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닫기 International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(1): 11-19

Published online March 25, 2023

https://doi.org/10.5391/IJFIS.2023.23.1.11

© The Korean Institute of Intelligent Systems

## Numerical Solution of Fuzzy Heat Equation with Complex Dirichlet Conditions

Hamzeh Zureigat1, Abd Ulazeez Alkouri2, Areen Al-khateeb1, Eman Abuteen3, and Sana Abu-Ghurra2

1Department of Mathematics, Faculty of Science and Technology, Jadara University, Irbid, Jordan
2Department of Mathematics, Faculty of Science, Ajloun National university, Ajloun, Jordan
3Department of Physics Basic Science, Faculty of Engineering Technology, Al-Balqa Applied University, Amman, Jordan

Correspondence to :

Received: August 27, 2022; Revised: February 2, 2023; Accepted: March 3, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Recently, complex fuzzy sets have become powerful tools for generalizing the range of fuzzy sets to wider ranges that lie on a unit disk in the complex plane. In this study, complex fuzzy numbers are discussed and applied for the first time to solve a complex fuzzy partial differential equation involving a complex fuzzy heat equation under Hukuhara differentiability. Subsequently, an explicit finite difference scheme, referred to as the forward time-center space (FTCS), was implemented to solve the complex fuzzy heat equations. The imprecision of the issue is evident in the initial and boundary conditions, as well as in the amplitude and phase terms’ coefficients, where the convex normalized triangular fuzzy numbers are extended to the unit disk in the complex plane. The proposed numerical methods utilized the properties and benefits of the complex fuzzy set theory. Furthermore, a new proof of consistency, stability, and convergence was established under this theory. A numerical example was provided to illustrate the reliability and feasibility of the proposed approach. The results obtained using the proposed approach are in adequate agreement with the exact solution and related theoretical aspects.

Keywords: Complex fuzzy sets, Finite difference methods, Fuzzy heat equations, Complex fuzzy numbers

The theory of fuzzy sets introduced by Zadeh handles uncertainty and vagueness in mathematical models to obtain a better understanding of real-life phenomena. Numerous real-world problems can be formulated as mathematical models involving differential equations. Classical (crisp) quantities in the differential equations that are uncertain and imprecise can be replaced by fuzzy quantities to reflect uncertainty. This leads to the following fuzzy differential equations. The interest in the analysis and applications of fuzzy differential equations has recently increased because of their considerable applicability in numerous fields, such as mathematical physics , engineering , and medicine .

A fuzzy partial differential equation has been used to describe the behavior of many time-dependent phenomena, including fuzzy heat conduction and fuzzy particle diffusion, in which uncertainty or vagueness exists. The fuzzy heat equation is considered one of the most significant fuzzy parabolic partial differential equations used to describe how a fuzzy quantity, such as heat, diffuses through a given region . In general, the exact analytical solution for the fuzzy heat equations is difficult to obtain. Therefore, numerical techniques are required to obtain the solution. In recent decades, several studies have been conducted to solve the fuzzy heat equation. Allahviranloo  developed and used finite difference methods to solve fuzzy heat and wave equations. Fuzzy numerical solutions were obtained using the schemes proposed under Seikkala derivative with complete error analysis. Numerical examples are presented to investigate the efficiency of the proposed schemes. An extension of the differential transformation method (DTM) was considered by Barkhordari and Kiani  to solve fuzzy partial differential equations under a strongly generalized differentiability. The DTM is an iterative method for obtaining an analytical-numerical series solution for fuzzy partial differential equations. The proposed method is investigated through several numerical experiments. The DTM was found to be a simple and effective method for obtaining analytical-numerical solutions of fuzzy partial differential equations. Wang and Qiu  proposed a fuzzy numerical technique based on the finite difference method to solve heat conduction problems that involve uncertainties in both the initial/boundary conditions and physical parameters. The fuzzy difference discrete equations are equivalently transformed into groups of interval equations based on Zadeh extension principle, including basic fuzzification and defuzzification concepts. A stability analysis of this method was also conducted. Two numerical experiments are conducted to demonstrate the feasibility of the proposed method. Recently, Bayrak and Can  considered the concept of generalized differentiability to solve fuzzy parabolic partial differential equations using the finite difference method. Here, fuzziness appeared in the initial and boundary conditions as well as the coefficients. The results obtained were compared with the exact solutions at different fuzzy levels. The finite difference method was found to be simple and efficient for solving parabolic partial differential equations.

The above studies solved the governing equations under the concept of fuzzy set theory, which has a range of values in [0, 1]. Recently, certain medical applications have been represented by fuzzy sets to solve complex biological systems and create reasonable algorithmic solutions . Fuzzy sets were developed into complex fuzzy sets (CFSs) by expanding the range of the membership function from [0, 1] to a unit disk in the complex plane. This expansion enables us to represent information in the human brain with greater detail without losing its full meaning. Normally, humans obtain meaningful information from large amounts of data and yield reasonable solutions. Human minds are affected by different phases and factors that lead to different changes in thinking and decisions. Therefore, our hypothesis was to use CFS to provide a suitable basis for the ability to summarize and extract large amounts of data related to the human brain affected by different phases/factors that are related to the performance of the task at hand.

In 1987, Buckley and his colleague  introduced the concept of fuzzy complex numbers (FCNs). Buckley’s definition incorporates complex numbers into the support of a fuzzy set to create FCNs. The fuzzy set represents the FCN as an ordinary fuzzy set with a range of membership functions within [0, 1]. However, the concept of generalizing the image of the membership function of fuzzy sets from [0, 1]to a unit disk in the complex plane crystallized a novel notion called CFSs in 2002 . CFSs differ from FCNs, a novel fuzzy set with complex-valued grades of membership functions. The concept of CFSs  has been extensively adopted, applied, and studied by numerous scholars . The innovation of the CFS appeared in further dimension membership, with the image function lying on the unit disk. The present idea of obtaining a wider range of CFS lies in its ability to denote both uncertainty and periodicity semantics simultaneously without losing its full meaning. The degree phase was developed to classify similar data measured at different phases or levels. Polar and Cartesian forms with two fuzzy components were used to represent complex fuzzy membership grades . The uncertainty and periodicity semantics are denoted by the amplitude r(x) and phase terms w(x) of the complex numbers, respectively. The values of both the amplitude and phase terms lie in [0, 1] . The CFS was reduced to a traditional fuzzy set without phase membership cite28. Tamir and Kandel  developed an axiomatic model for propositional complex fuzzy logic that imposes the strongest restrictions on complex fuzzy logic theory. CFS has certain limitations and restrictions that have forced researchers  to create an axiomatic for propositional complex fuzzy logic.

