research paper operational method for solving multi

RESEARCH PAPER OPERATIONAL METHOD FOR SOLVING MULTI-TERM FRACTIONAL DIFFERENTIAL EQUATIONS WITH THE GENERALIZED FRACTIONAL DERIVATIVES Myong-Ha Kim 1 , Guk-Chol Ri 2 , Hyong-Chol O

3

Abstract This paper provides results on the existence and representation of solution to an initial value problem for the general multi-term linear fractional differential equation with generalized Riemann-Liouville fractional derivatives and constant coefficients by using operational calculus of Mikusinski’s type. We prove that the initial value problem has the solution if and only if some initial values are zero. MSC 2010 : Primary 34A08, 34A25, 44A40; Secondary 26A33 Key Words and Phrases: generalized Riemann-Liouville fractional derivatives, fractional differential equations, operational calculus 1. Introduction In the recent years considerable interest in the theory and applications of fractional differential equations has been demonstrated, see for example ([1, 3, 15, 8, 17, 19]). Indeed, it is well known that the fractional derivatives are excellent tools for description of memory and hereditary effects, e.g. [14]. It is therefore important to understand the advantages of using fractional derivative in classical equations. The applications of the fractional calculus are based on fractional derivatives operators of different kinds, see e.g. [3]. There exist several definitions of differentiation of fractional order, as in [7, 16]. The (right-sided) c 2014 Diogenes Co., Sofia  pp. 79–95 , DOI: 10.2478/s13540-014-0156-6

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80

M.-Ha Kim, G.-Chol Ri, H.-Chol O

Riemann-Liouville fractional derivative is defined by α f (x) := D0+

where α f (x) I0+

1 := Γ(α)

dn n−α I f (x) , x > 0, dxn 0+

 0

x

(x − t)α−1 f (t)dt , x > 0,

(1.1)

(1.2)

0 f (x) := f (x) , x > 0 I0+

is the (right-sided) Riemann-Liouville fractional integral of order α with lower limit 0 (see e.g. [7, 16]). The (right-sided) Caputo fractional derivative is defined by dn f (x) , x > 0 (1.3) dxn whenever the right-hand side exists (see e.g. [7, 16]). In Hilfer [4], the generalized Riemann-Liouville fractional derivative (GRLFD) of order α and type β is defined as c

n−α α D0+ f (x) := I0+

dn (1−β)(n−α) (I f ))(x) , x > 0, (1.4) dxn 0+ where n − 1 < α ≤ n ∈ N and 0 ≤ β ≤ 1, whenever the right-hand side exists (see e.g. [6]). Operator 1.4 gives the classical Riemann-Liouville fractional differential operator (1.1) if β = 0. For β = 1 it reduces to the Caputo fractional differential operator (1.3). The concept of GRLFD has appeared in the theoretical modeling of broadband dielectric relaxation spectroscopy for glasses [5]. Some properties and applications of the GRLFD are given in [3, 4, 18, 20, 21, 22]. Several authors (see [2, 18, 22]) called (1.4) the Hilfer fractional derivative or composite fractional derivative operator. More recently, new results for fractional differential equations involving GRLFDs have been obtained in [2, 18, 22, 23]. In [23] the authors provided an approach based on the equivalence of a nonlinear Cauchy type problem with GRLFD to a nonlinear Volterra integral equation of the second kind in spaces of summable functions on a finite interval of the real axis and proved uniqueness and existence of the solution, using a variant of the Banach fixed point theorem. In [22] the authors investigated the solution of space-time fractional diffusion equations with a generalized Riemann-Liouville time fractional derivative. Operational calculus techniques have been alreday applied successfully to ordinary differential equations and partial differential equations (both of integer or fractional order), integral equations, and to the theory of special functions (see, for example [11, 24, 9, 10], etc.). For example, in [12] Mikusinski’s operational calculus has been applied to solve a Cauchy β(n−α)

