J. Nig. Soc. Phys. Sci. 1 (2019) 138–142

Journal of the
Nigerian Society

of Physical
Sciences

Original Research

Solution of Fractional Order Equations in the Domain of the
Mellin Transform

Sunday Emmanuel Fadugba∗

Department of Mathematics, Ekiti State University, Ado Ekiti, Nigeria

Abstract

This paper presents the Mellin transform for the solution of the fractional order equations. The Mellin transform approach occurs in many areas of
applied mathematics and technology. The Mellin transform of fractional calculus of different flavours; namely the Riemann-Liouville fractional
derivative, Riemann-Liouville fractional integral, Caputo fractional derivative and the Miller-Ross sequential fractional derivative were obtained.
Three illustrative examples were considered to discuss the applications of the Mellin transform and its fundamental properties. The results show
that the Mellin transform is a good analytical method for the solution of fractional order equations.

Keywords: Caputo fractional derivative, Fractional calculus, Mellin transform, Miller-Ross fractional derivative, Riemann-Liouville fractional
derivative, Riemann-Liouville fractional integral

Article History :
Received: 22 June 2019
Received in revised form: 24 August 2019
Accepted for publication: 18 September 2019
Published: 30 December 2019

c©2019 Journal of the Nigerian Society of Physical Sciences. All Rights Reserved.
Communicated by: T. Latunde

1. Introduction

The use of fractional derivatives and integrals provides a
powerful tool for incorporating history due to its non-local na-
ture [1].

Fractional calculus has gained much attention due to the fact
that fractional differential equations provide an excellent instru-
ment for the description of many practical dynamics phenom-
ena emanating from mathematical finance, economics, biology,
physics, just to mention a few, see Ref. [2].

The numerical solution of differential equations of integer
order has for a long time been a standard topic in numerical and
computational mathematics. However, in spite of a large num-
ber of recently formulated applied problems, the state of the art

∗Corresponding Author Tel. No: +2348067032044
Email address: sunday.fadugba@eksu.edu.ng (Sunday Emmanuel

Fadugba)

is far less advanced for fractional order differential equations
[3].

For various methods of solving fractional differential equa-
tions see; [4-9], just to mention a few. In this paper a survey
of the applications of the Mellin transform in the solution of
fractional calculus of different flavours is presented. It cov-
ers the widely known flavours, such as Riemann-Liouville frac-
tional derivative, Riemann-Liouville fractional integral, Caputo
fractional derivative and the Miller-Ross sequential fractional
derivative. The rest of the paper is organized as follows. Section
Two presents some definitions of terms. Section Three presents
the main results for the Mellin transform of fractional operators
of different flavours. In Section Four, three illustrative exam-
ples were presented. Section Five concludes the paper.

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Fadugba / J. Nig. Soc. Phys. Sci. 1 (2019) 138–142 139

2. Preliminaries

Definition 1 Let f (t) be locally Lebesgue integrable over
(0,∞). The Mellin transform of f (t) denoted by F(ω) is defined
as

F(ω) = M( f (t),ω) =
∫
∞

0
f (t)tω−1dt (1)

where ω ∈ C, ω is complex, such that a1 < <(ω) < a2. The
Mellin transform (1) exists if the function f (t) is piecewise con-
tinuous in every closed interval [a, b] ⊂ (0,∞) and∫ 1

0
| f (t) | ta1−1dt < ∞,

∫
∞

1
| f (t) | ta2−1dt < ∞ (2)

Definition 2 Let f (t) be integrable with fundamental strip (x, y).
If b is such that x < b < y and F(ω = b + it) = M( f (t); ω) is
integrable, the inverse Mellin transform is defined as

f (t) =
1

2πi

∫ b+i∞
b−i∞

F(ω)t−ωdω, t ∈ (0,∞) (3)

The integration contour in (3) is called the Bromwich contour.
The fundamental properties of the Mellin transform are pre-
sented in the following results.

