Generalized Multivariable Mittag – Leffler Function

On Certain Fractional Differential Equations Involving

Generalized Multivariable Mittag – Leffler Function

B.B. Jaimini

Department of Mathematics, Govt. Girls P.G. College, Kota -326001, India bbjaimini [email protected]

Jyoti Gupta

Department of Mathematics, Govt. College, Kota-324001,India [email protected]

Received: 17.11.2011; accepted: 16.10.2012.

Abstract. In this paper some fractional differential equations are solved by using the Laplace transform method. These fractional differential equations with Riemann – Liouville fractional derivative operators are very general in nature and involve the wide range of fractional differ- ential equations and their solutions in terms of various functions related to Mittag – Leffler function.

Keywords:Fractional differential equations, Riemann–Liouville fractional derivative opera- tor, Generalized Mittag–Leffler function, confluent hypergeometric function, Laplace transform

MSC 2000 classification:33E12, 34A08

1 Introduction

The entire function of the form E_α(z) =

X∞

k=0

z^k

Γ (kα + 1) (1)

where α∈ C; Re (α) > 0; z ∈ C defines the Mittag – Leffler function [2,3].

A generalization of Mittag – Leffler function E_α(z) of (1) is defined and studied by Wiman [11] as follows:

E_α,β(z) = X∞

k=0

z^k

Γ (αk + β) (2)

where α, β∈ C; Re (α) > 0; Re (β) > 0; z ∈ C.

http://siba-ese.unisalento.it/ c 2012 Universit`a del Salento

A generalization of Mittag – Leffler function E_α,β(z) of (2) is introduced by Prabhakar [5, p.7] as follows:

E_α,β^γ (z) = X∞

k=0

(γ)_k Γ (αk + β)

z^k

k! (3)

where α, β, γ ∈ C; Re (α) > 0; Re (β) > 0; z ∈ C and (λ)ndenotes the familiar Pochhammer symbol or the shifted factorial , since

(1)_n= n! (n∈ N0) and

(λ)_n= Γ (λ + n) Γ (λ) =

® 1

λ (λ + 1) ... (λ + n− 1)

(n = 0; λ∈ C\ {0})

(n∈ N; λ ∈ C) (4) ÄN₀ = N^[{0} = {0, 1, 2...}^ä

Recently generalization of Mittag–Leffler function E_α,β^γ (z) of (3) studied by Srivastava and Tomovski [10] is defined as follows:

E_α,β^γ,K(z) = X∞

n=0

(γ)_Kn Γ (αn + β)

zⁿ

n! (5)

(z, β, γ∈ C; Re (α) > max{0, Re(K) − 1}; Re(K) > 0) which, in the special case when

K = q (q∈ (0, 1) ∪ N) and min {Re (β) , Re (γ)} > 0 (6) was considered earlier by Shukla and Prajapati [8].

A multivariable analogue of Mittag–Leffler function defined in (3) is very re- cently studied by Gautam [1] and Saxena et al. [7, p.536, Eq. 1.14] in the following form:

E_(ρ^(γj)

j),λ[z₁, ..., z_r] = E_(ρ^{(γ1,...,γr)}

1,...,ρr),λ[z₁, ..., z_r] = X∞

k1,...,kr=0

(γ1)_k1...(γr)_kr Γ (λ + k₁ρ₁+ ... + k_rρ_r)

z^k₁¹...z_r^kr (k₁)!...(k_r)! (7) where λ, γ_j, ρ_j ∈ C; Re (ρj) > 0; j = 1, 2, ..., r.

If in (7) we take ρ₁ = ρ₂= ... = ρ_r = 1then it reduces to the following confluent hypergeometric series [9, p. 34, Eq. (1.4(8))]

Φ^(r)₂ [γ₁, ..., γ_r ; λ; z₁, ..., z_r] = 1 Γ (λ)

X∞

k1,...,kr=0

Qr i=1(γ_i)_ki (µ)_k1+...+kr

z₁^k1...z_r^kr

(k1)!...(kr)! (8) where λ, γj, zj ∈ C (j = 1, 2, ..., r) and max {|z1| , ..., |zr|} < 1;λ /∈ Z0⁻= {0, −1, −2, ....}.