The fuzzifications of the fuzzy heat equation are represented by values in (real numbers) [0, 1]. This study aims to fuzzify the complex fuzzy heat equation, represented by two values (amplitude and phase terms) in the unit disk in the complex plane, to provide generalization and increased accuracy in its solution by considering new periodicity semantics present in complex fuzzy information [28, 36].

Consider the general form of the one-dimensional complex fuzzy heat equation 

u˜(x,t)t=D˜(x,t)2u˜(x,t)x2+b˜(x,t),0<x<l,t>0,u˜(x,0)=f˜(x),u˜(0,t)=g˜(t),u˜(l,t)=z˜(t),

where ũ(x, t) is the complex fuzzy unknown function of crisp variables x and t. u˜(x,t)t and 2u˜(x,t)x2 are first and second complex fuzzy partial derivatives, respectively, and (x, t) and (x, t) are complex fuzzy functions. ũ(0, x): is the initial complex fuzzy condition. ũ(0, t) and ũ(l, t): are complex fuzzy boundary conditions. The ranges of the complex functions lie on a unit disk in the complex plane.

In Eq. (1) the complex fuzzy functions (x), (x), (x), (t) and (t) are complex-fuzzy convex numbers defined as follows :

{D˜(x,t)=q˜1eiαw1˜s1(x,t),b˜(x,t)=q˜2eiαw2˜s2(x,t),f˜(x)=q˜3eiαw3˜s3(x),g˜(t)=q˜4eiαw4˜s4(t),z˜(t)=q˜5eiαw5˜s5(t).

Here s1(x, t), s2(x, t), s3(x), s4(t), and s5(t) are the crisp functions of the crisp variables x and t, with 1, 2, 3, 4, 5, 1, 2, 3, 4, and 5 being the fuzzy convex numbers.

Fuzzification of Eq. (1) for all r ∈ [0, 1] is as follows 

[u˜(x,t)]r,gq=u_(x,t;r,θ),u¯(x,t;r,θ),[u˜(x,t)t]r,θ=u_(x,t;r,θ)t,u¯(x,t;r,θ)t,[2u˜(x,t)x2]r,θ=2u_(x,t;r,θ)x2,2u¯(x,t;r,θ)x2,[D˜(x,t)]r,θ=D_(x,t;r,θ),D¯(x,t;r,θ),[b˜(x,t)]r,θ=b_(x,t;r,θ),b¯(x,t;r,θ),[u˜(x,0)]r,θ=u_(x,0;r,θ),u¯(x,0;r,θ),[u˜(0,t)]r,θ=u_(0,t;r,θ),u¯(0,t;r,θ),[u˜(l,t)]r,θ=u_(l,t;r,θ),u¯(l,t;r,θ),[f˜(x)]r,θ=f_(x;r,θ),f¯(x;r,θ),{[g˜(t)]r,θ=g_(t;r,θ),g¯(t;r,θ),[z˜(t)]r,θ=z_(t;r,θ),z¯(t;r,θ),{[D˜(x,t)]r,θ=[q_1(r),q¯1(r)]eiα[w_1(θ),w¯1(θ)]s1(x,t),[b˜(x,t)]r,θ=[q_2(r),q¯2(r)]eiα[w_2(θ),w¯2(θ)]s2(x,t),[f˜(x)]r=[q_3(r),q¯3(r)]eiα[w_3(θ),w¯3(θ)]s3(x),[g˜(t)]r,θ=[q_4(r),q¯4(r)]eiα[w_4(θ),w¯4(θ)]s4(t),[z˜(t)]r,θ=[q_5(r),q¯5(r)]eiα[w_5(θ),w¯5(θ)]s5(t).

A complex membership function was defined using the fuzzy extension principle (reference based on a complex).

{u_(x,t;r,θ)=min{u˜(μ˜(r;θ),t))μ˜(r,θ)u˜(x,t;r,θ)},u¯(x,t;r,θ)=max{u˜(μ˜(r;θ),t)μ˜(r,θ)u˜(x,t;r,θ)}

Substituting Eqs. (3)(13) into Eq. (1) to obtain

{u_(x,t;r,θ)t=[q_1(r)eiαw_1(θ)s1(x,t)]2u_(x,t;r,θ)x2+q_2(r)eiαw_2(θ)s2(x,t),u_(x,0;r,θ)=q_3(r)eiαw_3(θ)s3(x),u_(0,t;r,θ)=q_4(r)eiαw_4(θ)s4(t),u_(l,t;r,θ)=q_5(r)eiαw_5(θ)s5(t),{u¯(x,t;r,θ)t=[q¯1(r)eiαw¯1(θ)s1(x,t)]2u¯(x,t;r,θ)x2+q¯2(r)eiαw¯2(θ)s2(x,t),u¯(x,0;r,θ)=q¯3(r)eiαw¯3(θ)s3(x),u¯(0,t;r,θ)=q¯4(r)eiαw¯4(θ)s4(t),u¯(l,t;r,θ)=q¯5(r)eiα   w¯5(θ)s5(t).

Eqs. (15) and (16) represent the general lower and upper forms of the complex fuzzy heat equations, respectively.

### 3. FTCS Scheme for Solution of Complex Fuzzy Heat Equation

This section adapts and uses a forward difference approximation for the first-order time derivative and a central difference approximation for the second-order space derivative to solve the complex fuzzy heat equation.

The partial time derivative u_(x,t;r,θ)t,u¯(x,t;r,θ)t is discretized as follows:

u_i,j(x,t;r,θ)t=u_i,j+1(x,t;r,θ)-u_i,j(x,t;r,θ)Δt,u¯i,j(x,t;r,θ)t=u¯i,j+1(x,t;r,θ)-u¯i,j(x,t;r,θ)Δt.

Furthermore, the second partial derivatives 2u_i,j(x,t;r,θ)dx2,2u¯i,j(x,t;r,θ)dx2 can be defined as follows:

2u_i,j(x,t;r,θ)dx2=u_i+1,j(x,t;r,θ)-2u_i,j(x,t;r,θ)+u_i-1,j(x,t;r,θ)Δx2,2u¯i,j(x,t;r,θ)dx2=u¯i+1,j(x,t;r,θ)-2u¯i,j(x,t;r,θ)+u¯i-1,j(x,t;r,θ)Δx2.