Theorem 1. Let f (t) be Mellin transformable function defined
on R+. If differentiation under integral sign is permitted, then

(a) M(td f (t); ω) = F(ω + d)
(b) M( f (t) ∗ g(t); ω) = F(ω)G(1 −ω)
(c) M(tλ

∫
∞

0
τµ f (τt)g(τ)dτ; ω) = F(ω + λ)G(1 −ω−λ + µ)

(d) M( f n(t); ω) = Γ(1−ω+n)
Γ(1−ω) F(ω− n)

(e) M(tn f n(t); ω) = (−ω)n F(ω)
(f)

(
d
dt

)n
M( f (t); ω) =

∫
∞

0
f (t)(log t)ntω−1

= M( f (t)(log t)n; ω− 1)
(g) M

(∫ t
0

f (τ)dτ; ω
)

= − 1
ω

F(ω + 1)

For more details see Refs. [3, 10, 11], just to mention a few.
Definition 3 A real function f (t), t > 0 is said to be in space
Cµ, µ ∈ R if there exists a real number ρ > µ, such that
f (t) = tρ f1(t) where f (t) ∈ C(0, 1), and it is said to be in space
Cnµ if f

n ∈ Cµ, n ∈ N
Definition 4 The Riemann-Liouville fractional derivative op-
erator of order α > 0 of a function f (t) is defined as

0 D
α
t f (t) =

1
Γ(n −α)

∫ t
0

(t −τ)n−α−1 f (τ)dτ,

α ∈ (n − 1, n) (4)

Definition 5 The Riemann-Liouville fractional integral oper-
ator of order α ≥ 0 of a function f ∈ Cµ, µ ≥ 1 is defined
as

Jα f (t) = 0 D
−α
t f (t)

=
1

Γ(α)

∫ t
0

(t −τ)α−1 f (τ)dτ,α > 0,τ > 0 (5)

The following result gives some of the properties of the Riemann-
Liouville fractional operator.

Theorem 2. If α,β ∈ R+, then

(a) 0 Dαt [0 D
−β
t f (t)] =0 D

α−β
t f (t)

(b) JαJβ f (t) = Jα+β f (t)
(c) JαJβ f (t) = JβJα f (t)
(d) J0 f (t) = f (t)

Definition 6 The Caputo fractional derivative of the function
f ∈ Cn

−1, n ∈ N is defined as

c
0 D

α
t f (t) =

1
Γ(n −α)

∫ t
0

(t −τ)n−α−1 f n(τ)dτ,

α ∈ (n − 1, n] (6)

The properties of the Caputo fractional derivative were summa-
rized in the following result.

Theorem 3. If α ∈ (n − 1, n], n ∈ N, f ∈ Cn
−1,µ > −1, then

(a) Jα[c0 D
α
t f (t)] = f (t) −

∑n−1
j=0 f

j(0)
(

t j
j!

)
(b) c0 D

α
t [J

α f (t)] = f (t)

Definition 7 The Miller-Ross sequential fractional derivative
is defined as

aDσnt ≡ aD
αn
t aD

αn−1
t ...aD

α1
t (7)

aDσn−1t ≡ aD
αn−1
t ...aD

α1
t (8)

where

σn =

n∑
j=1

α j, α j ∈ (0, 1], ( j = 1, 2, ..., n) (9)

3. Mellin transform of fractional operators of different flavours

This section presents the main results generated for the Mellin
transform of fractional operators of different flavours.

3.1. Mellin transform of the Riemann-Liouville fractional deriva-
tive and integral operators

The Mellin transform of the Riemann-Liouville fractional
derivative and integral operators are given in the following re-
sults.