A mild generalization of multivariable analogue of Mittag – Leffler function in (7) is also due to Saxena et al. defined as follows [7, p. 547, Eq.7.1]:

E_(ρ^(γr),(lr)

r),λ (z₁, ..., z_r) = X∞

k1,...,kr=0

(γ₁)_k1l1...(γ_r)_k

rlr

Γ (λ + k₁ρ₁+ ... + k_rρ_r)

z₁^k1...z^k_r^r

(k₁)!...(k_r)! (9) where λ, γ_j,l_j, ρ_j ∈ C; Re (ρj) > 0; Re (l_j) > 0; λ /∈ Z0⁻ ={0, −1, −2, ....};j = 1, 2, ..., r.

We consider the following integral operator involving the above generalized mul- tivariable Mittag – Leffler function in the kernel is defined and represented as follow:

E_(ρ^(γr),(lr)

r),µ,(ωr);a+Ψ^(x) = Z x

a

(x− t)^µ−1E_(ρ^(γr),(lr)

r),µ [ω₁(x− t)^ρ1, ..., ω_r(x− t)^ρr]Ψ (t) dt (10) with x > a; ω_j, ρ_j, γ_j, l_j, µ∈ C; Re (ρj) > 0;|ωj(x− t)^ρj| < 1; j = 1, 2, ..., r.

Remark: At l₁ = l₂ = ... = l_r = 1 the operator defined in (10) reduces to the integral operator studied by Gautam [1] and Saxena et al. [7].

In the present paper we consider the following type of differential equations D^α₀₊y
(x) = λ^E_(ρ^(γr),(lr)

r),µ,(ωr);0+

(x) + f (x) (11)

with the initial condition _Ä

I₀₊^1−αy^ä(0+) = c

where c is an arbitrary constant and (α, µ, ρj, γj, lj, ωj ∈ C; Re (α) > 0;

Re (ρ_j) > 0; Re (µ) > 0; Re(l_j) > 0;j = 1, 2, ..., r).

Here D^α₀₊ is Riemann – Liouville fractional derivative operator defined by [6]:

D^α_a+Ψ
(x) = Å d

dx ãnÄ

I_a+1^n−αΨ^ä(x) (12) α∈ C : Re(α) > 0 (n = |Re(α)| + 1) .

where I_a+^α Ψ
(x) is the Riemann – Liouville fractional integral operator defined by

I_a+^α Ψ
(x) = 1 Γ (α)

Z _x

a

Ψ (t)

(x− t)^1−αdt (13)

(α∈ C; Re (α) > 0)

For a = 0 the operator D_a+^α Ψ
(x) is represented by D^α₀₊Ψ
(x).

The Laplace transform of a function f (x) is defined by L [f (x); s] =

Z ∞

0

e^−sxf (x) dx = F (s) (14) The Laplace transform of fractional derivative D₀₊^α f
(x) is given by [4]:

L^D₀₊^α f ; s^= s^αF (s)− Xn

k=1

s^k−1D^α−k₀₊ f (0+) (n− 1 < α < n) (15)

Re(s) > 0 The Laplace transform of the function E_(ρ^(γr),(lr)

r),µ [.] defined in (9) is easily ob- tainable in the following form also required here

Lⁿx^µ−1E_(ρ^(γr),(lr)

r),µ [ω₁x^ρ1, ..., ω_rx^ρr]^o(s) = s^−µ Yr

i=1

®

1Ψ^∗₀

ñ (γ_i, l_i)

− ; ω_i s^ρi

ô´

(16)

(µ, ρ_j, γ_j, l_j, ω_j ∈ C; Re (s) > 0; Re (ρj) > 0; Re (µ) > 0; Re(l_j) > 0; j = 1, 2, ..., r) where1Ψ^∗₀ is the Fox – Wright hypergeometric function defined as follows [9]:

pΨ_q

ñ (a₁, A₁) , ..., (a_p, A_p) ; (b1, B1) , ..., (bq, Bq) ; z

ô

= X∞

n=0

Qp

i=1(ai)_Ain Qq

j=1(bj)_B

jn

zⁿ

n! (17)

The following integral involving the generalized Mittag – Leffler function in (9) is also required

1 Γ (σ)

Z _x

0

(x− t)^µ−1t^σ−1E_(ρ^(γr),(lr)

r),µ [ω₁(x− t)^ρ1, ..., ω_r(x− t)^ρr] dt

= x^µ+σ−1E_(ρ^(γr),(lr)

r),σ+µ[ω₁x^ρ1, ..., ω_rx^ρr] (18) The integral in (18) is established in view of (9) and elementary beta integral.

