EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6232
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Numerical Solution of the Intuitionistic Fuzzy Complex
Heat Equation with Intuitionistic Complex Dirichlet

Boundary Conditions via the Explicit Finite Difference
Method

Abd Ulazeez Alkouri1,∗, Sadeq Damrah2, Hamzeh Zureigat3, Osama Ogilat4,
Eman Hussein5

1 Department of Mathematics, Science college, Ajloun National University, P.O.43,
Ajloun-26810, Jordan
2 Department of Mathematics and Physics, College of Engineering, Australian University,
West Mishref, Safat 13015, Kuwait
3 Department of Mathematics, Faculty of Science, Jadara University, Irbid, Jordan
4 Department of Basic Sciences, Faculty of Arts and Science, Al-Ahliyya Amman University,
Amman 19328, Jordan
5 Department of Mathematics, Faculty of Arts and Science, Amman Arab University,
Amman, Jordan

Abstract. Recent developments in complex fuzzy (CF) sets have extended the classical fuzzy
framework from the unit interval [0,1] to the unit disk in the complex plane C, allowing for the
modeling of uncertainties in both magnitude and phase. Building upon this foundation, this study
introduces-for the first time-the use of complex intuitionistic fuzzy (CIF) numbers to solve partial
differential equations, specifically focusing on the CIF heat equation. The CIF framework integrates
both membership and non-membership functions with a complex-valued representation, enabling
a more expressive treatment of uncertainty, including hesitation. An explicit finite difference
method, namely the Forward Time Central Space (FTCS) scheme, is employed to discretize and
solve the CIF heat equation. The model considers fuzziness in the initial and boundary conditions,
where uncertainty impacts amplitude and phase terms. To represent this uncertainty, triangular
fuzzy numbers are used for the real and imaginary parts within the complex unit disk. The
proposed approach demonstrates numerical stability and achieves second-order spatial and first-
order temporal accuracy, validating its reliability and effectiveness. A numerical example confirms
the feasibility of the method, showing strong alignment with theoretical predictions. This work
generalizes existing CF heat equation models and provides a foundation for solving more complex
systems involving higher-order and bipolar uncertainties in future studies.

2020 Mathematics Subject Classifications: 34A07, 35R13, 03E72

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6232

Email addresses: alkouriabdulazeez@anu.edu.jo (A. U. Alkouri),
s.damrah@au.edu.kw (S. Damrah), hamzeh.zu@jadara.edu.jo (H. Zureigat),
o.oqilat@ammanu.edu.jo (O.Ogilat), e.hussein@aau.edu.jo (E. Hussein)

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



A. U. Alkouri et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6232 2 of 15

Key Words and Phrases: Finite difference methods, complex, fuzzy numbers, Intuitionistic
fuzzy sets, complex fuzzy numbers, fuzzy heat equations

1. Introduction

Zadeh founded the notion of fuzzy sets in 1965, and since then, numerous mathemati-
cians have written articles extending classical mathematical theories to fuzzy mathematics.
Subsequently, Atanassov developed and published the notion of intuitionistic fuzzy sets in
1986. Atanassov extended the notion of a fuzzy set to include a second degree, namely
degree of non-membership, and investigated the characteristics of sets with both degrees.
Several applications of interest [1],[2], [3], [4], [5] in the field of numerical solutions have
involved in both fuzzy and intuitionistic fuzzy sets. Recently, several fields—such as en-
vironmental science, fixed point theory, and algebra—have seen successful applications of
fuzzy set theory and its generalizations, including CF sets [6], [7], [8], [9], [10], [11], [12],
[13], [14], [15].

Recent advancements in fuzzy and intuitionistic fuzzy partial differential equations
(FPDEs, IFPDEs) enhance their applicability in solving complex heat equations under
uncertainty. Arshad et al. [16] proposed the Laplace-Residual Power Series Method
(LRPSM) to derive fast-converging analytical solutions for FPDEs, demonstrating its effec-
tiveness in handling boundary conditions using Laurent series and fractional calculus tech-
niques. This approach aligns with the need for precise numerical solutions in heat transfer
models with fuzzy uncertainty. Similarly, Rahman et al. [17] extended intuitionistic
fuzzy calculus to model uncertainty in decision-making, emphasizing acceptance-rejection
dynamics. Their use of intuitionistic fuzzy differential equations (IFDEs) in economic
systems highlights the broader applicability of intuitionistic complex Dirichlet conditions
in modeling imprecise systems. The combination of these methodologies supports the
explicit finite difference method used in this study for solving CIF heat equations, ensur-
ing improved computational accuracy under fuzzy and intuitionistic fuzzy environments.
Additionally, Payal et al. [6] investigated a fuzzy prey–predator model incorporating the
Holling Type II functional response, with a focus on mathematical analysis and quali-
tative behaviors relevant to biological systems such as pest control. In 2025, Rahaman
[7]constructed the Riemann–Liouville fractional integral within this framework and ex-
tended the analysis to include both Riemann–Liouville and Caputo fractional derivatives.
A Type 2 interval-valued Laplace transform was also developed. The theoretical results
were illustrated through an application to an economic lot maintenance model. Al-Qudah
et al. [8] proposed an efficient analytic-numeric method using fuzzy logic and general-
ized Taylor expansion to solve fractional-order differential equations with uncertain initial
data. The method showed high accuracy and reliability in obtaining fuzzy approximate
solutions, suitable for physics and engineering applications.

