)] = U (s) − , F (s) = B(α) s B(α) B(α) s where F (s) = (Na {f (t)}(s) and U (s) = (Na {u(t)}(s). Then, apply the inverse Laplace to get t 1−α α X f (t) = u(t) + u(s)ds. B(α) B(α) s=a+1 Therefore, we can define the corresponding left fractional integral of (CFR a D α ) by (CF a ∇ −α u)(t) =
t 1−α α X u(t) + u(s)ds. B(α) B(α) s=a+1
(35)
We define the right fractional integral (CF ∇b−α u)(t) =
b−1 α X 1−α u(t) + u(s)ds, B(α) B(α) s=t
(36)
so that (CF a ∇ −α Qu)(t) = Q(CF ∇b−α u)(t). Furthermore, by the action of the Qoperator we can easily show that (CF ∇b−α CF ∇bα u)(t) = u(t). Conversely, consider the equation (CF a ∇ −α u)(t) =
t 1−α α X u(s) = f (t). u(t) + B(α) B(α) a+1
22
T. ABDELJAWAD and D. BALEANU
Apply the nabla discrete Laplace transform to see that α 1−α U (s) = F (s). B(α) sB(α) It follows that U (s) =
B(α) sF (s) . 1−α s−λ
(37)
The right-hand side of (37) is nothing more but the Laplace of (CF a ∇ α f )(t). Hence, (CF a ∇ −α CF a ∇ α f )(t) = f (t).
Similarly or by the action of the Q-operator, we can show that (CF ∇b−α CF ∇bα f )(t) = f (t). PROPOSITION 4. One has (i) (CFC a ∇ α f )(t) = (CFR a ∇ α f )(t) − B(α) f (a)(1 − α)(t−a) . 1−α (ii) (CFC ∇bα f )(t) = (CFR ∇bα f )(t) − B(α) f (b)(1 − α)(b−t) . 1−α Proof : From the relations B(α) sF (s) , 1−α s−λ
(38)
B(α) 1 B(α) sF (s) − f (a)(1 − s)a , 1−α s−λ 1−α s−λ
(39)
Na {(CFR a ∇ α f )(t)}(s) = and
Na {(CFC a ∇ α f )(t)}(s) = we conclude that
Na {(CFC a D α f )(t)}(s) = Na {(CFR a D α f )(t)}(s) −
B(α) 1 f (a)(1 − s)a . 1−α s−λ
(40)
Applying the inverse Laplace to (40) and making use of the fact that Na {f (t−a)}(s) = (1 − s)a N {f (t)}(s) we reach our conclusion in the first part. The second part can be proved by the first part and the action of the Q-operator. 6. Integration by parts formula for fractional differences with discrete exponential kernels Above it was shown that the left fractional operator CFR a ∇ α and its associate fractional integral CF a ∇ −α satisfy (CFR a ∇ α CFa ∇ −α f )(t) = f (t) and that (CFR Dbα CF ∇b−α f )(t) = f (t) . Also it was shown that (CF a ∇ −α CFR a ∇ α f )(t) = f (t) and (CF ∇b−α CFR ∇bα f )(t) = f (t).
ON FRACTIONAL DERIVATIVES WITH EXPONENTIAL KERNEL AND. . .
23
THEOREM 5. Assume thaat 0 < α < 1 and the used functions are defined on Na ∩ b N, a ≡ b(mod1). Then (i) for functions ϕ(t) and ψ(t) we have b−1 X
CF
ϕ(t)(
t=a+1
a∇
−α
b−1 b−1 b−1 X 1−α X α X ϕ(s) ψ)(t) = ψ(t)ϕ(t) + ψ(t) B(α) t=a+1 B(α) t=a+1 s=t
=
b−1 X
t=a+1
ψ(t)CF ∇b−α ϕ(t),
(41)
and similarly b−1 X
t=a+1
ϕ(t)(CF ∇b−α ψ)(t) = =
b−1 b−1 t X α X 1−α X ψ(t)ϕ(t) + ψ(t)( ϕ(s)) B(α) t=a+1 B(α) t=a+1 s=a+1 b−1 X
t=a+1
ψ(t)(CF a ∇ −α ϕ)(t),
(42)
(ii) for functions f (t) and g(t) defined on Na we have b−1 X
t=a+1
f (t)(CFR a ∇ α g)(t) =
b−1 X
(CFR ∇bα f )(t)g(t).