We also need the following familiar derivative formula for Laplace transform here

dⁿ

dsⁿ[L[y(x)(s)] = (−1)ⁿL[xⁿy(x)](s) (19)

2 Main Results

Theorem 1. Let a∈ R+; α, µ, ρ_j, γ_j, l_j, ω_j ∈ C; Re (α) > 0; Re (ρj) > 0;

Re (µ) > 0; Re(lj) > 0; j = 1, 2, ..., r. Then for x > a, there hold the relations.

D_a+^{α h}(t− a)^µ−1E^(γ_(ρ^r),(lr)

r),µ [ω₁(t− a)^ρ1, ..., ω_r(t− a)^ρr]ⁱ(x)

= (x− a)^µ−α−1E_(ρ^(γr),(lr)

r),µ−α[ω₁(x− a)^ρ1, ..., ω_r(x− a)^ρr] (1) and

I_a+^{α h}(t− a)^µ−1E_(ρ^(γr),(lr)

r),µ [ω₁(t− a)^ρ1, ..., ω_r(t− a)^ρr]ⁱ(x)

= (x− a)^µ+α−1E_(ρ^(γr),(lr)

r),µ+α[ω₁(x− a)^ρ1, ..., ω_r(x− a)^ρr] (2) Theorem 2. The following fractional differential equation :

D^α₀₊y
(x) = λ^E_(ρ^(γr),(lr)

r),µ,(ωr);0+

(x) + f (x) (3)

(α, µ, ρ_j, γ_j, l_j, ω_j ∈ C; Re (α) > 0; Re (ρj) > 0; Re (µ) > 0; Re(l_j) > 0; j = 1, 2, ..., r.) with the initial condition ^ÄI₀₊^1−αy^ä(0+) = c has its solution in the space L (0,∞) given by

y(x) = cx^α−1

Γ (α)+λx^µ+αE_(ρ^(γr),(lr)

r),µ+α+1(ω₁x^ρ1, ..., ω_rx^ρr)+ 1 Γ (α)

Z _x

0

(x− t)^α−1f (t) dt (4) where c is an arbitrary constant .

Theorem 3. The following fractional differential equation:

D₀₊^α y
(x) = λ^E_(ρ^(γr),(lr)

r),µ,(ωr);0+

(x) + px^µE_(ρ^(γr),(lr)

r),µ+1[ω₁x^ρ1, ..., ω_rx^ρr] (5) (α, µ, ρ_j, γ_j, l_j, ω_j ∈ C; Re (α) > 0; Re (ρj) > 0; Re (µ) > 0; Re(l_j) > 0;

j = 1, 2, ..., r) with the initial condition ^ÄI₀₊^1−αy^ä (0+) = c has its solution in the space L (0,∞) given by

y(x) = cx^α−1

Γ (α) + (λ + p) x^µ+αE_(ρ^(γr),(lr)

r),µ+α+1[ω₁x^ρ1, ..., ω_rx^ρr] (6)

where c is an arbitrary constant .

Theorem 4. The following fractional differential equation:

x D₀₊^α y
(x) = λ^E_(ρ^(γr),(lr)

r),µ,(ωr);0+

(x) (7)

(α, µ, ρ_j, γ_j, l_j, ω_j ∈ C; Re (α) > 0; Re (ρj) > 0; Re (µ) > 0; Re(l_j) > 0; j = 1, 2, ..., r) with the initial condition ^ÄI₀₊^1−αy^ä (0+) = c has its solution in the space L (0,∞) given by

y(x) = cx^α−1 Γ(α)− λ

Γ (α) Z x

0

t^α−1(x− t)^µ−1E_(ρ^(γr),(lr)

r),µ+1[ω₁(x− t)^ρ1, ..., ω_r(x− t)^ρr] dt (8) where c is an arbitrary constant.

Theorem 5. The following fractional differential equation:

D₀₊^α y
(x) = λ^E_(ρ^(γr),(lr)

r),µ,(ωr);0+

(x)+

Xn

j=1

ï

p^jx^µjE^(γ

(j) r ),(l^(j)r )

(ρ^(j)r ),µj+1[ω^(j)₁ x^ρ(j)1 , .., ω^(j)_r x^ρ(j)r ] ò

(9) with the initial condition^ÄI₀₊^1−αy^ä(0+) = c has its solution in the space L (0,∞) given by

y(x) = c^x_Γ(α)^α−1 + λx^µ+αE_(ρ^(γr),(lr)

r),µ+α+1(ω1x^ρ1, ..., ωrx^ρr) + ^Pn

j=1

ï

p_jx^µj+αE^{(γ(j)r ),(l(j)r )}

(ρ^(j)r ),µj+α+1

ω₁^(j)x^ρ(j)1 , ..., ω^(j)r x^{ρ(j)r ò} (10)

where c is an arbitrary constant.