Buckley introduced a novel definition of fuzzy complex numbers (FCNs) [18], [19], [20],
[21] by incorporating complex-valued fuzzy numbers to extend classical fuzzy set theory. In
this formulation, FCNs are represented as ordinary fuzzy sets with a membership function



A. U. Alkouri et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6232 3 of 15

ranging between [0,1]. Subsequently, a new idea emerged in 2002 to extend the membership
function’s image from [0, 1] to the unit disk in the complex plane C. This gave rise to
CF sets (CFSs) [22]. Unlike FCNs, CFSs feature complex-valued membership grades,
which represent a significant advancement in fuzzy set theory. The concept of CFS has
since garnered extensive attention with widespread applications and research contributions
[23], [24], [25] Incorporating complex numbers into the fuzzy theory and intuitionistic fuzzy
theory, Ramote et al. [26]and Alkouri and Salleh [27], introduced the useful notions of
the CFS and Complex intuitionistic fuzzy set (CIFS). These notions extended the range
of membership and (non-membership) functions from [0, 1] to unit disk in the C. This
extension allows mathematicians to convey and represent a special type of information
to mathematical formulation and vice versa without losing the interpretability provided
by human experts at various levels or stages. Moreover, several real-life applications
of CFS and CIFS have been reported in the literature [28], [29], [30], [31], [32]. The
first combination of CFS with numerical methods for solving FPDE involving fuzzy heat
equation (FHE) has been introduced by Zureigat et al. in 2023 [33]

The research gaps and motivations for extending the CF heat equation to the novel
notion of a CIF heat equation come from the inability of CF heat equation to account
for hesitation degrees, which are often present in real-world physical or engineering sys-
tems. These systems may involve not only uncertainty in measurement but also incom-
plete trust in that uncertainty due to imprecise, conflicting, or missing information. The
CIFS framework enriches the model by incorporating both the degree of membership and
non-membership in complex-valued form, along with a residual hesitation margin. This
provides a more expressive and reliable modeling framework for environments where data
vagueness and conflict coexist with complex-valued parameters. Therefore, the novelty of
our work is that it is among the first to systematically extend the complex fuzzy heat equa-
tion to a CIF setting, combining the strengths of intuitionistic fuzzy theory and complex
representation. Moreover, the proposed formulation introduces new solution procedures
under CIF boundary conditions, which have not been widely investigated in the literature.

This paper is considered an extension of the work presented in [25] which combined
CIFSs with numerical methods to solve FPDEs involving the FHE. In this study, we ad-
dress the case involving two complex-valued functions: membership and nonmembership.
The key contributions of this work are threefold. First, it introduces a novel extension
from the classical heat equation and the CF heat equation to the CIF heat equation.
To the best of our knowledge, this is the first attempt to integrate complex-valued and
intuitionistic fuzzy uncertainty into a heat conduction modeling framework. Second, the
paper formulates the CIF heat equation under CIF boundary conditions and proposes a
numerical solution using a modified Crank–Nicolson scheme adapted to this new setting. It
also generalizes the concept of Hukuhara differentiability to CIF-valued functions. Third,
the proposed model effectively captures multiple layers of uncertainty-namely magnitude
uncertainty, phase variation, and hesitation (incompleteness)-offering a richer and more
expressive framework for modeling real-world thermal systems under deep uncertainty.

