EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 6186
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A Computational Analysis of Convection-Diffusion
Model with Memory Using the Caputo-Fabrizio

Derivative and Cubic Trigonometric B-Spline Functions

Mehwish Saleem1, Arshed Ali1, Imtiaz Ahmad2,3,∗, Aziz Khan4, Thabet Abdeljawad4,∗

1 Department of Mathematics, Islamia College Peshawar, Khyber Pakhtunkhwa, Pakistan
2 Department of Mathematics, Saveetha School of Engineering (SIMATS), Thandalam 600124,
Chennai, Tamil Nadu, India
3 Institute of Informatics and Computing in Energy (IICE), Universiti Tenaga Nasional
(UNITEN), Kajang 43000, Selangor, Malaysia
4 Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833,
11586 Riyadh, Saudi Arabia

Abstract. This paper presents the first study on computational solutions for time-fractional par-
tial integro-differential equations (PIDE) arising from convection-diffusion processes with memory,
employing the Caputo-Fabrizio fractional derivative and the cubic trigonometric B-spline differ-
ential quadrature method. The first order backward finite difference formula is used to evaluate
the Caputo-Fabrizio derivative, converting the PIDE into an integro-differential equation (IDE).
The cubic trigonometric B-spline-based differential quadrature method is applied to approximate
the spatial derivatives, transforming the IDE into a system of algebraic equations by expressing
spatial derivatives as a weighted sum of function values. The weighting coefficients are obtained
using an efficient tridiagonal solver. The method is validated through three test problems, and its
computational efficiency, stability,and numerical convergence are analyzed. The results are also
compared with those obtained using the cubic B-spline collocation method.

2020 Mathematics Subject Classifications: 35R11, 65M70,45J05, 65M06, 35K05

Key Words and Phrases: Caputo-Fabrizio fractional derivative, Cubic trigonometric B-spline
functions, Fractional partial integro-differential equation, Convection-Diffusion process

1. Introduction

Fractional calculus has gained significant attention from researchers over the last two
decades because of its ability to model non-classical phenomena accurately and reliably
across various scientific fields. These include applications in areas such as viscoelastic

∗Corresponding author.
∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.6186

Email addresses: imtiazkakakhil@gmail.com (I. Ahmad), tabdeljawad@psu.edu.sa (T. Abdeljawad)

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



M. Saleem et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6186 2 of 22

materials in polymers, motion of an immersed plate, gas flow in fluids, heat flux inten-
sity in blast furnace walls, radiation cooling of a semi-infinite body, acoustic dissipation,
rheology, neutron transport in nuclear reactors, self-focusing in laser pulses, materials
with thermal resistance, electrical circuits with fractance, dynamical system controllers,
and the mechanical and electrical properties of real materials [1–5]. Similarly to classi-
cal (integer-order) calculus, fractional (noninteger-order) calculus also has a long-standing
history. In fact, fractional calculus generalizes classical calculus. The key advantage of
fractional differential operators is that, unlike integer-order differential operators, they do
not overlook the effects of memory and inherited properties within the underlying model,
thereby providing a more accurate definition of the process. Additionally, classical deriva-
tives are local in nature, failing to capture processes and systems with dynamic memory
and spatial nonlocality. In contrast, fractional derivatives inherently possess the feature
of nonlocality [6–10].
The Caputo fractional derivative, introduced by Caputo in 1960 [11], is one of the most
widely used fractional differentiation operators that depends on the singular kernel.The
advantage of this definition is that it facilitates the inclusion of easily interpretable initial
conditions in addition that the derivative of a constant is zero. However, the singular
kernel in Caputo definition is a vital complexity and cognitive complication in the solu-
tion of fraction order differential equations [12–16]. In 2015, Caputo and Fabrizio [17]
proposed a novel fractional derivative with a non-singular kernel to circumvent this diffi-
culty. In addition to this property, the Caputo-Fabrizio derivative possesses several notable
characteristics, including its ability to model material heterogeneities configurations with
different scales which are similar to Caputo derivative. There has been observed consider-
able progress in realization of the Caputo-Fabrizio fractional derivative, leading to several
significant advancements in its theory and applications in real wold problems. Losada and
Nieto [18] founded the Caputo-Fabrizio integral, and application of this integral in the
solution of linear and nonlinear fractional differential equations involving Caputo-Fabrizio
and fractional falling body problem. Atangana and Alkahtani [19] successfully translated a
resistance, inductance, capacitance circuit from ordinary equation to fractional differential
equation by using Caputo-Fabrizio derivative in order to include the parameter to describe
the low or high flow of electricity. Gómez-Aguilar et al. [20] modeled and analyzed the
mass-spring-damper systems with different source terms to examine the displacement of
the oscillator in fractal geometries, by using the Caputo-Fabrizio approach. Atangana
and Alqahtani [21] developed numerical methods for time fractional and space fractional
Caputo-Fabrizio derivatives and implemented by solving fractional advection diffusion
model. Abdeljawad and Baleanu [22] provided the integration by parts formula and dis-
crete version for Caputo-Fabrizio derivative using discrete exponential kernel. Al-Refai [23]
derived a closed form reduction order formula for Caputo-Fabrizio derivative to get second
linearly independent solution of linear fractional differential equations. Ypez-Martnez and
Gómez-Aguilar [24] presented a modified definition of Caputo-Fabrizio fractional-order op-
erator and some properties of this definition to solve non-linear polynomial type fractional
differential equations using analytical methods. Qureshi et al. [25] obtained a first-order
formula for Capto-Fabrizio derivative and applied to diffusion-wave equation. Jassim and