α,β f (x) := (I0+ D0+

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OPERATIONAL METHOD FOR SOLVING MULTI-TERM . . . 81 boundary-value problem for a certain linear equation involving RiemannLiouville fractional derivatives. In [13], by using the operational calculus of Mikusinski’s type for the Caputo fractional differential operator, the authors obtained exact solutions of an initial value problem for linear fractional differential equations with constant coefficients and Caputo fractional derivatives. In [6] an operational calculus of Mikusinski’s type was introduced for the generalized Riemann-Liouville fractional derivative operator and it was applied to solve an initial value problem for the general n-term linear fractional differential equation with constant coefficients and GRLFD of arbitrary orders and types. The authors of [6] considered initial values dependent on the type of generalized Riemann-Liouville fractional derivatives in n-term linear fractional differential equation (see problem (36)-(37) in [6]) and defined the space of solutions dependent on the type (see Definition 1 in [6]). They provided a result on existence and uniqueness of the solution of n-terms linear fractional differential equation, when the orders of the fractional derivatives are all between n − 1 and n and proposed an explicit formula for the solution (see formula (42) of [6]). Let us consider the following initial value problem 0.7,0 0.5,0 0.3 0.5 y(x) − D0+ y(x) = 0, I0+ y(0) = C0 ∈ R, I0+ y(0) = C1 ∈ R. D0+

This initial value problem does not belong to the class of the problems that were solved in [6], because this problem has 2 terms and the orders of the derivatives in this equation are between 0 and 1 and not between 1 and 2. The aim of this paper is to extend the results obtained in Hilfer-LuchkoTomovski [6] to a wider class of the equations including the above mentioned equation by using the operational calculus method developed in [6]. We present the existence and representation of solution for the n-term linear initial value problem with generalized Riemann-Liouville fractional derivatives and constant coefficients. According to our results, the solution to the above mentioned problem exists if and only if the initial value C1 is zero. In Section 2 we start with some preliminaries and operational calculus for the generalized fractional derivative, following mainly Hilfer-LuchkoTomovski [6]. In Section 3, we provide a necessary and sufficient condition for the existence of the solution of the initial value problem and give some examples. 2. Preliminaries and operational calculus for generalized fractional derivatives The background and the basic knowledge on the results in this paper are contained as essence in Luchko, Yakubobich and Luchko [11, 24]. In this

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82

M.-Ha Kim, G.-Chol Ri, H.-Chol O

section, the stuff is taken from Hilfer-Luchko-Tomovski [6], unless specially indicated. Definition 2.1. ([6]) A function f (x), x > 0, is said to be in the space Cαm , m ∈ N0 = N∪{0}, α ∈ R, if there exists a real number p, p > α, such that (2.1) f (m) (x) = xp f1 (x) with a function f1 (x) in C[0, ∞). Especially the space Cα0 is denoted by Cα . By [6], Cα is a vector space and the set of spaces Cα is ordered by inclusion according to (2.2) Cβ ⊂ Cα ⇔ α ≥ β. α , α≥ Lemma 2.1. ([13]) The Riemann-Liouville fractional integral I0+ 0, is a linear map of the space Cγ , γ ≥ −1, in to itself , that is, α : Cγ → Cα+γ ⊂ Cγ . I0+

(2.3)

α , α ≥ 0 has the following convoIt is well known, that the operator I0+ lutional representation in the space Cγ , γ ≥ −1.  x α f (x)hα (x − t)dt , x > 0, (2.4) I0+ f (x) = hα (x)  f (x) = 0

, x > 0, α > 0, Γ(α)  x g(x − t)f (t)dt , x > 0, (g  f )(x) := hα (x) :=

where

xα−1

(2.5) (2.6)

0

is the Laplace convolution. Moreover, the following properties of the RiemannLiouville fractional integral are well known: β α α+β α β I0+ f (x) = I0+ I0+ f (x) = I0+ , α, β ≥ 0, I0+ β α nα · · · I0+ f )(x) = (I0+ f )(x). (I0+   

(2.7)

n

m , m ∈ N , f (0) = · · · = f (m−1) (0) = 0 Lemma 2.2. ([13]) Let f ∈ C−1 0 1 . Then the Laplace convolution and g ∈ C−1  x f (t)g(x − t)dt (2.8) h(x) := 0

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OPERATIONAL METHOD FOR SOLVING MULTI-TERM . . . 83 m and h(0) = · · · = h(m) (0) = 0. is in the space C−1

Now according to [6], we define the following space of functions.

Ωαβ ,

Definition 2.2. ([6]) A function y ∈ C−1 is said to be in the space (1−β)(n−α) n , n − 1 < α ≤ n ∈ N for α > 0, 0 ≤ β ≤ 1. if I0+ y ∈ C−1 Then the following lemmas hold.