Theorem 4. Let f (t) be Mellin transformable function, where
0 ≤ n − 1 < α < n, t > 0, then the Mellin transform of the
Riemann-Liouville fractional derivative is obtained as

M(0 D
α
t f (t); ω) =

Γ(1 −ω + α)
Γ(1 −ω)

F(ω−α) (10)

Proof: Take 0 ≤ n−1 < α < n, t > 0, where α is the order. From
the definition of the Riemann-Liouville fractional derivative in
(4), one can write that

0 D
α
t f (t) =

1
Γ(n −α)

∫ t
0

(t −τ)n−α−1 f (τ)dτ

=
dn

dtn 0
D−(n−α)t f (t) (11)

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Fadugba / J. Nig. Soc. Phys. Sci. 1 (2019) 138–142 140

Taking the Mellin transform of (11) yields

M(0 D
α
t f (t); ω) =

∫
∞

0

dn

dtn
h(t)tω−1dt

=

∫
∞

0
hn(t)tω−1dt (12)

with

h(t) = 0 D
−(n−α)
t f (t) (13)

By means of the derivative property of the Mellin transform and
simplifying further, (12) becomes

M(0 D
α
t f (t); ω) =

n−1∑
j=0

Γ(1 −ω + j)
Γ(1 −ω)

[
0 D

α− j−1
t f (t)t

ω− j−1
]∞
0

+
Γ(1 −ω + α)

Γ(1 −ω)
F(ω−α) (14)

Equation (14) can be written in a simplified form as

M(0 D
α
t f (t); ω) =

[
0 D

α−1
t f (t)t

ω−1
]∞
0

+
Γ(1 −ω + j)

Γ(1 −ω)
F(ω−α) (15)

for α ∈ (0, 1). Since the first term in the RHS of (15) becomes
zero, (10) is established. This completes the proof.

Theorem 5. Let f (t) be Mellin transformable function, where
0 ≤ n − 1 < α < n, t > 0, then the Mellin transform of the
Riemann-Liouville fractional integral is obtained as

M(Jα f (t); ω) = M(0 D
−α
t f (t); ω) =

Γ(1 −ω−α)
Γ(1 −ω)

F(ω + α) (16)

Proof: By means of variable transformation τ = tρ, dτ = tdρ,
the Riemann-Liouville fractional integral operator given by (5)
as

Jα f (t) =
1

Γ(α)

∫ t
0

(t −τ)α−1 f (τ)dτ,α > 0,τ > 0

yields

Jα f (t) =
tρ

Γ(α)

∫
∞

0
f (tρ)g(ρ)dρ (17)

where

g(t) =
{

(1 − t)α−1, 0 ≤ t < 1
0, t ≥ 1

(18)

The Mellin transform of (18) is the Euler Beta function of the
form

M(g(t); ω) = B(α,ω) =
Γ(α)Γ(ω)
Γ(α + ω)

(19)

By means of (17) and (19), the Mellin transform of the Riemann-
Liouville fractional integral operator is obtained as

M(Jα f (t); ω) =
1

Γ(α)
F(ω + α)B(α, 1 −ω−α) (20)

with

B(α, 1 −ω−α) =
Γ(α)Γ(1 −ω−α)

Γ(1 −ω)
(21)

Hence, (16) is established.

3.2. Mellin transform of the Caputo fractional derivatives

The following result gives the Mellin transform of the Ca-
puto fractional derivatives.

Theorem 6. Let f (t) be Mellin transformable function, where
0 ≤ n − 1 < α ≤ n, t > 0 and α is the order, then the Mellin
transform of the Caputo fractional derivatives is given by

M(c0 D
α
t f (t); ω) =

Γ(1 −ω + α)
Γ(1 −ω)

F(ω−α) (22)

Proof: The Caputo fractional derivative given by (6) can be
written as

c
0 D

α
t f (t) = J

n−αDnt f (t)

=
1

Γ(n −α)

∫ t
0

(t −τ)n−α−1 f n(τ)dτ (23)

Let

L(t) = f n(t) (24)

Taking the Mellin transform of (23) yields

M(c0 D
α
t f (t); ω) = M(J

n−αDnt f (t))

=
Γ(1 −ω− n + α)

Γ(1 −ω)
L(ω + n −α) (25)

where

L(ω + n −α) =
n−1∑
j=0

Γ(1 −ω− n + α + j)
Γ(1 −ω− n + α)[

f n− j−1(t)tw+n−α− j−1
]∞
0

+
Γ(1 −ω + α)

Γ(ω−α)
F(ω−α) (26)

Simplifying (25) further and for α ∈ (0, 1), yields

M(c0 D
α
t f (t); ω) =

Γ(α−ω)
Γ(1 −ω)

[
f (t)tω−α

]∞
0

+
Γ(1 −ω− n + α)

Γ(1 −ω)
F(ω−α) (27)

If the function f (t) and <(ω) are such that all substitutions of
the limit t = 0 and t = ∞ in (27) equals zero, then (22) is
established. Hence, this completes the proof.

