# A Novel Approach for Solving Fuzzy Differential Equations Using Cubic Spline Method

Authors: S. Karpagappriya; Nazek Alessa; P. Jayaraman; K. Loganathan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Ambiguity in real-world problems can be modeled into fuzzy differential equations. The main objective of this work is to introduce a new class of cubic spline function approach to solve fuzzy initial value problems efficiently. Further, the convergence of this method is shown. As it is a single-step method that converges faster, the complexity of the proposed method is too low. Finally, a numerical example is illustrated in order to validate the effectiveness and feasibility of the proposed method, and the results are compared with the exact as well as Taylor’s method of order two.


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## Body

## 1. Introduction

The entire real world is complex; it is found that the complexity arises from uncertainty in the form of ambiguity. Uncertainties in the real-world problem can be modeled easily with the help of fuzzy set theory when one lacks complete information about the variables and parameters [1]. This concept of fuzzy set theory was first introduced by Zadeh [2] in 1965. Chang and Zadeh explicated the concept of fuzzy derivatives [3]. The term fuzzy differential equation was formulated by Kandal and Byatt [4] in 1978. These equations help in modeling the propagation of epistemic uncertainty in a dynamical environment [5]. Kaleva [6], Seikkala [7], and Song and Wu [8] have extensively studied the existence and uniqueness of solutions of these equations. A general formulation of the first-order fuzzy initial value problem was given by Buckley and Feuring [9]. Later, the fuzzy initial and boundary value differential equation was given by O’Regan et al. [10].First-order linear fuzzy differential equations have inspired several authors to focus on solving them numerically since they appear in many real-world applications. These applications include different fields of science such as medical diagnosis, biology, and civil engineering and also in the field of economics [11] where the information are not given in the crisp set [12]. Based on Zadeh’s extension principle, a new fuzzy version of Euler’s method was developed by Ahamed and Hasan [13]. Solving of these equations by the Taylor method of order p has been studied by Abbasbandy and Viranloo [14], and the same was discussed by Allahviranloo et al. [15] by using the predictor-corrector algorithm. Finally, the authors concluded that a fuzzy differential equation can be modified into a system of ordinary differential equations (ODEs). Also, they found out that there are two solutions for a fuzzy differential equation by solving the associated ODEs. The convergence, consistency, and stability for approximating the solution of fuzzy differential equations with initial value conditions have been studied by Ezzati et al. [16]. All the numerical results of these equations and their applications were summarized by Chakraverty et al. [12].In this paper, the fuzzy initial value problem is solved numerically by using a new class of function approximation called cubic spline, for better accuracy of the solution.


## 2. Preliminaries

LetX′=x where X′ is the space of points and x is the generic element of X′.Definition 1 (see [2]).
A fuzzy subsetμA′ of the set A′ in X′ is a function μA′:A′⟶0,1.Definition 2 (see [17]).
Theα-level set of the fuzzy set A′ of X′ is a crisp set A′α=x∈X′|μA′x≥α if α∈0,1.Definition 3 (see [17]).
LetA′ be a triangular fuzzy number (TFN) which is defined as l,m,n where l,n is the support, m is the core, and the membership function is(1)μA′x=x−lm−l,ifx∈l,m,n−xn−m,ifx∈m,n,0,ifx∉l,n,where l<m<n.Let us denote the set of all fuzzy numbers onℝ as F which is a fuzzy number such that μ:ℝ⟶0,1.Definition 4 (see [18]).
Letlandm∈F. If there exists n∈F such that l=m+n, then n is the Hukuhara difference of l and m. This can be denoted as n=l⊖m. To define the differentiability of a fuzzy function, we can make use of this difference as follows.
LetH:u,v⟶F be differentiable at t0∈u,v. If there exists some element H′t0∈F such that(2)limh⟶0+Ht0+h⊖Ht0h=limh⟶0+Ht0⊖Ht0−hh=H′t0,then H is said to be Hukuhara differentiable at t0.
SupposeH is differential at the point t0∈u,v, then all its α-level sets, Hαt=Htα, are Hukuhara differentiable at t0 and H′t0α=DHαt0, where DHα denotes the Hukuhara derivatives of Hα and Hα as the multivalued mapping.Theorem 1 (see [19]).
Letqx∈C3u,v and Δk be a sequence of partitions on u,v, with limk⟶∞Δk=0; then, for the interpolate cubic spline SΔkx, uniformly for u≤x≤v,(3)qpx−SΔkpx=OΔk3−p,forp=0,1,2,and3.
Ifq‴x satisfies the Holder condition on u,v with 0<α<1, then(4)ypx−SΔkpx=OΔk3+α−p.Proof.
This theorem has been proved in the work by Ahlberg et al. [19] (p. 29).


