# Approximate Solution of Intuitionistic Fuzzy Differential Equations with the Linear Differential Operator by the Homotopy Analysis Method

Authors: S. Melliani; Z. Belhallaj; M. Elomari; L. S. Chadli
Journal: Advances in Fuzzy Systems
 (2021)

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## Abstract

In this work, the purpose is to discuss the homotopy analysis method (HAM) for the use of intuitionistic fuzzy differential equations with the linear differential operator. Furthermore, a numerical example is presented to shed light on the capability of the present method, and the numerical results illustrated by adopting the homotopy perturbation method (HPM) are compared with the exact solution to ensure the validity of our outcomes.


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## Body

## 1. Introduction

Intuitionistic fuzzy sets theory plays a major key role in different domains such as industry, audiovisual systems, robotics, the control of complex processes, the transmission of energy in a material medium in various forms, and the evolution of certain populations and organisms. The notion of intuitionistic fuzzy theory (IFT) was first mentioned by Atanassov [1–3] as a generalization of Zadeh’s fuzzy sets [4]. The concept of the intuitionistic fuzzy metric space was introduced by Melliani et al. [5]. The authors in [6] constructed the existence and uniqueness theorem of a solution to the nonlocal intuitionistic fuzzy differential equation. The study of numerical methods for solving intuitionistic fuzzy differential equations has been rapidly growing in recent years. It is difficult to obtain exact solutions for intuitionistic fuzzy DEs, and hence, some numerical methods presented in [7–11]. In [12–14], the authors gave a thorough and systematic introduction to the latest research achievement on the theories of interval-valued intuitionistic fuzzy sets and their applications to multiattribute group decision-making (MAGDM). Also, some arithmetic aggregation operators for the triangular Atanassov intuitionistic fuzzy number (TAIFN) are defined in [15]. The first publication on intuitionistic fuzzy partial differential equations was [23].In this paper, we well resolve the intuitionistic fuzzy differential equation using an analytical method called homotopy analysis method (HAM). This approach was first set by Liao in 1992 [18, 19]. Numerous authors used this method to resolve different linear and nonlinear differential equations for the benefit of many practical use cases in scientific and engineering problems [19–23], and the homotopy analysis method rapidly converges in many linear and nonlinear problems. The principal benefit of the homotopy analysis method (HAM) is the applicability to give an approximate and exact solution to linear and nonlinear problems, without the necessity of discretization and linearization as in the numerical methods. The structure of this paper is organized as follows.After discussing the motivation behind this research in the introduction section, Section2 is intended to give the basic notion of intuitionistic fuzzy sets (IFS) and intuitionistic fuzzy numbers (IFN). Section 3 is dedicated to present some basic notions about the homotopy method. For the sake of clarity, the homotopy perturbation method for the use of resolving the intuitionistic fuzzy differential equations with the linear differential operator is presented. In Section 4, we give an example to illustrate the capability and flexibility of the proposed method, and finally, conclusion is given in Section 5.


## 2. Preliminaries

In this section, we present the necessary definitions and notations that will be used in this work as follows.


### 2.1. Intuitionistic Fuzzy Sets

An intuitionistic fuzzy setA∈X is given by(1)A=x,κAx,ωAx|x∈X,where the function κAx,ωAx:X⟶0,1 defines, respectively, the degree of membership and degree of nonmembership of the element x∈X to set A, which is a subset of X, which satisfies for every x∈X, 0≤κx+ωx≤1.For the sake of clarity, every fuzzy set has the form(2)x,κAx,κAcx|x∈X.For each intuitionistic fuzzy setA∈X, we will call(3)πAx=1−κAx−ωAx.The intuitionistic fuzzy index ofx∈A verifies that 0≤πAx≤1.