Outline of Proofs

Proof of (1): To prove the assertion (1) of Theorem-1, we denote its left hand side by ∆₁ i.e.

∆₁= D_a+^{α h}(t− a)^µ−1E_(ρ^(γr),(lr)

r),µ [ω₁(t− a)^ρ1, ..., ω_r(t− a)^ρr]ⁱ(x) On using the definition of E_(ρ^(γr),(lr)

r),µ [.] given in (9), we obtain

∆₁ = D^α_a+



 (t− a)^µ−1

X∞

k1,...,kr=0

Yr

i=1

ñ(γ_i)_k

ili

(k_i)! (ω_i)^ki

ô (t− a)^Pri=1ρiki Γ (µ +^Pr_i=1ρ_ik_i)

# (x)

i.e.

∆1 = X∞

k1,...,kr=0

Yr

i=1

ñ(γ_i)_kili (k_i)! (ωi)^ki

ô 1

Γ (µ +^Pr_i=1ρ_ik_i)D^α_a+^h(t− a)^{µ+Pri=1ρiki−1i}(x)

On using the fractional derivative [6] D^α_a+^î(t− a)^λó(x) = _{Γ(λ−α+1)}^Γ(λ+1) (x− a)^λ−α we have

∆₁= X∞

k1,...,kr=0

Yr

i=1

ñ(γi)_kili (ki)! (ω_i)^ki

ô(x− a)^{µ+Pri=1ρiki−α−1} Γ (µ− α +^Pri=1ρiki) On interpreting in view of the definition of E_(ρ^(γr)(lr)

r),µ [.] given in (9), we at once arrive at the desired result in (1).

Proof of (2): The assertion (2) of Theorem-2 is proved similarly following the lines as to prove the result in (1) and using the definition of fractional integral operator I_a+^α therein.

Proof of (4): On using the definition of operator ^E_(ρ^(γr),(lr)

r),µ,(ωr);a+Ψ^ (x) given in (10) with (a = 0 and Ψ(x) = 1) and formula (18) with (σ = 1) in equation (3), it takes the following form

D^α₀₊y
(x) = λx^µE_(ρ^(γr),(lr)

r),µ+1[ω₁x^ρ1, ..., ω_rx^ρr] + f (x) (11) By applying Laplace transform on both sides of (11) and then using formulae (15) with (n = 1), (16) and (14) we obtain

s^αy (s) = c + λs^−µ−1 Yr

i=1

®

1Ψ^∗₀

ñ (γ_i, l_i)

− ; ωi

s^ρi ô´

+ F (s) It yields

y(s) = cs^−α+ λs^{−µ−α−1} Yr

i=1

®

1Ψ^∗₀

ñ (γ_i, l_i)

− ; ω_i s^ρi

ô´

+ F (s)· s^−α

in view of (17) , we have

y(s) = cs^−α+λ X∞

k1,...,kr=0

Yr

i=1

ñ(γ_i)_k

ili

(k_i)! (ω_i)^ki ô

s^{−µ−α−Pri=1ρiki−1}+F (s)·s^−α (12)

Now taking the inverse Laplace transform on both sides of (12), we find by means of the Laplace convolution theorem that

y(x) = cx^α−1 Γ (α)+λ

X∞

k1,...,kr=0

Yr

i=1

ñ(γi)_kili (ki)! (ω_i)^ki

ô

L^−1îs^{−µ−α−Pri=1ρiki−1ó}+L⁻¹[F (s)· s^−α] i.e.

y(x) = cx^α−1 Γ (α) + λ

X∞

k1,...,kr=0

Yr

i=1

ñ(γi)_kili (k_i)! (ωi)^ki

ô (x)^µ+

Pr i=1

ρiki+α

Γ Å

µ + α + ^Pr

i=1ρiki+ 1 ã

+ 1

Γ (α) Z _x

0

(x− t)^α−1f (t) dt Now on interpreting second term in view of the definition of E^(γ_(ρ^r),(lr)

r),µ [.] given in (9), we at once arrive at the desired result in (4).