Subsequently, substitute Eqs. (17) and (20) into Eqs. (15) and (16) respectively to obtain

u_i,j+1(x,t;r,θ)-u_i,j(x,t;r,θ)Δt=D_(x,t;r,θ)×u_i+1,j(x,t;r,θ)-2u_i,j(x,t;r,θ)+u_i-1,j(x,t;r,θ)Δx2+b_(x,t;r,θ),u¯i,j+1(x,t;r,θ)-u¯i,j(x,t;r,θ)Δt=D¯(x,t;r,θ)×u¯i+1,j(x,t;r,θ)-2u¯i,j(x,t;r,θ)+u¯i-1,j(x,t;r,θ)Δx2+b¯(x,t;r,θ).

By assuming that s˜(r,θ)=D˜(x,t;r,θ)ΔtΔx2, Eqs. (21)(22) are simplified to obtain the general lower and upper solution for the complex fuzzy heat equation for all r, θ ∈ [0, 1] as follows:

u_i,j+1(x,t;r,θ)=(1-2s)u_i,j(x,t;r,θ)+s(u_i+1,j(x,t;r,θ)+u_i-1,j(x,t;r,θ)+Δtb_(x,t;r,θ),u¯i,j+1(x,t;r,θ)=(1-2s)u¯i,j(x,t;r,θ)+s(u¯i+1,j(x,t;r,θ)+u¯i-1,j(x,t;r,θ)+Δtb¯(x,t;r,θ).

For each spatial grid point, Eqs. (23) and (24) were evaluated to obtain linear equations. At the end of each period, a system of linear equations is obtained. This system is then solved to obtain the values ũi,n+1(x, t; r, θ) for that particular time level.

### Theorem 1

The forward time-centered space (FTCS) scheme in Eq. (23) for complex fuzzy heat equation is stable under the condition s12.

Proof

Although the proof is for a crisp heat equation is standard, we describe it as a fuzzy complex heat equation for completeness.

Let ɛ˜i0 represent the fuzzy error of the discretization of the initial condition.

Let u˜i0=u˜i0-ɛ˜i0,u˜in and u˜in refer to a numerical solution of Eq. (15) in terms of the initial data f˜i0 and f˜i0, respectively.

Let [u˜i+1n(x,t)]r,θ=u¯(r,θ)-u_(r,θ), where r, θ ∈ [0, 1].

The fuzzy absolute error is established by the following form:

The fuzzy error equations for Eq. (12) are as follows:

[ɛ˜in]r,θ=[u˜in-u˜in]r,θ,n=1,2,,X×M,i=1,2,,X-1,ɛ˜in+1=(1-2s)ɛ˜in+s(ɛ˜i+1n+ɛ˜i-1n).

ɛ˜0n=ɛ˜Xn=0,n=1,2,,T×M. Let ɛ˜in=[ɛ˜1n,ɛ˜2n,,ɛ˜X-1n], and introduce the following fuzzy norm:

ɛ˜n22=i=1X-1hɛ˜in2.

Suppose that ɛ˜in can be expressed as follows:

ɛ˜in=λ˜ne-θi,   where θ˜i=qih.

Substituting Eq. (28) into Eq. (26) to obtain

λ˜n+1e-θi=(1-2s˜)λ˜ne-θi+s˜(λ˜ne-θi+1-λ˜ne-θi-1).

Divide Eq. (29) on λ˜ne-θi to obtain

λ˜=(1-2s˜)+s˜(e-θi+e--θi).

Since (e-θi+e--θi)=cosθ-1-2sin2(θ2) and substituted into Eq. (30) to obtain

λ˜=(1-2s˜)+s˜(2-4sin2(θ2)).

By simplifying Eq. (31) we obtain

λ˜=1-4s˜sin2(θ2).

For the Fourier method, a scheme is stable if and only if |λ̃| ≤ 1. So 1-4s˜sin2(θ2)1. Since the maximum of sin2(θ2)=1, we obtain s˜12.

The fuzzy heat equation in  is generalized by adding the phase term, as discussed in the previous sections, as a complex heat equation as follows:

U˜(x,t)t-2U˜(x,t)x2=0,0<x<1,t>0.

The boundary conditions are ũ(0, t) = ũ(1, t) = 0, and the initial condition is ũ(x, 0) = (x), where the complex fuzzy function (x) is defined as follows:

f˜(x)=k˜ei2πw˜cos (πx-π2).

Here,

[k˜]r=[-1,0,1]r=[r-1,1-r],[w˜]θ=[-1,0,1]θ=[θ-1,1-θ],

and

[E˜]r,θ=U˜(t,x;r,θ)-u˜(t,x;r,θ)={[E_]r,θ=U_(t,x;r,θ)-u_)(t,x;r,θ),[E¯]r,θ=U¯(t,x;r,θ)-u¯)(t,x;r,θ).

The exact solution of Eq. (33) is defined as follows:

u˜(x,t;r)=α˜ei2πw˜-πtcos (πx-π2).

In Figures 13 and Tables 12, the numerical solutions obtained by the explicit FTCS method possess a high degree of congruence with the exact solution at x = 0.9 and t = 0.05 for all r, θ ∈ [0, 1]. Furthermore, exact and numerical solutions to the proposed schemes assume the shape of triangular fuzzy numbers for both the real and imaginary parts, which satisfies the properties of complex fuzzy numbers. Tables 12 indicate that accuracy of the numerical results is dependent on the value of phase term θ, confirming our theoretical analysis and illustrating the importance of including the phase term. It considers that the numerical solution of the fuzzy heat equation is obtained from the complex fuzzy heat by substituting θ = 0 and 1.

In this study, a FCN was applied to solve the complex fuzzy heat equation based on the FTCS scheme. The fuzziness of the problem appears in the initial and boundary conditions, as well as the coefficients in both the amplitude and phase terms, simultaneously. The obtained results using the FTCS scheme satisfy the complex fuzzy number properties by assuming the triangular fuzzy number shape for both the real part and imaginary part and have an accuracy of order Otx2). The stability of the proposed approach indicated that the FTCS scheme was conditionally stable. Furthermore, the complex fuzzy approach was found to be general and computationally efficient for the periodic transfer of information. The presented approach can be extended to solve several linear and nonlinear complex fuzzy partial differential equations, which will be investigated in detail. Fig. 1.

Exact solution of Eq. (33) at θ = 0.2 and for all r ∈ [0, 1]. Fig. 2.

Exact and numerical solution of Eq. (33) by FTCS at t = 0.05, x = 0.9, and θ = 0.2 for all r ∈ [0, 1]. Fig. 3.

Exact and numerical solution of Eq. (33) by FTCS at t = 0.05, x = 0.9, and θ = 0.4 for all r ∈ [0, 1].

