International Journal of Analysis and Applications

Volume 17, Number 3 (2019), 396-405

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-17-2019-396

EXISTENCE OF TIME-SCALE CLASS OF THREE DIMENSIONAL FRACTIONAL

DIFFERENTIAL EQUATIONS

RABHA W. IBRAHIM1,∗, AND MASLINA DARUS2

1IEEE:94086547

2Centre of Modelling and Data Science, Faculty of Sciences and Technology,

Universiti Kebangsaan Malaysia,43600 Bangi, Selangor, Malaysia; maslina@ukm.edu.my

∗Corresponding author: rabhaibrahim@yahoo.com

Abstract. The holomorphic results for fractional differential operator formals have been established. The

analytic continuation of these outcomes has been studied for the fractional differential formal




∂αυ(℘,z)
∂℘α

= H(℘,z,υ, ∂υ
∂z
, ∂

2υ
∂z2

), α ∈ [0, 1)

υ(a,z) = ψ(z), in a proximity to z ∈ U,

where U is the open unit disk. The benefit of such a problem is that a generalization of two significant

problems: the Cauchy problem and the diffusion problem. Moreover, the analytic solution is given inside the

open unit disk, this leads to discuss the solution geometrically. The upper bound of outcomes is determined

by suggesting a majorant analytic function in U (for two functions characterized by a power series, a majorant

is the summation of a power series with positive coefficients which are not less than the absolute values of

the conforming coefficients of the assumed series). This technique is very useful in approximation theory.

Received 2019-01-13; accepted 2019-02-25; published 2019-05-01.

2010 Mathematics Subject Classification. 34A12,45C30.

Key words and phrases. fractional calculus; fractional differential equation; fractional operator.

c©2019 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

396

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-17-2019-396


Int. J. Anal. Appl. 17 (3) (2019) 397

1. Introduction

Time scales calculus [1] cards us to teaching the dynamic equations, which contains both differences and

differential equations, both of which are substantial in understanding applications. The dynamical behavior

of different classes of fractional operating formals on time scales is presently experiencing active studies.

Several authors considered the existence and uniqueness solutions for problems involving classical fractional

derivative (see [2]- [10]). Holomorphic solution for some complex fractional classes is given in [5]- [7]. In this

work, we use a majorant technique of analytic functions to prove the convergent of outcomes. We generalize

some properties by applying the concept of classic fractional derivative formal operator.

Our construction is furnished by the Riemann-Liouville fractional operators.

Definition 1. The Riemann-Liouville fractional integral formal of the function φ of arbitrary order α > 0

is given by

Iαa φ(℘) =

∫ ℘
a

(℘− τ)α−1

Γ(α)
φ(τ)dτ.

Definition 2. The Riemann-Liouville fractional differential formal of the function φ of arbitrary order

α ∈ [0, 1) is given by

Dαaφ(℘) =
d

d℘

∫ ℘
a

(℘− τ)−α

Γ(1 −α)
φ(τ)dτ =

d

d℘
I1−αa φ(℘).

Definition 3. [8] The majorant formula is given by : σ(χ) =
∑
σiχ

i and Λ(χ) =
∑

Λiχ
i, then σ(χ) � Λ(χ)

if and only if |σi| ≤ |Λi| for each i. Similarly, if

ρ(℘,χ) =
∑

ρik(℘−ε)iχk

and

Θ(t,χ) =
∑

Θik(℘−ε)iχk,

then ρ(℘,χ) �ε Θ(℘,χ) if and only if |ρik| ≤ Θik for all i and k.

Define the family of majorant functions: for each k ∈ N, we set

Ξ(k)ν (z) =

∞∑
n=0

zn

ν(n + 1)k+2
, (|z| < 1,ν ≥ 1). (1.1)

Clearly that for every k ∈ N,ν ≥ 1, the functional Ξ(k)ν converges for all values |z| < 1. Further, this

functional has some significant majorant correlations as follows:



Int. J. Anal. Appl. 17 (3) (2019) 398

Proposition 1. The following inequalities hold.

