# Adams Predictor-Corrector Systems for Solving Fuzzy Differential Equations

Authors: Dequan Shang; Xiaobin Guo
Journal: Mathematical Problems in Engineering
 (2013)

---

## Abstract

A predictor-corrector algorithm and an improved predictor-corrector (IPC) algorithm based on Adams method are proposed to solve first-order differential equations with fuzzy initial condition. These algorithms are generated by updating the Adams predictor-corrector method and their convergence is also analyzed. Finally, the proposed methods are illustrated by solving an example.


---

## Body

## 1. Introduction

Fuzzy differential equations (FDEs), which are utilized for the purpose of the modeling problems in science and engineering, have been studied by many researchers. Most of the practical problems require the solutions of fuzzy differential equations (FDEs) which are satisfied with fuzzy initial conditions; therefore a fuzzy initial problem occurs and should be solved. However, for the vast majority of fuzzy initial value problems, their exact solutions are difficult to be obtained. Thus it is necessary to consider their numerical methods.The concept of a fuzzy derivative was first introduced by Chang and Zadeh [1]; it was followed up by Dubois and Prade [2] who used the extension principle in their approach. Other fuzzy derivative concepts have been proposed by Puri and Ralescu [3] and Goetschel Jr. and Voxman [4] as an extension of the Hukuhara derivative of multivalued functions. In the past decades, many works have been appeared on the aspects of theories and applications on fuzzy differential equations; see [5–12]. The notation of fuzzy differential equation was initially introduced by Kandel and Byatt [13, 14] and later they applied the concept of fuzzy differential equation to the analysis of fuzzy dynamical problems [7, 15]. A thorough theoretical research of fuzzy Cauchy problems was given by Kaleva [16, 17], Wu and Song [18], Ouyang and Wu [19], Kim and Sakthivel [20], and M. D. Wu [21]. A generalization of fuzzy differential equation was given by Aubin [22, 23], Baĭdosov [6], Kloeden [24], and Colombo and Křivan [25].For a fuzzy Cauchy problem(1)y′(t)=f(t,y),t0≤t≤T,y(t0)=α0,
in 1999, Friedman et al. [26] firstly treated it and obtained its numerical solution by Euler method. In recent years, some researchers such as Abbasbandy and Allahviranloo applied the Taylor series method, the Runge-Kutta method, and the linear multistep method to solve fuzzy differential equations [5, 20, 27–32]. They proposed some numerical methods and discussed the convergence and stability of their methods under the fuzzy numbers background. However, their methods always have some of low convergence order.In this paper, based on Adams-Bashforth four-step method and Adams-Moulton three-step method, two Adams predictor-corrector algorithms are proposed to solve fuzzy initial problems. The convergence of the proposed methods is also presented in detail. Finally, an example is given to illustrate our methods. The structure of this paper is organized as follows.In Section2, some basic definitions and results are recalled. An explicit Adams-Bashforth method and an implicit Adams-Moulton method for solving FDEs are mentioned in Section 3. The predictor-corrector method and the improved predictor-corrector (IPC) systems algorithm are introduced in Section 4. The convergence of the proposed methods is discussed in Section 5. An illustrating example is given in Section 6 and the conclusion is drawn in Section 7.


## 2. Preliminaries

### 2.1. Fuzzy Numbers

Definition 1 (see [1]).
A fuzzy number is a fuzzy set likeu:R→I=[0,1] which satisfies the following:(1)
uis upper semicontinuous;(2)
u is fuzzy convex; that is, u(λx+(1-λ)y)≥min{u(x),u(y)}, for all x, y∈R, λ∈[0,1];(3)
u is normal; that is, there exists x0∈R such that u(x0)=1;(4)
suppu={x∈R∣u(x)>0} is the support of the u, and its closure cl(supp u) is compact.
LetE1 be the set of all fuzzy numbers on R.Definition 2 (see [2]).
A fuzzy numberu in parametric form is a pair (u_,u¯) of functions u_(r), u¯(r), 0≤r≤1, which satisfies the following requirements:(1)
u_(r) is a bounded monotonic increasing left continuous function;(2)
u¯(r) is a bounded monotonic decreasing left continuous function;(3)
u_(r)≤u¯(r), 0≤r≤1.
LetI be a real interval. A mapping y:I→E1 is called a fuzzy process and its α-level set is denoted by
(2)[y(t)]α=[y_α(t),y¯α(t)],t∈I,α∈(0,1].Definition 3.
A triangular fuzzy number is a fuzzy setu in E1 that is characterized by an ordered triple (xl,xc,xr)∈R3 with xl≤xc≤xr such that [u]0=[xl,xr] and [u]1={xc}.
Theα-level set of a triangular fuzzy number u is given by
(3)[u]α=[xc-(1-α)(xc-xl),xc+(1-α)(xr-xc)],
for any α∈I.Definition 4 (see [9]).
The supremum metricd∞ on E1 is defined by
(4)d∞(u,v)=sup{dH{[u]α,[v]α}:α∈I}.
With the spermium metric, the space(E1,d∞) is a complete metric space.Definition 5 (see [9]).
A mappingF:T→E1 is Hukuhara differentiable at t0∈T⊂R if for some h0>0 the Hukuhara difference
(5)F(t0+Δt)~hF(t0),F(t0)~hF(t0-Δt)
exists in E1, for all 0<Δt<h0, and if there exists an F′(t0)∈E1 such that
(6)limh→0+d∞(F(t0+Δt)~hF(t0)Δt,F′(t0))=0,limh→0+d∞(F(t0)~hF(t0-Δt)Δt,F′(t0))=0.
The fuzzy set F′(t0) is called the Hukuhara derivative of F at t0.Recall thatu~hv=w∈E1 is defined on α-level sets, where [u]α~h[v]α=[w]α, for all α∈I. By the definition of the metric d∞, all the α-level set mappings [F(·)]α are Hukuhara differentiable at t0 with Hukuhara derivative [F′(t0)]α for each α∈I when F:T→E1 is Hukuhara differentiable at t0 with Hukuhara derivative F′(t0).Remark 6.
IfF:T→E1 is Hukuhara differentiable and its Hukuhara derivative F′ is integrable over [0,1], then
(7)F(t)=F(t0)+∫t0t‍F′(s)ds,
for all 0≤t0≤t≤1.Definition 7.
A mappingy:I→E1 is called a fuzzy process. We designate
(8)[y(t)]α=[y1α(t),y2α(t)],t∈I,0≤α≤1.
The Seikkala derivative y′(t) of a fuzzy process y is defined by
(9)[y′(t)]α=[(y1α)′(t),(y2α)′(t)],0≤α≤1,
provided that this equation in fact defines a fuzzy number y′(t)∈E1.Remark 8.
Ify:I→E1 is Seikkala differentiable and its Seikkala derivative y′ is integrable over [0,1], then
(10)y(t)=y(t0)+∫t0t‍y′(s)ds,
for all t0,t∈I.