Proof of (6): On using the definition of operator ^E_(ρ^(γr),(lr)

r),µ,(ωr);a+Ψ^ (x) given in (10) with (a = 0 and Ψ(x) = 1) and formula (18) with (σ = 1) in equation (5), it takes the following form

D₀₊^α y
(x) = (λ + p)x^µE^(γ_(ρ^r),(lr)

r),µ+1[ω₁x^ρ1, ..., ω_rx^ρr] (13) By applying Laplace transform on both sides of (13) and then using formulae (15) with (n = 1) and (16), we get

y (s) = cs^−α+ (λ + p) s^{−µ−α−1} Yr

i=1

®

1Ψ^∗₀

ñ (γ_i, l_i)

− ; ωi

s^ρi ô´

in view of (17) we obtain

y (s) = cs^−α+ (λ + p) X∞

k1,...,kr=0

Yr

i=1

ñ(γ_i)_kili

(k_i)! (ωi)^{kiô î}s^{−µ−α−Pri=1ρiki−1ó} (14) On taking inverse Laplace transform on both sides of (14), we have

y(x) = cx^α−1

Γ (α)+ (λ + p) X∞

k1,...,kr=0

Yr

i=1

ñ(γ_i)_k

ili

(k_i)! (ω_i)^ki

ô x^{µ+α+Pri=1ρiki} Γ (µ + α +^Pr_i=1ρ_ik_i+ 1) Now on interpreting with help of definition of E^(γ_(ρ^r),(lr)

r),µ [.] given in (9), we at once arrive at the desired result in (6).

Proof of (8): On using the definition of operator ^E_(ρ^(γr),(lr)

r),µ,(ωr);a+Ψ^ (x) given in (10) with (a = 0 and Ψ(x) = 1) and formula (18) with (σ = 1) in equation (7), it takes the following form

x D₀₊^α y
(x) = λx^µE_(ρ^(γr),(lr)

r),µ+1[ω₁x^ρ1, ..., ω_rx^ρr] (15) By applying Laplace transform on both sides of (15) and then using formulae (16) and (19), (15) with (n = 1), we get

d

dsy (s) +α

sy (s) =−λs^{−µ−α−1} Yr

i=1

®

1Ψ^∗₀

ñ (γ_i, l_i)

− ; ω_i s^ρi

ô´

in view of (17), we have

d

dsy (s) +α

sy (s) =−λ X∞

k1,...,kr=0

Yr

i=1

ñ(γ_i)_kili (k_i)! (ωi)^ki

ô

s^{−µ−α−Pri=1ρiki−1}

This is a linear differential equation of first order and first degree. Hence

y (s) =−λ X∞

k1,...,kr=0

Yr

i=1

ñ(γi)_kili (ki)! (ωi)^ki

ôs^{−α−µ−Pri=1ρiki}

(µ +^Pr_i=1ρiki) + cs^−α (16) By applying inverse Laplace transform on both sides of (16), we obtain

y (x) =−λ X∞

k1,...,kr=0

Yr

i=1

ñ(γ_i)_k

ili

(k_i)! (ω_i)^ki ô

· 1

(µ +^Pr_i=1ρ_ik_i)L^−1îs^{−µ−Pri=1ρiki}· s^−αó+ CL⁻¹(s^−α) On using Laplace convolution theorem, we obtain

y (x) =−λ X∞

k1,...,kr=0

Yr

i=1

ñ(γ_i)_k

ili

(k_i)! (ω_i)^ki ô

· 1

Γ (µ + 1 +^Pr_i=1ρ_ik_i) 1 Γ (α)

Z _x

0

t^α−1(x− t)^{µ+Pri=1ρiki−1}dt + C 1 Γ (α)x^α−1 Now on changing the order of integration and summation, and then on inter- preting the so obtained result in view of definition of E_(ρ^(γr),(lr)

r),µ [.] given in (9), we at once arrive at the desired result in (8).

Proof of (10): On using the definition of operator ^E_(ρ^(γr),(lr)

r),µ,(ωr);a+Ψ^ (x) given in (10) with (a = 0 and Ψ(x) = 1) and formula (18) with (σ = 1) in equation (9), it takes the following form

D₀₊^α y
(x) = λx^µE^(γ_(ρ^r),(lr)

r),µ+1[ω₁x^ρ1, ..., ω_rx^ρr] + Xn

j=1

ï

pjx^µjE^(γ

(j) r ),(l^(j)r ) (ρ^(j)r ),µj+1

Å

ω^(j)₁ x^ρ(j)1 , ..., ω_r^(j)x^ρ(j)r ãò

(17)