### 2.2. Intuitionistic Fuzzy Numbers

An elementκ,ω of F1 is said to be an intuitionistic fuzzy number if it satisfies the following conditions:(i)
κ,ω is normal, i.e., there exist x0, x1∈ℝ such that κx0=1 and ωx1=1(ii)
The membership functionκ is fuzzy convex, i.e., κηx1+1−ηx2≥minκx1,κx2(iii)
The nonmembership functionω is fuzzy concave, i.e., ωηx1+1−ηx2≤maxωx1,ωx2(iv)
κ is upper semicontinuous, and ω is lower semicontinuous(v)
Suppκ,ω=clx∈ℝ:|ωx<1 is boundedSo, we denote the collection of all intuitionistic fuzzy numbers byF1.Forα∈0,1 and κ,ω∈F1, the upper and lower α-cuts of κ,ω are defined by(4)κ,ωα=x∈ℝ:ωx≤1−α,κ,ωα=x∈ℝ:κx≥α.Remark 1.
Ifκ,ω∈F1, we can see κ,ωα as κα and κ,ωα as 1−ωα in the fuzzy case.We define01,0∈F1 as(5)01,0t=1,0,t=0,0,1,t≠0.Letκ,ω, κ′,ω′∈F1 and η∈ℝ; we define the following operations by(6)κ,ω⊕κ′,ω′z=supz=x+yminκx,κ′y,infz=x+ymaxωx,ω′y,ηκ,ω=ηκ,ηωifη≠0,01,0ifη=0.Forκ,ω, χ,Ψ∈F1 and η∈ℝ, the addition and scalar multiplication are defined as follows:(7)κ,ω⊕χ,Ψα=κ,ωα+χ,Ψα,ηχ,Ψα=ηχ,Ψα,κ,ω⊕χ,Ψα=κ,ωα+χ,Ψα,ηχ,Ψα=ηχ,Ψα.Definition 1.
Letκ,ω be an element of F1 and α∈0.1; we define the following sets:(8)κ,ωl+α=infx∈ℝ|κx≥α,κ,ωr+α=supx∈ℝ|κx≥ακ,ωl−α=infx∈ℝ|ωx≤1−α,κ,ωr−α=supx∈ℝ|ωx≤1−α.Remark 2.
κ,ωα=κ,ωl+α,κ,ωr+α and κ,ωα=κ,ωl−α,κ,ωr−α.Proposition 1.
For allα,β∈0,1 and κ,ω∈F1,(i)
κ,ωα⊂κ,ωα(ii)
κ,ωα and κ,ωα are nonempty compact convex sets in ℝ(iii)
Ifα≤β, then κ,ωβ⊂κ,ωα and κ,ωβ⊂κ,ωα(iv)
Ifαn↗α, then κ,ωα=∩nκ,ωαn and κ,ωα=∩nκ,ωαn
LetH be any set and α∈0,1; we denote by(9)Hα=x∈ℝ:κx≥α,Hα=x∈ℝ:ωx≤1−α.Lemma 1.
LetHα,α∈0,1 and Hα,α∈0,1 be two families of ℝ satisfying (i)–(iv) in Proposition 1; if κ and ω are defined by(10)κx=0,ifx∉H0,supα∈0,1:x∈Hα,ifx∈H0,ωx=1,ifx∉H0,1−supα∈0,1:x∈Hα,ifx∈H0,then κ,ω∈F1.Lemma 2.
A mappingd:F1×F1⟶ℝ is said to be an intuitionistic fuzzy metric on F1 if it satisfies the following conditions:(1)
dκ1,ω1,κ2,ω2≥0, ∀κ1,ω1,κ2,ω2∈F1(2)
dκ1,ω1,κ2,ω2=0 iff κ1,ω1=κ2,ω2(3)
dκ1,ω1,κ2,ω2=dκ2,ω2,κ1,ω1∀κ1,ω1,κ2,ω2∈F1(4)
dκ1,ω1,κ3,ω3≤dκ1,ω1,κ2,ω2+dκ2,ω2,κ3,ω3, ∀κ1,ω1,κ2,ω2,κ3,ω3
On the spaceF1, we will consider the following metric:(11)d∞κ,ω,χ,Ψ=14sup0<α≤1κ,ωr+α−χ,Ψr+α+14sup0<α≤1κ,ωl+α−χ,Ψl+α+14sup0<α≤1κ,ωr−α−χ,Ψr−α+14sup0<α≤1κ,ωl−α−χ,Ψl−α,where ⋅ denotes the usual Euclidean norm in ℝn.Proposition 2.
(see [24]). F1,dp is a metric space.Definition 2.
The generalized Hukuhara difference of two fuzzy numbersκ,ω, κ′,ω′∈F1 is defined as follows:(12)κ,ω⊖gHκ′,ω′=Ψ,χ⇔κ,ω=κ′,ω′+Ψ,χorκ′,ω′=κ,ω+−1Ψ,χ.Definition 3 (see [25]).
LetF:a,b⟶W1 and x0∈a,b. It is said that F is strongly generalized differentiable on x0 if ∃F′+x0, ∃F′−x0∈E1 such that(i)
For allh>0 sufficiently small, ∃F+x0+h−F+x0, F+x0−F+x0−h, and the limits (in metric D).(13)limh⟶0F+x0+h−F+x0h=limh⟶0F+x0−F+x0−hh=F′+x0,or(ii)
For allh>0 sufficiently small, ∃F+x0−F+x0+h, F+x0−h−F+x0, and the limits(14)limh⟶0F+x0−F+x0+h−h=limh⟶0F+x0−h−F+x0−h=F′+x0,or(iii)
For allh>0 sufficiently small, ∃F+x0+h−F+x0, F+x0−h−F+x0, and the limits(15)limh⟶0F+x0−F+x0−hh=limh⟶0F+x0−h−F+x0−h=F′+x0.(iv)
For allh>0 sufficiently small, ∃F+x0−F+x0+h, F+x0−F+x0−h, and the limits(16)limh⟶0F+x0−F+x0+h−h=F+x0−F+x0−h−h=F′+x0.