Lemma 2.3. ([6]) Let y ∈ Ωαβ , n−1 < α ≤ n ∈ N. Then the RiemannLiouville fractional integral (1.2) and the generalized fractional derivative (1.4) are connected by the relation α,β α D0+ y)(x) = y(x) − yα,β (x), x > 0, (I0+

(2.9)

where yα,β (x) :=

n−1 

dk (1−β)(n−α) xk−n+α−βα+βn (I y)(0+), x > 0. Γ(k − n + α − βα + βn + 1) dxk 0+ k=0 (2.10)

As in the case of Mikusinski’s type operational calculus for the RiemannLiouville or for the Caputo fractional derivatives, we have the following lemmas (see e.g. [6, 12, 13]). Lemma 2.4. ([6]) The space C−1 with the operations of the Laplace convolution and ordinary addition becomes a commutative ring (C−1 , , +) without divisors of zero. This ring can be extended to the field M−1 of convolution quotients by following the lines of the classical Mikusinski’s operational calculus: M−1 := (C−1 × (C−1 \ {0}))/ ∼,

(2.11)

where the relation ”∼” is defined as usual by (f, g) ∼ (f1 , g1 ) ⇔ (f  g1 )(x) = (g  f1 )(x). This relation is an equivalence relation in terms of Lemma 2.4. For the sake of convenience, the elements of the field M−1 can be formally considered as convolution quotients fg . The operations of addition and multiplication are the defined in M−1 as usual: f1 f  g1 + g  f1 f + := (2.12) g g1 g  g1

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84

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and f  f1 f f1 · := . (2.13) g g1 g  g1 Then the space M−1 with the operations of addition (2.12) and multiplication (2.13) becomes a commutative field (M−1 , ·, +), where the unit element is defined by I := ff , and the zero element is defined by 0 := f0 , f = 0, see [6]. The field (M−1 , ·, +) is a quotient field of the ring (C−1 , , +). The ring C−1 can be embedded into the field M−1 by the mapping: hα  f , α > 0, f → (f, I) = hα I hα hα = = , α > 0. (2.14) sα := hα hα  hα h2α Lemma 2.5. ([6]) For the function y ∈ C−1 , the Riemann-Liouville fractional integral of y can be represented by the multiplication in the field M−1 : I α y)(x) = · y. (2.15) (I0+ sα α,β y of the function y ∈ Ωαβ , α > And the generalized fractional derivative D0+ 0, 0 ≤ β ≤ 1, can be represented as multiplication in the field M−1 of convolution quotients α,β y)(x) = sα · y − sα · yα,β . (D0+

(2.16)

Formula (2.7) means that for α > 0, n ∈ N hnα (x) := hα  · · ·  hα = hnα (x).    n

This relation can be extended to an arbitrary positive real power exponent: hλα (x) := hλα (x). For any λ > 0, then inclusion hλα (x) ∈ C−1 hold true and the following relations can be easily proved β γ hβα  hγα = hβα  hγα = h(β+γ)α = hβ+γ α , hα1 = hα2 ⇔ α1 β = α2 γ. (2.17)

The above relations motivate the following for the element sα with an arbitrary real power exponent λ: ⎧ −λ if λ < 0 ⎨ hα λ I if λ = 0 . (2.18) sα := ⎩ I if λ > 0 −λ hα

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OPERATIONAL METHOD FOR SOLVING MULTI-TERM . . . 85 For any α, α1 , α2 , β, γ > 0, it follows from this definition and from relations (2.17) that β γ sβα · sγα = s(β+γ)α = sβ+γ α , sα1 = sα2 ⇔ α1 β = α2 γ.

For the application of the operational calculus to solutions of differential equations with generalized fractional derivatives it is important to identify the elements of the field M−1 , which can be represented by the elements of the ring C−1 . One useful class of such representations is given by the following lemma (see e.g. [6, 13]). Lemma 2.6. ([6, 13]) Let β > 0, αi > 0, i = 1, · · · , n. Then I−

sβα n

i=1

= xβα1 E(α1 α,···αn α),βα (λ1 xα1 α , · · · , λn xαn α )

(2.19)

with the multivariate vector-index Mittag-Leffler function E(a1 ,··· ,an ),b (z1 , · · · , zn ) := n lj ∞   j=1 zj  , (k, l1 , · · · , ln ) = Γ(b + nj=1 aj lj ) k=0 l1 +···+ln =k ; l1 ,··· ,ln ≥0

aj > 0, b > 0, zj ∈ C

(2.20)

and multinomial coefficients (k, l1 , · · · , ln ) :=

k! . l1 ! × · · · × ln !