### 2.1. Cubic Spline Function Approximation for Initial Value Problems

Let the givenn+1 data points be ui,vi, i=0,1,2,…,n, where u0<u1<u2<…<un. Let us define the cubic spline Piu, which is defined in the interval ui−1,ui as follows.(i)
Foru<u0 and u>un, Piu is a polynomial whose degree is one(ii)
Piu is at most a cubic polynomial in each subinterval ui−1,ui, where i=1,2,…,n(iii)
Piu,Pi′x, and Pi″x are continuous at each point ui,vi, where i=0,1,2,…,n(iv)
Piui=vi, where i=0,1,2,…,nIfPi″u0=Pi″un=0 and Piu,Pi′u, and Pi″u are all continuous in u0,un, then this cubic spline is called as natural spline [20].Many applications make use of slopes. So let us denote the cubic spline function that is obtained in terms of first derivatives to bemi. The cubic spline Pu formula for an initial value problem in ui−1≤u≤ui in terms of its first derivatives P′ui=mi can be obtained by using Hermite’s interpolation formula as follows [21, 22]:(5)Pu=mi−1ui−u2u−ui−1h2−miu−ui−12ui−uh2+vi−1ui−u22u−ui−1+hh3+viu−ui−122ui−u+hh3,where h=ui−ui−1 for all i:(6)P′u=mi−1h2ui−u2ui−1+ui−3u−mih2u−ui−1ui−1+2ui−3u+6h3vi−vi−1ui−uu−ui−1,(7)P″u=−2mi−1h2ui−1+2ui−3u−2mih22ui−1+ui−3u+6h3vi−vi−1ui−1+ui−2u.Settingu=ui and Pui=vi for all i in (7), we have(8)P″ui=2mi−1h+4mih−6h2Pi−Pi−1.Now consider a differential equation of first order with the initial condition as follows:(9)dvdu=fu,vandvu0=v0.On differentiating (9) twice with respect to u,(10)v″u=fuu,v+fvu,vfu,v.Takingu=ui and Pui=vi, the above equation becomes(11)P″ui=fuui,Pi+fvui,Pifui,Pi.On equating (8) and (11), we obtain(12)2mi−1h+4mih−6h2Pi−Pi−1=fuui,Pi+fvui,Pifui,Pi.From this, we can computePi’s. Substituting these Pi’s in (5) gives the required solution. The convergence of this method has been proved by Patricio [21].


## 2.1. Cubic Spline Function Approximation for Initial Value Problems

Let the givenn+1 data points be ui,vi, i=0,1,2,…,n, where u0<u1<u2<…<un. Let us define the cubic spline Piu, which is defined in the interval ui−1,ui as follows.(i)
Foru<u0 and u>un, Piu is a polynomial whose degree is one(ii)
Piu is at most a cubic polynomial in each subinterval ui−1,ui, where i=1,2,…,n(iii)
Piu,Pi′x, and Pi″x are continuous at each point ui,vi, where i=0,1,2,…,n(iv)
Piui=vi, where i=0,1,2,…,nIfPi″u0=Pi″un=0 and Piu,Pi′u, and Pi″u are all continuous in u0,un, then this cubic spline is called as natural spline [20].Many applications make use of slopes. So let us denote the cubic spline function that is obtained in terms of first derivatives to bemi. The cubic spline Pu formula for an initial value problem in ui−1≤u≤ui in terms of its first derivatives P′ui=mi can be obtained by using Hermite’s interpolation formula as follows [21, 22]:(5)Pu=mi−1ui−u2u−ui−1h2−miu−ui−12ui−uh2+vi−1ui−u22u−ui−1+hh3+viu−ui−122ui−u+hh3,where h=ui−ui−1 for all i:(6)P′u=mi−1h2ui−u2ui−1+ui−3u−mih2u−ui−1ui−1+2ui−3u+6h3vi−vi−1ui−uu−ui−1,(7)P″u=−2mi−1h2ui−1+2ui−3u−2mih22ui−1+ui−3u+6h3vi−vi−1ui−1+ui−2u.Settingu=ui and Pui=vi for all i in (7), we have(8)P″ui=2mi−1h+4mih−6h2Pi−Pi−1.Now consider a differential equation of first order with the initial condition as follows:(9)dvdu=fu,vandvu0=v0.On differentiating (9) twice with respect to u,(10)v″u=fuu,v+fvu,vfu,v.Takingu=ui and Pui=vi, the above equation becomes(11)P″ui=fuui,Pi+fvui,Pifui,Pi.On equating (8) and (11), we obtain(12)2mi−1h+4mih−6h2Pi−Pi−1=fuui,Pi+fvui,Pifui,Pi.From this, we can computePi’s. Substituting these Pi’s in (5) gives the required solution. The convergence of this method has been proved by Patricio [21].