M. Saleem et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6186 3 of 22

Hussain [26] compared the approximate analytical solutions of the fractional system of
differential equations with Caputo-Fabrizio fractional derivative operator obtained using
two approximate analytical methods. Jin et al. [27] examined the dynamic behavior of
the drinking population through the Caputo-Fabrizio fractional drinking model. Jamil et
al. [28] analyzed Caputo-Fabrizio based non-Newtonian magnetic Casson blood flow in an
inclined stenosed artery. Adel et al. [29] discussed full-number solicitation model for the
spread of corona virus in COVID-19 pandemic. Ahmad et al. [30] investigate the dynam-
ical behavior of the Hepatitis B virus using the Caputo-Fabrizio fractional derivative. De-
hingia et al. [31] investigate the dynamics of a nutrient-plankton system by incorporating
Caputo and Caputo-Fabrizio fractional operators. Chauhan et al [32] provided Caputo-
Fabriio based system for the dynamics and behavior of a financial model. Kausar et al. [33]
introduced a nonlinear fractional circuits modeling using a stochastic neuro-computational
artificial intelligent-based Caputo-Fabrizio stiff electric circuit model. Güngör [34] studied
the Caputo-Fabrizio fractional Cahn-Allen equation using the Caputo-Fabrizio q-Elzaki
homotopy analysis transform method.
Physical phenomena that involve memory and inherited characteristics are frequently mod-
eled in physics and engineering through partial integro-differential equations (PIDEs) (see
[35–37] and references therein). With the growing interest in fractional derivatives, nu-
merous linear and nonlinear PIDEs have been investigated using these derivatives. These
include PIDE models that describe heat transfer in materials with memory [38], vis-
coelasticity [39], transonic multiphase flows [40] and thin beam and plate structures [41].
Finding exact analytical solutions for fractional PIDEs is generally difficult, thus numer-
ical methods play a crucial role in solving these equations [36]. Hussain et al. [38] used
the variational iteration method to numerically solve fractional PIDEs. Saima [42] used
the cubic B-spline collocation method to solve a second-order time-fractional PIDE. Avaz-
zadeh et al. [39] applied Legendre wavelets to solve a fractional viscoelastic PIDE. Bazgir
and Ghazanfari [41] presented a solution for fourth-order fractional PIDEs with a singular
kernel through a mixed spectral method. Arqub [40] solved various time-fractional PIDEs
using the iterative reproducing kernel method. Loha et al. [43] provided an approximate
solution for a class of time-fractional PIDEs. Guo et al. [44] solved a nonlinear time-
fractional PIDE using the iterative finite difference method. Tayabba et al. [45] developed
an extended cubic B-spline collocation method along with Newton’s linearization tech-
nique for nonlinear time-fractional PIDEs. Atta and Youssri [46] proposed the spectral
Chebyshev collocation method to solve a time-fractional nonlinear PIDE with a weakly
singular kernel. Luo et al. [47] introduced a new compact difference scheme based on the
shifted and weighted Grünwald formula for time-fractional PIDEs. Mohapatra et al. [48]
found the solution of time-fractional PIDE using the Adomian decomposition method and
the homotopy perturbation method. Recently, Panda and Mohapatra implemented three
semi-analytical techniques to solve a time-fractional PIDE. In this paper, the following
PIDE with the Caputo-Fabrizio fractional derivative is considered:



M. Saleem et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6186 4 of 22

CF
0 Dα

t v(ξ, t)+λ
∂v(ξ, t)

∂ξ
−η∂

2v(ξ, t)