Table. 1.

Table 1. Numerical solution of Eq. (33) by FTCS at t = 0.05 and x = 0.9 for all r, θ ∈ [0, 1].

Lower solutionUpper solution

θrũ(0.9, 0.5; r, θ)(0.9, 0.5; r, θ)ũ(0.9, 0.5; r, θ)(0.9, 0.5; r, θ)
θ = 0.20−0.09091 − 0.279796i0.0030 + 0.0095i0.09091 − 0.279796i0.0030 + 0.0095i
0.2−0.07273 − 0.22383i0.0025 + 0.0076i0.07273 − 0.22383i0.0025 + 0.0076i
0.4−0.05455 − 0.16787i0.0019 + 0.0057i0.05455 − 0.16787i0.0019 + 0.0057i
0.6−0.03636 − 0.11191i0.0012 + 0.0038i0.03636 − 0.11191i0.0012 + 0.0038i
0.8−0.01818 − 0.05596i0.00061 + 0.0019i0.01818 − 0.05596i0.00061 + 0.0019i
10000

θ = 0.400.23800 − 0.17292i0.0081 + 0.0059i−0.23800 − 0.17292i0.0081 + 0.0059i
0.20.19040 − 0.13833i0.0065 + 0.0047i−0.19040 − 0.13833i0.0065 + 0.0047i
0.40.14280 − 0.10375i0.0049 + 0.0035i−0.14280 − 0.10375i0.0049 + 0.0035i
0.60.09520 − 0.06917i0.0032 + 0.0023i−0.09520 − 0.06917i0.0032 + 0.0023i
0.80.04760 − 0.03458i0.0016 + 0.0012i−0.04760 − 0.03458i0.0016 + 0.0012i
10000

Table. 2.

Table 2. Numerical solution of Eq. (33) by FTCS at t = 0.05 and x = 0.9 for all r, θ ∈ [0, 1].

Lower solutionUpper solution

θrũ(0.9, 0.5; r, θ)(0.9, 0.5; r, θ)ũ(0.9, 0.5; r, θ)(0.9, 0.5; r, θ)
θ = 0.600.23800 + 0.17292i0.0081 + 0.0059i−0.23800 + 0.17292i0.0081 + 0.0059i
0.20.19040 + 0.13833i0.0065 + 0.0047i−0.19040 + 0.13833i0.0065 + 0.0047i
0.40.14280 + 0.10375i0.0049 + 0.0035i−0.14280 + 0.10375i0.0049 + 0.0035i
0.60.09520 + 0.06917i0.0032 + 0.0023i−0.09520 + 0.06917i0.0032 + 0.0023i
0.80.04760 + 0.03458i0.0016 + 0.0012i−0.04760 + 0.03458i0.0016 + 0.0012i
10000

θ = 0.80−0.09091 + 0.279796i0.0030 + 0.0095i0.09091 + 0.279796i0.0030 + 0.0095i
0.2−0.07273 + 0.22383i0.0025 + 0.0076i0.07273 + 0.22383i0.0025 + 0.0076i
0.4−0.05455 + 0.16787i0.0019 + 0.0057i0.05455 + 0.16787i0.0019 + 0.0057i
0.6−0.03636 + 0.11191i0.0012 + 0.0038i0.03636 + 0.11191i0.0012 + 0.0038i
0.8−0.01818 + 0.05596 i0.00061 + 0.0019i0.01818 + 0.05596 i0.00061 + 0.0019i
10000

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23. Bayrak, MA, and Can, E (2020). Numerical solution of fuzzy parabolic differential equations by a finite difference methods. TWMS Journal of Applied and Engineering Mathematics. 10, 886-896.
24. Buckley, J . Fuzzy complex numbers., Proceedings of the International Symposium on Fuzzy Systems and Knowledge Engineering (ISFK), 1987, Guangzhou, China, pp.597-700.
25. Buckley, JJ (1989). Fuzzy complex numbers. Fuzzy Sets and Systems. 33, 333-345. https://doi.org/10.1016/0165-0114(89)90122-X
26. Buckley, JJ, and Qu, Y (1991). Fuzzy complex analysis I: differentiation. Fuzzy Sets and Systems. 41, 269-284. https://doi.org/10.1016/0165-0114(91)90131-9
27. Buckley, JJ (1992). Fuzzy complex analysis II: integration. Fuzzy Sets and Systems. 49, 171-179. https://doi.org/10.1016/0165-0114(92)90322-U
28. Ramot, D, Friedman, M, Langholz, G, and Kandel, A (2003). Complex fuzzy logic. IEEE Transactions on Fuzzy Systems. 11, 450-461. https://doi.org/10.1109/TFUZZ.2003.814832
29. Alkouri, AUM, Massa’deh, MO, and Ali, M (2020). On bipolar complex fuzzy sets and its application. Journal of Intelligent & Fuzzy Systems. 39, 383-397. https://doi.org/10.3233/JIFS-191350
30. Alkouri, AUM, and Salleh, AR (Array). Complex Atanassov’s intuitionistic fuzzy relation. Abstract and Applied Analysis. article no 287382
31. Alkouri, AUM, and Salleh, AR (2014). Linguistic variable, hedges and several distances on complex fuzzy sets. Journal of Intelligent & Fuzzy Systems. 26, 2527-2535. https://doi.org/10.3233/IFS-130923
32. Tamir, DE, and Kandel, A (2011). Axiomatic theory of complex fuzzy logic and complex fuzzy classes. International Journal of Computers Communications & Control. 6, 562-576. https://doi.org/10.15837/ijccc.2011.3.2135
33. Tamir, DE, Jin, L, and Kandel, A (2011). A new interpretation of complex membership grade. International Journal of Intelligent Systems. 26, 285-312. https://doi.org/10.1002/int.20454
34. Tamir, DE, Last, M, and Kandel, A (). Generalized complex fuzzy propositional logic.
35. Tamir, DE, Teodorescu, HN, Last, M, and Kandel, A. Discrete complex fuzzy logic, 1-6. https://doi.org/10.1109/NAFIPS.2012.6291020
36. Yazdanbakhsh, O, and Dick, S (2018). A systematic review of complex fuzzy sets and logic. Fuzzy Sets and Systems. 338, 1-22. https://doi.org/10.1016/j.fss.2017.01.010
37. Bertone, AM, Jafelice, RM, de Barros, LC, and Bassanezi, RC (2013). On fuzzy solutions for partial differential equations. Fuzzy Sets and Systems. 219, 68-80. https://doi.org/10.1016/j.fss.2012.12.002
38. Zureigat, HH, and Ismail, AIM. Numerical solution of fuzzy heat equation with two different fuzzifications, 85-90. https://doi.org/10.1109/SAI.2016.7555966
39. Allahviranloo, T, and Taheri, N (2009). An analytic approximation to the solution of fuzzy heat equation by Adomian decomposition method. International Journal of Contemporary Mathematical Science. 4, 105-114. Hamzeh Zureigat is an assistant professor in the Department of Mathematics at Jadara University in Jordan. He received his M.A. and Ph.D. from the University of Science Malaysia (USM). His research interests focus on numerical analysis and computational optimization fields in fuzzy fractional partial differential equations. Abd Ulazeez Alkouri is an associate professor in the Department of Mathematics at Ajloun National University in Jordan. He received his Ph.D. degree from the National University of Malaysia (UKM). His research interests focus on Intuitionistic fuzzy set, complex fuzzy set, and fuzzy algebra.

