Nonlinear multi-order fractional differential equations with periodic/anti-periodic boundary...

$: Nonlinear multi-order fractional differential equations with periodic/anti-periodic boundary conditions$
RESEARCH PAPER

NONLINEAR MULTI-ORDER FRACTIONAL

DIFFERENTIAL EQUATIONS WITH PERIODIC/

ANTI-PERIODIC BOUNDARY CONDITIONS

Sangita Choudhary 1, Varsha Daftardar-Gejji 2

Abstract

In the present manuscript we analyze non-linear multi-order fractionaldifferential equation

L(D)u(t) = f(t, u(t)), t ∈ [0, T ], T > 0,

where

L(D) = λncDαn + λn−1

cDαn−1 + · · · + λ1cDα1 + λ0

cDα0,

λi ∈ R (i = 0, 1, · · · , n), λn �= 0, 0 ≤ α0 < α1 < · · · < αn−1 < αn < 1,

and cDα denotes the Caputo fractional derivative of order α. We find theGreens functions for this equation corresponding to periodic/ anti-periodicboundary conditions in terms of the two-parametric functions of Mittag-Leffler type. Further we prove existence and uniqueness theorems for thesefractional boundary value problems.

MSC 2010 : Primary 26A33; Secondary 33E12, 34A08, 34K37, 35R11

Key Words and Phrases: fractional calculus, multi order Mittag-Lefflerfunctions, fractional differential equations, periodic/ anti-periodic bound-ary conditions

c© 2014 Diogenes Co., Sofiapp. 333–347 , DOI: 10.2478/s13540-014-0172-6

334 S. Choudhary, V. Daftardar-Gejji

1. Introduction

Fractional Calculus deals with differentiation and integration of arbi-trary orders. Fractional order derivatives provide a new approach for mod-elling many complex phenomena in Physics, Chemistry, Biology and Engi-neering sciences [8],[13], especially when dealing with memory effects. Com-pared to integer order models, the fractional order models provide betterdescription of underlying processes. Fractional order differential equations,therefore, have attracted attention of researchers and analysis of fractionaldifferential equations has widely been studied during last decade [6]-[8], [12]-[14]. The existence of periodic solutions is one of the important aspects ofdifferential systems. Several results in this context have been reported inthe literature. In this respect, Kaslik and Sivasundaram [10] have recentlyproved the non-existence of exact periodic solutions in a wide class of frac-tional order dynamical systems. Extensive work has been carried on linear/non linear boundary value problems for fractional differential equations, e.g.[7], [9], [16]. Belmekki et. al [4] have introduced and studied a new con-cept of periodic boundary conditions in the context of Riemann-Liouvillederivatives. Existence results for non-linear fractional differential equationswith integral boundary conditions [5] and anti-periodic fractional bound-ary conditions [1], [15] have been investigated. For partially ordered metricspaces, Baleanu et. al [2] recently have proved existence and uniqueness ofsolutions in case of non-linear fractional differential equations.

In this article we analyze solutions for the multi-order fractional differ-ential equations. This equation is a generalization of the classical relaxationequation, and governs some fractional relaxation processes. In this regard,Bazhlekova [3] has studied linear (non homogeneous) initial value problemand derived fundamental solution [corresponding to f(t) = 0, u(0) = 1] andimpulse response solution [f(t) = δ(t), u(0) = 0]. Further asymptotic prop-erties of these solutions are studied. In the present paper we consider anonlinear case along with periodic and anti-periodic boundary conditions.We find the Greens functions in terms of multi-order Mittag-Leffler func-tions. Further we prove existence and uniqueness theorems for solutionsunder various conditions.

The organization of the present work is as follows. In Section 2, pre-liminaries and notations are presented. In Section 3 we derive the Greenfunction for multi-order fractional differential equations followed by exis-tence and uniqueness theorems for the fractional boundary value problems.

2. Preliminaries

In this section, we introduce notations, definitions and preliminarieswhich are used in the paper [8, 13].

NONLINEAR MULTI-ORDER FRACTIONAL . . . 335

C(J,R) denotes the Banach space of all continuous functions from J =[0, T ] into R with the norm

‖ u ‖∞:= sup{|u(t)| : t ∈ J}, T > 0.

Cn(J,R) denotes the class of all real valued functions defined on J =[0, T ], T > 0 which have continuous nth order derivative.

