EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6238
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Finite Rank Solution for Conformable Second-Order
Abstract Cauchy Problem in Hilbert Space

Huda Odetallah1,∗, Mayada Abualhomos1, Tala Sasa1, Lubaba Shaikh1,
Omniya Miri2

1 Department of Mathematics, Applied Science Private University, Amman 11931, Jordan
2 Department of Basic Science, Deanship of Preparatory Year and Supporting Studies,
Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam 34212, Saudi Arabia

Abstract. This paper presents a comprehensive analytical framework for constructing finite-
rank solution to second-order conformable fractional abstract Cauchy problem. We examine the
mathematical structure:

Eu(2α)(t) +Au(α)(t) +Bu(t) = f(t)

subject to prescribed initial conditions u(0) = u0 and u(α)(0) = u
(α)
0 , where A, B, and E

represent closed linear operators acting on a Banach space X, f : [0,∞) → X is continuous, and
u is continuously differentiable on [0,∞). Our analytical methodology exploits tensor product
decomposition techniques to transform the problem into finite-dimensional systems. This work
proves solution existence and uniqueness under specific conditions, and provides computational
methods for many types of this problem.

2020 Mathematics Subject Classifications: 34G10, 26A33, 34A08, 46M05
Key Words and Phrases: Abstract Cauchy problem, Conformable fractional derivative, Ten-
sor product of Banach spaces, Finite-rank function

1. Introduction

For a Banach space X and I = [0, 1] or [0,∞), C(I) is the Banach space of all
real-valued continuous functions defined on I with the supremum norm, C(I,X) is the
space of all continuous functions defined on I taking values in X, and C(2α) (I,X) is the
space of functions on I taking values in X with continuous conformable derivatives up
to order 2α.

The abstract Cauchy problem represents one of the most fundamental classes of dif-
ferential equations in applied mathematics, with applications spanning from heat con-
duction and wave propagation to population dynamics and financial modeling. The

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6238

Email addresses: h odetallah@asu.edu.jo (H. Odetallah),
abuhomos@asu.edu.jo (M. Abualhomos), t sasa@asu.edu.jo (T. Sasa),
l shaikh@asu.edu.jo (L. Shaikh), ormiri@iau.edu.sa (O. Miri)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



H. Odetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6238 2 of 11

general form of the second-order conformable fractional abstract Cauchy problem under
investigation is:

Eu(2α)(t) +Au(α)(t) +Bu(t) = f(t)z (1)

u(0) = u0

u(α)(0) = u
(α)
0

where A, B and E are closed linear operators on a Banach space X, u ∈ C(2α)(I,X)

is the unknown function, f ∈ C(I) and z, u0, u
(α)
0 ∈ X.

If f = 0 or z = 0, then the equation is homogeneous otherwise it is called non-
homogeneous. If f ̸= 0 and z ̸= 0, then we have two cases. If f is given, then the
problem is called a direct problem, otherwise the problem is called an inverse problem.

In this paper, we find a finite-rank solution of the second-order fractional type of the
abstract Cauchy problem with some conditions on A, B and E.

Among various fractional derivative definitions available in the literature [12,15], this
paper employs the conformable fractional derivative introduced by Khalil et al. [10]
due to its advantageous properties.

Definition 1. Let f : [0,∞) → R be a function. The conformable fractional derivative
of f of order α, where 0 < α ≤ 1 is defined by:

f (α)(t) = lim
ϵ→0

f(t+ ϵt1−α)− f(t)

ϵ

for all t > 0. If f is α−differentiable on (0, c) and limt→0+ f (α)(t) exists, then we
define f (α)(0) = limt→0+ f (α)(t).

The power of this definition is that it satisfies the most of the properties of the usual
derivatives such as product rule, quotient rule and chain rule, etc.

To read more about the conformable fractional derivatives see [1,2,9].

First of all, let us define what the tensor product and the finite-rank function are.

Definition 2. Let X and Y be Banach spaces, and T ∈ X∗ (the dual of X). For x ∈ X
and y ∈ Y, the tensor product of x and y is the map x⊗y : X∗ → Y as x⊗y(T ) = T (x)y
for all T ∈ X∗.

The operator x ⊗ y is bounded and linear with ∥x⊗ y∥ = ∥x∥ ∥y∥ (see [13]). Such
operators are called atoms, and every atom has rank 1.

The span of all atoms forms a subspace of L(X∗, Y ), denoted by X ⊗ Y. A finite
sum of atoms: n

i=1xi ⊗ yi constitutes a finite-rank function, which forms the basis of our
solution approach.