t=a+1
Proof : (i) From the definition and the identity b−1 X a+1
ψ(t)(
t X
s=a+1
ϕ(s)) =
b−1 X a+1
b−1 X ϕ(s)( ψ(t)),
(43)
t=s
the first part follows. The other case follows similarly by the definition of the right fractional difference in (36) and the identity (43). (ii) From the statement before Theorem 5 and the first part we have b−1 X
t=a+1
f (t)(CFR a ∇ α g)(t) = = =
b−1 X
(CF ∇b−α CFR ∇bα f )(t)(CFR a ∇ α g)(t)
b−1 X
(CFR ∇bα f )(t)(CF a ∇ −α CFR a ∇ α g)(t)
b−1 X
(CFR ∇bα f )(t)g(t).
t=a+1
t=a+1
t=a+1
Before proving the by part formula for Caputo fractional derivatives, we use the following notation:
24
T. ABDELJAWAD and D. BALEANU
(a) the (left) discrete exponential integral operator (t−a) t X 1 (Eλ,a + ϕ)(t) = ϕ(s), 1−λ s=a+1 (b) the (right) discrete exponential integral operator (b−t) b−1 X 1 ϕ(t), (Eλ,b− ϕ)(x) = t=s 1 − λ
t ∈ Na ,
(44)
t ∈ b N.
(45)
THEOREM 6. One has b−1 b−1 X X B(α) ρ α (i) (CFC a−1 ∇ α f )(t)g(t) = f (t)(CFR ∇b−1 g)(t) + f (t)E −α ,b− g)(t)|ba , 1−α 1 − α t=a t=a (ii)
b X
α (CFC ∇b+1 f )(t)g(t)
t=a+1
=
b X
t=a+1
f (t)(CFR a+1 ∇ α g)(t) −
B(α) σ f (t)E −α ,a + g)(t)|ba . 1−α 1−α
Proof : The proof of the first part follows by Theorem 5 and the first part of Proposition 4. The proof of the second part can be established by Theorem 5 and the second part of Proposition 4. 7. New discrete fractional Euler–Lagrange equations THEOREM 7. Let 0 < α < 1 be noninteger, c, d ∈ R, c < d, c ≡ d(mod1). Assume that d−1 X J (f ) = L(t, f ρ (t), CFC c−1 ∇ α f (t)) t=c
possesses a local extremum in S = {z : (Nc−1 ∩ d−1 N) → R : z(c − 1) = C, z(d − 1) = D} at some f ∈ S, where L : (Nc−1 ∩ d−1 N) × R × R → R. Then where
α [L1 (s) + CFR ∇c−1 L2 (s)] = 0, for all s ∈ (Nc−1 ∩ d−1 N),
L1 (s) =
∂L (s) ∂f ρ
and
L2 (s) =
∂L ∂ CFC
c−1 ∇
αf
(46)
(s).
Proof : We suppose that J reaches a local maximum in S at f . As a result, there exists an ǫ > 0 fulfilling J (fb) − J (f ) ≤ 0 for all fb ∈ S such that kfb − f k = supt∈Nc ∩d N |fb(t) − f (t)| < ǫ. For any fb ∈ S there is an η ∈ H = {y : (Nc−1 ∩ d−1 N) → R, z(a − 1) = z(b − 1) = 0} such that fb = f + ǫη. The ǫ-Taylor’s theorem and the assumption lead us to the conclusion that δJ (η, z) =
d−1 X [ηρ (t)L1 (t) + (CF C c−1 ∇ α η)(t)L2 (t)]dt = 0 t=c
ON FRACTIONAL DERIVATIVES WITH EXPONENTIAL KERNEL AND. . .