3.3. Mellin transform of the the Miller-Ross sequential frac-
tional derivatives

The Mellin transform of the Miller-Ross sequential frac-
tional derivatives is given in the following result.

Theorem 7. By means of induction, the Mellin transform of the
Miller-Ross sequential fractional derivative is given by

M(0 D
σn
t f (t); ω) =

Γ(1 −ω−σn)
Γ(1 −ω)

F(ω−σn) (28)

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Fadugba / J. Nig. Soc. Phys. Sci. 1 (2019) 138–142 141

Proof: For n = 2, setting h(t) = 0 D
β
t f (t) and by means of the

property of the Mellin transform, one gets

M(0 D
σ2
t f (t); ω) =

[
0 D

α2
t h(t)t

ω−1
]∞
0

+
Γ(1 −ω + α2)

Γ(1 −ω)
H(ω−α2) (29)

Solving (29) further and rearranging terms yields

M(0 D
σ2
t f (t); ω) =

[
0 D

σ2−1
t f (t)t

ω−1
]∞
0

+
Γ(1 −ω + α2)

Γ(1 −ω)

[
0 D

σ1−1
t f (t)t

ω−α2−1
]∞
0

+
Γ(1 −ω + σ2)

Γ(1 −ω)
F(ω−σ2) (30)

In general, by means of induction, the Mellin transform of the
Miller-Ross sequential fractional derivative is obtained as

M(0 D
σn
t f (t); ω) =

n∑
i=1

Γ(1 −ω + σn −σi)
Γ(1 −ω)

×
[
0 D

σi−1
t f (t)t

ω−σn−σi−1
]∞
0

+
Γ(1 −ω + σn)

Γ(1 −ω)
F(ω−σn) (31)

Since the first term of (31) yields zero, therefore (31) reduces
to (28). This completes the proof.

4. Applications

This section presents three illustrative examples.

4.1. Application 1
Solve the following problem via the Mellin transform

t2 f 2(t) + t3 f 3(t) = δ(t − a) (32)

By applying the Mellin transform to both sides of (32), one
obtains

M((t2 f 2(t) + t3 f 3(t)); ω) = M(δ(t − a); ω) (33)

Using the property of the Mellin transform of the form

M(tα f α(t); ω) =
Γ(ω + α)

Γ(ω)
F(ω) (34)

and the fact that

M(δ(t − a); ω) = aω−1 (35)

Equation (34) becomes
Γ(ω + 2)

Γ(ω)
F(ω) +

Γ(ω + 3)
Γ(ω)

F(ω) = aω−1 (36)

Thus,

F(ω) =
aω−1Γ(ω)

Γ(ω + 2) + Γ(ω + 3)
(37)

Applying the inverse Mellin transform to (37) by using com-
plex inversion integral to cover the function f (t) explicitly as
the solution of (32)

f (t) =
1

2πi

∫ b+i∞
b−i∞

Γ(ω)
Γ(ω + 2) + Γ(ω + 3)

aω−1t−ωdω (38)

4.2. Application 2
Solve the following problem via the Mellin transform

t
3
2 f

3
2 (t) + t

5
2 f

5
2 (t) = (log t)αδ(t − a) (39)

Taking the Mellin transform of (39) yields

M((t
3
2 f

3
2 (t) + t

5
2 f

5
2 (t)); ω) = M((log t)αδ(t − a); ω) (40)

Once again, using (34) and the fact that

M((log t)αδ(t − a); ω) = aω−1(log a)α (41)

(40) becomes

Γ
(
ω + 32

)
Γ(ω)

F(ω) +
Γ
(
ω + 52

)
Γ(ω)

F(ω) = aω−1(log a)α (42)

Rearranging terms in (41), one gets

F(ω) =
aω−1(log a)αΓ(ω)

Γ
(
ω + 32

)
+ Γ

(
ω + 52

) (43)
Once again, applying the inverse Mellin transform to (43) by
using complex inversion integral to cover the function f (t) ex-
plicitly as the solution

f (t) =
1

2πi

∫ b+i∞
b−i∞

Γ(ω)

Γ
(
ω + 32

)
+ Γ

(
ω + 52

)
×aω−1(log a)αt−ωdω (44)

Equation (40) is the required explicit solution of (39).