(i) Ξ(0)ν (z)Ξ
(0)
ν (z) � Ξ

(0)
ν (z);

(ii) Ξ(0)ν (z) � Ξ
(1)
ν (z) � Ξ

(2)
ν (z) �··· ;

(iii)
1

2i+2
Ξ(i−1)ν (z) �

d

dz
Ξ(k)ν (z) � Ξ

(k−1)
ν (z) and

1

2k+2
Ξ(k−2)ν (z) �

d2

dz2
Ξ(k)ν (z) � Ξ

(k−2)
ν (z);

(iv) Ξ(k)ν (z)Ξ
(k)
ν (z) · · ·Ξ

(k)
ν (z) � Ξ

(k)
ν (z);

(v)
1

1 −εz
Ξ(k)ν (z) � Ci,εΞ

(k)
ν (z),

(ε ∈ (0, 1), and Ck,ε ∈ (0,∞));

(vi)
Ξ

(k−2)
ν (z)

(2ν)3(k+2)
� Dαa Ξ

(k)
ν (z),

for sufficient large ν ≥ 1.

Proof. By employing the formula expansion of Ξ
(k)
ν (z), inequalities (i) and (ii) are achieved. According to

the following inequalities:

1

ν2k+2(n + 1)k+1
=

n + 1

ν2k+2(n + 1)k+2

=
n + 1

ν(2n + 2)k+2

≤
n + 1

ν(n + 2)k+2

and

n + 1

ν(n + 2)k+2
<

n + 2

ν(n + 2)k+2

=
1

ν(n + 2)k+1

≤
1

ν(n + 1)k+1
,

we get (iii). Similarly for (iv). To show (v), by arbitrary choice of ε, we assume that

εn ≤
Ck,ε

ν(n + 1)k+2

which leads to

1

1 −εz
=

∞∑
n=0

εnzn � Ci,εΞ(k)ν (z).

This implies that for all n

1

1 −εz
Ξ(k)ν (z) � Ci,εΞ

(k)
ν (z).



Int. J. Anal. Appl. 17 (3) (2019) 399

Finally, by using the relation

Dαa Ξ
(k)
ν (z) =

∞∑
n=0

Γ(n + 1)

Γ(n + 1 −α)(n + 1)k+2
zn−α, (|z| < 1)

we get inequality (vi) for sufficient large ν ≥ 1. �

In the same manner of Proposition 1, we have the following result:

Proposition 2. If φ(z) is holomorphic in a proximity of |z| ≤ r0, then φ(z) is majorized by

φ(z) �
M

1 − ( z
r0

)2
�

M

1 − (εz
r

)2
× Ξ(k)ν (

z

r
) � MCk,εΞ(k)ν (

z

r
),

for any 0 < r < εr0.

2. Fractional operator formal

Assume that H(℘,z,υ,v,w), ℘ ∈ J = [a,T] is a holomorphic function in a proximity of the four dim.

point (a,b,c,d,e) ∈ J×C4, and suppose that ψ(z) is a holomorphic function in a proximity of z = b achieving

ψ(b) = c,
∂ψ

∂z
(b) = d and

∂2ψ

∂z2
(b) = e.

Consider the initial value problem




∂αυ(℘,z)
∂℘α

= H(℘,z,υ, ∂υ
∂z
, ∂

2υ
∂z2

),

υ(a,z) = ψ(z), in a proximity of z = b.

(2.1)

(
α ∈ [0, 1), z ∈ U, ℘ ∈ J

)
Eq.(2.1) has Cauchy problem when

H(℘,z,υ,
∂υ

∂z
,
∂2υ

∂z2
) ≡ Θ(℘,z,υ,

∂υ

∂z
)

and diffusion problem of fractional order when

H(℘,z,υ,
∂υ

∂z
,
∂2υ

∂z2
) ≡ Θ(℘,z,υ,

∂2υ

∂z2
).

Theorem 3. Consider the initial value problem (2.1), then it has a unique holomorphic outcome (υ(℘,z)),

in a proximity of (a,b) ∈ J ×C.