The implications of the current research can be viewed from both theoretical and
methodological perspectives. From a theoretical standpoint, the study contributes to



A. U. Alkouri et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6232 4 of 15

the advancement of fuzzy differential equations by introducing the CIF heat equation,
which integrates the strengths of CFS with intuitionistic fuzzy logic. Furthermore, it
presents a new class of solutions involving CIF-valued functions and extends the concept
of Hukuhara differentiability to a more expressive and generalized setting. Additionally,
it opens new research directions for addressing higher-order uncertainties, where ampli-
tude, phase, and hesitation coexist—offering a mathematically richer framework for future
developments. From a methodological perspective, the study demonstrates how classical
numerical schemes can be adapted to solve partial differential equations under complex
intuitionistic fuzzy conditions, thereby broadening the applicability of traditional solvers
in uncertain environments. Moreover, this work lays the foundation for integrating the
proposed framework with time-fractional derivatives, stochastic modeling, and hybrid in-
telligent systems in future research.

It is well known that values in the interval [0, 1] represent the fuzzification used in fuzzy
heat or diffusion equations. The objective of the current research is to extend this concept
by focusing on the fuzzification of the CIF heat equation, which is characterized by two
distinct components: the amplitude and phase terms within the unit disk in the complex
plane C. This approach aims to provide a generalization and improved accuracy for solving
intuitionistic fuzzy heat equations, considering the novel semantics of periodicity embedded
within CIF information, as discussed in [27].The following flowchart Figuer 1 illustrates
the modeling framework proposed in this research.

Figure 1: Modeling framework



A. U. Alkouri et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6232 5 of 15

2. Complex Intuitionistic Fuzzy Heat Equation

At this stage, we consider the general form of the one-dimensional fuzzy heat equation
[34]:

∂ũ(ξ, t)

∂t
= D̃(ξ, t)

∂2ũ(ξ, t)

∂ξ2
+ b̃(ξ, t), 0 < ξ < l, t > 0

ũ(ξ, 0) = f̃(ξ), ũ(0, t) = g̃(t), ũ(l, t) = z̃(t) (1)

Such that:
ũ(ξ, t): is the unknown fuzzy function at ξ and t variables.
∂ũ(ξ,t)

∂t , ∂
2ũ(ξ,t)
∂ξ2

: are the first order and the second order fuzzy partial spatial derivatives

with involving D̃(ξ, t) and b̃(ξ, t) are known fuzzy functions. Also, ũ(0, t) and ũ(l, t) are
the fuzzy known boundary conditions and the ũ(0, ξ) is the fuzzy known initial condition,
where the range of the given functions is confined to the unit disk in C.

The fuzzy heat equation in Eq. 1 is transform to CF heat equation: [33]:

∂ũ(ξ,t;r,θ)
∂t =

[
q̃1(r) e

iα w̃1(θ) s1(ξ, t)
] ∂2ũ(ξ,t;r,θ)

∂ξ2
+ q̃2(r) e

iα w̃2(θ) s2(ξ, t)

ũ(ξ, 0; r, θ) = q̃3(r) e
iα w̃3(θ) s3(ξ)

ũ(0, t; r, θ) = q̃4(r) e
iα w̃4(θ) s4(t) , ũ(l, t; r, θ) = q̃5(r) e

iα w̃5(θ) s5(t)

(2)

where in Eq. 1 the CF functions or fuzzy parameters D̃(ξ), b̃(ξ), f̃(ξ), g̃(t) and z̃(t)
are being CF convex normalized numbers that defined as following [33].

D̃(ξ, t) = q̃1 eiα w̃1 s1(ξ, t)

b̃(ξ, t) = q̃2 eiα w̃2 s2(ξ, t)

f̃(ξ) = q̃3 eiα w̃3 s3(ξ)

g̃(t) = q̃4 eiα w̃4 s4(t)

z̃(t) = q̃5 eiα w̃5 s5(t)

where s1(ξ, t), s2(ξ, t), s3(ξ), s4(t)and s5(t) are known crisp functions of x and t vari-
ables with q̃1, q̃2, q̃3, q̃4, q̃5, w̃1, w̃2, w̃3, w̃4, and w̃5 are fuzzy normalized convex numbers.