S denotes the class of functions α : R+ → [0, 1) satisfying the condition

α(tn) → 1 implies tn → 0.Φ denotes the class of increasing functions φ : [0,∞) → [0,∞) such

that φ(x) < x for all x > 0 and φ(x)x ∈ S.

Definition 2.1. The fractional integral of order α > 0 of the functionf ∈ C(J,R) is defined as

Iαf(t) =1

Γ(α)

t∫0

f(s)(t− s)1−α

ds, 0 < t < T.

Definition 2.2. The Caputo fractional derivative of order α > 0 ofthe function f ∈ Cn(J,R) is defined as

cDαf(t) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

In−αDnf(t) =1

Γ(n− α)

t∫0

f (n)(s)(t− s)α−n+1

ds,

n− 1 < α < n, n ∈ N,

f (n)(t), α = n.

The Laplace transform of a function f(t) of exponential order α isdefined as

f(s) = L{f(t); s} =

∞∫0

e−stf(t)dt, s ∈ R.

The Laplace transform of the Caputo derivative of order α is

L{cDαf(t); s} = sαf(s) −n−1∑k=0

sα−k−1f (k)(0), n− 1 < α ≤ n.

The two-parametric Mittag-Leffler function is defined by (see e.g. [13])

Eα,β(z) =∞∑

k=0

zk

Γ(kα+ β), α > 0, β > 0.

The Laplace transform of the Mittag-Leffler function in two param-eters and more generally, of the so-called function of Mittag-Leffler type


Ek(z, a;α, β) := zαk+β−1E(k)α,β(azα) related to its derivatives, has the form

(see Podlubny [13, (1.80), (9.8)-(9.9)])

L{tαk+β−1E(k)α,β(±atα)} =

k!λα−β

(λα ∓ a)k+1, Re(λ) > |a| 1

α ,

where E(k)α,β(y) =

dk

dykEα,β(y) =

∞∑j=0

(j + k)!yj

j!Γ(αj + αk + β), k = 0, 1, 2, . . ..

Theorem 2.1. (see [11]) Let (M,d) be a complete metric space andlet T : M →M . Suppose there exists α ∈ S such that for each x, y ∈M,

d(T (x), T (y))) ≤ α(d(x, y))d(x, y),

then T has a unique fixed point z ∈ M and {T n(x)} converges to z, foreach x ∈M.

3. Main Results

In this section we study the existence and uniqueness of solution forthe non-linear multi-order fractional differential equation

L(D)u(t) = f(t, u(t)), t ∈ J = [0, T ], T > 0, (3.1)

where L(D) = λncDαn + λn−1

cDαn−1 + · · ·+ λ1cDα1 +λ0

cDα0, λi ∈ R (i =0, 1, · · · , n), λn �= 0, 0 ≤ α0 < α1 < · · · < αn−1 < αn < 1, with periodicboundary condition

u(0) = u(T ), (3.2)

and anti-periodic boundary condition

u(0) = −u(T ). (3.3)

We prove the results in two steps.Case (i): n = 1 and α0 = 0.

Lemma 3.1. The non-linear fractional differential equation

λ1cDα1u(t)+λ0u(t) = f(t, u(t)), t ∈ J = [0, T ], T > 0, λ0, λ1 ∈ R, λ1 �= 0,

(3.4)with periodic boundary condition u(0) = u(T ), is equivalent to the frac-tional integral equation

u(t) =

T∫0

G(t, s)f(s, u(s))ds,


where G(t, s) is the following Green’s function:

G(t, s) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎡⎢⎢⎣

Eα1,1

(−λ0

λ1tα1

)

1−Eα1,1

(−λ0

λ1Tα1

)⎤⎥⎥⎦ 1λ1

(T−s)α1−1Eα1,α1

(−λ0

λ1(T−s)α1

)

+(

1λ1

)(t− s)α1−1Eα1,α1

(−λ0

λ1(t− s)α1

), for 0 ≤ s < t,⎡

⎢⎢⎣Eα1,1

(−λ0

λ1tα1

)

1−Eα1,1

(−λ0

λ1Tα1

)⎤⎥⎥⎦ 1λ1

(T−s)α1−1Eα1,α1

(−λ0

λ1(T−s)α1

),

for t ≤ s < T .