## 2.1. Intuitionistic Fuzzy Sets

An intuitionistic fuzzy setA∈X is given by(1)A=x,κAx,ωAx|x∈X,where the function κAx,ωAx:X⟶0,1 defines, respectively, the degree of membership and degree of nonmembership of the element x∈X to set A, which is a subset of X, which satisfies for every x∈X, 0≤κx+ωx≤1.For the sake of clarity, every fuzzy set has the form(2)x,κAx,κAcx|x∈X.For each intuitionistic fuzzy setA∈X, we will call(3)πAx=1−κAx−ωAx.The intuitionistic fuzzy index ofx∈A verifies that 0≤πAx≤1.


## 2.2. Intuitionistic Fuzzy Numbers

An elementκ,ω of F1 is said to be an intuitionistic fuzzy number if it satisfies the following conditions:(i)
κ,ω is normal, i.e., there exist x0, x1∈ℝ such that κx0=1 and ωx1=1(ii)
The membership functionκ is fuzzy convex, i.e., κηx1+1−ηx2≥minκx1,κx2(iii)
The nonmembership functionω is fuzzy concave, i.e., ωηx1+1−ηx2≤maxωx1,ωx2(iv)
κ is upper semicontinuous, and ω is lower semicontinuous(v)
Suppκ,ω=clx∈ℝ:|ωx<1 is boundedSo, we denote the collection of all intuitionistic fuzzy numbers byF1.Forα∈0,1 and κ,ω∈F1, the upper and lower α-cuts of κ,ω are defined by(4)κ,ωα=x∈ℝ:ωx≤1−α,κ,ωα=x∈ℝ:κx≥α.Remark 1.
Ifκ,ω∈F1, we can see κ,ωα as κα and κ,ωα as 1−ωα in the fuzzy case.We define01,0∈F1 as(5)01,0t=1,0,t=0,0,1,t≠0.Letκ,ω, κ′,ω′∈F1 and η∈ℝ; we define the following operations by(6)κ,ω⊕κ′,ω′z=supz=x+yminκx,κ′y,infz=x+ymaxωx,ω′y,ηκ,ω=ηκ,ηωifη≠0,01,0ifη=0.Forκ,ω, χ,Ψ∈F1 and η∈ℝ, the addition and scalar multiplication are defined as follows:(7)κ,ω⊕χ,Ψα=κ,ωα+χ,Ψα,ηχ,Ψα=ηχ,Ψα,κ,ω⊕χ,Ψα=κ,ωα+χ,Ψα,ηχ,Ψα=ηχ,Ψα.Definition 1.
Letκ,ω be an element of F1 and α∈0.1; we define the following sets:(8)κ,ωl+α=infx∈ℝ|κx≥α,κ,ωr+α=supx∈ℝ|κx≥ακ,ωl−α=infx∈ℝ|ωx≤1−α,κ,ωr−α=supx∈ℝ|ωx≤1−α.Remark 2.
κ,ωα=κ,ωl+α,κ,ωr+α and κ,ωα=κ,ωl−α,κ,ωr−α.Proposition 1.
For allα,β∈0,1 and κ,ω∈F1,(i)