Now, the CF heat equation in Eq. 2 is generalized to represent the general form of the
CIF heat equation using the single parametric form based on the r, β-cut approach for all
r, β, θ ∈ [0, 1] as follows:

[ũ(ξ, t)]r, θ,β = [ u(ξ, t; r, θ), u(ξ, t; r, θ)] ,
[
u

′
(ξ, t;β, θ), u′(ξ, t;β, θ)

]
(3)

[
∂ũ(ξ, t)

∂t

]
r,θ,β

=

[
∂u(ξ, t; r, θ)

∂t
,
∂u(ξ, t; r, θ)

∂t

]
,

[
∂u

′
(ξ, t;β, θ)

∂t
,

∂u′(ξ, t;β, θ)

∂t

]
(4)



A. U. Alkouri et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6232 6 of 15

[
∂2ũ(ξ, t)

∂ξ2

]
r,θ,β

=

[
∂2u(ξ, t; r, θ)

∂ξ2
,
∂2u(ξ, t; r, θ)

∂ξ2

]
,

[
∂2u

′
(ξ, t;β, θ)

∂ξ2
,

∂2u′(ξ, t;β, θ)

∂ξ2

]
(5)

[
D̃(ξ, t)

]
r,θ,β

=
[
D(ξ, t; r, θ), D(ξ, t; r, θ)

]
,
[
D

′
(ξ, t;β, θ), D′(ξ, t;β, θ)

]
(6)

[
b̃(ξ, t)

]
r,θ,β

=
[
b(ξ, t; r, θ), b(ξ, t; r, θ)

]
,
[
b
′
(ξ, t;β, θ), b′(ξ, t;β, θ)

]
(7)

[ũ(ξ, 0)]r,θ,β = [u(ξ, 0; r, θ), u(ξ, 0; r, θ)] ,
[
u

′
(ξ, 0;β, θ), u′(ξ, 0;β, θ)

]
(8)

[ũ(0, t)]r,θ,β = [u(0, t; r, θ), u(0, t; r, θ)] ,
[
u

′
(0, t;β, θ), u′(0, t;β, θ)

]
(9)

[ũ(l, t)]r,θ,β = [u(l, t; r, θ), u(l, t; r, θ)] ,
[
u

′
(l, t;β, θ), u′(l, t;β, θ)

]
(10)

[
f̃(ξ)

]
r,θ,β

=
[
f(x; r, θ), f(x; r, θ)

]
,

[
f ,(x; r, θ), f ,(x; r, θ)

]
(11)

[g̃ (t)]r,θ,β =
[
g(t; r, θ), g(t; r, θ)

]
,
[
g
′
(t; r, θ), g′(t;β, θ)

]
[z̃(t)]r,θ,β = [z(t; r, θ), z(t; r, θ)] ,

[
z
′
(t; r, θ), z′(t;β, θ)

] (12)



[
D̃(ξ, t)

]
r,θ,β

=
[[
q
1
(r), q1(r)

]
eiα [w1(θ),w1(θ)]s1(ξ, t)

]
,
[[
q
′

1
(β), q′

1(β)
]
eiα [w1(θ),w1(θ)]s1(ξ, t)

]
[
b̃(ξ, t)

]
r,θ,β

=
[[
q
2
(r), q2(r)

]
eiα[w2(θ),w2(θ)] s2(ξ, t)

]
,
[[
q
′

2
(β), q′

2(β)
]
eiα[w2(θ),w2(θ)] s2(ξ, t)

]
[
f̃(ξ)

]
r,θ,β

=
[[
q
3
(r), q3(r)

]
eiα[w3(θ),w3(θ)] s3(ξ)

]
,
[[
q
′

3
(β), q′

3(β)
]
eiα[w3(θ),w3(θ)] s3(ξ)

]
[g̃(t)]r,θ,β =

[[
q
4
(r), q4(r)

]
eiα [w4(θ),w4(θ)] s4(t)

]
,
[[
q
′

4
(β), q′

4(β)
]
eiα [w4(θ),w4(θ)] s4(t)

]
[z̃(t)]r,θ,β =

[[
q
5
(r), q5(r)

]
eiα [w5(θ),w5(θ)] s5(t)

]
,
[[
q
′

5
(β), q′

5(β)
]
eiα [w5(θ),w5(θ)] s5(t)

]
(13)

The complex membership and non-membership functions are defined by using the fuzzy
extension principle [33]as follows:{

u(ξ, t; r, θ) = [min {ũ (µ̃(r, θ), t))|µ̃(r, θ) ∈ ũ(ξ, t; r, θ)}]
u(ξ, t; r, θ) = [max {ũ (µ̃(r, θ), t)|µ̃(r, θ) ∈ ũ(ξ, t; r, θ)}] (14)

{
u

′
(ξ, t;β, θ) = [min {ũ (µ̃(β, θ), t))|µ̃(r, θ) ∈ ũ(ξ, t;β, θ)}]

u′(ξ, t;β, θ) = [max {ũ (µ̃(β, θ), t)|µ̃(r, θ) ∈ ũ(ξ, t;β, θ)}]
(15)