By applying Laplace transform on both sides of (17), and then using formulae (15) with (n = 1) and (16) we obtain

y(s) = Cs^−α+ λs^{−µ−α−1 Qr}

i=1

®

1Ψ^∗₀

ñ (γ_i, l_i)

− ;_s^ω_ρiⁱ ô´

+ ^Pn

j=1

ñ

p_js^{−α−µj−1 Qr}

i=1

®

1Ψ^∗₀

ñ (γ_i^(j), l^(j)_i )

− ; ^ω

(j) i

s^ρ

(j) i

ô´ô

in view of (17), we obtain

y(s) = cs^−α+ λ X∞

k1,..,kr=0

Yr

i=1

ñ(γ_i)_kili (ki)! (ω_i)^ki

ô

s^{−µ−α−Pri=1ρiki−1}

+ Xn

j=1

p_j



 X∞

k1,..,kr=0

Yr

i=1

Ö(γ_i^(j))

kil^(j)_i

(k_i)! (ω_i^(j))^ki è

s^{−µj−α−Pri=1ρ(j)i ki−1} (18)

On applying the inverse Laplace transform on both sides of (18) we have

y (x) = c^x_Γ(α)^α−1 + λ ^P∞

k1,...,kr=0

Qr i=1

ï(γi)_kili (ki)! (ω_i)^ki

ò

x

µ+α+

Pr i=1

ρiki

Γ

Å

µ+α+

Pr i=1

ρiki+1

ã

+ ^Pn

j=1p_j



 P^∞

k1,..,kr=0

Qr i=1

Ñ(γ_i^(j))

kil(j) i

(ki)! (ω_i^(j))^ki é

 ^x

α+µj +

Pr i=1

ρ(j) i ki

Γ

Å

µ+α+

Pr i=1

ρ^(j)_i ki+1

ã

Now on interpreting with the help of definition of E_(ρ^(γr),(lr)

r),µ [.] given in (9) we at once arrive at the desired result in (10).

3 Special Cases

(1) If in Theorems-1 to 4, we take r = 1 then these reduce to the following results involving generalized Mittag – Leffler function due to Srivastava and Tomovski [10] as follows:

(i) Let a ∈ R+; α, µ, γ, ρ, l, ω ∈ C; Re (α) > 0; Re(ρ) > 0; Re (µ) > 0;

Re(l) > 0. Then for x > a, there hold the relations

D_a+^{α î}(t− a)^µ−1E_ρ,µ^γ,l[ω(t− a)^ρ]^ó(x) = (x− a)^µ−α−1E_ρ,µ−α^γ,l [ω(x− a)^ρ] (1) and

I_a+^{α î}(t− a)^µ−1E_ρ,µ^γ,l[ω(t− a)^ρ]^ó(x) = (x− a)^µ+α−1E_ρ,µ+α^γ,l [ω(x− a)^ρ] (2) Corollary 1. The following fractional differential equation

D₀₊^α y
(x) = λ^ÄE_0+,ρ,µ^{ω,γ,l ä}(x) + f (x) (3) (0 < α < 1; ω∈ C; Re (ρ) > max{0, Re (l) − 1}; min {Re (µ) , Re (γ) , Re (l)} >

0 )

With the initial condition

ÄI₀₊^1−αy^ä(0+) = c has its solution in the space L (0,∞) given by

y (x) = cx^α−1

Γ (α) + λx^µ+αÄE_ρ,µ+α+1^{γ,l ä}(ωx^ρ) + 1 Γ (α)

Z x

0

(x− t)^α−1f (t) dt (4) where c is an arbitrary constant and^ÄE_0+,ρ,µ^{ω,γ,l ä}(x) is the integral operator studied by Srivastava and Tomovski [10, p.202, Eq. (12)]:

ÄE_a+;α,β^ω,γ,K φ^ä(x) = Z _x

a

(x− t)^β−1E_α,β^γ,K[ω(x− t)^α]φ(t)dt(x > a) (5)

(γ, ω ∈ C; Re(α) > max{0, Re(K) − 1}; min{Re(β), Re(K)} > 0) . Corollary 2. The following fractional differential equation

D^α₀₊y
(x) = λ^ÄE_0+,ρ,µ^{ω,γ,l ä}(x) + px^µE_ρ,µ+1^γ,l [ωx^ρ] (6) (0 < α < 1; ω∈ C; Re (ρ) >max{0, Re (l) − 1}; min {Re (µ) , Re (γ) , Re (l)} >

0)