When a ∈ Z and b ∈ Z satisfy a ≤ b, we denote by Za,b the set of all integers i satisfying a ≤ i ≤ b. 3. Existence and uniqueness of the solution In this section the operational calculus constructed in Hilfer-LuchkoTomovski [6] is applied to solve linear fractional differential equations with generalized derivatives and constant coefficients. We consider the linear differential equation α0 ,β0 y(x) − D0+

n  i=1

αi ,βi ai D0+ y(x) = g(x)

(3.1)

and the initial conditions dk (1−βi )(mi −αi ) I y(0+) = yk−(1−βi )(mi −αi ) ∈ R, i ∈ Z0,n , k ∈ Z0,mi −1 , dxk 0+ (3.2)

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86

M.-Ha Kim, G.-Chol Ri, H.-Chol O

where αi ≥ 0, mi − 1 < αi ≤ mi ∈ N, 0 ≤ βi ≤ 1, i ∈ Z0,n and ai ∈ R, i ∈ Z1,n . The ordering m0 − (1 − β0 )(m0 − α0 ) ≥ · · · ≥ mn − (1 − βn )(mn − αn )

(3.3)

is assumed without lost of generality.

Definition 3.1. The function ni=0 Ωαβii is called the solution of the initial value problem (3.1) - (3.2), if y satisfies the equation (3.1) and the initial conditions (3.2). Theorem 3.1. The initial value problem (3.1) and (3.2) with homogeneous initial conditions dk (1−βi )(mi −αi ) I y(0+) = 0, i ∈ Z0,n , k ∈ Z0,mi −1 , dxk 0+ where



(3.4)

α0 ∈ N , α0 ∈ /N

has a solution y(x) which is unique in the space ni=0 Ωαβii and is provided by the formula  x tα0 −1 E(α0 −α1 ,··· ,α0 −αn ),α0 (a1 tα0 −α1 , · · · , an tα0 −αn )g(x − t)dt. y(x) = g(x) ∈

C−1 if 1 if C−1

0

(3.5)

P r o o f. Let y(x) ∈ ni=0 Ωαβii and satisfy (3.1) and (3.4). Then the following algebraic equation in the field M−1 of the convolution quotients is obtained n  ai sαi · y = g (3.6) sα0 · y − i=1

from homogeneous initial value problem (3.1) and (3.4) by using Lemma 2.5. This linear equation has a unique solution in the field M−1 : y=

sα0 −

I n

i=1 ai sαi

· g.

(3.7)

From (2.19) we get sα0 −

I n

i=1 ai sαi

=

I−

α0 −1

=x

s−α0 i=1 ai s−(α0 −αi )

n

E(α0 −α1 ,···α0 −αn ),α0 (a1 xα0 −α1 , · · · , an xα0 −αn ). (3.8)

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OPERATIONAL METHOD FOR SOLVING MULTI-TERM . . . 87 Therefore we obtain y(x) = xα0 −1 E(α0 −α1 ,··· ,α0 −αn ),α0 (a1 xα0 −α1 , · · · , an xα0 −αn )  g(x). Thus, we have the solution (3.5).

Now we consider y(x) ∈ ni=0 Ωαβii . γk y for γk = (1 − βk )(mk − αk ), k ∈ Z0,n . Then we have Let yγk := I0+ γk ((xα0 −1 E(α0 −α1 ,··· ,α0 −αn ),α0 (a1 xα0 −α1 , · · · , an xα0 −αn ))  g(x)) yγk = I0+

=hγk (x)  (xα0 −1 E(α0 −α1 ,··· ,α0 −αn ),α0 (a1 xα0 −α1 , · · · , an xα0 −αn ))  g(x) =(xα0 −1 E(α0 −α1 ,··· ,α0 −αn ),α0 (a1 xα0 −α1 , · · · , an xα0 −αn ))  (hγk (x)  g(x))  x tα0 −1 E(α0 −α1 ,··· ,α0 −αn ),α0 (a1 tα0 −α1 , · · · , an tα0 −αn )gγk (x − t)dt, = 0

where gγk (x) :=

γk I0+ g(x)

 ∈

C−1 if 1 if C−1

α0 ∈ N, α0 ∈ / N.