Here, substituting all Equations 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, and 13 into equation 2,
the following is obtained:



A. U. Alkouri et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6232 7 of 15


∂u(ξ,t;r,θ)

∂t =
[
q
1
(r) eiα w1(θ) s1(ξ, t)

]
∂2u(ξ,t;r,θ)

∂ξ2
+ q

2
(r) eiα w2(θ) s2(ξ, t)

u(ξ, 0; r, θ) = q
3
(r) eiα w3(θ) s3(ξ)

u(0, t; r, θ) = q
4
(r) eiα w4(θ) s4(t) , u(l, t; r, θ) = q

5
(r) eiα w5(θ) s5(t)

(16)


∂u(ξ,t;r,θ)

∂t =
[
q1(r) e

iα w1(θ) s1(ξ, t)
] ∂2u(ξ,t;r,θ)

∂ξ2
+ q2(r) e

iα w2(θ) s2(ξ, t)

u(ξ, 0; r, θ) = q3(r) e
iα w3(θ) s3(ξ)

u(0, t; r, θ) = q4(r) e
iα w4(θ) s4(t) , u(l, t; r, θ) = q5(r) e

iα w5(θ) s5(t)

(17)


∂u

′
(ξ,t;β,θ)
∂t =

[
q
′

1
(β) eiα w1(θ) s1(ξ, t)

]
∂2u

′
(ξ,t;β,θ)
∂ξ2

+ q
′

2
(β) eiα w2(θ) s2(ξ, t)

u
′
(ξ, 0;β, θ) = q

′

3
(β) eiα w3(θ) s3(ξ)

u
′
(0, t;β, θ) = q

′

4
(β) eiα w4(θ) s4(t) , u

′
(l, t;β, θ) = q

′

5
(β) eiα w5(θ) s5(t)

(18)


∂u′ (ξ,t;β,θ)

∂t =
[
q′

1(β) e
iα w1(θ) s1(ξ, t)

]
∂2u′ (ξ,t;β,θ)

∂ξ2
+ q′

2(β) e
iα w2(θ) s2(ξ, t)

u′(ξ, 0;β, θ) = q′
3(β) e

iα w3(θ) s3(ξ)

u′(0, t;β, θ) = q′
4(β) e

iα w4(θ) s4(t) , u′(l, t;β, θ) = q′
5(β) e

iα w5(θ) s5(t)

(19)

The Eq. 16 and Eq. 17 display the lower and upper bounds of membership function
respectively of the general form of CIF heat equation while the Eq. 18 and Eq. 19 present
the lower and upper bounds of non-membership function respectively of the general form
of CIF heat equation.

3. Forward Time Centered Space Scheme for solution of complex
intuitionistic fuzzy heat equation

This section highlights the adaptation and utilization of a forward difference for the
first order derivative in time with centre difference for the second-order derivative in space
to solve the CIF heat equation,

The partial time derivative
∂u(ξ, t; r, θ)

∂t
,
∂u(ξ, t; r, θ)

∂t
,
∂u

′
(ξ, t;β, θ)

∂t
,
∂u′(ξ, t;β, θ)

∂t
is

discretised as follows:

∂ui,j(ξ, t; r, θ)

∂t
=

ui,j+1(ξ, t; r, θ)− ui,j(ξ, t; r, θ)

∆t
(20)

∂ui,j(ξ, t; r, θ)

∂t
=

ui,j+1(ξ, t; r, θ)− ui,j(ξ, t; r, θ)

∆t
(21)

∂u
′
i,j(ξ, t;β, θ)

∂t
=

u
′
i,j+1(ξ, t;β, θ)− u

′
i,j(ξ, t;β, θ)

∆t
(22)



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∂u′
i,j(ξ, t;β, θ)

∂t
=

u′
i,j+1(ξ, t;β, θ)− u′

i,j(ξ, t;β, θ)

∆t
(23)

Furthermore, the second fuzzy partial derivatives

∂2ui,j(ξ, t; r, θ)

dξ2
,
∂2ui,j(ξ, t; r, θ)

dξ2
,
∂2u

′
i,j(ξ, t;β, θ)

dξ2
,
∂2u′

i,j(ξ, t;β, θ)

dξ2

can be formulated as:

∂2ui,j(ξ, t; r, θ)

dξ2
=

ui+1,j(ξ, t; r, θ)− 2ui,j(ξ, t; r, θ) + ui−1,j(ξ, t; r, θ)

∆ξ2
(24)