There are many norms on X ⊗ Y, but the most important one is that called the
injective norm. Furthermore, for any T =n

i=1 xi ⊗ yi ∈ X ⊗ Y, the injective norm is
defined as

∥T∥∨ = sup {ni=1x
∗(xi).y

∗(yi) : x
∗ ∈ X∗, y∗ ∈ Y ∗, ∥x∗∥ = ∥y∗∥ = 1}



H. Odetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6238 3 of 11

It should be noted that the space (X ⊗ Y, ∥.∥∨) does not necessarily exhibit com-

pleteness. We denote by X
∨
⊗ Y the completion of (X ⊗ Y, ∥.∥∨) , which is referred to as

the completed injective tensor product of X with Y.
A fundamental result with significant applications in differential equation theory can

be formulated as follows:

Theorem 1. [16] For any compact hausdorff space K and any Banach space X, the

space C (K,X) is isometrically isomorphic to C (K)
∨
⊗X∗.

This theorem yields the important corollary that for any two compact metric spaces

I and J, we have C (I × J) ∼= C (I)
∨
⊗ C (J) .

For more on tensor product we refer to [6,7,13,16].

2. Direct Problem

Let u be an 2α−differentiable on I = [0, 1] into the Hilbert space X = ℓ2, where
ℓ2 = {(xn) :∞n=1 |xn|2 < ∞}. The standard basis of ℓ2 is denoted by {δ1, δ2, ...}. Let A,
B be two closed operators on ℓ2 such that domains of A and B contain the elements of
the standard basis of ℓ2.

In this section we look for a solution to the direct problem (1) among finite-rank

functions of the form u(t) =n
i=1 ui(t)δi, where u

(2α)
i (t) ∈ C(I), i = 1, 2, ..., n.

Before analyzing the main problem, we establish the theoretical foundation using
fundamental matrices.

Definition 3. The conformable fractional fundamental matrix ϕα (t) is the unique n×
n matrix-valued function that satisfies the fractional differential equation ϕ

(α)
α (t) =

Aϕα (t) , t > 0 and α ∈ (0, 1] with initial condition ϕα (0) = I, where I is the n × n
identity matrix.

For the conformable fractional derivative, the fundamental matrix can be expressed
as:

ϕα (t) = exp

(
Atα

α

)
The essential properties of the fundamental matrix are:

1- ϕα (t) is continuously differentiable in the conformable sense for t > 0.
2- ϕα (t) is continuous at t = 0.
3- ϕα (t) is invertible for all t ≥ 0.
4- The inverse satisfies ϕ−1

α (t) = ϕα (−t) .
5- d

dtϕα (t) =
1

t1−αAϕα (t) .
To read more about ϕα (t) see [10].
We now present the main theorem.



H. Odetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6238 4 of 11

Theorem 2. In problem (1), let E = I (the identity operator) and u(t) =n
i=1 ui(t)δi,

where u
(2α)
i (t) ∈ C(I), i = 1, 2, ..., n. Then, the problem has a unique solution.

Proof. Since u(t) =n
i=1 ui(t)δi, the derivatives are u

(α)(t) =n
i=1 u

(α)
i (t)δi, u

(2α)(t) =n
i=1

u
(2α)
i (t)δi. Substituting into problem (1):

n
i=1u

(2α)
i (t)δi +

n
i=1 u

(α)
i (t)A(δi) +

n
i=1 ui(t)B(δi) = f(t)z (2)

Taking the inner product with δj (2):

n
i=1u

(2α)
i (t) ⟨δi, δj⟩+n

i=1 u
(α)
i (t) ⟨A(δi), δj⟩+n

i=1 ui(t) ⟨B(δi), δj⟩ = f(t) ⟨z, δj⟩ (3)

Using orthonormality of the standard basis:

u
(2α)
j (t) +n

i=1 u
(α)
i (t) ⟨A(δi), δj⟩+n

i=1 ui(t) ⟨B(δi), δj⟩ = f(t) ⟨z, δj⟩ (4)

This yields a system of second-order conformable fractional differential equations.
Converting to first-order form by introducing new variables:

vi = ui, vn+i = u
(α)
i , i = 1, 2, ..., n

Then the system can be written as:

v
(α)
i = vn+i, i = 1, 2, ..., n

v
(α)
n+j(t) = −

n∑
i=1

vn+i(t) ⟨A(δi), δj⟩ −
n∑

i=1

vi(t) ⟨B(δi), δj⟩+ f(t) ⟨z, δj⟩ (5)