κ,ωα⊂κ,ωα(ii)
κ,ωα and κ,ωα are nonempty compact convex sets in ℝ(iii)
Ifα≤β, then κ,ωβ⊂κ,ωα and κ,ωβ⊂κ,ωα(iv)
Ifαn↗α, then κ,ωα=∩nκ,ωαn and κ,ωα=∩nκ,ωαn
LetH be any set and α∈0,1; we denote by(9)Hα=x∈ℝ:κx≥α,Hα=x∈ℝ:ωx≤1−α.Lemma 1.
LetHα,α∈0,1 and Hα,α∈0,1 be two families of ℝ satisfying (i)–(iv) in Proposition 1; if κ and ω are defined by(10)κx=0,ifx∉H0,supα∈0,1:x∈Hα,ifx∈H0,ωx=1,ifx∉H0,1−supα∈0,1:x∈Hα,ifx∈H0,then κ,ω∈F1.Lemma 2.
A mappingd:F1×F1⟶ℝ is said to be an intuitionistic fuzzy metric on F1 if it satisfies the following conditions:(1)
dκ1,ω1,κ2,ω2≥0, ∀κ1,ω1,κ2,ω2∈F1(2)
dκ1,ω1,κ2,ω2=0 iff κ1,ω1=κ2,ω2(3)
dκ1,ω1,κ2,ω2=dκ2,ω2,κ1,ω1∀κ1,ω1,κ2,ω2∈F1(4)
dκ1,ω1,κ3,ω3≤dκ1,ω1,κ2,ω2+dκ2,ω2,κ3,ω3, ∀κ1,ω1,κ2,ω2,κ3,ω3
On the spaceF1, we will consider the following metric:(11)d∞κ,ω,χ,Ψ=14sup0<α≤1κ,ωr+α−χ,Ψr+α+14sup0<α≤1κ,ωl+α−χ,Ψl+α+14sup0<α≤1κ,ωr−α−χ,Ψr−α+14sup0<α≤1κ,ωl−α−χ,Ψl−α,where ⋅ denotes the usual Euclidean norm in ℝn.Proposition 2.
(see [24]). F1,dp is a metric space.Definition 2.
The generalized Hukuhara difference of two fuzzy numbersκ,ω, κ′,ω′∈F1 is defined as follows:(12)κ,ω⊖gHκ′,ω′=Ψ,χ⇔κ,ω=κ′,ω′+Ψ,χorκ′,ω′=κ,ω+−1Ψ,χ.Definition 3 (see [25]).
LetF:a,b⟶W1 and x0∈a,b. It is said that F is strongly generalized differentiable on x0 if ∃F′+x0, ∃F′−x0∈E1 such that(i)
For allh>0 sufficiently small, ∃F+x0+h−F+x0, F+x0−F+x0−h, and the limits (in metric D).(13)limh⟶0F+x0+h−F+x0h=limh⟶0F+x0−F+x0−hh=F′+x0,or(ii)
For allh>0 sufficiently small, ∃F+x0−F+x0+h, F+x0−h−F+x0, and the limits(14)limh⟶0F+x0−F+x0+h−h=limh⟶0F+x0−h−F+x0−h=F′+x0,or(iii)
For allh>0 sufficiently small, ∃F+x0+h−F+x0, F+x0−h−F+x0, and the limits(15)limh⟶0F+x0−F+x0−hh=limh⟶0F+x0−h−F+x0−h=F′+x0.(iv)
For allh>0 sufficiently small, ∃F+x0−F+x0+h, F+x0−F+x0−h, and the limits(16)limh⟶0F+x0−F+x0+h−h=F+x0−F+x0−h−h=F′+x0.