P r o o f. Laplace transform of Eq. (3.4) yields

λ1sα1u(s) − λ1s

α1−1u(0) + λ0u(s) = f(s, u(s)).

Now taking inverse Laplace transform and using u(0) = c, we obtain

u(t) = L−1

{f(s, u(s))λ1sα1 + λ0

}+ cL−1

{λ1s

α1−1

λ1sα1 + λ0

}

=(

1λ1

)tα1−1Eα1,α1

(−λ0

λ1tα1

)∗ f(t, u(t)) + c Eα1,1

(−λ0

λ1tα1

).

(3.5)By applying periodic boundary condition u(0) = u(T ) to Eq. (3.5), wehave

c =

1λ1

T∫0

(T − s)α1−1Eα1,α1

(−λ0

λ1(T − s)α1

)f(s, u(s))ds

1 − Eα1,1

(−λ0

λ1Tα1

) .

Substituting above value of c in Eq. (3.5) gives the solution

u(t) =1λ1

t∫0

(t− s)α1−1Eα1,α1

(−λ0

λ1(t− s)α1

)f(s, u(s))ds

+Eα1,1

(−λ0

λ1tα1

)

1−Eα1,1

(−λ0

λ1Tα1

) 1λ1

T∫0

[(T−s)α1−1Eα1,α1

(−λ0

λ1(T−s)α1

)f(s, u(s))ds

]


=

T∫0

G(t, s)f(s)ds.

Hence the proof is over. �

Case (ii): n ≥ 2.


L(D)u(t) = f(t, u(t)), (3.6)with the periodic boundary value condition u(0) = u(T ), is equivalent tothe fractional integral equation

u(t) =

T∫0


where G(t, s) is the following Green’s function: For 0 ≤ s < t,

G(t, s) =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

n∑r=0

(λr

λn

) ∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)

1 −n∑

r=0

(λr

λn

) ∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)

n−2∏i=0

(λi

λn

)ki

tαm+β−1E(m)α,β

(−λn−1

λntα

)n−2∏i=0

(λi

λn

)ki

Tαm+β−1E(m)α,β

(−λn−1

λnTα

)⎤⎥⎥⎥⎦

× 1λn

∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)n−2∏i=0

(λi

λn

)ki

(T − s)αm+γ−1E(m)α,γ

(−λn−1

λn(T − s)α

)

+1λn

∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)n−2∏i=0

(λi

λn

)ki

(t− s)αm+γ−1E(m)α,γ

(−λn−1

λn(t− s)α

).


For t ≤ s < T ,

G(t, s) =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

n∑r=0

(λr

λn

) ∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)

1 −n∑

r=0

(λr

λn

) ∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)

n−2∏i=0

(λi

λn

)ki


(−λn−1

λntα

)n−2∏i=0

(λi

λn

)ki


(−λn−1

λnTα

)⎤⎥⎥⎥⎦

× 1λn

∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)n−2∏i=0

(λi

λn

)ki


(−λn−1

λn(T − s)α

),

where (m; k0, k1, . . . , kn−2) are the multinomial coefficients,

α = αn − αn−1, β = αn +n−2∑j=0

(αn−1 − αj)kj − αr + 1,

γ = αn +n−2∑j=0

(αn−1 − αj)kj .

P r o o f. Laplace transform of Eq. (3.6) yields

n∑k=0

λksαk u(s) −

n∑k=0

λksαk−1u(0) = f(s, u(s)).

Now taking inverse Laplace transform [13] and using u(0) = c, we obtain

u(t) = L−1

{f(s, u(s))

λnsαn + λn−1sαn−1 + · · · + λ0sα0

}

+ c

n∑r=0

L−1

{λrs

αr−1

λnsαn + λn−1sαn−1 + · · · + λ0sα0

}


=

t∫0

1λn

∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)n−2∏i=0

(λi

λn

)ki

(t− s)(αn−αn−1)m+(αn+

n−2∑j=0

(αn−1−αj)kj)−1

×E(m)

αn−αn−1,αn+n−2∑j=0

(αn−1−αj)kj

(−λn−1

λn(t− s)αn−αn−1

)f(s, u(s))ds

+ c

⎧⎪⎪⎨⎪⎪⎩

n∑r=0

(λr

λn

) ∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)n−2∏i=0

(λi

λn

)ki

t(αn−αn−1)m+(αn+

n−2∑j=0

(αn−1−αj)kj−αr+1)−1

× E(m)

αn−αn−1,αn+n−2∑j=0

(αn−1−αj)kj−αr+1

(−λn−1

λntαn−αn−1

)⎫⎪⎬⎪⎭ .