In matrix form:

V (α)(t) = ĢV (t) + F(t) (6)

where:

V =



v1
v2
...
vn
vn+1

vn+2
...

v2n


=



u1
u2
...
un

u
(α)
1

u
(α)
2
...

u
(α)
n


, Ģ=

(
0
−B

I
−A

)
, F(t) =



0
...
0

f(t) ⟨z, δ1⟩
...

f(t) ⟨z, δn⟩



Here, 0 is a zero matrix, I is the identity matrix, A and B are n × n coefficient
matrices:



H. Odetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6238 5 of 11

A = (aij)n×n , B =(bij)n×n

where aij = ⟨A(δi), δj⟩ , bij = ⟨B(δi), δj⟩ for all i, j = 1, 2, ..., n.
This system has a unique solution of the form:

V (t) = ϕα (t)V (0) + ϕα (t)

∫ t

0

ϕ−1
α (s)F (s)

s1−α
ds (7)

where ϕα (t) is the n × n conformable fundamental matrix, and V (0) is the initial

condition vector containing the initial value of u0 and u
(α)
0 .

Now, the following theorem takes a special case on the operators A and B.

Theorem 3. Consider problem (1) with E = I and u(t) =n
i=1 ui(t)δi, where u

(2α)
i (t) ∈

C(I), i = 1, 2, ..., n. If A(δi) = λiδi and B(δi) = βiδi, then the problem has a unique
solution.

Proof. We have u(t) =n
i=1 ui(t)δi, then u(α)(t) =n

i=1 u
(α)
i (t)δi, and u(2α)(t) =n

i=1

u
(2α)
i (t)δi. Thus

n
i=1u

(2α)
i (t)δi +

n
i=1 u

(α)
i (t)A(δi) +

n
i=1 ui(t)B(δi) = f(t)z (8)

If we take the inner product of δj with both sides of equation (8), then the equation
becomes

n
i=1u

(2α)
i (t) ⟨δi, δj⟩+n

i=1 u
(α)
i (t) ⟨A(δi), δj⟩+n

i=1 ui(t) ⟨B(δi), δj⟩ = f(t) ⟨z, δj⟩ (9)

But since the standard basis is orthonormal and A(δi) = λiδi, B(δi) = βiδi, then

u
(2α)
j (t) + λju

(α)
j (t) + βjuj(t) = f(t) ⟨z, δj⟩ (10)

Each equation represents a second-order linear conformable fractional differential
equation with constant coefficients. Given appropriate initial conditions uj (0) and

u
(α)
j (0) for all j = 1, 2, ..., n, each equation admits a unique solution.

Now, in the following theorem we take the case where E ̸= I.

Theorem 4. Consider problem (1) where En is orthogonally diagonalizable with An|ker(En)

invertible. If u(t) =n
i=1 ui(t)δi with u

(2α)
i (t) ∈ C(I) for i = 1, 2, ..., n, then problem (1)

has a unique solution.

Proof. Let {θ1, θ2, ..., θn} be an orthonormal basis such that the matrix represen-
tation of En with respect to this basis is D = diag (λ1, λ2, ..., λn) with λ1, λ2, ..., λn

corresponding eigenvalues. Now, if λi ̸= 0, for all i = 1, 2, ..., n, then problem (1) reduces
to:

u(2α)(t) + E−1
n Anu

(α)(t) + E−1
n Bnu(t) = E−1

n f(t)



H. Odetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6238 6 of 11

This has a unique solution by Theorem 3.
Suppose λi ̸= 0, for i = 1, 2, ..., r, and λi = 0, for i = r + 1, r + 2, ..., n. Let

u (t) =
∑n

i=1 vi (t) θi. The system becomes:

n∑
i=1

v
(2α)
i (t)En (θi) +

n∑
i=1

v
(α)
i (t)An (θi) +

n∑
i=1

vi (t)Bn (θi) = f (t) z (11)

Taking the inner product of θj with both sides of equation (11), we obtain

n∑
i=1

v
(2α)
i (t) ⟨En (θi) , θj⟩+

n∑
i=1

v
(α)
i (t) ⟨An (θi) , θj⟩+

n∑
i=1

vi (t) ⟨Bn (θi) , θj⟩ = f (t) ⟨z, θj⟩

(12)
Since θi is an eigenvector of En with corresponding eigenvalue λi for every i and