## 3. Homotopy Analysis Method

In this section, we are interested to resolve the partial differential equations with the intuitionistic fuzzy approach by adopting the homotopy analysis method. Therefore, we describe the basic idea of the homotopy analysis method by considering the following differential equation.We consider the following differential equation:(17)ℕϕx,t=fx,t,where ϕ is an unknown intuitionistic fuzzy function, ℕ is an intuitionistic fuzzy linear or nonlinear differential operator, and fx,t is an intuitionistic fuzzy function.Here,(18)ϕα,βx,t=ϕαx,t,ϕβx,t=ϕαlx,t,ϕαrx,t,ϕβlx,t,ϕβrx,t,ℕα,βx,t=ℕx,t,ϕαlx,t,α,ϕαrx,t,α,ϕβlx,t,β,ϕβrx,t,β=ℕαlx,t,ϕαl,ϕαr,ℕαrx,t,ϕαl,ϕαr,ℕβlx,t,ϕβl,ϕβr,ℕβrx,t,ϕβl,ϕβr,fα,βx,t=fαx,t,fβx,t=fαlx,t,fαrx,t,fβlx,t,fβrx,t.Now, from (17), we get(19)ℕx,t,ϕαlx,t,α,ϕαrx,t,α,ϕβlx,t,β,ϕβrx,t,β=fαlx,t,fαrx,t,fβlx,t,fβrx,t.Therefore,(20)ℕx,t,ϕαlx,t,α,ϕαrx,t,α=fαlx,t,fαrx,t,ℕx,t,ϕβlx,t,β,ϕβrx,t,β=fβlx,t,fβrx,t.In order to generalize the traditional method of homotopy, we will treat the original differential equation for the objective to construct a family of zero-order deformation equations as follows:(21)1−qΓϕαlx,t,q−uαl0x,t=CHx,tqℕϕαlx,t;q,(22)1−qΓϕαrx,t,q−uαr0x,t=CHx,tqℕϕαrx,t;q,(23)1−qΓϕβlx,t,q−uβl0x,t=CHx,tqℕϕβlx,t;q,(24)1−qΓϕβrx,t,q−uβr0x,t=CHx,tqℕϕβrx,t;q.The last equations are called the zero-order deformation equations whose solutions vary continuously with respect to the parameterq∈0,1, where q is the deformation parameter, C is the nonzero convergence control parameter, Γ is the linear operator, Hx is the nonzero auxiliary function, and u0x is the initial approximation of the desired solution.It is well known that ifq = 0, then Γϕαβx,t,0−uαβ,0x,t=0 since Γ is linear; therefore, ϕα,βx,t,0=uαβ0x,t; this is the initial condition of the problem ℕϕx,t=fx,t.And ifq = 1, then CHx,tℕϕα,βx,t,1=0 since C≠0 and Hx,t≠0; thus, ℕϕα,βx,t,1=0 such that ϕα,βx,t,1 is a solution of the problem ℕϕα,βx,t=fα,βx,t.Asq increases from 0 to 1, the solution ϕα,βx,t,q will vary from the initial condition uαβ,0x,t to the solution uαβx,t.Using Taylor’s development forϕα,βx,q with respect to q, we have(25)ϕα,βx,t,q=uα,β0x,t+∑m=1∞uα,βmx,tqm,where(26)uα,βmx,t=1m!∂mϕα,β∂qmx,t,q|q=0,and when the linear operator, the initial approximation, the auxiliary function, and the convergence control parameter are well selected, therefore, (21)–(24) converge for q = 1 and(27)ϕα,βx,t,q=uα,β0x,t+∑m=1∞uα,βmx,t.ForHx,t=1 and C=−1, equations (21)–(24) turn into(28)1−qΓϕαlx,t,q−u0x+qℕϕαlx,t,q=0,1−qΓϕαrx,t,q−u0x+qℕϕαrx,t,q=0,1−qΓϕβlx,t,q−u0x+qℕϕβlx,t,q=0,1−qΓϕβrx,t,q−u0x+qℕϕβrx,t,q=0,which are mainly used in the homotopy perturbation method (HPM), proving that this method is a special case of the homotopy analysis method (HAM).Differentiating equations (21)–(24) m times with respect to the integrated parameter q, then setting q = 0, and finally dividing them by m!, we have the mth order of deformation equations:(29)Γuαlmx,t−χmuαlm−1x,t=CHx,tRmu⟶αlm−1,(30)Γuαrmx,t−χmuαrm−1x,t=CHx,tRmu⟶αrm−1,(31)Γuβlmx,t−χmuβlm−1x,t=CHx,tRmu⟶βlm−1,(32)Γuβrmx,t−χmuβrm−1x,t=CHx,tRmu⟶βrm−1,where(33)χm=0,m≤1,1,m>1,(34)Rmu⟶αlm−1=1m−1∂m−1∂qm−1ℕϕαlx,t,q|q=0,Rmu⟶αrm−1=1m−1∂m−1∂qm−1ℕϕαrx,t,q|q=0,Rmu⟶βlm−1=1m−1∂m−1∂qm−1ℕϕβlx,t,q|q=0,Rmu⟶βrm−1=1m−1∂m−1∂qm−1ℕϕβrx,t,q|q=0.Theorem 1.
The seriesϕα,βx,t=∑m=0∞uα,βmx,t converge to uα,βx,t or uα,βmx,t which is ruled by high-order deformation equations (29)–(32) with definitions (33) and (34), and it must be the exact solution of equation (17).Proof.
Liao [18].