Whereas, from (3.7) we have γ0 g(x) + y(x) = I0+

n 

and γ0 gγk (x) + yγk (x) = I0+

i=1

n  i=1

By (3.10) we obtain yγk (x) =

γ0 γ1 (I0+ ψ1 )(x),

γ0 −γi ai I0+ y(x)

 ψ1 ∈

γ0 −γi ai I0+ yγk (x).

C−1 if 1 if C−1

α0 ∈ N, α0 ∈ / N.

(3.9)

(3.10)

(3.11)

Combining now the relations (3.10) and (3.11) and repeating the same arguments p times (p = [α0 /(α0 − α1 )] + 1), we arrive to the representation  C−1 if α0 ∈ N, γ0 (3.12) yγk (x) = (I0+ ψp )(x), ψp ∈ 1 if α0 ∈ / N. C−1 In the case α0 = m0 ∈ N, it follows from (3.12) that 0) 0 −1) = ψp ∈ C−1 , yγ0 (0) = · · · = yγ(m (0) = 0. yγ(m 0 0

/ N, then since If α0 ∈ hγ0 (x) =

xγ0 −1 m0 −1 0 −2) ∈ C−1 , hα0 (0) = · · · = h(m (0) = 0, α0 Γ(γ0 )

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88

M.-Ha Kim, G.-Chol Ri, H.-Chol O (m −1)

m0 , yγ0 (0) = · · · = yγ0 0 (0) = 0 by using Lemma 2.2. we obtain yγ0 ∈ C−1 In the case k = 1, · · · , n, with arguments similar to the above, we have mk k −1) , yγk (0) = · · · = yγ(m (0) = 0. yγk ∈ C−1 k

Hence y(x) is a unique solution of homogeneous initial value problem (3.1) and (3.4). The proof of the theorem is completed. 2 Theorem 3.2. For the initial value problem (3.1) - (3.2) with inhomogeneous initial conditions we have the following conclusions: i) If (1 − β0 )(m0 − α0 ) = · · · = (1 − βn )(mn − αn ) = γ, then the initial value problem (3.1) - (3.2) has the unique solution ⎫ ⎧ m n 0 −1 ⎬ ⎨   xk−γ + yk−γ ai xk−γ+α0 −αi E1 (k, i) y(x) = yg (x) + ⎭ ⎩ Γ(k − γ + 1) k=0

in the space

i=lk +1

(3.13)

n

αi i=0 Ωβi ,

where

E1 (k, i) := E(α0 −α1 ,··· ,α0 −αn ),k+1−γ+α0 −αi (a1 xα0 −α1 , · · · , an xα0 −αn ) (3.14) and lk ∈ N0 for every fixed k = 0, · · · , m0 − 1 is the integer such that mlk ≥ k + 1

and mlk +1 ≤ k.

(3.15)

(If mi ≤ k, 0 = 1, · · · , n we set lk := 0, and if mi ≥ k + 1, i = 0, · · · , n, then we let lk := n.) ii) If γ = (1 − β0 )(m0 − α0 ) = · · · = (1 − βl−1 )(ml−1 − αl−1 ) = (1 − )(m β

l n l α−i αl ), 0 < l ≤ n, then there exists a solution y(x) in the space i=0 Ωβi if and only if yk−(1−βi )(mi −αi ) = 0, i ∈ Zl,n , k ∈ Z0, mi −1 , yk−(1−βi )(mi −αi ) = 0, i ∈ Z0,l−1 , k ∈ Z0, M (where M = [ml − (1 − βl )(ml − αl ) − 1 + γ])

(3.16)

and the unique solution y(x) is provided by ⎫ ⎧ m n 0 −1 ⎬ ⎨   xk−γ + yk−γ ai xk−γ+α0 −αi E1 (k, i) , y(x) = yg (x) + ⎭ ⎩ Γ(k − γ + 1) k=M

i=lk +1

(3.17) where E1 (k, i) is given by (3.14) and lk ∈ N0 , k = 0, · · · , m0 − 1 are determined from the condition mlk ≥ k + 1

and mlk +1 ≤ k.