∂ξ2
=

∫ t

0
(t−p)−βv(ξ, p)dp+q(ξ, t), ξ ∈ Ω, t > 0, 0 < α < 1,

(1)
subject to initial condition

v(ξ, 0) = δ1(ξ), ξ ∈ Ω, t > 0, (2)

and boundary conditions

v(γ1, t) = λ1(t), v(γ2, t) = λ2(t), t > 0, (3)

where Ω = [γ1, γ2], q(ξ, t) is a known function, and
∫ t
0 (t − p)−βv(ξ, p)dp is the memory

term, λ1, λ2 are boundary operators, CF
0 Dα

t is the Caputo-Fabrizio fractional derivative,
λ, η are convection and diffusion coefficients respectively. For the integer-order case that
is for α = 1, Eq.(1) is solved by different methods including cubic B-spline collocation
method (CBSCM) [49], quartic B-spline collocation technique [50], radial basis colloca-
tion method [51], sinc collocation method product trapezoidal [52], cubic B-spline Galerkin
method with quadratic weight function [53] and cubic trigonometric B-spline collocation
method [54].
Computational methods are widely used to provide a systematic and efficient way to solve
complex problems that arise in several fields, including engineering, science, social sciences,
and economics. These methods utilize mathematical models, algorithms, and numerical
techniques to simulate, analyze, and predict the behavior of systems that may be difficult
or impossible to solve analytically. Cubic trigonometric B-spline (CTBS) functions spans
a variety of fields that require smooth, flexible, and computationally efficient solutions for
approximating, interpolating, or modeling data. Some important features of these func-
tions are smoothness, nonnegativity, design and analysis of shape of curve, partition of
unity, extension of solution to the entire spatial domain, C2 continuity, and recurrence
relation. Recently, different fractional differential equations have been solved using CTBS
based methods, including the fractional diffusion wave equation [55], the fractional Burger
equation [56], the fractional telegraph equation [57], the fractional diffusion equation [58],
time-fractional Schrödinger equation [59], and the time-fractional gas dynamics problem
[60].
Upto the best of our knowledge, computational solution of the time-fractional PIDEs using
Caputo-Fabrizio fractional derivative is not yet studied, which is the significance of the
present work.
The remaining portion of the paper is outlined as: Section 1 provides an introduction
and some preliminaries on the Caputo-Fabrizio fractional derivative and CTBS functions.
Section 2 presents the development of the proposed method using the Caputo-Fabrizio frac-
tional derivative, backward difference formula, differential quadrature, and CTBS func-
tions. Section 3 focuses on the stability analysis of the proposed schemes. Section 4
presents numerical results, including error norms, computational efficiency, spectral radii,
numerical convergence, and a comparison with an existing method to evaluate the perfor-
mance of the current approach. The paper’s findings are summarized in Section 5.



M. Saleem et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6186 5 of 22

In this work we use the following notions to obtain the proposed scheme:
Definition [17, 21, 61]: The fractional order Caputo-Fabrizio derivative CF

0 Dα
t f(t) is

defined by:

CF
0 Dα

t f(t) =
M(α)

1− α

∫ t

0
exp

(
−α
1− α

(t− p)

)
f ′(p)dp, 0 < α < 1,

where f ∈ H1(0, t) and M(α) is a normalization function such that M(0) =M(1) = 1.
Theorem [17, 21]: Let function f(t) be a function in C2[0, T ], then the Caputo-Fabrizio
derivative has first-order approximation at a point tj = jδt, with f j = f(tj), j = 0, 1, 2, ..., s
is defined as:

CF
0 Dα

t f(t
s) =

M(α)

α

s∑
j=1

(
f j+1 − f j

δt

)
dj,δt +O(δt2), 0 < α < 1, (4)

with dj,δt = exp
(
−α δt

1−α(s− j − 1)
)
− exp

(
−α δt

1−α(s− j)
)
.

2. The Proposed Method

2.1 Temporal Approximation
We approximate the problem in time, taking t = ts+1, s = 0, 1, 2, ...,M, in Eq. (1) as,

∂αv(ξ, ts+1)

∂tα
+λ

∂v(ξ, ts+1)

∂ξ
−η∂

2v(ξ, ts+1)

∂ξ2
=

∫ ts+1

0
(ts+1−p)−βv(ξ, p)dp+ q(ξ, ts+1). (5)

The time fractional derivative ∂αv(ξ,ts+1)
∂tα in (5) is approximated using (4) as follows:

∂αv(ξ, ts+1)

∂tα
=
M(α)

αδt

s∑
j=0

(
v(ξ, ts+1)− v(ξ, ts)

)(
exp

(
−α δt

1− α
(s− j + 1)

)
− exp

(
−α δt

1− α
(s− j)

))

=
M(α)

αδt

s∑
j=0

µj(v(ξ, t
s−j+1)− v(ξ, ts−j)),

=
M(α)

αδt

µ0v(ξ, ts+1) +
s−1∑
j=0

(µj+1 − µj)v(ξ, t
s−j) + µsv(ξ, t

0)

 ,



M. Saleem et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6186 6 of 22

where µj =
(
exp

(
−α
1−α(jδt)

)
− exp

(
−α
1−α((j + 1)δt)

))
, j = 0, 1, 2, ..., s.