E-mail: alkouriabdulazeez@gmail.com Areen Al-Khateeb is an assistant professor in the Department of Mathematics at Jadara University in Jordan. She received her M.A. and Ph.D. degrees from the Eastern Mediterranean University. In addition to being a faculty member at the College of Science and Information Technology, he is currently the head of mathematics department at Jadara University. Here research interests focus on existence and stability for fractional differential equations. Eman Abuteen is an assistant professor in the Department of Mathematics at Al-Balqa Applied University in Jordan. She received her Ph.D. degree from the Jordan University. Here research interests focus on fractional calculus, numerical analysis, and differential equations.

E-mail: Dr.eman.abuteen@bau.edu.jo Sana Abu-Ghurra is an assistant professor in the Department of Mathematics at Ajloun National University in Jordan. He received his Ph.D. from the Jordan University. Here research interests focus on fractional calculus, numerical analysis, and differential equations.

E-mail: sana3j@yahoo.com

### Article

#### Original Article International Journal of Fuzzy Logic and Intelligent Systems 2023; 23(1): 11-19

Published online March 25, 2023 https://doi.org/10.5391/IJFIS.2023.23.1.11

## Numerical Solution of Fuzzy Heat Equation with Complex Dirichlet Conditions

Hamzeh Zureigat1, Abd Ulazeez Alkouri2, Areen Al-khateeb1, Eman Abuteen3, and Sana Abu-Ghurra2

1Department of Mathematics, Faculty of Science and Technology, Jadara University, Irbid, Jordan
2Department of Mathematics, Faculty of Science, Ajloun National university, Ajloun, Jordan
3Department of Physics Basic Science, Faculty of Engineering Technology, Al-Balqa Applied University, Amman, Jordan

Received: August 27, 2022; Revised: February 2, 2023; Accepted: March 3, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

Recently, complex fuzzy sets have become powerful tools for generalizing the range of fuzzy sets to wider ranges that lie on a unit disk in the complex plane. In this study, complex fuzzy numbers are discussed and applied for the first time to solve a complex fuzzy partial differential equation involving a complex fuzzy heat equation under Hukuhara differentiability. Subsequently, an explicit finite difference scheme, referred to as the forward time-center space (FTCS), was implemented to solve the complex fuzzy heat equations. The imprecision of the issue is evident in the initial and boundary conditions, as well as in the amplitude and phase terms’ coefficients, where the convex normalized triangular fuzzy numbers are extended to the unit disk in the complex plane. The proposed numerical methods utilized the properties and benefits of the complex fuzzy set theory. Furthermore, a new proof of consistency, stability, and convergence was established under this theory. A numerical example was provided to illustrate the reliability and feasibility of the proposed approach. The results obtained using the proposed approach are in adequate agreement with the exact solution and related theoretical aspects.

Keywords: Complex fuzzy sets, Finite difference methods, Fuzzy heat equations, Complex fuzzy numbers

### 1. Introduction

The theory of fuzzy sets introduced by Zadeh handles uncertainty and vagueness in mathematical models to obtain a better understanding of real-life phenomena. Numerous real-world problems can be formulated as mathematical models involving differential equations. Classical (crisp) quantities in the differential equations that are uncertain and imprecise can be replaced by fuzzy quantities to reflect uncertainty. This leads to the following fuzzy differential equations. The interest in the analysis and applications of fuzzy differential equations has recently increased because of their considerable applicability in numerous fields, such as mathematical physics , engineering , and medicine .

A fuzzy partial differential equation has been used to describe the behavior of many time-dependent phenomena, including fuzzy heat conduction and fuzzy particle diffusion, in which uncertainty or vagueness exists. The fuzzy heat equation is considered one of the most significant fuzzy parabolic partial differential equations used to describe how a fuzzy quantity, such as heat, diffuses through a given region . In general, the exact analytical solution for the fuzzy heat equations is difficult to obtain. Therefore, numerical techniques are required to obtain the solution. In recent decades, several studies have been conducted to solve the fuzzy heat equation. Allahviranloo  developed and used finite difference methods to solve fuzzy heat and wave equations. Fuzzy numerical solutions were obtained using the schemes proposed under Seikkala derivative with complete error analysis. Numerical examples are presented to investigate the efficiency of the proposed schemes. An extension of the differential transformation method (DTM) was considered by Barkhordari and Kiani  to solve fuzzy partial differential equations under a strongly generalized differentiability. The DTM is an iterative method for obtaining an analytical-numerical series solution for fuzzy partial differential equations. The proposed method is investigated through several numerical experiments. The DTM was found to be a simple and effective method for obtaining analytical-numerical solutions of fuzzy partial differential equations. Wang and Qiu  proposed a fuzzy numerical technique based on the finite difference method to solve heat conduction problems that involve uncertainties in both the initial/boundary conditions and physical parameters. The fuzzy difference discrete equations are equivalently transformed into groups of interval equations based on Zadeh extension principle, including basic fuzzification and defuzzification concepts. A stability analysis of this method was also conducted. Two numerical experiments are conducted to demonstrate the feasibility of the proposed method. Recently, Bayrak and Can  considered the concept of generalized differentiability to solve fuzzy parabolic partial differential equations using the finite difference method. Here, fuzziness appeared in the initial and boundary conditions as well as the coefficients. The results obtained were compared with the exact solutions at different fuzzy levels. The finite difference method was found to be simple and efficient for solving parabolic partial differential equations.