(3.7)

Applying boundary condition u(0) = u(T ) to Eq. (3.7) and substituting

αn − αn−1 = α, αn +n−2∑j=0

(αn−1 − αj)kj − αr + 1 = β,

αn +n−2∑j=0

(αn−1 − αj)kj = γ, we get

c =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

T∫o

1λn

∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)

1 −n∑

r=0

(λr

λn

) ∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)

n−2∏i=0

(λi

λn

)ki


(−λn−1

λn(T − s)α

)n−2∏i=0

(λi

λn

)ki


(−λn−1

λnTα

)⎤⎥⎥⎥⎦ .


Substituting the above value of c in Eq. (3.7) gives the solution

u(t) =

t∫0

1λn

∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)n−2∏i=0

(λi

λn

)ki

(t− s)αm+β−1E(m)α,γ

(−λn−1

λn(t− s)λn−λn−1

)f(s, u(s))ds

+

⎡⎢⎢⎢⎢⎢⎢⎢⎣

T∫0

1λn

∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)

1 −n∑

r=0

(λr

λn

) ∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)

n−2∏i=0

(λi

λn

)ki


(−λn−1

λn(T − s)α

)n−2∏i=0

(λi

λn

)ki


(−λn−1

λnTα

)⎤⎥⎥⎥⎦

×n∑

r=0

(λr

λn

) ∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)

n−2∏i=0

(λi

λn

)ki


(−λn−1

λntαn−αn−1

)

=T∫0

G(t, s)f(s)ds.

Hence the proof is over. �

Now we prove results for anti-periodic condition.

Case (i): n = 1, α0 = 0.


λ1cDα1u(t)+λ0u(t) = f(t, u(t)), t ∈ J = [0, T ], T > 0, λ0, λ1 ∈ R, λ1 �= 0,

with anti-periodic boundary condition u(0) = −u(T ), is equivalent to thefractional integral equation


u(t) =

T∫0


where G(t, s) is the following Green’s function:

G(t, s) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎡⎢⎢⎣

(−1)Eα1,1

(−λ0

λ1tα1

)

1 + Eα1,1

(−λ0

λ1Tα1

)⎤⎥⎥⎦ 1λ1

(T − s)α1−1Eα1,α1

(−λ0

λ1(T − s)α1

)

+(

1λ1

)(t− s)α1−1Eα1,α1

(−λ0

λ1(t− s)α1

), for 0 ≤ s < t,⎡

⎢⎢⎣(−1)Eα1,1

(−λ0

λ1tα1

)

1 + Eα1,1

(−λ0

λ1Tα1

)⎤⎥⎥⎦ 1λ1

(T − s)α1−1Eα1,α1

(−λ0

λ1(T − s)α1

),

for t ≤ s < T .

Case (2): n ≥ 2.


L(D)u(t) = f(t, u(t)),

with the anti-periodic boundary value condition u(0) = −u(T ), is equiva-lent to the fractional integral equation

u(t) =

T∫0

G(t, s)f(s, u(s))ds, (3.8)

where G(t, s) is the following Green’s function: For 0 ≤ s < t,

G(t, s) =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

(−1)n∑

r=0

(λr

λn

) ∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)

1 +n∑

r=0

(λr

λn

) ∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)


n−2∏i=0

(λi

λn

)ki


(−λn−1

λntα

)n−2∏i=0

(λi

λn

)ki


(−λn−1

λnTα

)⎤⎥⎥⎥⎦

× 1λn

∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)n−2∏i=0

(λi

λn

)ki


(−λn−1

λn(T − s)α

)

+1λn

∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)n−2∏i=0

(λi

λn

)ki


(−λn−1

λn(t− s)α

).