Int. J. Anal. Appl. 17 (3) (2019) 400

Proof. Move the point (a,b) into the origin (0, 0) and change the variable as follows:

ϕ(℘,z) = υ(℘,z) −ψ(z),

where ϕ(℘,z) is the new variable, then we get




∂αϕ(t,z)
∂℘α

= Θ(℘,z,ϕ, ∂ϕ
∂z
, ∂

2ϕ
∂z2

),

ϕ(0, 0) = 0, in a proximity of z = 0.

(2.2)

Here, the functional Θ(℘,z,ϕ, ∂ϕ
∂z
, ∂

2ϕ
∂z2

) is holomorphic in a proximity of the origin in I ×C4, ℘ ∈ I = [0, 1].

Therefore, it is sufficient to consider (2.2). Let the above equation has a unique outcome:

ϕ(℘,z) =

∞∑
k=0

ϕk(z) ℘
k, (℘ ∈ I).

We show that ϕ(℘,z) converges.

Let r0 > 0 and ρ > 0 be small enough and the function Θ(℘,z,ϕ,v,w) be holomorphic in a proximity of the

set

S =
{

(℘,z,ϕ,v,w) ∈ I ×C4; ℘ < τ ≤ 1, |z| ≤ r0, |ϕ| ≤ ρ, |v| ≤ ρ and |w| ≤ ρ
}
.

Assume that Θ is bounded by M in this domain. Since Θ is holomorphic, then it has the following construc-

tion:

Θ(℘,z,ϕ,v,w) =
∑
p,q,s,l

ap,q,s,l(z)℘
pϕqvswl,

(
℘ ∈ I, (ϕ,v,w) ∈ (C×C×C)

)
.

In virtue of the Cauchy’s inequality and the certainty that the coefficient ap,q,s(z) is holomorphic in a

proximity of {z ∈ C; |z| ≤ r0}, implies that

ap,q,s,l(z) �
M

τpρq+s+l
1

1 − ( z
r0

)2
. (2.3)

In this case, the problem turns to evaluate a function ϑ(℘,z) satisfying the majorant inequalities



Int. J. Anal. Appl. 17 (3) (2019) 401




∂αϑ(℘,z)
∂℘α

�
∑
p,q,s,l

M
τpρq+s+l

1
1−( z

r0
)2
℘pϑq(∂ϑ

∂z
)s(∂

2ϑ
∂z2

)l, ℘ ∈ I

ϑ(0, 0) � 0,

(2.4)

then the function ϑ(℘,z) majorizes the formal solution ϕ(℘,z). Assume 0 < r < r0 and define

ϑ(℘,z) = LΞ(2)ν

(
℘ + (

z

r
)2
)
, (L > 0 ). (2.5)

Operating by the fractional differential formal with respect to ℘ we get

∂αϑ(℘,z)

∂℘α
= L

∂αΞ
(2)
ν

(
℘ + (z

r
)2
)

∂℘α
, (L > 0 ). (2.6)

Then by Proposition 1 (vi) we get

∂αϑ(℘,z)

∂℘α
�

L

Cν
Ξ(0)ν

(
℘ + (

z

r
)2
)
, (2.7)

where Cν := (2ν)
12. For a constant K0 > 0 again in virtue of Proposition 1 (ii) and (iii) we find

∑
p,q,s,l

M

ρq+s+l
1

1 − ( z
r0

)2
1

1 − ℘
τ

ϑq(
∂ϑ

∂z
)s(

∂2ϑ

∂z2
)l

�
∑
p,q,s

M

ρq+s
1

1 − ( z
r0

)2 − ℘
τ

{LΞ(2)ν
(
℘ + (

z

r
)2
)
}q

×{
2r0L

r2
Ξ(0)ν

(
℘ + (

z

r
)2
)
}s{

2r0L

r2
Ξ(0)ν

(
℘ + (

z

r
)2
)
}l

�
∑
p,q,s

M

ρq+s
1

1 − ( z
r0

)2 − ℘
τ

{LΞ(0)ν
(
℘ + (

z

r
)2
)
}q

×{
2r0L

r2
Ξ(0)ν

(
℘ + (

z

r
)2
)
}s{

2r0L

r2
Ξ(0)ν

(
℘ + (

z

r
)2
)
}l

�
MK0

1 −L/ρ− 4Lr0/ρr2
Ξ(0)ν

(
℘ + (

z

r
)2
)
,

(2.8)

whenever,

L

ρ
+

4r0L

r2
< 1.