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(3.18)

OPERATIONAL METHOD FOR SOLVING MULTI-TERM . . . 89 (If mi ≤ k, i = 0, · · · , n we set lk := 0, and if mi ≥ k + 1, i = 0, · · · , n, then we let lk := n.) In (3.13) and (3.17) yg (k) is the solution (3.5) of homogeneous initial value problem (3.1) and (3.4). P r o o f. By using Lemma 2.4 and Lemma 2.5 we get the following algebraic equation in the field M−1 of convolution quotients sα0 · y −

n 

ai sαi · y = g + sα0 · yα0 ,β0 −

i=1

n 

ai sαi · yαi ,βi

(3.19)

i=1

from inhomogeneous initial value problem (3.1) - (3.2), where yαi ,βi is represented by (2.10). This linear equation has a unique solution in the field M−1 as follows:   n  I I   ·g+ ai sαi · yαi ,βi . sα0 · yα0 ,β0 − y= sα0 − ni=1 ai sαi sα0 − ni=1 ai sαi i=1 (3.20) I  Here yg := sα − n ai sα · g is represented by (3.5) and is the solution of i=1 0 i the homogeneous initial value problem (3.1) and (3.4). Let   n  I  · sα0 · yα0 ,β0 − ai sαi · yαi ,βi . (3.21) yh := sα0 − ni=1 ai sαi i=1

Then we have yh =

−

I−

n  i=1

n

m −1 0 

I

i=1 ai sαi −α0

ai sαi −α0

m 0 −1 

yk−(1−β0 )(m0 −α0 ) s−(k−(1−β0 )(m0 −α0 )+1)

k=0

 yk−(1−β0 )(m0 −α0 ) s−(k−(1−β0 )(m0 −α0 )+1)

.

(3.22)

k=0

If (1 − β0 )(m0 − α0 ) = · · · = (1 − βn )(mn − αn ) = γ, from (3.22) we have k m 0 −1  I − li=1 ai sαi −α0 I n yk−γ · yh = sk−γ+1 I − i=1 ai sαi −α0 k=0   n m 0 −1  a s I i α −α 0 i i=lk +1  = yk−γ · I+ sk−γ+1 I − ni=1 ai sαi −α0 k=0 ⎫ ⎧ m n 0 −1 ⎬ ⎨   xk−γ + yk−γ ai xk−γ+α0 −αi E·,k+1−γ+α0 −αi (x) , = ⎭ ⎩ Γ(k − γ + 1) k=0

i=lk +1

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90

M.-Ha Kim, G.-Chol Ri, H.-Chol O

where lk , k = 0, · · · , m0 − 1 are determined from condition (3.15). According to the definition of the numbers lk , we have mi ≤ k for i ∈ Zlk +1,n . this It follows then that k − γ + α0 − αi ≥ α0 − γ, i ∈ Zlk +1,n . Using

γ m0 yh ∈ C−1 and y(x) ∈ ni=0 Ωαβii . inequality, we readily get the inclusion I0+  xk−γ + ni=lk +1 ai xk−γ+α0 −αi E(·),k+1−γ+α0 −αi (x) we have For uk := Γ(k−γ+1)  1 if k = l γ (l) (I0+ uk ) (0) = , k, l = 0, · · · , m0 − 1. 0 if k = l Therefore yh satisfies the initial condition (3.2) and hence y(x) = yg (x) + yh (x) is a unique solution of the initial value problem (3.1) and (3.2). If γ = (1− β0 )(m0 − α0 ) = · · · = (1− βl−1 )(ml−1 − αl−1 ) = (1− βl )(ml − αl ), 0 < l ≤ n, then from (3.22) we have m −1 0  I n · yk−γ s−(k−γ+1) yh = I − i=1 ai sαi −α0 k=0

−

l−1 

ai sαi −α0

i=1

−

n 

ai sαi −α0

i=l

=

m 0 −1  k=0

−

I−

m i −1 

m i −1 

yk−γ s−(k−γ+1)

k=0



yk−(1−βi )(mi −αi ) s−(k−(1−βi )(mi −αi )+1)

k=0

k I − li=1 ai sαi −α0 n yk−γ s−(k−γ+1) I − i=1 ai sαi −α0

n

n 

I

i=1 ai sαi −α0 i=l

ai sαi −α0

m i −1 

yk−(1−βi )(mi −αi ) s−(k−(1−βi )(mi −αi )+1) ,

k=0

where lk is defined by (3.18). From the above equation we have ⎫ ⎧ m n 0 −1 ⎬ ⎨ xk−γ   + yk−γ ai xk−γ+α0 −αi E(·),k+1−γ+α0 −αi (x) yh (x) = ⎭ ⎩Γ(k−γ +1) k=0 n 

−

i=l

ai

m i −1 

i=lk +1

yk−(1−βi )(mi −αi ) xα0 −αi +k−(1−βi )(mi −αi ) E2 (i, k).

k=0

Here E2 (i, k) := E(·),α0 −αi +k+1−(1−βi )(mi −αi ) (x).