## 4. Numerical Application

Let us consider the fuzzy intuitionistic equation of the following form:(35)dudt−d2udx2=aex,where a=−1,0,1,−1,0,2, i.e., aα,β=α−1,1−α−β,2β with initial conditions(36)uα,β0=1+α,3−α2−2β,2+2β,uα,β′0=α,2−α1−2β,1+β.The exact solution, given by the classical solution method, is(37)uαlx,t,α=2−tet+α−1te−t+α−1ex,uαrx,t,α=2−tet−α−1te−t−α−1ex,uβlx,t,α=124−β−2tet−β+2βte−t−βex,uβrx,t,α=124−β−2tet+β+2βte−t+2βex.According to the homotopy analysis method (HPM), we are looking forϕx,t,q that has the form(38)ϕα,βx,t,q=∑m≥0uα,βmx,tqm.Thus, according to the deformation equations of orderm, (29)–(32), with(39)Rm−1u⟶αlm−1=utm−1−uxxm−1−α−1ex,Rm−1u⟶αrm−1=utm−1−uxxm−1+α−1ex,Rmu⟶βlm−1=utm−1−uxxm−1+βex,Rmu⟶βrm−1=utm−1−uxxm−1−2βex,we find that(40)Γuαlmx,t−χmuαlm−1x,t=CHx,tutm−1−uxxm−1−α−1ex,Γuαrmx,t−χmuαrm−1x,t=CHx,tutm−1−uxxm−1+α−1ex,Γuβlmx,t−χmuβlm−1x,t=CHx,tutm−1−uxxm−1+βex,Γuβrmx,t−χmuβrm−1x,t=CHx,tutm−1−uxxm−1−2βex.If we takeC=−1 and Hx,t=1, we have(41)uαlmx,t−χmuαlm−1x,t=−Γ−1utm−1−uxxm−1−α−1ex,uαrmx,t−χmuαrm−1x,t=−Γ−1utm−1−uxxm−1+α−1ex,uβlmx,t−χmuβlm−1x,t=−Γ−1utm−1−uxxm−1+βex,uβrmx,t−χmuβrm−1x,t=−Γ−1utm−1−uxxm−1−2βex,and we choose the operator Γ=d/dt, then Γ−1∗=∫∗dt, and by using the initial values, we get(i)
uαlmx,t.Form = 1,(42)uαl1x,t=−Γ−1uαlt0−uαlxx0−α−1ex+χ1uαl0x,t=−Γ−1−1+αex−α−1ex=−Γ−1−2αex=2αtex.Form = 2,(43)uαl2x,t=−Γ−1uαlt1−uαlxx1−α−1ex+χ2uαl1x,t=−Γ−12αex−2αtex−αex+ex+2αtex=−Γ−1α+1ex−2αtex+2αtex=α−1tex+αt2ex.Form = 3,(44)uαl3x,t=−Γ−1uαlt2−uαlxx2−α−1ex+χ3uαl2x,t=−Γ−1α−1ex+2αtex−α−1tex−αt2ex−α−1ex+α−1tex+αt2ex=−Γ−1α+1tex−αt2ex+α−1tex+αt2ex=α−1tex+α+1t22ex+αt33ex.Form = 4,(45)uαl4x,t=−Γ−1uαlt3−uαlxx3−α−1ex+χ4uαl3x,t=−Γ−1αt2ex+α+1tex+α−1ex−α−1tex−α+1t22ex−αt33ex−α−1ex+α−1tex+α+1t22ex+αt33ex=−Γ−1−αt33ex+α−1t22ex+2tex+α−1tex+α+1t22ex+αt33ex=α−1tex+α−1t22!ex+α+1t33!ex+2αt44!.After five iterations, we get(46)uαl0x,t+uαl1x,t+uαl2x,t+uαl3x,t+uαl4x,t=α+1ex+5α−3tex+4αt22!ex+3α+1t33!ex+2αt44!.(ii)