Comparing (2.7) and (2.8) with the inequality

L

Cµ
≥

MK0
1 −L/ρ− 4Lr0/ρr2

(2.9)

then we obtain the majorant inequalities in (2.4) are achieved.

Note that relation (2.9) holds by choosing a sufficiently small L,r0, such that

L

ρ
+

4r0L

r2
< 1.



Int. J. Anal. Appl. 17 (3) (2019) 402

Hence, ϑ(℘,z) in (2.5) majorizes the formal solution ψ(℘,z). This now implies that ϕ(℘,z) converges in a

domain containing {(℘,z) ∈ I ×C; |℘ + (z
r
)2| < 1}. �

3. Continuation outcomes

Suppose that Ω is a proximate of the origin (0, 0) and H(℘,z,υ,v,w), ℘ ∈ I, is a holomorphic function

in Ω ×Cυ ×Cv ×Cw. Consider the following equation:

∂αυ

∂℘α
= H(℘,z,υ,

∂υ

∂z
,
∂2υ

∂z2
). (3.1)

Then we have

H(℘,z,υ,v) =
∑
j,p,q

aj,p,q(℘,z)υ
jvpwq. (3.2)

Define the following two sets:

S0 = {(j,p,q) ∈ N3; aj,p,q(℘,z) 6= 0}

and

S = {(j,p,q) ∈ S0; j + p + q ≥ 3}.

Clearly, H is linear if and only if S = ∅; and it is nonlinear otherwise.

Suppose that H is nonlinear, this implies that S is nonempty with the coefficients formal

aj,p,q(℘,z) = ℘
kj,p,qbj,p,q(℘,z), (3.3)

where kj,p,q is a non-negative integer and bj,p,q(0,z) = 0. Applying (3.1), we get

∂αυ

∂℘α
=
∑
j,p,q

℘kj,p,qbj,p,q(℘,z)υ
j(
∂υ

∂z
)p(

∂2υ

∂z2
)q. (3.4)

For κ ∈ R, we have

δ(κ) := inf
(j,p,q)∈S

(
kj,p,q + 1 + κ(j + p + q − 1)

)
. (3.5)

It is clear that, when κ = 0, we get δ(κ) ≥ 1. Also, if

κ > sup
(j,p,q)∈S

∣∣∣∣ (kj,p,q + 1)j + p + q − 1
∣∣∣∣ ,

then δ(κ) is positive.

Next, we show that υ(℘,z) = O(℘κ) is analytically continued up to a proximate of the origin.



Int. J. Anal. Appl. 17 (3) (2019) 403

Theorem 4. Let υ(℘,z) be a holomorphic solution of (3.4) in Ω. Then for some κ ∈ R achieving δ(κ) > 0,

such that

sup
z∈C
|υ(℘,z)| = O(℘κ), (℘ → 0),

and the solution υ(℘,z) can be expanded analytically as a holomorphic outcome of Eq. (3.4) up to a approx-

imate of the origin.

Proof. Suppose that υ(℘,z) is a solution for Eq. (3.4), which is holomorphic in the domain Ω. In addition,

we let the expansion (3.2) is valid in the domain ∆ where

∆ := {(℘,z,υ,v,w) : ℘ ≤ 2τ, |z| ≤ 2r, |υ| ≤ ρ, |v| ≤ ρ, |w| ≤ ρ},

such that τ < 1, 2r < 1 and ρ is positive number. Let M be a bound of F in ∆.

Now we suggest the following initial value problem in

χ(℘,z) :=

∞∑
k=0

χk(z)(℘−ε)k :




∂αχ(℘,z)
∂℘α

=
∑
j,p,q ℘

kj,p,qbj,p,q(℘,z)χ
j(∂χ
∂z

)p(∂
2χ
∂z2

)q, ℘ ∈ I

χ(ε,z) = υ(ε,z).