{θ1, θ2, ..., θn} is an orthonormal basis, we get

λjv
(2α)
j (t) +

n∑
i=1

v
(α)
i (t) ⟨An (θi) , θj⟩+

n∑
i=1

vi (t) ⟨Bn (θi) , θj⟩ = f (t) ⟨z, θj⟩ (13)

Introducing new variables to represents the first derivatives of vi :

wi = vi, wn+i = v
(α)
i , i = 1, 2, ..., n

w
(α)
i = wn+i, i = 1, 2, ..., n

Then the system can be written as:

λjw
(α)
n+j(t) = −

n∑
i=1

wn+i(t) ⟨An(θi), θj⟩ −
n∑

i=1

wi(t) ⟨Bn(θi), θj⟩+ f(t) ⟨z, θj⟩ (14)

So, we get the following system(
In

0

0

D̃

)
W(α)(t) =

(
0
−B

In

−A

)
W (t) + F(t) (15)

where W(t) =



w1

w2
...
wn

wn+1

wn+2
...

w2n


, D̃ =

(
D
0

0
0

)
, A =

(
G1

G3

G2

Ĝ

)
, B =

(
H1

H3

H2

H4

)

where Ĝ = An|ker(En) = [⟨An(θj), θi⟩]i,j=r+1,...,n.



H. Odetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6238 7 of 11

Now, multiplying (15) by

(
In

0
0
K

)
2n×2n

where K =

(
D−1

0

0

Ĝ−1

)
, we get

(
In

0

0

KD̃

)
W(α)(t) =

(
0

−KB
In

−KA

)
W(t) +

(
In
0

0
K

)
F(t) (16)

whereKD̃ =

(
Ir

0
0
0

)
, KA =

(
D−1G1

Ĝ−1G3

D−1G2

In−r

)
andKB =

(
D−1H1

Ĝ−1H3

D−1H2

Â−1H4

)

Let U1 =


w1

w2
...
wr

 , U2 =


wr+1

wr+2
...
wn

 , U3 =


wn+1

wn+2
...

wn+r

, U4 =


wn+r+1

wn+r+2
...

w2n

, F1(t) =

f (t)


⟨z, θ1⟩
⟨z, θ2⟩

...
⟨z, θr⟩

 and F2 (t) = f (t)


⟨z, θr+1⟩
⟨z, θr+2⟩

...
⟨z, θn⟩

 . Then, we have

U
(α)
1 (t) = U3 (t) (17)

U
(α)
2 (t) = U4 (t) (18)

U
(α)
3 (t) = −D−1H1U1(t)−D−1H2U2(t)−D−1G1U3(t)−D−1G2U4(t) +D−1F1(t) (19)

and

0 = −Ĝ−1H3U1(t)− Ĝ−1H4U2(t)− Ĝ−1G3U3(t)− U4(t) + Ĝ−1F2(t) (20)

From the last equation, we get

U4(t) = −Ĝ−1H3U1(t)− Ĝ−1H4U2(t)− Ĝ−1G3U3(t) + Ĝ−1F2(t) (21)

Substituting (21) in (18) and (19), we get

U
(α)
2 (t) = −Ĝ−1H3U1(t)− Ĝ−1H4U2(t)− Ĝ−1G3U3(t) + Ĝ−1F2(t) (22)

U
(α)
3 (t) = D−1

(
G2Ĝ

−1H3 −H1

)
U1(t) +D−1

(
G2Ĝ

−1H4 −H2

)
U2(t) +D−1

(
G2Ĝ

−1G3 −G1

)
U3(t)(23)

+D−1F1(t)−D−1G2Ĝ
−1F2 (t)

If we define a combined state vector U (t) =

 U1 (t)
U2 (t)
U3 (t)

 that includes all variables

except U4 (which has been eliminated), then the merged system can be written as:

U (α) (t) = MU (t) +G (t) (24)



H. Odetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6238 8 of 11

whereM =

 0

−Ĝ−1H3

D−1
(
G2Ĝ

−1H3 −H1

) 0

−Ĝ−1H4

D−1
(
G2Ĝ

−1H4 −H2

) Ir
−Ĝ−1G3

D−1
(
G2Ĝ

−1G3 −G1

)


and

G (t) =

 0
0

D−1

F1 (t) +

 0

Ĝ−1

−D−1G2Ĝ
−1

F2 (t) .

Thus, the system (24) has a unique solution and then the problem (1) has a unique
solution as required.