To evaluate the integral term in Eq. (5), we proceed as follows:∫ ts+1

0
(ts+1 − p)−βv(ξ, p)dp =

s∑
j=0

∫ tj+1

tj
p−βv(ξ, ts+1 − p) dp

=
s∑

j=0

v(ξ, ts−j+1)

∫ tj+1

tj
p−β dp

=
δt1−β

1− β

s∑
j=0

νjv(ξ, t
s−j+1),

where νj =
(
(j + 1)1−β − j1−β

)
, j = 1, 2, ..., s, ν0 = 1.

Substituting these approximations in Eq. (5), we have

M(α)

αδt

µ0v(ξ, ts+1) +
s−1∑
j=0

(µj+1 − µj)v(ξ, t
s−j)− µsv(ξ, t

0)

+ λ
∂v(ξ, ts+1)

∂ξ
− η

∂2v(ξ, ts+1)

∂ξ2

=
δt1−β

1− β

s∑
j=0

νjv(ξ, t
s−j+1) + q(ξ, ts+1).

(6)

Rearranging the terms in Eq. (6),(
1− αδt2−β

µ0M(α)(1− β)

)
v(ξ, ts+1) +

αδt

µ0M(α)

(
λ
∂v(ξ, ts+1)

∂ξ
− η

∂2v(ξ, ts+1)

∂ξ2

)
=

(
µ0 − µ1
µ0

)
v(ξ, ts)+

s−1∑
j=1

(
µj − µj+1

µ0

)
v(ξ, ts−j) +

(
µs
µ0
v(ξ, t0)

)
+

αδt2−β

µ0M(α)(1− β)

 s∑
j=1

νsv(ξ, t
s−j+1) + q(ξ, ts+1)

 .

(7)

2.2 Approximation in Space
We now approximate the solution in space dimension. The differential quadrature method
[36] provides kth order derivative of the function v(ξ, t) from the values of v(ξ, t) at
ξr, r ∈ I1 = {1, 2, ..., N}, as

∂kv(ξr, t)

∂ξk
=

N∑
r=1

a(k)rs v(ξs, t), (8)

where a
(k)
rs , k = 1, 2, ... are kth order weighting coefficients which are determined by test



M. Saleem et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6186 7 of 22

functions (for detail see [36]). Now taking ξ = ξr and using (8) in (7), we get(
1− αδt2−β

µ0M(α)(1− β)

)
v(ξr, t

s+1) +
αδt

µ0M(α)

(
λ

N∑
m=1

a(1)rmv(ξm, t
s+1)− η

N∑
m=1

a(2)rmv(ξm, t
s+1)

)

=

(
µ0 − µ1
µ0

)
v(ξr, t

s) +

s−1∑
j=1

(
µj − µj+1

µ0

)
v(ξr, t

s−j) +

(
µs
µ0
v(ξr, t

0)

)

+
αδt2−β

µ0M(α)(1− β)

 s∑
j=1

νsv(ξr, t
s−j+1) + q(ξr, t

s+1)

 .

(9)

To obtain the weighting coefficients a
(1)
rm, a

(2)
rm, r,m = 1, 2, ..., N in Eq. (2), the following

modified CTBS functions are used [62]:

S1(ξ) = T1(ξ) + 2T0(ξ),
S2(ξ) = T2(ξ)− T0(ξ),
Sl(ξ) = Tl(ξ), for l = 3, 4, ..., N − 2,

SN−1(ξ) = TN−1(ξ)− TN+1(ξ),

SN (ξ) = TN (ξ) + 2TN+1(ξ),

which form a basis in region Ω and the CTBS functions are defined by

Tr(ξ) =
1

θ



sin3( ξ−ξr−2

2 ), ξ ∈ Ωr−1,

[sin( ξ−ξr−2

2 ) sin( ξr−ξ
2 )+

sin( ξr+1−ξ
2 ) sin( ξ−ξr−1

2 )] sin( ξ−ξr−2

2 ) + sin2( ξ−ξr−1

2 ) sin( ξr+1−ξ
2 ), ξ ∈ Ωr,

sin( ξ−ξr−2

2 ) sin2( ξr+1−ξ
2 )+

[sin( ξ−ξr−1

2 ) sin( ξr+1−ξ
2 ) + sin( ξr+2−ξ

2 ) sin( ξ−ξr
2 )] sin( ξr+2−ξ

2 ), ξ ∈ Ωr+1,

sin3( ξr+2−ξ
2 ), ξ ∈ Ωr+2,

0, otherwise,

(10)
where θ = sin(h2 ) sin(

3h
2 ) sin(h).