In order that yh (x) ∈ ni=0 Ωαβii , the following relation should be satisfied: α0 − αi + k + 1 − (1 − βi )(mi − αi ) + γ − m0 = = k + 1 − mi + βi (mi − αi ) − β0 (m0 − α0 ) ≥ 0

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OPERATIONAL METHOD FOR SOLVING MULTI-TERM . . . 91 for every i ∈ Zl,n , k ∈ Z0,mi −1 . This relation holds only in the case k = mi − 1 and βi (mi − αi ) ≥ β0 (m0 − α0 ). Therefore yk−(1−βi )(mi −αi ) = 0 for every i ∈ Zl,n , k ∈ Z0,mi −2 and, as yh (x) must satisfy the initial condition (3.2), we get ymi −1−(1−βi )(mi −αi ) = 0 for every i ∈ Zl,n . Also, as the relation k − γ + (1 − βl )(ml − αl ) − k1 ≥ 0, k = 0, · · · , m0 −1, k1 = 0, · · · , ml −1 must be satisfied, the following relations should hold: yk−γ = 0, k = 0, · · · , M, where M = [ml − (1 − βl )(ml − α1 ) − 1 + γ]. Thus, we obtain yh (x) =

m 0 −1 

 yk−γ

k=M

+

xk−γ Γ(k − γ + 1)  ai xk−γ+α0 −αi E(·),k+1−γ+α0 −αi (x) .

n  i=lk +1

Hence the solution is y(x) = yg (x) + yh (x). The proof of Theorem 3.2 is completed. 2 Example 3.1. If β0 = · · · = βn = 1, then the initial value problem (3.1) and (3.2) becomes the initial value problem with Caputo fractional derivatives n  αi c α0 D0+ y(x) − aci D0+ y(x) = g(x), (3.23) i=1

dk y(0+) = yk ∈ R, k ∈ Z0, m0 −1 , (3.24) dxk where α0 > α1 > · · · > αn ≥ 0, mi − 1 < αi ≤ mi ∈ N, i ∈ Z0,n , and the m0 . From Theorem 3.2, the initial value space ni=0 Ωαβii coincides with C−1 problem (3.23) and (3.24) has a unique solution  x tα0 −1 E(α0 −α1 ,··· ,α0 −αn ),α0 (a1 tα0 −α1 , · · · , an tα0 −αn )g(x − t)dt y(x) = 0

+ +

m 0 −1 

yk

k=0 n 



xk Γ(k + 1)

 ai xk+α0 −αi E(α0 −α1 ,··· ,α0 −αn ),k+1+α0 −αi (a1 xα0 −α1 , · · · , an xα0 −αn ) ,

i=lk +1

(3.25) where lk , k = 0, · · · , m0 − 1 are determined from condition (3.15).

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92

M.-Ha Kim, G.-Chol Ri, H.-Chol O

Example 3.2. If β0 = · · · = βn = 0, then the initial value problem (3.1) and (3.2) becomes the initial value problem with Riemann-Liouville fractional derivatives: n  α0 αi y(x) − ai D0+ y(x) = g(x), (3.26) D0+ i=1

dk

I mi −αi y(0+) = yk−(mi −αi ) ∈ R, i ∈ Z0,n , k ∈ Z0,mi −1 , (3.27) dxk 0+ where α0 > α1 > · · · > αn ≥ 0, mi − 1 < αi ≤ mi ∈ N, i ∈ Z0,n . If m0 − α0 = · · · = mn − αn = γ, then the initial value problem (3.26) and (3.27) has unique solution  x tα0 −1 E(α0 −α1 ,··· ,α0 −αn ),α0 (a1 tα0 −α1 , · · · , an tα0 −αn )g(x − t)dt y(x) = 0 ⎤ ⎡ m n 0 −1 k−γ   x + yk−γ ⎣ ai xk−γ+α0 −αi E1 (k, i)⎦ (3.28) + Γ(k − γ + 1) k=0

n

αi i=0 Ω0

i=lk +1

Ωα0 i .