(3.6)

We aim to show that the formal χ(℘,z) converges in a domain including the origin. This leads to υ(℘,z)

can be continued analytically by χ(℘,z) up to some neighborhood of the origin.

But υ(℘,z) = O(℘κ) as ℘ → 0, this implies that there exists a positive constant C0 > 0 such that |u(ε,z)| ≤

C0ε
κ uniformly in z. Thus, by Proposition 2, for some positive constant C1 > 0, we get

υ(ε,z) � C0εκC1Ξ(2)ν (
℘−ε
cτ

+ (
z

r
)2). (3.7)

Assume that 0 < κ < 1 (without lose generality). To formulate an inequality satisfying the majorant

function, we majorize the expression ℘kj,pbj,p(℘,z) by using Ξ
(0)
ν (z). Let

Λ := (
z

r
)2 +

℘−ε
cτ

,

then ℘ is majorized by

℘ = ε + (℘−ε) �ε
(
ε + 4cτ

)(
1 +

℘−ε
4cτ

)
�ε

(
ε + 4cτ

)
Ξ(0)ν (Λ).

(3.8)

We proceed to extend the function bj,p,q(℘,z) as follows



Int. J. Anal. Appl. 17 (3) (2019) 404

bj,p,q(℘,z) =

∞∑
m=0

b
(m)
j,p,q(z)℘

m,

where each b
(m)
j,p,q is holomorphic in some domain of {|z| ≤ 2r} and achieves

|b(m)j,p,q(z)| ≤
M

ρj+p+q(2τ)m+kj,p,q
.

This estimate poses,

b
(m)
j,p,q(z) �

C0C1Ξ
(0)
ν (Λ)

ρj+p+q(2τ)m+kj,p,q
(3.9)

where C1 is a positive constant satisfying (3.7). Joining relations (3.8) and (3.9) and using Proposition 1 (i),

we get

℘kj,p,qbj,p,q(℘,z) �ε
∞∑
m=0

[
(ε + 4cτ)Ξ(0)ν (Λ)

]m+kj,p,q[ C0C1Ξ(0)ν (Λ)
ρj+p+q(2τ)m+kj,p,q

]
�ε

C0C1
ρj+p+q

Ξ(0)ν (Λ)

∞∑
m=0

(4c)m+kj,p,q.

(3.10)

Putting ε = cτ
2

and 0 < c ≤ 1 and fixing r so that cr < 1/4, 0 < c ≤ 1, we finally get the formal

℘kj,p,qbj,p,q(℘,z) �ε
2C0C1
ρj+p+q

Ξ(0)ν (Λ).

Therefore, the function W(℘,z) achieves the majorant conclusion




∂αW
∂℘α

�ε
∑
j,p,q

2C0C1
ρj+p+q

Ξ
(0)
ν (Λ)W

j(∂W
∂Λ

)p(∂
2W
∂Λ2

)q, ℘ ∈ I

W(ε,z) �ε εκC0C1 Ξ
(2)
ν (Λ)

(3.11)

is one majorant function for the formal solution χ(℘,z). In the similar manner of the proof of Theorem 3 by

choosing suitable values for ρ > 0, c > 0, and letting ε = cτ
2
, we have

W(℘,z) = εκC0C1Ξ
(2)
ν (Λ)

achieves the majorant formal given in (3.11). Thus, W(℘,z) is holomorphic in a domain involving (0,0);

consequently must be true for χ(℘,z). �

Acknowledgments

The authors would like to express their thanks to the reviewers for their important and useful comments

to improve the paper. The work here is partially supported by the Universiti Kebangsaan Malaysia grant:

GUP (Geran Universiti Penyelidikan)-2017-064.



Int. J. Anal. Appl. 17 (3) (2019) 405

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	1. Introduction
	2. Fractional operator formal
	3.  Continuation outcomes
	Acknowledgments
	References