The following lemma provides Tr(ξm), T ′
r (ξm) and T ′′

k (ξm).
Lemma [55]: The CTBS functions Tr(ξm) (10) and their derivatives at ξ = ξm are given



M. Saleem et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6186 8 of 22

as:

Tr(ξm) =


σ1, if m = r + 1 or r − 1,

σ2, if m = r,

0, elsewhere,

T ′
r (ξm) =


σ3, if m = r − 1,

−σ3, if m = r + 1,

0, elsewhere,

and

T ′′
r (ξm) =


σ4, if m = r + 1 or r − 1,

σ5, if m = r,

0, elsewhere,

.

where σ1 =
sin2(h

2
)

sin( 3h
2
) sin(h)

, σ2 =
2

2 cos(h)+1 , σ3 =
3/4

sin( 3h
2
)
, σ4 =

(3+9 cos(h))

16 sin2(h
2
)(2 cos(h

2
)+cos( 3h

2
))

and

σ5 = −3 cot2(h2 )4 cos(h) + 2).
Thus using CTBS functions in Eq.(8), we get

∂kSm
∂ξk

(ξr) =

N∑
i=1

a
(k)
ri Sm(ξi), for m, r ∈ I1, and k = 1, 2,

which leads to the following matrix form:

P a⃗(k)r = Q(k)
r , (11)

where a⃗
(k)
r = [a

(k)
rm : m ∈ I1], Q

(k)
r = [∂

kSl

∂ξk
(ξr) : l ∈ I1]

T , P = [prm : r,m ∈ I1] and

prm =


2σ1 + σ2, if r = m = 1 or N,

σ2, if r = m and 1 < r,m < N,

σ1, if (m = r − 1 and 2 < r ≤ N) or (r = m− 1 and 1 ≤ m < N − 1),

0, elsewhere.

The weighting coefficients a
(k)
rm, r,m ∈ I1, are obtained by solving the system (11) through

the well known efficient tridiagonal solver “Thomas algorithm”. Eq. (11) gives to the
following matrix form:

vs+1 = Cvs + F, s ≥ 0, (12)



M. Saleem et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6186 9 of 22

where C = A−1B, F = A−1D

A =



a+ ba
(1)
11 − ca

(2)
11 ba

(1)
12 − ca

(2)
12 ... ba

(1)
1N − ca

(2)
1N

ba
(1)
21 − ca

(2)
22 a+ ba

(1)
22 − ca

(2)
22 ... ba

(1)
2N − ca

(2)
2N

. . ... .

. . ... .

. . ... .

ba
(1)
N1 − ca

(2)
N1 ba

(1)
N2 − ca

(2)
N2 ... a+ ba

(1)
NN − ca

(2)
NN


,

B =



d 0 ... 0
0 d ... 0
. . ... .
. . ... .
. . ... .
0 0 ... d

 ,

a =
(
1− αδt2−β

µ0M(α)(1−β)

)
, b = λαδt

µ0M(α) , c =
ηαδt

µ0M(α) , d =
(
µ0−µ1

µ0
+ αν1δt2−β

µ0M(α)(1−β)

)
,

vs = [vs0, v
s
1, ..., v

s
N ]T D = [d1, d2, ..., dN ]T ,

dr =
∑s−1

j=1

(
µj−µj+1

µ0

)
v(ξr, t

s−j)+
(
µs

µ0
v(ξr, t

0)
)
+ αδt2−β

µ0M(α)(1−β)

(∑s
j=2 νsv(ξr, t

s−j+1) + q(ξr, t
s+1)

)
.

3. Stability

Here, we assess the stability of the present method given in (12). Let Es be the er-
ror between the approximate solution v̂s and the exact solution vs at sth time level i.e.
Es = vs − v̂s. Then error equation for the method (12) is given by

Es+1 = CEs,

where C is known as amplification matrix. If ∥ C ∥≤ 1 (see [63] and Lax-Richtmyer
condition for stability) then method (12) will be stable. This condition is equivalent to
σ(C) ≤ 1, where ρ(C) represents spectral radius of the matrix C [63] and is defined
as ρ(C)=max1≤i≤N | ζi | and ζi is an eigenvalue of C. In the next section, we provide
computational values of ρ(C) for the parameters δt, N , and α to demonstrate that the
method (12) satisfies the stability condition.

4. Test Problems

This section is devoted to numerical simulation of the proposed method. Three exam-
ples are implemented using the spatial domain with γ1 = 0 and γ2 = 1 and M(α) = 1 for
validation of the present method (12). Error norms L∞, L2, pointwise absolute error, spec-
tral radius ρ(C) and rate of convergence (RoC) [64] are computed to assess performance
and reliability of the current technique. To compare our results with CBSCM [49], the



M. Saleem et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6186 10 of 22

values of convection-diffusion coefficients λ, η, and β are used which are given in [49] and
α ≈ 1. Initial/boundary conditions (2)-(3) are taken from the exact analytical solution.
Corei3, 2.4GHz processor and 2GB RAM are used for simulation while computer run time
(RT) is measured in seconds.
4.1 Example-1
In the first example, we take Eqs.(1)-(3) and the exact solution u(ξ, t) = (t+1)2 sin(πξ)[49].
Then

f(ξ, t) =

(
M(α)(exp( αt

α−1)(1− 2α) + α(t+ 2)− 1)

α2
− t1−β(2t(t+ 3)− β(2t+ 5) + β2 + 6)

(β − 1)(β − 2)(β − 3)

)
sin(πξ) + π(t+ 1)2 (κ cos(πξ)− νπ sin(πξ)) .