∂2ui,j(ξ, t; r, θ)

dξ2
=

ui+1,j(ξ, t; r, θ)− 2ui,j(ξ, t; r, θ) + ui−1,j(ξ, t; r, θ)

∆ξ2
(25)

∂2u
′
i,j(ξ, t;β, θ)

dξ2
=

u
′
i+1,j(ξ, t;β, θ)− 2u

′
i,j(ξ, t;β, θ) + u

′
i−1,j(ξ, t;β, θ)

∆ξ2
(26)

∂2u′
i,j(ξ, t;β, θ)

dξ2
=

u′
i+1,j(ξ, t;β, θ)− 2u′

i,j(ξ, t;β, θ) + u′
i−1,j(ξ, t;β, θ)

∆ξ2
(27)

Now substitute Eq. 20, 21, 22, 23, 24, 25, 26, and 27 in Eq. 16, 17, 18, and 19
respectively in order to obtain the following:

ui,j+1(ξ, t; r, θ)− ui,j(ξ, t; r, θ)

∆t
=

D(ξ, t; r, θ)
ui+1,j(ξ, t; r, θ)− 2ui,j(ξ, t; r, θ) + ui−1,j(ξ, t; r, θ)

∆ξ2
+ b(ξ, t; r, θ)

(28)

ui,j+1(ξ, t; r, θ)− ui,j(ξ, t; r, θ)

∆t
=

D(ξ, t; r, θ)
ui+1,j(ξ, t; r, θ)− 2ui,j(ξ, t; r, θ) + ui−1,j(ξ, t; r, θ)

∆ξ2
+ b(ξ, t; r, θ)

(29)

u
′
i,j+1(ξ, t;β, θ)− u

′
i,j(ξ, t;β, θ)

∆t
=

D
′
(ξ, t;β, θ)

u
′
i+1,j(ξ, t;β, θ)− 2u

′
i,j(ξ, t;β, θ) + u

′
i−1,j(ξ, t;β, θ)

∆ξ2
+ b

′
(ξ, t;β, θ)

(30)

u′
i,j+1(ξ, t;β, θ)− u′

i,j(ξ, t;β, θ)

∆t
=

D′(ξ, t;β, θ)
u′

i+1,j(ξ, t;β, θ)− 2u′
i,j(ξ, t;β, θ) + u′

i−1,j(ξ, t;β, θ)

∆ξ2
+ b′(ξ, t;β, θ)

(31)



A. U. Alkouri et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6232 9 of 15

By assume that s̃(r, θ) = D̃(ξ,t;r,θ)∆t

∆ξ2
and s̃(β, θ) = D̃(ξ,t;β,θ)∆t

∆ξ2
then the Eq. 28, 29,

30, and 31 are simplified to get lower and upper bounds of membership function and non-
membership function respectively of the general form of CIF heat equation. for all r, θ, β ∈
[0, 1] as follows:

ui,j+1(ξ, t; r, θ) = (1−2s)ui,j(ξ, t; r, θ)+s (ui+1,j(ξ, t; r, θ)+ui−1,j(ξ, t; r, θ)+∆t b(ξ, t; r, θ)
(32)

ui,j+1(ξ, t; r, θ) = (1−2s)ui,j(ξ, t; r, θ)+s(ui+1,j(ξ, t; r, θ)+ui−1,j(ξ, t; r, θ)+∆t b(ξ, t; r, θ)
(33)

u
′
i,j+1(ξ, t;β, θ) = (1−2s)u

′
i,j(ξ, t;β, θ)+s (u

′
i+1,j(ξ, t;β, θ)+u

′
i−1,j(ξ, t;β, θ)+∆t b

′
(ξ, t;β, θ)

(34)

u′
i,j+1(ξ, t;β, θ) = (1−2s)u′

i,j(ξ, t;β, θ)+s(u′
i+1,j(ξ, t;β, θ)+u′

i−1,j(ξ, t;β, θ)+∆t b′(ξ, t;β, θ)
(35)

4. Numerical Simulation

The CIF heat equation is generalized by adding the complex non-membership term to
CF heat equation as discussed in [33] as the following:

∂Ũ(ξ, t)

∂t
− ∂2Ũ(ξ, t)

∂ξ2
= 0, 0 < ξ < 1, t > 0 (36)

where the boundary conditions ũ(0, t) = ũ(1, t) = 0 and initial condition
ũ(ξ, 0) = f̃(ξ), with CF function f̃(ξ) is defined as follows

f̃(ξ) = k̃ ei2π w̃Cos
(
πξ − π

2

)
(37)

In r-cut and β-cut formula of the intuitionistic fuzzy number is shown as below:[
k̃
]
r,β