The above studies solved the governing equations under the concept of fuzzy set theory, which has a range of values in [0, 1]. Recently, certain medical applications have been represented by fuzzy sets to solve complex biological systems and create reasonable algorithmic solutions . Fuzzy sets were developed into complex fuzzy sets (CFSs) by expanding the range of the membership function from [0, 1] to a unit disk in the complex plane. This expansion enables us to represent information in the human brain with greater detail without losing its full meaning. Normally, humans obtain meaningful information from large amounts of data and yield reasonable solutions. Human minds are affected by different phases and factors that lead to different changes in thinking and decisions. Therefore, our hypothesis was to use CFS to provide a suitable basis for the ability to summarize and extract large amounts of data related to the human brain affected by different phases/factors that are related to the performance of the task at hand.

In 1987, Buckley and his colleague  introduced the concept of fuzzy complex numbers (FCNs). Buckley’s definition incorporates complex numbers into the support of a fuzzy set to create FCNs. The fuzzy set represents the FCN as an ordinary fuzzy set with a range of membership functions within [0, 1]. However, the concept of generalizing the image of the membership function of fuzzy sets from [0, 1]to a unit disk in the complex plane crystallized a novel notion called CFSs in 2002 . CFSs differ from FCNs, a novel fuzzy set with complex-valued grades of membership functions. The concept of CFSs  has been extensively adopted, applied, and studied by numerous scholars . The innovation of the CFS appeared in further dimension membership, with the image function lying on the unit disk. The present idea of obtaining a wider range of CFS lies in its ability to denote both uncertainty and periodicity semantics simultaneously without losing its full meaning. The degree phase was developed to classify similar data measured at different phases or levels. Polar and Cartesian forms with two fuzzy components were used to represent complex fuzzy membership grades . The uncertainty and periodicity semantics are denoted by the amplitude r(x) and phase terms w(x) of the complex numbers, respectively. The values of both the amplitude and phase terms lie in [0, 1] . The CFS was reduced to a traditional fuzzy set without phase membership cite28. Tamir and Kandel  developed an axiomatic model for propositional complex fuzzy logic that imposes the strongest restrictions on complex fuzzy logic theory. CFS has certain limitations and restrictions that have forced researchers  to create an axiomatic for propositional complex fuzzy logic.

The fuzzifications of the fuzzy heat equation are represented by values in (real numbers) [0, 1]. This study aims to fuzzify the complex fuzzy heat equation, represented by two values (amplitude and phase terms) in the unit disk in the complex plane, to provide generalization and increased accuracy in its solution by considering new periodicity semantics present in complex fuzzy information [28, 36].

### 2. Heat Equation in Complex Fuzzy Environment

Consider the general form of the one-dimensional complex fuzzy heat equation 

$∂u˜(x,t)∂t=D˜(x,t)∂2u˜(x,t)∂x2+b˜(x,t), 00,u˜(x,0)=f˜(x), u˜(0,t)=g˜(t), u˜(l,t)=z˜(t),$

where ũ(x, t) is the complex fuzzy unknown function of crisp variables x and t. $∂u˜(x,t)∂t$ and $∂2u˜(x,t)∂x2$ are first and second complex fuzzy partial derivatives, respectively, and (x, t) and (x, t) are complex fuzzy functions. ũ(0, x): is the initial complex fuzzy condition. ũ(0, t) and ũ(l, t): are complex fuzzy boundary conditions. The ranges of the complex functions lie on a unit disk in the complex plane.

In Eq. (1) the complex fuzzy functions (x), (x), (x), (t) and (t) are complex-fuzzy convex numbers defined as follows :

${D˜ (x,t)=q˜1eiαw1˜s1(x,t),b˜ (x,t)=q˜2eiαw2˜s2(x,t),f˜(x)=q˜3eiαw3˜s3(x),g˜(t)=q˜4eiαw4˜s4(t),z˜(t)=q˜5eiαw5˜s5(t).$

Here s1(x, t), s2(x, t), s3(x), s4(t), and s5(t) are the crisp functions of the crisp variables x and t, with 1, 2, 3, 4, 5, 1, 2, 3, 4, and 5 being the fuzzy convex numbers.

Fuzzification of Eq. (1) for all r ∈ [0, 1] is as follows 

$[u˜(x,t)]r,gq=u_(x,t;r,θ),u¯(x,t;r,θ),$$[∂u˜(x,t)∂t]r,θ=∂u_(x,t;r,θ)∂t,∂u¯(x,t;r,θ)∂t,$$[∂2u˜(x,t)∂x2]r,θ=∂2u_(x,t;r,θ)∂x2,∂2u¯(x,t;r,θ)∂x2,$$[D˜(x,t)]r,θ=D_(x,t;r,θ),D¯(x,t;r,θ),$$[b˜(x,t)]r,θ=b_(x,t;r,θ),b¯(x,t;r,θ),$$[u˜(x,0)]r,θ=u_(x,0;r,θ),u¯(x,0;r,θ),$$[u˜(0,t)]r,θ=u_(0,t;r,θ),u¯(0,t;r,θ),$$[u˜(l,t)]r,θ=u_(l,t;r,θ),u¯(l,t;r,θ),$$[f˜(x)]r,θ=f_(x;r,θ),f¯(x;r,θ),$${[g˜ (t)]r,θ=g_(t;r,θ),g¯(t;r,θ),[z˜ (t)]r,θ=z_(t;r,θ),z¯(t;r,θ),$${[D˜(x,t)]r,θ=[q_1(r),q¯1(r)]eiα[w_1(θ),w¯1(θ)]s1(x,t),[b˜(x,t)]r,θ=[q_2(r),q¯2(r)]eiα[w_2(θ),w¯2(θ)]s2(x,t),[f˜(x)]r=[q_3(r),q¯3(r)]eiα[w_3(θ),w¯3(θ)]s3(x),[g˜(t)]r,θ=[q_4(r),q¯4(r)]eiα[w_4(θ),w¯4(θ)]s4(t),[z˜(t)]r,θ=[q_5(r),q¯5(r)]eiα[w_5(θ),w¯5(θ)]s5(t).$

A complex membership function was defined using the fuzzy extension principle (reference based on a complex).