## 3. Fuzzy Initial Value Problem

Consider the first-order fuzzy differential equation as(13)u′ξ=fξ,uξ,ξ∈ξ0,T,T≥0,with the initial condition uξ0=u0∈F, where u is a fuzzy function of the crisp variable ξ; that is, u∈F, which is unknown. f:ξ0,T×F⟶F, which is a fuzzy function. u′ is the fuzzy derivative of u, and uξ0 is a fuzzy number. Here, let us assume the fuzzy number to be a triangular fuzzy number.Forα∈0,1, let us denote the α-level sets:(14)uξα=uξlα,uξrα,anduξ0α=uξ0lα,uξ0rα.Also,fξ,uξα=f1ξ,uξ;α,f2ξ,uξ;α,where(15)f1ξ,uξ;α=minfξ,s|s∈uξlα,uξrα=Gξ,uξlα,uξrα,(16)f2ξ,uξ;α=maxfξ,s|s∈uξlα,uξrα=Hξ,uξlα,uξrα.The mappingf:ξ0,T×F⟶F is a fuzzy process, and the derivatives fi∈F, for i=1,2,…,p, are defined as(17)fiξ,uξα=f1iξ,uξ;α,f2iξ,uξ;α,where(18)f1iξ,uξ;α=minfiξ,s|s∈uξlα,uξrα,f2iξ,uξ;α=maxfiξ,s|s∈uξlα,uξrα.Equation (13) can be replaced by an equivalent system of equations, and hence,(19)u′ξα=u′ξlα,u′ξrα,where(20)u′ξlα=f1ξ,uξ;α=Gξ,uξlα,uξrα,by15,(21)u′ξrα=f2ξ,uξ;α=Hξ,uξlα,uξrα,by16.The system of equations (20) and (21) will have a unique solution, uξlα,uξrα∈J=Cξ0,F×Cξ0,F. Thus, given fuzzy differential equation (13) possesses a unique solution on J.Usually, equations (20) and (21) can be solved analytically. Yet, in most of the cases, this becomes tedious, and hence, a numerical approach to these systems of equations has to be considered.


## 4. Cubic Spline Method for Solving Fuzzy Initial Value Problem

Assume that(22)Uξnα=Uξnlα,Uξnrα,as the exact solution of (13):(23)Pξnα=Pξnlα,Pξnrα,as the approximated solution of (13) at ξn where 0≤n≤N.Now let us calculate the solutions by mesh points atξ0<ξ1<…<ξN=T, h=T−ξ0/N, and ξn=ξ0+nh, where n=0,1,2,…,N.The cubic spline functionPξ;α for a fuzzy initial value problem in ξi−1≤ξ≤ξi in terms of its first derivatives P′ξi;α=mi is given as(24)Pξ;α=Pξα=mi−1ξi−ξ2ξ−ξi−1h2−miξ−ξi−12ξi−ξh2+uξi−1αξi−ξ22ξ−ξi−1+hh3+uξiαξ−ξi−122ξi−ξ+hh3,where h=ξi−ξi−1. But, we know that(25)Pξα=Pξlα,Pξrα,where(26)Pξlα=mi−1ξi−ξ2ξ−ξi−1h2−miξ−ξi−12ξi−ξh2+uξli−1αξi−ξ22ξ−ξi−1+hh3+uξliαξ−ξi−122ξi−ξ+hh3,(27)Pξrα=mi−1ξi−ξ2ξ−ξi−1h2−miξ−ξi−12ξi−ξh2+uξri−1αξi−ξ22ξ−ξi−1+hh3+uξriαξ−ξi−122ξi−ξ+hh3.By carrying out simple and similar calculations for (26) and (27) as given in “cubic spline function approximation for initial value problems” (especially equations from (6)–(12)), we obtain the following set of equations:(28)2mi−1h+4mih−6h2Pli−Pli−1=Gξξi,Pli;α+Guξξi,Pli;αGξi,Pli;α,(29)2mi−1h+4mih−6h2Pri−Pri−1=Hξξi,Pri;α+Huξξi,Pri;αHξi,Pri;α,where i=1,2,…,n and h=ξi−ξi−1. From (28), Pli’s can be computed, and they are substituted in (26) to obtain the solution, Pξlα. Similarly, Pri’s can be evaluated from (29) and are substituted in (27) to yield Pξrα. Each Pi value depends on Pi−1th value, for i=1,2,…,n.Both these solutions collectively yield the desired solutionPξα of (13) at a fixed ξ∈ξi−1,ξi, i=1,2,…,n.