4.3. Application 3
Consider the solution of the Caputo fractional derivative of

the form
1∑

k=0

tα+k c0 D
α+ky(t) = f (t) (45)

subject to the initial conditions

y(0) = y′(0) = 0, y(∞) = y′(∞) = 0 (46)

via the Mellin transform.
Applying the Mellin transform to both sides of (45), yields

Γ(1 −ω)(1 −ω−α)
Γ(1 −ω−α)

Y (ω) = F(ω) (47)

Therefore,

Γ(1 −ω−α)F(ω)
Γ(1 −ω)(1 −ω−α)

= Y (ω) (48)

Equation (48) can be written in a compact form as

F(ω)H(1 −ω) = Y (ω) (49)

where

H(1 −ω) =
Γ(1 −ω−α)

Γ(1 −ω)(1 −ω−α)
(50)

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Fadugba / J. Nig. Soc. Phys. Sci. 1 (2019) 138–142 142

In the same manner,

H(ω) =
Γ(ω−α)

Γ(ω)(ω−α)
(51)

Equation (51) can be split as follows

H(ω) = A(ω)B(ω) (52)

A(ω) =
Γ(ω−α)

Γ(ω)
(53)

and

B(ω) =
1

ω−α
(54)

respectively. The inverse Mellin transform of (53) and (54) are
obtained as

A(t) =
 t−α(1−t)α−1Γ(α) , t ∈ (0, 1)0, t > 1 (55)

and

B(t) =
{

t−α, t ∈ (0, 1)
0, t > 1

(56)

respectively. The inverse Mellin transform of (51) is obtained
as

h(t) =
∫
∞

0
A

( t
τ

)
B(τ)

dτ
τ

(57)

From (55) and (56), it follows that

h(t) =
∫ 1

t
A

( t
τ

)
B(τ)

dτ
τ
, t ∈ (0, 1) (58)

and that

h(t) = 0, t > 1 (59)

The function H(ω) has a double pole at ω = α and ordinary
poles at ω = α − n − 1, n ∈ Z+. By means of the residue
theorem of the form

1
2πi

∫
δω

f (ω)dω =
k∑

j=0

Res( f,ω j),ω ∈ C

one obtains

h(t) =
∫
Pτ

H(ω)t−ωdω

=
1

2πi

∫ b+i∞
b−i∞

H(ω)t−ωdω

=
∑

[Res H(ω)t−ω]ω=α

+

∞∑
n=0

[Res H(ω)t−ω]ω=α−n−1 (60)

Simplifying (60) further, yields

h(t) =
t−α

Γ(α)
(ln t + Ξ(α) − Φ)

+

∞∑
n=0

(−1)ntn−α+1

(n + 1)Γ(n + 2)Γ(α− n − 1)

(61)

where Φ is the logarithmic derivative of the gamma function
and the Euler constant Ξ = 0.577215. The solution of the Ca-
puto fractional differential equation (45) subject to (46) is ob-
tained

y(t) =
∫
∞

0
f (tτ)h(τ)dτ (62)

Equation (62) is known as the Mellin convolution of the func-
tions f (t) and h(t).

5. Conclusion

This paper presents the solutions of fractional order equa-
tions in the domain of the Mellin transform. The Mellin trans-
form of the Riemann-Liouville fractional derivative, Riemann-
Liouville fractional integral, Caputo fractional derivative and
the Miller-Ross sequential fractional derivative were obtained.
Moreover, three fractional order equations were solved via the
Mellin transform. Hence, the solutions obtained were recovered
by means of the Mellin transform inversion formula.

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