${u_(x,t;r,θ)=min{u˜(μ˜(r;θ),t))∣μ˜(r,θ)∈u˜(x,t;r,θ)},u¯(x,t;r,θ)=max{u˜(μ˜(r;θ),t)∣μ˜(r,θ)∈u˜(x,t;r,θ)}$

Substituting Eqs. (3)(13) into Eq. (1) to obtain

${∂u_(x,t;r,θ)∂t=[q_1(r)eiαw_1(θ)s1(x,t)]∂2u_(x,t;r,θ)∂x2+q_2(r)eiαw_2(θ)s2(x,t),u_(x,0;r,θ)=q_3(r)eiαw_3(θ)s3(x),u_(0,t;r,θ)=q_4(r)eiαw_4(θ)s4(t),u_(l,t;r,θ)=q_5(r)eiαw_5(θ)s5(t),$${∂u¯(x,t;r,θ)∂t=[q¯1(r)eiαw¯1(θ)s1(x,t)]∂2u¯(x,t;r,θ)∂x2+q¯2(r)eiαw¯2(θ)s2(x,t),u¯(x,0;r,θ)=q¯3(r)eiαw¯3(θ)s3(x),u¯(0,t;r,θ)=q¯4(r)eiαw¯4(θ)s4(t),u¯(l,t;r,θ)=q¯5(r)eiα w¯5(θ)s5(t).$

Eqs. (15) and (16) represent the general lower and upper forms of the complex fuzzy heat equations, respectively.

### 3. FTCS Scheme for Solution of Complex Fuzzy Heat Equation

This section adapts and uses a forward difference approximation for the first-order time derivative and a central difference approximation for the second-order space derivative to solve the complex fuzzy heat equation.

The partial time derivative $∂u_(x,t;r,θ)∂t,∂u¯(x,t;r,θ)∂t$ is discretized as follows:

$∂u_i,j(x,t;r,θ)∂t=u_i,j+1(x,t;r,θ)-u_i,j(x,t;r,θ)Δt,$$∂u¯i,j(x,t;r,θ)∂t=u¯i,j+1(x,t;r,θ)-u¯i,j(x,t;r,θ)Δt.$

Furthermore, the second partial derivatives $∂2u_i,j(x,t;r,θ)dx2,∂2u¯i,j(x,t;r,θ)dx2$ can be defined as follows:

$∂2u_i,j(x,t;r,θ)dx2=u_i+1,j(x,t;r,θ)-2u_i,j(x,t;r,θ)+u_i-1,j(x,t;r,θ)Δx2,$$∂2u¯i,j(x,t;r,θ)dx2=u¯i+1,j(x,t;r,θ)-2u¯i,j(x,t;r,θ)+u¯i-1,j(x,t;r,θ)Δx2.$

Subsequently, substitute Eqs. (17) and (20) into Eqs. (15) and (16) respectively to obtain

$u_i,j+1(x,t;r,θ)-u_i,j(x,t;r,θ)Δt=D_(x,t;r,θ)×u_i+1,j(x,t;r,θ)-2u_i,j(x,t;r,θ)+u_i-1,j(x,t;r,θ)Δx2+b_(x,t;r,θ),$$u¯i,j+1(x,t;r,θ)-u¯i,j(x,t;r,θ)Δt=D¯(x,t;r,θ)×u¯i+1,j(x,t;r,θ)-2u¯i,j(x,t;r,θ)+u¯i-1,j(x,t;r,θ)Δx2+b¯(x,t;r,θ).$

By assuming that $s˜(r,θ)=D˜(x,t;r,θ)ΔtΔx2$, Eqs. (21)(22) are simplified to obtain the general lower and upper solution for the complex fuzzy heat equation for all r, θ ∈ [0, 1] as follows:

$u_i,j+1(x,t;r,θ)=(1-2s)u_i,j(x,t;r,θ)+s(u_i+1,j(x,t;r,θ)+u_i-1,j(x,t;r,θ)+Δtb_(x,t;r,θ),$$u¯i,j+1(x,t;r,θ)=(1-2s)u¯i,j(x,t;r,θ)+s(u¯i+1,j(x,t;r,θ)+u¯i-1,j(x,t;r,θ)+Δtb¯(x,t;r,θ).$

For each spatial grid point, Eqs. (23) and (24) were evaluated to obtain linear equations. At the end of each period, a system of linear equations is obtained. This system is then solved to obtain the values ũi,n+1(x, t; r, θ) for that particular time level.

### Theorem 1

The forward time-centered space (FTCS) scheme in Eq. (23) for complex fuzzy heat equation is stable under the condition $s≤12$.

Proof

Although the proof is for a crisp heat equation is standard, we describe it as a fuzzy complex heat equation for completeness.

Let $ɛ˜i0$ represent the fuzzy error of the discretization of the initial condition.

Let $u˜i0=u˜i′0-ɛ˜i0,u˜in$ and $u˜i′n$ refer to a numerical solution of Eq. (15) in terms of the initial data $f˜i0$ and $f˜i0$, respectively.

Let $[u˜i+1n(x,t)]r,θ=u¯(r,θ)-u_(r,θ)$, where r, θ ∈ [0, 1].

The fuzzy absolute error is established by the following form:

The fuzzy error equations for Eq. (12) are as follows:

$[ɛ˜in]r,θ=[u˜i′n-u˜in]r,θ,n=1,2,…,X×M,i=1,2,…,X-1,$$ɛ˜in+1=(1-2s)ɛ˜in+s(ɛ˜i+1n+ɛ˜i-1n).$

$ɛ˜0n=ɛ˜Xn=0,n=1,2,…,T×M$. Let $ɛ˜in=[ɛ˜1n,ɛ˜2n,…,ɛ˜X-1n]$, and introduce the following fuzzy norm:

$‖ɛ˜n‖22=∑i=1X-1h∣ɛ˜in∣2.$

Suppose that $ɛ˜in$ can be expressed as follows:

$ɛ˜in=λ˜n e-θi, where θ˜i=qih.$

Substituting Eq. (28) into Eq. (26) to obtain

$λ˜n+1e-θi=(1-2s˜)λ˜ne-θi+s˜ (λ˜ne-θi+1-λ˜ne-θi-1).$

Divide Eq. (29) on $λ˜ne-θi$ to obtain

$λ˜=(1-2s˜)+s˜ (e-θi+e--θi).$

Since $(e-θi+e--θi)=cosθ-1-2sin2 (θ2)$ and substituted into Eq. (30) to obtain

$λ˜=(1-2s˜)+s˜ (2-4 sin2 (θ2)).$

By simplifying Eq. (31) we obtain

$λ˜=1-4 s˜ sin2 (θ2).$

For the Fourier method, a scheme is stable if and only if |λ̃| ≤ 1. So $∣1-4s˜ sin2 (θ2)∣ ≤1$. Since the maximum of $sin2 (θ2)=1$, we obtain $s˜≤12$.