=
{ [

k(r), k(r)
]
,
[
k(β), k(β)

] }
= {[r − 1, 1− r], [−1.25β, 1.25β]} (38)

[w̃]θ,β = { [w(θ), w(θ)] , [w(β), w(β)] } = {[θ − 1, 1− θ], [−1.25β, 1.25β]} (39)

The exact complex intuitionistic solution of Eq. 36 is described as below:

ũ(ξ, t; r) = k̃ ei2π w̃ −πtCos
(
πξ − π

2

)
(40)



A. U. Alkouri et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6232 10 of 15

Figure 2: The exact solution of Eq. 33 at θ = 0.2 and for all r, β ∈ [0, 1]

Figure 3: The numerical and exact solutions by FTCS of Equation 33 at t = 0.05, ξ = 0.9 and θ = 0.2 for
all r, β ∈ [0, 1]



A. U. Alkouri et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6232 11 of 15

Table 1: Numerical lower solutions of Eq. 33 by FTCS at t = 0.05, ξ = 0.9, for every r, β, θ ∈ [0, 1].

θ r, β ũ(0.9, 0.5; r, θ) Ẽ(0.9, 0.5; r, θ) ũ(0.9, 0.5;β, θ) Ẽ(0.9, 0.5;β, θ)

0.2

0 −0.09091− 0.279796i 0.0030 + 0.0095i 0 0
0.2 −0.07273− 0.22383i 0.0025 + 0.0076i −0.021818− 0.067149i 0.00074276 + 0.002286i
0.4 −0.05455− 0.16787i 0.0019 + 0.0057i −0.043636− 0.1343i 0.0014855 + 0.0045719i
0.6 −0.03636− 0.11191i 0.0012 + 0.0038i −0.065454− 0.20145i 0.0022283 + 0.0068579i
0.8 −0.01818− 0.05596i 0.00061 + 0.0019i −0.087272− 0.2686i 0.002971 + 0.0091439i
1 0 0 −0.10909− 0.33574i 0.0037138 + 0.01143i

0.4

0 0.23800− 0.17292i 0.0081 + 0.0059i 0 0
0.2 0.19040− 0.13833i 0.0065 + 0.0047i 0.057120− 0.041500i 0.00194456 + 0.00141280i
0.4 0.14280− 0.10375i 0.0049 + 0.0035i 0.114241− 0.083001i 0.00388912 + 0.00282562i
0.6 0.09520− 0.06917i 0.0032 + 0.0023i 0.171361− 0.124501i 0.00583369 + 0.00423842i
0.8 0.04760− 0.03458i 0.0016 + 0.0012i 0.228481− 0.166001i 0.00777825 + 0.00565123i
1 0 0 0.285602− 0.207502i 0.00972282 + 0.00706404i

Table 2: Numerical lower solutions of Eq. 33 by FTCS at t = 0.05, ξ = 0.9, for every r, β, θ ∈ [0, 1].

θ r, β ũ(0.9, 0.5; r, θ) Ẽ(0.9, 0.5; r, θ) ũ(0.9, 0.5;β, θ) Ẽ(0.9, 0.5;β, θ)

0.6

0 0.23800 + 0.17292i 0.0081 + 0.0059i 0 0
0.2 0.19040 + 0.13833i 0.0065 + 0.0047i 0.057120 + 0.041500i 0.00194456 + 0.00141280i
0.4 0.14280 + 0.10375i 0.0049 + 0.0035i 0.114241 + 0.083001i 0.00388912 + 0.00282562i
0.6 0.09520 + 0.06917i 0.0032 + 0.0023i 0.171361 + 0.124501i 0.00583369 + 0.00423842i
0.8 0.04760 + 0.03458i 0.0016 + 0.0012i 0.228481 + 0.166001i 0.00777825 + 0.00565123i
1 0 0 0.285602 + 0.207502i 0.00972282 + 0.00706404i

0.8

0 −0.09091− 0.279796i 0.0030 + 0.0095i 0 0
0.2 −0.07273− 0.22383i 0.0025 + 0.0076i −0.021818 + 0.067149i 0.00074276 + 0.002286i
0.4 −0.05455− 0.16787i 0.0019 + 0.0057i −0.043636 + 0.1343i 0.0014855 + 0.0045719i
0.6 −0.03636− 0.11191i 0.0012 + 0.0038i −0.065454 + 0.20145i 0.0022283 + 0.0068579i
0.8 −0.01818− 0.05596i 0.00061 + 0.0019i −0.087272 + 0.2686i 0.002971 + 0.0091439i
1 0 0 −0.10909 + 0.33574i 0.0037138 + 0.01143i