Computations are carried out with different values of α, h, δt and results of the present
method are provided in Tables 1-4 using λ = 0.05, η = 0.4, β = 1/2 [49]. In Table 1,
pointwise absolute errors at various values of ξ are presented for α = 0.25, 0.5, 0.75, 0.95,
with δt = 10−4, h = 0.002, and t = 0.1. Tables 2–3 report the L2, L∞ errors, spec-
tral radius ρ(C), runtime (RT), and rates of convergence of the proposed method at
t = 0.1 for α = 0.25, 0.5. Table 4 presents a comparison of the L2 and L∞ errors for
M = 10, 50, 100, 500 and α = 1 − 10−4, with δt = 10−4, 10−5 and h = 0.01, alongside
the results obtained using the CBSCM method [49]. Fig. 1 shows exact and approximate
solutions using α = 0.25, 0.5, 0.75, 0.9, δt = 0.005 and h = 0.05 over the space interval
[0, 1]. Fig. 2 depicts behavior of the solution plotted in Fig. 1 over a small space interval.
Fig. 3 shows surface plot of the computed solutions at different times t, 0 ≤ t ≤ 0.1 with
α = 0.5, δt = 0.005 and h = 0.05.

Table 1: Absolute error versus α at t = 0.1 using δt = 10−4 and h = 0.002

ξ α = 0.25 α = 0.5 α = 0.75 α = 0.95

0.1 2.3090× 10−6 2.4380× 10−6 2.1814× 10−6 1.2954× 10−6

0.2 5.7425× 10−6 5.4219× 10−6 4.4510× 10−6 2.5014× 10−6

0.3 8.5385× 10−5 7.8344× 10−6 6.2730× 10−6 3.4645× 10−6

0.4 1.0446× 10−5 9.4527× 10−6 7.4733× 10−6 4.0897× 10−6

0.5 1.1301× 10−5 1.0131× 10−5 7.9380× 10−6 4.3150× 10−6

0.6 1.1040× 10−5 9.8133× 10−6 7.6252× 10−6 4.1179× 10−6

0.7 9.7122× 10−6 8.5442× 10−6 6.5689× 10−6 3.5173× 10−6

0.8 2.5714× 10−6 6.4595× 10−6 4.8763× 10−6 3.4744× 10−6

0.9 4.5521× 10−6 3.7768× 10−6 2.7173× 10−6 1.3717× 10−6



M. Saleem et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6186 11 of 22

Table 2: L2, L∞ and RoC with h = 0.002, t = 0.1

α δt L2 L∞ RoC(L∞) ρ(C) RT

0.5 0.002 7.4147× 10−6 2.3453× 10−4 −−− 1.0 2.0723

0.5 0.001 3.7382× 10−6 1.1824× 10−4 0.9880 1.0 2.0408

0.5 0.0005 1.8575× 10−6 5.8753× 10−5 1.0090 1.0 2.5511

0.5 0.00025 9.0164× 10−7 2.8519× 10−5 1.0427 1.0 4.7765

0.25 0.002 8.2754× 10−6 2.6184× 10−4 −−− 1.0 1.8673

0.25 0.001 4.1728× 10−6 1.3203× 10−4 0.9878 1.0 2.0043

0.25 0.0005 2.0737× 10−6 6.5612× 10−5 1.0088 1.0 2.5864

0.25 0.00025 1.0068× 10−6 3.1855× 10−5 1.0424 1.0 4.8788

Table 3: L2, L∞ and RoC at t = 0.1using δt = 0.000125

α h L2 L∞ RoC(L∞) ρ(C) RT

0.5 0.05 2.2139× 10−4 1.3833× 10−3 −−− 1.0 0.4582

0.5 0.025 3.7852× 10−5 3.2332× 10−4 2.0971 1.0 0.8893

0.5 0.0125 5.7741× 10−6 5.7753× 10−5 2.4850 1.0 1.6376

0.5 0.00625 3.7145× 10−7 9.7241× 10−6 2.5703 1.0 3.4969

Table 4: L∞ and L2 for α = 1− 10−4, h = 0.01

CTBS-DQ CBSCM[49]