25
for all η ∈ H . Thus, to make η free, we use Theorem 6 to obtain δJ (η, f ) =
d−1 X s=c
α ηρ (s)[L1 (s) + CFR ∇d−1 L2 (s)] + ηρ (t)
B(α) (E −α − L2 )(t)|dc = 0, 1 − α 1−α ,d
for all η ∈ H , and hence the result comes as a result of the discrete FLCV.
(E −α ,d − L2 )(t)|dc 1−α
Here = 0 denotes the natural boundary condition. For the case when the Lagrangian contains the discrete right Caputo fractional derivative, we obtain the theorem. THEOREM 8. Let 0 < α ≤ 1 be noninteger, c, d ∈ R, c < d, c ≡ d(mod1). Suppose J , that d X α J (f ) = L(t, f σ (t), CFC ∇d+1 f (t)), c+1
admits a local extremum in S = {z : (Nc+1 ∩ d+1 N) → R : z(c +1) = A, z(d +1) = B} at some f ∈ S, where L : (Nc+1 ∩ d+1 N) × R × R → R. Then where
[L1 (s) + CFR c+1 ∇ α L2 (s)] = 0, for all s ∈ (Nc+1 ∩ d+1 N), L1 (s) =
∂L (s) ∂f σ
and
L2 (s) =
∂L α f ∂ CF C ∇d+1
(47)
(s).
Proof : The proof looks similar to the one of Theorem 7 by using Theorem 6 to obtain the natural boundary condition of the form (E −α ,c+ L2 )(t)|dc = 0. 1−α
EXAMPLE 2. Below we provide a discrete example of physical interest under Theorem 7, namely d−1 X 1 J (z) = [ (CFC c−1 ∇ α z(t))2 − V (zρ (t))], t=c 2
where 0 < α < 1 and with z(c − 1), z(d − 1) given or with the natural boundary condition (E −α ,d − CFC c−1 ∇ α z(t))(t)|dc = 0. 1−α
Thus, the related Euler–Lagrange equation becomes dV α oCFC c−1 ∇ α z)(s) − (CFR ∇d−1 (s) = 0 for all s ∈ (Nc−1 ∩ d−1 N). dz To see the difference with the classical discrete case within nabla we suggest the reader [29]. 8.
Conclusions The Caputo–Fabrizio derivative was already applied successfully since it was recently suggested. Several applications of it were implemented in some areas
26
T. ABDELJAWAD and D. BALEANU
where the nonlocality appears in real world phenomena. However, the properties of this derivative should be deeply explored and applied to describe the real world phenomena. In this manuscript we elaborated the discrete version of the Caputo–Fabrizio derivative. By proving the integration by parts we open the gate for application in discrete variational principles and their applications in control theory. The reported discrete fractional Euler–Lagrange will play a crucial role in constructing the related Lagrangian and Hamiltonian dynamics. REFERENCES [1] I. Podlubny: Fractional Differential Equations, Academic Press: San Diego, CA 1999. [2] G. Samko, A. A. Kilbas and S. Marichev: Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon 1993. [3] A. A. Kilbas, M. H. Srivastava and J. J. Trujillo: Theory and Application of Fractional Differential Equations, North Holland Mathematics Studies 204, 2006. [4] R. L. Magin: Fractional Calculus in Bioengineering, Begell House Publishers 2006. [5] D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo: Fractional Calculus Models and Numerical Methods (Series on Complexity, Nonlinearity and Chaos), World Scientific, Singapore 2012. [6] Y. Zhou: Basic Theory of Fractional Differential Equations, World Scientific, Singapore 2014. [7] Y. Zhou: Fractional Evolution Equations and Inclusions: Analysis and Control, Academic Press 2016. [8] T. Abdeljawad: On Riemann and Caputo fractional differences, Comput. Math. Appl. 