### 4.1. Convergence of Fuzzy Cubic Spline Method

Let us consider the equations:(30)P″ξilα=2mi−1h+4mih−6h2Pli−Pli−1,(31)u″ξilα=Gξξi,Pli;α+Guξξi,Pli;αGξi,Pli;α.According to the results given in the work by Ahlberg et al. [19] (p. 34) and Theorem 1, if Gξ,uξ;α∈C3ξ0,T, we have(32)upξ−Ppξ=Oh4−p,p=0,1,2,and3.Ifp=2, then the above equation can be written as(33)u″ξ−P″ξ=Oh2.Atξ=ξi, for i=1,2,…,n, we have(34)u″ξi−P″ξi=Oh2, whereh=maxihioru″ξilα=Oh2+P″ξilα,⇒h2u″ξilα=2hmi−1+4hmi−6Pli−Pli−1+Oh4.Again, with (31), we obtain Pli explicitly or not according to the linearity or nonlinearity of Gξ,uξ;α in uξ. Then, we can write(35)Pli=c1+c2α+Oh4,where c1 and c2 are constants and α∈0,1 or(36)rPli=Oh4.Hence, the order of the method is sustained, and it is true forξ∈ξ0,T.From (26), we have(37)Pξlα<mj−1h−mjh+uξlj−1α+uξljα,where ξj−1≤ξ≤ξj, ∀j=0,1,…,n.Similarly, by considering the equations,(38)P″ξirα=2mi−1h+4mih−6h2Pri−Pri−1,u″ξirα=Hξξi,Pri;α+Huξξi,Pri;αHξi,Pri;α,we get(39)Pξrα<mj−1h−mjh+uξrj−1α+uξrjα.Thus, from (24), we obtain(40)Pξα<mj−1h−mjh+uξj−1α+uξjα,ξ∈ξj−1,ξj,∀j=0,1,…,n.


## 4.1. Convergence of Fuzzy Cubic Spline Method

Let us consider the equations:(30)P″ξilα=2mi−1h+4mih−6h2Pli−Pli−1,(31)u″ξilα=Gξξi,Pli;α+Guξξi,Pli;αGξi,Pli;α.According to the results given in the work by Ahlberg et al. [19] (p. 34) and Theorem 1, if Gξ,uξ;α∈C3ξ0,T, we have(32)upξ−Ppξ=Oh4−p,p=0,1,2,and3.Ifp=2, then the above equation can be written as(33)u″ξ−P″ξ=Oh2.Atξ=ξi, for i=1,2,…,n, we have(34)u″ξi−P″ξi=Oh2, whereh=maxihioru″ξilα=Oh2+P″ξilα,⇒h2u″ξilα=2hmi−1+4hmi−6Pli−Pli−1+Oh4.Again, with (31), we obtain Pli explicitly or not according to the linearity or nonlinearity of Gξ,uξ;α in uξ. Then, we can write(35)Pli=c1+c2α+Oh4,where c1 and c2 are constants and α∈0,1 or(36)rPli=Oh4.Hence, the order of the method is sustained, and it is true forξ∈ξ0,T.From (26), we have(37)Pξlα<mj−1h−mjh+uξlj−1α+uξljα,where ξj−1≤ξ≤ξj, ∀j=0,1,…,n.Similarly, by considering the equations,(38)P″ξirα=2mi−1h+4mih−6h2Pri−Pri−1,u″ξirα=Hξξi,Pri;α+Huξξi,Pri;αHξi,Pri;α,we get(39)Pξrα<mj−1h−mjh+uξrj−1α+uξrjα.Thus, from (24), we obtain(40)Pξα<mj−1h−mjh+uξj−1α+uξjα,ξ∈ξj−1,ξj,∀j=0,1,…,n.