For t ≤ s < T ,

G(t, s) =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

(−1)n∑

r=0

(λr

λn

) ∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)

1 +n∑

r=0

(λr

λn

) ∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)

n−2∏i=0

(λi

λn

)ki


(−λn−1

λntα

)n−2∏i=0

(λi

λn

)ki


(−λn−1

λnTα

)⎤⎥⎥⎥⎦

× 1λn

∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)n−2∏i=0

(λi

λn

)ki


(−λn−1

λn(T − s)α

)


+1λn

∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=m

k0≥0,··· ,kn−2≥0

(m; k0, k1, . . . , kn−2)n−2∏i=0

(λi

λn

)ki


(−λn−1

λn(t− s)α

),

where (m; k0, k1, . . . , kn−2) are the multinomial coefficients,

α = αn − αn−1, β = αn +n−2∑j=0

(αn−1 − αj)kj − αr + 1,

γ = αn +n−2∑j=0

(αn−1 − αj)kj .

Remark 3.1. The function t ∈ J �→T∫0

|G(t, s)|ds is continuous on J ,

and hence is bounded. Let

G = sup

⎧⎨⎩

T∫0

|G(t, s)|ds, t ∈ J

⎫⎬⎭ . (3.9)

Define the operator F : C(J,R) → C(J,R)

F (u) =

T∫o

G(t, s)f(s, u(s))ds, (3.10)

where G(t, s) is the Green function corresponding to boundary conditions(3.2) and (3.3), respectively.

The boundary value problems (3.1)-(3.2) and (3.1)-(3.3) have solutionsif and only if the operator F has fixed points.

Theorem 3.1. The boundary value problems (3.1)-(3.2) and (3.1)-(3.3) have unique solution if the following conditions hold:

(1) The function f : J × R → R is continuous.

(2) There exists φ ∈ Φ such that |f(t, y) − f(t, x)| ≤ 1Gφ(|y − x|),

∀x, y ∈ R.

P r o o f. Let M = C(J,R). Then (M,d) is a complete metric space,where d(g, h) = sup

t∈J|g(t) − h(t)|.


Consider F : M →M as defined in (3.10). For u �= v,

d(Fu,Fv) = supt∈J

|(Fu)(t) − (Fv)(t)|

≤ supt∈J

T∫o

|G(t, s)||f(s, u(s)) − f(s, v(s))|ds

≤ supt∈J

T∫o

|G(t, s)| 1Gφ(|u(s) − v(s)|)ds

≤ φ(d(u, v))1G

supt∈J

T∫o

|G(t, s)|ds (Since φ is increasing)

= φ(d(u, v))1GG = φ(d(u, v))

=φ(d(u, v))d(u, v)

d(u, v) = α(d(u, v))d(u, v).

Therefore, there exists α ∈ S such that d(Fu,Fv) ≤ α(d(u, v))d(u, v),∀u, v ∈M. Thus by Theorem 2.1, F has a unique fixed point.

Hence the boundary value problems (3.1)-(3.2) and (3.1)-(3.3) haveunique solution. �

In view of Remark 3.1, we can also prove the following existence anduniqueness theorems for the boundary value problems (3.1)-(3.2) and (3.1)-(3.3), in the same lines as given in Ref. [5].

Theorem 3.2. The boundary value problems (3.1)-(3.2) and (3.1)-(3.3) have at least one solution if the following conditions hold:

(1) The function f : J × R → R is continuous.(2) There exists p ∈ C(J,R+) and ψ : [0,∞) → (0,∞) continuous and

nondecreasing such that |f(t, u)| ≤ p(t)ψ(|v|), for t ∈ J and v ∈ R.

(3) There exists a constant M > 0 such that M > p∗ψ(M)G,where p∗ = sup {p(s), s ∈ J} .

Theorem 3.3. Assume that there exists K > 0 such that

|f(t, u) − f(t, v)| ≤ K|u− v|,for t ∈ J and every u, v ∈ R. If KG < 1, then there exist unique solution

for the boundary value problems (3.1)-(3.2) and (3.1)-(3.3).


Acknowledgements

The authors acknowledge the Department of Science and Technology,N. Delhi, India for the Research Grants [Project No SR/S2/HEP-024/2009].

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Dept. of Mathematics, University of Pune

Ganeshkhind, Pune - 411007, INDIA1 e-mail: [email protected] e-mail: [email protected], [email protected]

Received: November 14, 2013

Please cite to this paper as published in:Fract. Calc. Appl. Anal., Vol. 17, No 2 (2014), pp. 333–347;DOI: 10.2478/s13540-014-0172-6


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