δt M L∞ L2 L∞ L2
10−4 10 2.3777× 10−7 1.6813× 10−8 9.4351× 10−6 1.3253× 10−7

50 1.1041× 10−6 7.8070× 10−8 1.1446× 10−5 3.3175× 10−7

100 2.1475× 10−6 1.5185× 10−7 1.1943× 10−5 5.7312× 10−7

500 9.6180× 10−6 6.8010× 10−7 3.5394× 10−5 2.4398× 10−6

10−5 10 4.7593× 10−8 3.3653× 10−9 4.3771× 10−6 4.4106× 10−8

50 1.4499× 10−7 1.0252× 10−8 8.2580× 10−6 9.5928× 10−8

100 2.6654× 10−7 1.8847× 10−8 9.5573× 10−6 1.2493× 10−7

500 1.2329× 10−6 8.7181× 10−8 1.1579× 10−5 2.2516× 10−7



M. Saleem et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6186 12 of 22

Figure 1: Exact and approximate solutions for δt = 0.005, h = 0.05

Figure 2: Exact and approximate solutions for δt = 0.005, h = 0.05

Figure 3: Approximate solutions for δt = 0.005, h = 0.05, 0 ≤ t ≤ 0.1

4.2 Example-2
Consider Eqs.(1)-(3) with exact solution u(ξ, t) = (t + 1)2[1 − cos 2πξ + 2π2ξ(1 − ξ)][49].
Then



M. Saleem et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6186 13 of 22

f(ξ, t) =

(
2M(α)(exp( αt

α−1)(2α− 1)− α(t+ 2) + 1)

α2
− t1−β(2t(t+ 3)− β(2t+ 5) + β2 + 6)

(β − 1)(β − 2)(β − 3)

)
(
cos(2πξ) + 2π2ξ(ξ − 1)− 1

)
+ (t+ 1)2

(
κ(2π sin(2πξ) + 2π2(1− 2ξ))− ν(4π2 cos(2πξ)− 4π2)

)
.

Simulation is done using various values of α, δt, h and results of the present method are
provided in Tables 5-8 using λ = 0.005, η = 0.5, β = 1/3 [49]. In Table 5, pointwise
absolute errors at various values of ξ for α = 0.25, 0.5, 0.75, 0.95, δt = 10−4, h = 0.002, t =
0.1 are provided. In Tables 6-7, L2, L∞, ρ(C), RT, and rates of convergence of the suggested
method at t = 0.1 for α = 0.5 are given. Table 8 compares the error norms L2, L∞ at
different time levels M for α = 1 − 10−6, δt = 10−4, 10−5, h = 0.01 with the results
of CBSCM [49]. Fig. 1 shows exact and approximate solutions using α = 0.75, δt =
0.005, h = 0.05. Fig. 2 depicts error in solution obtained using α = 0.75, δt = 0.005, h =
0.05. Fig. 3 displays surface plot of the numerical solutions at different times t, 0 ≤ t ≤ 0.1
with α = 0.75, δt = 0.005 and h = 0.05.

Table 5: Absolute error versus α at t = 0.1 using δt = 10−4 and h = 0.002

ξ α = 0.25 α = 0.5 α = 0.75 α = 0.95

0.1 1.1203× 10−5 1.0707× 10−5 9.5337× 10−6 7.0939× 10−6

0.2 1.6870× 10−5 1.5778× 10−5 1.3460× 10−5 8.8207× 10−6

0.3 1.8148× 10−5 1.6618× 10−5 1.3477× 10−5 7.2238× 10−6

0.4 1.7521× 10−5 1.5718× 10−5 1.2089× 10−5 4.8659× 10−6

0.5 1.7107× 10−5 1.5188× 10−5 1.1375× 10−5 3.8166× 10−6

0.6 1.7731× 10−5 1.5846× 10−5 1.2140× 10−5 4.8675× 10−6

0.7 1.8574× 10−5 1.6880× 10−5 1.3585× 10−5 7.2317× 10−6

0.8 1.7521× 10−5 1.6184× 10−5 1.3633× 10−5 8.8413× 10−6

0.9 1.2086× 10−5 1.1264× 10−5 9.7799× 10−6 7.1312× 10−6

Table 6: L2, L∞ and RoC at t = 0.1 with h = 0.002

α δt L2 L∞ RoC(L∞) ρ(C) RT

0.5 0.002 2.1309× 10−5 6.1749× 10−4 −−− 1.0 2.1026

0.5 0.001 1.0491× 10−5 3.2831× 10−4 1.0323 1.0 2.2626

0.5 0.0005 5.0286× 10−6 1.5500× 10−4 1.0828 1.0 2.9344

0.5 0.00025 2.2819× 10−6 6.7775× 10−5 1.1936 1.0 5.2883



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Table 7: L2, L∞ and RoC at t = 0.1 with δt = 0.000125

α h L2 L∞ RoC(L∞) ρ(C) RT

0.5 0.05 1.6001× 10−3 1.2404× 10−2 −−− 1.0 0.5416

0.5 0.025 2.8046× 10−4 3.0793× 10−3 2.0101 1.0 1.0159

0.5 0.0125 4.7168× 10−5 7.3755× 10−4 2.0618 1.0 1.8947

0.5 0.00625 6.6359× 10−6 1.5141× 10−4 2.2839 1.0 3.9057

Table 8: L∞ and L2 for α = 1− 10−6, h = 0.01

CTBS-DQ CBSCM[49]