## 5. Numerical Illustration (Exponential Decay Problem with Decay Constant as 1)

Consider the fuzzy differential equation(41)y′ξ=−yξ,with y0=0.5,1,1.5 as its fuzzy initial condition. Let us find the solution of (41) at ξ=0.2 and 0.3.Equation (41) can be modified into a system of ordinary differential equations as follows:(42)y′ξlα=−yξlα,y0lα=0.5α+0.5,(43)y′ξrα=−yξrα,y0rα=1.5−0.5α.The solution of these two equations collectively gives the solution of (41). Therefore, the exact solution of (41) is(44)Yξα=Ylξα,Yrξα=0.5α+0.5exp−ξ,1.5−0.5αexp−ξ.Now let us compute the numerical solution of (41) by using the cubic spline method.For simplicity, assumeh = 0.1.Consider equation (42), here Gξ,y;α=−y and so Gξξ,y;α=0 and Gyξ,y;α=−1.Also,Gξi,Pli;α=−Pli.Using (28) at i = 1 and ξ=0.1, we get(45)20m0+40m1−600Pl1−Pl0=Pl1.Sincemi=Pl′ξi,m0=−0.5α−0.5andm1=−Pl1, the above equation on simplification gives(46)Pl1=290α+290641.Similarly, ati = 2 and ξ=0.2, (28) becomes(47)Pl2=16412168200α+168200.This is the approximate solution of (42) at ξ=0.2.By using (42) at i = 3, we obtain(48)20m2+40m3−600Pl3−Pl2=Pl3,where m2=−Pl2 and m3=−Pl3. This equation gives the approximate solution of (42) at ξ=0.3.Now consider equation (43), here Hξ,y;α=−y, and hence, Hξξ,y;α=0, Hyξ,y;α=−1, and Hξi,Pri;α=−Pi.By using (29) at i = 1 and ξ=0.1, we obtain(49)Pr1=−290α+870641,where mi=Pr′ξi, for all i.From (29), taking i = 2 and ξ=0.2, we have(50)20m1+40m2−600Pr2−Pr1=Pr2,∴Pr2=16412−168200α+504600on simplification.Similarly, fori = 3 and ξ=0.3 in (29), we get(51)20m2+40m3−600Pr3−Pr2=Pr3,where m2=−Pr2 which is given by (50) and m3=−Pr3. This equation on further simplification gives the approximate solution of (43) at ξ=0.3.Tables1 and 2 represent the comparison of the solutions for equation (41) that are obtained by exact, cubic spline method and Taylor’s method of order, p=2 at ξ=0.2 with h=0.1. Comparison of exact and cubic spline solutions at ξ=0.2 is graphically given in Figure 1. Similarly, Figure 2 interprets the compared results of exact and cubic spline at ξ=0.3 of step length h=0.1.Table 1
Comparison of the results (approximated to 9 decimals) obtained by exact, cubic spline method and Taylor’s method of order,p=2 at ξ=0.2 with h = 0.1 for equation (42).
α-cutExact solutionYlCubic splinePlTaylor solutionTl for p=20.00.4093653770.4093642680.4095125000.10.4503019140.4503006950.4504637500.20.4912384520.4912371220.4914150000.30.5321749900.5321735490.5323662500.40.5731115270.5731099760.5733175000.50.6140480650.6140464030.6142687500.60.6549846020.6549828300.6552200000.70.6959211400.6959192560.6961712500.80.7368576780.7368556830.7371225000.90.7777942150.7777921100.7780737501.00.8187307530.8187285370.819025000Table 2
Comparison of the results (approximated to 9 decimals) obtained by exact, cubic spline method and Taylor’s method of order,p=2 at ξ=0.2 with h = 0.1 for equation (43).
α-cutExact solutionYrCubic splinePrTaylor solutionTr for p=20.01.2280961301.2280928051.2285375000.11.1871595921.1871563791.1875862500.21.1462230541.1462199521.1466350000.31.1052865171.1052835251.1056837500.41.0643499791.0643470981.0647325000.51.0234134411.0234106711.0237812500.60.9824769040.9824742440.9828300000.70.9415403660.9415378180.9418787500.80.9006038280.9006013910.9009275000.90.8596672910.8596649640.8599762501.00.8187307530.8187285370.819025000Figure 1
Comparison of exact and cubic spline solutions atξ=0.2.Figure 2
Comparison of exact and cubic spline solutions atξ=0.3 for h = 0.1.In general, the numerical solution of the fuzzy differential equation by using the cubic spline method can be given as(52)Piα=Pliα,Priα,where i=1,2,…,n, i.e., Pξα=Plξα,Prξα, for a fixed ξ.


## 6. Conclusion

In this article, a new class of cubic spline function method is introduced for solving fuzzy differential equations subject to fuzzy initial conditions. The desired solution which is obtained is ofOh4 convergence based on certain conditions on the derivatives. This numerical method is verified with an example, and the results are compared with the exact as well as with the solution obtained by Taylor’s method of order, p=2. From the comparison of results, one can conclude that the proposed method is a single-step method that converges faster and has greater accuracy than the Taylor method of order two. In future, one can extend this method to solve higher-order linear and nonlinear fuzzy initial value problems.



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