= Here E1 (k, i) is given in (3.14), and lk , k = in the space 0, · · · , m0 − 1 are determined from condition (3.18). If γ = m0 − α0 = · · · = ml−1 − αl−1 = ml − αl , 0 < l ≤ n, then there exists a solution y(x) in the space ni=0 Ωα0 i = Ωα0 i , if and only if yk−(mi −αi ) = 0, i ∈ Zl,n , k ∈ Z0,mi −1 yk−(mi −αi ) = 0, i ∈ Z0,l−1 , k ∈ Z0,M , M = [αl − 1 + γ].

(3.29)

Then the unique solution of the initial value problem (3.26) and (3.27) is given by  x tα0 −1 E(α0 −α1 ,··· ,α0 −αn ),α0 (a1 tα0 −α1 , · · · , an tα0 −αn )g(x − t)dt y(x) = 0 ⎤ ⎡ m n 0 −1 k−γ   x + yk−γ ⎣ ai xk−γ+α0 −αi E1 (k, i)⎦ . (3.30) + Γ(k − γ + 1) k=M

i=lk +1

Here E1 (k, i) and lk , k = 0, · · · , m0 − 1 are the same as in (3.28). Example 3.3. We consider the so-called composite fractional relaxation equation (see [5]) d α,μ f (t) + τ2α D0+ f (t) + f (t) = 0, dt (1−μ)(1−α) f )(0+) = f(1−μ)(1−α) , f (0+) = 1, (I0+

τ1

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(3.31) (3.32)

OPERATIONAL METHOD FOR SOLVING MULTI-TERM . . . 93 where 0 < α < 1, 0 ≤ μ ≤ 1, 0 < τ1 , < ∞. By using Theorem 3.2, the initial value (3.31) and (3.32) has a unique solution f (t) in the

problem

space Ω10 Ωαμ Ω00 , if and only if (1−μ)(1−α)

(I0+

f )(0+) = f(1−μ)(1−α) = 0.

(3.33)

Then this unique solution is given by f (t) = E(1−α,

1), 1 (−

τ2α 1−α 1 t), t > 0. t , − τ1 τ −1

(3.34)

Acknowledgements The authors would like to thank Prof. Virginia Kiryakova and Prof. Yuri Luchko for their great help and valuable advices for the improvement and publication of this article. References [1] R. Caponetto, G. Dongola, L. Fortuna, I. Petr´ aˇs, Fractional Order Systems: Modeling and Control Applications. World Scientific Ser. on Nonlinear Science, Vol. 72, World Scientific, Singapore (2010). [2] K.M. Furati, M.D. Kassim, N.e.-Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative. Comp. Math. Appl. 64, No 6 (2012), 1616–1626. [3] R. Hilfer, Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000). [4] R. Hilfer, Fractional time evolution. In: Applications of Fractional Calculus in Physics (Ed. R. Hilfer). World Scientific, Singapore (2000), 87–130. [5] R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials. Chem. Phys. 284 (2002), 399–408. ˇ Tomovski, Operational method for the solution [6] R. Hilfer, Y. Luchko, Z. of fractional differential equations with generalized Riemann-Liouville fractional derivatives. Fract. Calc. Appl. Anal. 12, No 3 (2009), 299– 318; at http://www.math.bas.bg/∼fcaa. [7] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Vol. 204, Elsevier, North Holland (2006). [8] M.H. Kim, Hyong-Chol O, Explicit representations of Green’s function for linear fractional differential operator with variable coefficients. J. of Fractional Calculus and Applications 5, No 1 (2014), 26–36. [9] V. Kiryakova, Generalized Fractional Calculus and Applications. Longman & J. Wiley, Harlow & N. York (1994).

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OPERATIONAL METHOD FOR SOLVING MULTI-TERM . . . 95 [24] S. Yakubovich, Yu. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions. Ser. Mathematics and Its Applications 287, Kluwer Acad. Publ., Dordrecht-Boston-London (1994). Faculty of Mathematics Kim Il Sung University Kumsong Street, Taesong District Pyongyang, D.P.R. KOREA 1 2 3

e-mail: myongha [email protected] e-mail: [email protected] e-mail: [email protected]

Received: April 18, 2013

Please cite to this paper as published in: Fract. Calc. Appl. Anal., Vol. 17, No 1 (2014), pp. 79–95; DOI: 10.2478/s13540-014-0156-6

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