δt M L∞ L2 L∞ L2
10−4 10 1.0000× 10−4 1.4380× 10−6 3.0131× 10−5 1.1709× 10−6

50 1.0000× 10−4 1.7304× 10−6 3.3049× 10−4 2.0104× 10−5

100 1.3501× 10−4 2.1011× 10−6 1.0609× 10−3 7.2658× 10−5

500 2.3642× 10−4 3.9788× 10−6 2.1712× 10−2 1.5005× 10−3

10−5 10 1.0000× 10−5 1.4295× 10−7 3.7552× 10−7 1.4690× 10−8

50 1.0000× 10−5 1.7332× 10−7 5.0060× 10−6 2.5430× 10−7

100 1.6378× 10−5 2.3661× 10−7 1.6837× 10−5 9.2099× 10−7

500 2.6842× 10−5 1.7990× 10−6 2, 8412× 10−4 1.9062× 10−5

Figure 4: Exact and approximate solutions for α = 0.75, δt = 0.005, h = 0.05



M. Saleem et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6186 15 of 22

Figure 5: Exact and approximate solutions for α = 0.75, δt = 0.005, h = 0.05

Figure 6: Approximate solutions for α = 0.75, δt = 0.005, h = 0.05, 0 ≤ t ≤ 0.1

4.3 Example-3

In this example we take Eqs.(1)-(3) with exact solution u(ξ, t) = (t + 1) cos(πξ) [49].
Then
f(ξ, t) =

(
M(α)
α (exp( αt

α−1)− 1)− t1−β(t−β+2)
(β−1)(β−2)

)
cos(πξ)− π(t+ 1)(κ sin(πξ)− νπ cos(πξ)).

Computations are performed and the results are recorded in Table 8 for L2, L∞ errors at
M = 10, 50, 100, 500 using α = 1 − 10−6, δt = 10−4, 10−5, h = 0.01, λ = 0.5, η = 0.005,
β = 1/3 along with the results of CBSCM [49]. In Fig. 7, plot of exact solution and
approximate solutions at t = 0.1 for α = 0.25, 0.5, 0.75, 0.9 using δt = 0.005, h = 0.05 are
shown. In Fig. 8, surface plot of the approximate solutions at various times t, 0 ≤ t ≤ 0.1
using α = 0.5, δt = 0.005 and h = 0.05 is depicted.



M. Saleem et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6186 16 of 22

Table 9: L∞ and L2 for α = 1− 10−6, h = 0.01

CTBS-DQ CBSCM[49]

δt M L∞ L2 L∞ L2
10−4 10 1.0000× 10−4 1.4380× 10−6 3.0131× 10−5 1.1709× 10−6

50 1.0000× 10−4 1.7304× 10−6 3.3049× 10−4 2.0104× 10−5

100 1.3501× 10−4 2.1011× 10−6 1.0609× 10−3 7.2658× 10−5

500 2.3642× 10−4 3.9788× 10−6 2.1712× 10−2 1.5005× 10−3

10−5 10 1.0000× 10−5 1.4295× 10−7 3.7552× 10−7 1.4690× 10−8

50 1.0000× 10−5 1.7332× 10−7 5.0060× 10−6 2.5430× 10−7

100 1.6378× 10−5 2.3661× 10−7 1.6837× 10−5 9.2099× 10−7

500 6.0528× 10−5 6.9515× 10−7 2.8412× 10−4 1.9062× 10−5

Figure 7: Exact and approximate solutions for δt = 0.005, h = 0.05

.

Figure 8: Exact and approximate solutions for δt = 0.005, h = 0.05, 0 ≤ t ≤ 0.1

5. Conclusion



M. Saleem et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 6186 17 of 22

The differential quadrature method based on cubic trigonometric B-spline functions is
applied to solve convection-diffusion type fractional partial integro-differential equations,
using the first-order method for the Caputo-Fabrizio derivative. The method is validated
through three test problems. It demonstrates computational efficiency, and its accuracy
and numerical convergence are analyzed for various time-step sizes, space-step sizes, and
fractional orders. The stability of the method is discussed via spectral radius analysis. The
proposed technique provided better accuracy compared to the cubic B-spline collocation
method. Based on its excellent agreement with the exact solution, this approach is efficient,
accurate, economical and stable to obtain approximate solutions of fractional PIDEs. The
proposed method can be extended to multi-dimensional convection-diffusion models with
variable coefficients. Additionally, incorporating other non-singular fractional operators
could further enhance the model’s flexibility and accuracy.

Declarations

Competing interests:
Authors declare no conflict of interest.
Availability of data and materials:
Data will be provided on request to the corresponding author.
Acknowledgements and Funding:
Aziz Khan and Thabet Abdeljawad would like to thank Prince Sultan University for paying
the APC and supporting through TAS research lab.

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