3. Inverse Problem Case

In this section, we consider the inverse problem where both solution u (t) and function
f (t) have finite-rank representations: u (t) =

∑n
i=1 ui (t) δi, f (t) =

∑n
i=1 fi (t) δi, with

u
(2α)
i , fi ∈ C (I) for i = 1, 2, ..., n.

Theorem 5. Consider problem (1) with u (t) =
∑n

i=1 ui (t) δi and f (t) =
∑n

i=1 fi (t) δi,

where u
(2α)
i , fi ∈ C (I) for i = 1, 2, ..., n. If the following conditions hold:

1) There exists x ∈ ℓ2 such that ⟨ui (t) δi, x⟩ = hi (t) where h
(2α)
i ∈ C (I) and ⟨δi, x⟩ ≠

0
2) Aδi = λiδi and Bδi = βiδi for all i = 1, 2, ..., n.
Then the problem has a unique solution.

Proof. Under the given representations, problem (1) becomes

n∑
i=1

u
(2α)
i (t) δi +

n∑
i=1

u
(α)
i (t)Aδi +

n∑
i=1

ui (t)Bδi =
n∑

i=1

fi (t) δi (25)

But Aδi = λiδi and Bδi = βiδi for all i = 1, 2, ..., n by condition 2, so we get

n∑
i=1

u
(2α)
i (t) δi +

n∑
i=1

λiu
(α)
i (t) δi +

n∑
i=1

βiui (t) δi =
n∑

i=1

fi (t) δi (26)

Taking inner products with δj :

n∑
i=1

u
(2α)
i (t) ⟨δi, δj⟩+

n∑
i=1

λiu
(α)
i (t) ⟨δi, δj⟩+

n∑
i=1

βiui (t) ⟨δi, δj⟩ =
n∑

i=1

fi (t) ⟨δi, δj⟩ (27)

Moreover, the basis {δi}ni=1 is orthonormal so then the equation (27) becomes

u
(2α)
j (t) + λju

(α)
j (t) + βjuj (t) = fj (t) (28)

Multiplying (28) by δj and taking inner product x :

g
(2α)
j (t) + λjg

(α)
j (t) + βjgj (t) = fj (t) ⟨δj , x⟩ (29)



H. Odetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6238 9 of 11

Therefore, fj (t) is uniquely determined by

fj (t) =
g
(2α)
j (t) + λjg

(α)
j (t) + βjgj (t)

⟨δj , x⟩
(30)

since ⟨δj , x⟩ ̸= 0 by assumption. Hence, f (t) has been completely determined. Each

differential equation admits a unique solution given initial conditions uj (0) and u
(α)
j (0).

4. Physical Applications and Engineering Relevance

1- Viscoelastic Material Modeling:
In viscoelastic material analysis, stress-stain relationships naturally incorporate mem-

ory effects through fractional derivative formulations:

E
∂2αu

∂t2α
+A

∂αu

∂tα
+Bu = f (t)

where u represents displacement field distribution, E captures internal effects (po-
tentially degenerate in quasi-static scenarios), A models viscous damping mechanisms,
and B represents elastic restoring forces.

2- Anomalous Diffusion Phenomena:
In systems exhibiting anomalous diffusion characteristics, finite-rank approaches nat-

urally capture dominant transport modes:

E
∂2αc

∂t2α
−D

∂αc

∂tα
+Rc = S (x, t)

where c denotes concentration distribution, D represents diffusion operator, and R
models reaction mechanisms.

5. Conclusion

This study presents a comprehensive framework for analyzing conformable fractional
abstract Cauchy problems through finite-rank solution techniques. The main contribu-
tions include:

1- Theoretical Achievements
Existence and Uniqueness Theorems: Established under various operator conditions

including degenerate cases.
Solution Methodology: Developed systematic approach using tensor product decom-

position.
Computational Framework: Provided constructive algorithms for solution computa-

tion.
2- Practical Impact
The finite-rank approach offers significant computational advantages by reducing

infinite-dimensional problems to finite-dimensional systems. Applications in viscoelastic
materials and anomalous diffusion demonstrate practical relevance.



H. Odetallah et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6238 10 of 11

3- Novel Contributions
Extension of tensor product techniques to conformable fractional derivatives.
Treatment of degenerate operators through spectral decomposition.
Unified approach handling both direct and inverse problems.
4- Future Research Directions
Higher-Order Problems: Extension to conformable fractional problems of order greater

than 2.
The framework established here provides a solid foundation for further research in

fractional differential equations and their applications to real-world phenomena exhibit-
ing memory effects and anomalous behavior.

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