### 5. Numerical Example

The fuzzy heat equation in  is generalized by adding the phase term, as discussed in the previous sections, as a complex heat equation as follows:

$∂U˜(x,t)∂t-∂2U˜(x,t)∂x2=0, 00.$

The boundary conditions are ũ(0, t) = ũ(1, t) = 0, and the initial condition is ũ(x, 0) = (x), where the complex fuzzy function (x) is defined as follows:

$f˜(x)=k˜ei2πw˜ cos (πx-π2).$

Here,

$[k˜]r=[-1, 0, 1]r=[r-1, 1-r],$$[w˜]θ=[-1, 0, 1]θ=[θ-1, 1-θ],$

and

$[E˜]r,θ=∣U˜(t,x;r,θ)-u˜(t,x;r,θ)∣={[E_]r,θ=∣U_(t,x;r,θ)-u_)(t,x;r,θ)∣,[E¯]r,θ=∣U¯(t,x;r,θ)-u¯)(t,x;r,θ)∣.$

The exact solution of Eq. (33) is defined as follows:

$u˜(x,t;r)=α˜ei2πw˜-πt cos (πx-π2).$

In Figures 13 and Tables 12, the numerical solutions obtained by the explicit FTCS method possess a high degree of congruence with the exact solution at x = 0.9 and t = 0.05 for all r, θ ∈ [0, 1]. Furthermore, exact and numerical solutions to the proposed schemes assume the shape of triangular fuzzy numbers for both the real and imaginary parts, which satisfies the properties of complex fuzzy numbers. Tables 12 indicate that accuracy of the numerical results is dependent on the value of phase term θ, confirming our theoretical analysis and illustrating the importance of including the phase term. It considers that the numerical solution of the fuzzy heat equation is obtained from the complex fuzzy heat by substituting θ = 0 and 1.

### 6. Conclusion

In this study, a FCN was applied to solve the complex fuzzy heat equation based on the FTCS scheme. The fuzziness of the problem appears in the initial and boundary conditions, as well as the coefficients in both the amplitude and phase terms, simultaneously. The obtained results using the FTCS scheme satisfy the complex fuzzy number properties by assuming the triangular fuzzy number shape for both the real part and imaginary part and have an accuracy of order Otx2). The stability of the proposed approach indicated that the FTCS scheme was conditionally stable. Furthermore, the complex fuzzy approach was found to be general and computationally efficient for the periodic transfer of information. The presented approach can be extended to solve several linear and nonlinear complex fuzzy partial differential equations, which will be investigated in detail.

### Fig 1. Figure 1.

Exact solution of Eq. (33) at θ = 0.2 and for all r ∈ [0, 1].

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 11-19https://doi.org/10.5391/IJFIS.2023.23.1.11

### Fig 2. Figure 2.

Exact and numerical solution of Eq. (33) by FTCS at t = 0.05, x = 0.9, and θ = 0.2 for all r ∈ [0, 1].

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 11-19https://doi.org/10.5391/IJFIS.2023.23.1.11

### Fig 3. Figure 3.

Exact and numerical solution of Eq. (33) by FTCS at t = 0.05, x = 0.9, and θ = 0.4 for all r ∈ [0, 1].

The International Journal of Fuzzy Logic and Intelligent Systems 2023; 23: 11-19https://doi.org/10.5391/IJFIS.2023.23.1.11

Numerical solution of Eq. (33) by FTCS at t = 0.05 and x = 0.9 for all r, θ ∈ [0, 1].

Lower solutionUpper solution

θrũ(0.9, 0.5; r, θ)(0.9, 0.5; r, θ)ũ(0.9, 0.5; r, θ)(0.9, 0.5; r, θ)
θ = 0.20−0.09091 − 0.279796i0.0030 + 0.0095i0.09091 − 0.279796i0.0030 + 0.0095i
0.2−0.07273 − 0.22383i0.0025 + 0.0076i0.07273 − 0.22383i0.0025 + 0.0076i
0.4−0.05455 − 0.16787i0.0019 + 0.0057i0.05455 − 0.16787i0.0019 + 0.0057i
0.6−0.03636 − 0.11191i0.0012 + 0.0038i0.03636 − 0.11191i0.0012 + 0.0038i
0.8−0.01818 − 0.05596i0.00061 + 0.0019i0.01818 − 0.05596i0.00061 + 0.0019i
10000

θ = 0.400.23800 − 0.17292i0.0081 + 0.0059i−0.23800 − 0.17292i0.0081 + 0.0059i
0.20.19040 − 0.13833i0.0065 + 0.0047i−0.19040 − 0.13833i0.0065 + 0.0047i
0.40.14280 − 0.10375i0.0049 + 0.0035i−0.14280 − 0.10375i0.0049 + 0.0035i
0.60.09520 − 0.06917i0.0032 + 0.0023i−0.09520 − 0.06917i0.0032 + 0.0023i
0.80.04760 − 0.03458i0.0016 + 0.0012i−0.04760 − 0.03458i0.0016 + 0.0012i
10000

Numerical solution of Eq. (33) by FTCS at t = 0.05 and x = 0.9 for all r, θ ∈ [0, 1].

Lower solutionUpper solution

θrũ(0.9, 0.5; r, θ)(0.9, 0.5; r, θ)ũ(0.9, 0.5; r, θ)(0.9, 0.5; r, θ)
θ = 0.600.23800 + 0.17292i0.0081 + 0.0059i−0.23800 + 0.17292i0.0081 + 0.0059i
0.20.19040 + 0.13833i0.0065 + 0.0047i−0.19040 + 0.13833i0.0065 + 0.0047i
0.40.14280 + 0.10375i0.0049 + 0.0035i−0.14280 + 0.10375i0.0049 + 0.0035i
0.60.09520 + 0.06917i0.0032 + 0.0023i−0.09520 + 0.06917i0.0032 + 0.0023i
0.80.04760 + 0.03458i0.0016 + 0.0012i−0.04760 + 0.03458i0.0016 + 0.0012i
10000

θ = 0.80−0.09091 + 0.279796i0.0030 + 0.0095i0.09091 + 0.279796i0.0030 + 0.0095i
0.2−0.07273 + 0.22383i0.0025 + 0.0076i0.07273 + 0.22383i0.0025 + 0.0076i
0.4−0.05455 + 0.16787i0.0019 + 0.0057i0.05455 + 0.16787i0.0019 + 0.0057i
0.6−0.03636 + 0.11191i0.0012 + 0.0038i0.03636 + 0.11191i0.0012 + 0.0038i
0.8−0.01818 + 0.05596 i0.00061 + 0.0019i0.01818 + 0.05596 i0.00061 + 0.0019i
10000

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