62 (2011), 1602–1611. [9] F. M. Atıcı and P. W. Eloe: A Transform method in discrete fractional calculus, Int. J. Differ. Equ. 2 (2007), 165–176. [10] F. M. Atıcı and P. W. Eloe: Initial value problems in discrete fractional calculus, Proc. Amer. Math. Soc. 137 (2009), 981–989. [11] F. M. Atıcı and P. W. Eloe: Discrete fractional calculus with the nabla operator, Electr. J. Qualit. Theor. Differ. Equ. 1–12 (2009). [12] F. M. Atıcı and S. S¸eng¨ul: Modelling with fractional difference equations, J. Math. Anal. Appl. 369 (2010), 1–9. [13] K. S. Miller and B. Ross: Fractional difference calculus, Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, Nihon University, Koriyama, Japan, (1989) 139–152. [14] T. Abdeljawad and D. Baleanu: Fractional differences and integration by parts, J. Comput. Anal. Appl. 13 (2011), 574–582. [15] N. R. O. Bastos, R. A. C. Ferreira and D. F. M. Torres: Discrete-time fractional variational problems, Signal Proc. 91 (2011), 513–524. [16] H. L. Gray and N. F. Zhang: On a new definition of the fractional difference, Math. Comput. 50 (1988) , 513–529. [17] G. A. Anastassiou: Principles of delta fractional calculus on time scales and inequalities, Math. Comput. Model. 52 (2010), 556–566. [18] G. A. Anastassiou: Nabla discrete calculus and nabla inequalities, Math. Comput. Model., 51 (2010), 562–571. [19] T. Abdeljawad and F. Atici: On the definitions of nabla fractional differences, Abstr. Appl. Anal. 2012 (2012), Article ID 406757, 13 pages, doi:10.1155/2012/406757. [20] T. Abdeljawad: On Delta and Nabla Caputo fractional differences and dual identities, Discr. Dyn. Nat. Soc. 2013 (2013), Article ID 406910, 12 pages. [21] G. C. Wu, D. Baleanu, S. D. Zeng and Z. G. Deng: Discrete fractional diffusion equation, Nonlinear Dyn. 80 (2015), 281–286. [22] G. C. Wu, D. Baleanu and S. D. Zeng: Several fractional differences and their applications to discrete maps, J. Appl. Nonlinear Dyn. 4 (2015), 339–348.
ON FRACTIONAL DERIVATIVES WITH EXPONENTIAL KERNEL AND. . .
27
[23] T. Abdeljawad: Dual identities in fractional difference calculus within Riemann, Adv. Differ. Equ. 2013 (2013), 36. [24] M. Bohner and A. Peterson: Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston-BaselBerlin 2003. [25] Bohner M. and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkh¨auser, Boston 2003. [26] T. Abdeljawad, F. Jarad and D. Baleanu: A semigroup-like property for discrete Mittag-Leffler functions, Adv. Differ. Equ. 2012 (2012), 72. [27] T. Abdeljawad and D. Baleanu: Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels, Adv. Differ. Equ. 2016 (2016), 232. [28] M. Caputo and M. Fabrizio: A new definition of fractional derivative without singular kernel, Progr. Frac. Differ. Appl. 1 (2015), 73–85. [29] T. Abdeljawad: Nabla Euler–Lagrange equations in discrete fractional variational calculus within Riemann and Caputo, Int. J. Math. Comput.22 (2014), 144-153. [30] T. Abdeljawad (Maraaba), D. Baleanu and F. Jarad: Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives, J. Math. Phys. 49, (2008). [31] M. Caputo and M. Fabrizio: A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 1 (2) (2015), 73-85.