Figure 4: The numerical and exact solutions by FTCS of Equation 33 at t = 0.05, ξ = 0.9 and θ = 0.4 for
all r, β ∈ [0, 1]



A. U. Alkouri et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6232 12 of 15

The numerical solutions derived using the explicit FTCS method demonstrate strong
agreement with the exact solution at ξ = 0.9, t = 0.05 ∀r, β, θ ∈ [0, 1], as demonstrated in
Figures 2, 3, and 4 and Tables 1, 2. Moreover, the numerical solution and exact solution
obtained from our proposed schemes assume the structure of triangular fuzzy numbers in
both the amplitude and phase parts, aligning with the characteristics of a C. In addition,
as can be seen from tables 1, 2, The accuracy of the numerical outcomes is contingent
upon the phase term value θ. This supports our theoretical framework and illustrates the
important influence of including the complex non-membership function. It is assumed that
the numerical solution of the intuitionistic fuzzy heat equation is derived from the CIF
heat equation through substitution θ = 0 and 1 . In more detail, Tables 1 and 2 shows
the complex intuitionistic lower and complex intuitionistic upper solutions for membership
and non-membership function for different values of r, θ, β ∈ [0, 1]. The results show that
the proposed scheme consistently maintains small errors. For instance, in Table 1 at r = 0
and θ = 0, the approximate value is −0.09091−0.279796i, closely matching the exact value
with an error of 0.0030 +0.0095i . Then, as r increases, the errors remain relatively small,
indicating that the proposed scheme maintains good accuracy across different r values.
Similarly, the upper solutions in Table 2 also show high accuracy. These results show that
the proposed scheme accurately models the behavior of CIF heat equation. Fig. 3 and
Fig. 4 illustrate the intuitionistic fuzzy lower and upper solutions for membership and non-
membership functions, respectively. These visual representations support the numerical
findings in Tables 1 and 2, showing good agreement between the proposed scheme results
and the exact solutions.

In addition to its mathematical contribution, the proposed model based on CIFSs
offer meaningful physical interpretations and practical relevance. CIFSs enable the si-
multaneous representation of uncertainty in amplitude, phase, and degree of hesitation,
which are critical in many real-world systems where uncertainty is multi-faceted and not
purely scalar. This makes the CIF heat equation framework particularly useful for mod-
eling and solving problems in electromagnetic signal processing, where uncertain phase
and intensity variations are common; biomedical imaging, such as MRI or EEG, where
data uncertainty includes both measurement noise and interpretative ambiguity; and ther-
mal analysis in composite or anisotropic materials, where heat conduction properties are
not precisely known. Furthermore, the ability of CIFSs to capture membership, non-
membership, and hesitation information allows for more nuanced modeling in decision-
making systems, robotics, and intelligent control under uncertainty. These applications
demonstrate the broader utility of the proposed CIF heat equation model beyond pure
mathematical theory.

4. Conclusions

In this study, we proposed and analyzed a solution approach for the CIF heat equation
using the FTCS finite difference scheme. The model incorporates CIF numbers to repre-
sent uncertainties in both the amplitude and phase of membership and non-membership
functions. This dual representation captures a higher degree of uncertainty compared to



A. U. Alkouri et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6232 13 of 15

classical or even complex fuzzy models. The results demonstrate that the FTCS scheme
preserves the structural features of CIF numbers when triangular fuzzy numbers are used
for both amplitude and phase parts. The scheme achieves a computational accuracy of
order O(∆t+∆ξ2), confirming its numerical stability and convergence under complex intu-
itionistic fuzzy conditions. Moreover, the proposed method proves to be computationally
efficient and generalizable for modeling periodic heat transfer and other dynamic systems
involving complex uncertainty. The findings support the viability of CIF heat equation for
a wide range of real-world applications, such as thermal systems, material modeling, and
engineering processes under uncertainty. This work makes a significant contribution by
extending the classical and complex fuzzy heat equation models to a more expressive CIF
framework. The integration of both membership and non-membership uncertainties—
along with phase information—offers a richer mathematical tool for handling high-order
fuzzy uncertainty. For future research, we recommend exploring the use of alternative nu-
merical schemes, such as explicit methods for time-fractional heat equations, to enhance
modeling flexibility. Additionally, further generalization toward complex bipolar fuzzy
and complex bipolar intuitionistic fuzzy heat equations is proposed to model systems.

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