uαrmx,t.Form = 1,(47)uαr1x,t=−Γ−1ut0−uxx0−1−αex+χ1uαr0x,t=−Γ−1−3−αex−1−αex=−Γ−1−4ex+2αex=4−2αtex.Form = 2,(48)uαr2x,t=−Γ−1ut1−uxx1−1−αex+χ2uαr1x,t=−Γ−14−2αex−4−2αtex−1−αex+4−2αtex=−Γ−13−αex−22−αtex+4−2αtex=1−αtex+2−αt2ex.Form = 3,(49)uαr3x,t=−Γ−1ut2−uxx2−1−αex+χ3uαr2x,t=−Γ−11−αex+22−αtex−1−αtex−2−αt2ex−1−αex+1−αtex+2−αt2ex=−Γ−1−2−αt2ex+3−αtex+1−αtex+2−αt2ex=1−αtex+7−3αt22ex+2−αt33ex.Form = 4,(50)uαr4x,t=−Γ−1ut3−uxx3−1−αex+χ4uαr3x,t=−Γ−11−αex+7−3αtex+2−αt2ex−1−αtex−7−3αt22ex−2−αt33ex−1−αex+1−αtex+7−3αt22ex+2−αt33ex=−Γ−12−αt33ex+α−3t22ex+6−2αtex+1−αtex+7−3αt22ex+2−αt33ex=1−αtex+13−5αt22ex+7−3αt33!ex+2−αt44!ex.After five iterations, we get(51)uαr0x,t+uαr1x,t+uαr2x,t+uαr3x,t+uαl4x,t=3−αex+7−5αtex+24−10αt22ex+11−5αt33!ex+22−αt44!ex.In the same way foruβlmx,t and uβrmx,t, we find that(52)uβl1x,t=2−3βtex,uβl2x,t=−βtex+2−3βt22ex,uβl3x,t=−βtex−βt22ex+2−3βt33!ex,uβl4x,t=−βtex−βt22ex+4−5βt33!ex+2−3βt44!ex,anduβr1x,t=2+4βtex,uβr2x,t=2βtex+2+4βt22ex,uβr3x,t=2βtex+βt2ex+2+4βt33!ex,uβr4x,t=2βtex+2βt22ex+2βt33!ex−2+4βt44!ex,so after five iterations, we get(53)uβr1x,t+uβr2x,t+uβr3x,t+uβr4x,t=2+2βex+2+10βtex+2+8βt22ex+2+6βt33!ex−2+4βt44!ex.To ensure the validity of the present model, we illustrate in Figures1 and 2 the comparison of the numerical solutions with the exact ones for the membership and nonmembership functions at t = 1 and m = 4 for α∈0,1; we have calculated all the data by using MATLAB.Figure 1
Exact and approximate solutions for membership and nonmembership functions att = 1 and m = 4.
(a)(b)Figure 2
The surface of intuitionistic fuzzy solutions.
(a)(b)From the figures, we can see that the results of the homotopy perturbation method (HPM) are close to the exact solution which confirms the validity of our method.


## 5. Conclusion

In this work, we have presented the procedure for simulating and computing an approximate solution for intuitionistic fuzzy differential equations with the linear differential operator by using the homotopy analysis method, which can also be used to solve some linear and nonlinear problems that cannot be solved by classical methods. Moreover, in the homotopy analysis method, we can chooseh appropriately to ensure the convergence of the series solution for highly nonlinear problems. The basic ideas of this approach should be used to solve other intuitionistic fuzzy problems in many practical domains such as fluid mechanics.



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