### 2.2. A Fuzzy Cauchy Problem

Consider the first-order fuzzy differential equationy′=f(t,y), where y is a fuzzy function of t, f(t,y) is a a fuzzy function of the crisp variable t and the fuzzy variable y, and y′ is the Hukuhara or Seikkala fuzzy derivative of y. Given an initial value y(t0)=α0, we can define a first-order fuzzy Cauchy problem as follows:
(11)y′(t)=f(t,y),t0≤t≤T,y(t0)=y0.
The existence theorem is obtained for the Cauchy problem (11).Let[y(t)]α=[y_α(t),y¯α(t)]; if y(t) is Hukuhara differentiable then [y′(t)]α=[(y_α(t))′,(y¯α(t))′]. So (11) translates into the following system of ODEs:
(12)(y_α(t))′=f_α(t,y_α(t),y¯α(t)),(y¯α(t))′=f¯α(t,y_α(t),y¯α(t)),y_α(t0)=y_0α,y¯α(t0)=y¯0α.Theorem 9 (see [33]).
Let one consider the FCP (11) where f:[t0,T]×E1→E1 is such that(i)
[f(t,y)]α=[f_α(t,y_α(t),y¯α(t)),f¯α(t,y_α(t),y¯α(t))];(ii)
f_α and f¯α are equicontinuous and uniformly bounded on any bounded set;(iii)
f_α and f¯α satisfy the Lipschitz conditions.
Then the FCP (11) and the system of ODEs (12) are equivalent.


### 2.3. Interpolation for Fuzzy Numbers

The problem of interpolation for fuzzy sets is as follows.Suppose that at various time instantt information f(t) is presented as fuzzy set. The aim is to approximate the function f(t), for all t in the domain of f. Let t0<t1<⋯<tn be n+1 distinct points in R and let u0,u1,…,un be n+1 fuzzy sets in E1. A fuzzy polynomial interpolation of the data is a fuzzy-value function F:R→E1 satisfying the following conditions:(1)
f(ti)=ui, for all i=0,1,…,n;(2)
f is continuous;(3)
if the data is crisp, then the interpolationf is a crisp polynomial.A functionf fulfilling these conditions may be constructed as follows. Let Cαi=[ui]α, for any α∈[0,1], i=0,1,…,n. For each x=(x0,x1,…,xn)∈Rn+1, the unique polynomial of degree ≤n is denoted by PX such that
(13)PX(ti)=xi,i=0,1,…,n,PX(t)=∑i=0n‍xi(∏i≠j‍t-tjti-tj).Finally, for allt, ξ∈R are defined by f(t)∈E1 such that
(14)f(t)(ξ)=sup{α∈[0,1]:∃X∈Cα0211111×Cα1×⋯×Cαn,PX(t)=ξ}.The interpolation polynomial can be written level setwise as(15)[f(t)]α={y∈R:y=PX(t),x∈Cαi,i=0,1,…,n},
for 0≤α≤1.Theorem 10 (see [34]).
Let(ti,ui), i=0,1,…,n be the observed data and suppose that each of the ui=(uil,uic,uir) is an element of E1. Then for each t∈[t0,tn], f(t)=(fl(t),fc(t),fr(t))∈E1,
(16)fl(t)=∑li(t)≥0‍li(t)uil+∑li(t)<0‍li(t)uir,fc(t)=∑i=0n‍li(t)uic,fr(t)=∑li(t)≥0‍li(t)uir+∑li(t)<0‍li(t)uil,
where li(t)=∏i≠j‍((t-tj)/(ti-tj)), i=0,1,…,n.


## 2.1. Fuzzy Numbers

Definition 1 (see [1]).
A fuzzy number is a fuzzy set likeu:R→I=[0,1] which satisfies the following:(1)
uis upper semicontinuous;(2)
u is fuzzy convex; that is, u(λx+(1-λ)y)≥min{u(x),u(y)}, for all x, y∈R, λ∈[0,1];(3)
u is normal; that is, there exists x0∈R such that u(x0)=1;(4)
suppu={x∈R∣u(x)>0} is the support of the u, and its closure cl(supp u) is compact.
LetE1 be the set of all fuzzy numbers on R.Definition 2 (see [2]).
A fuzzy numberu in parametric form is a pair (u_,u¯) of functions u_(r), u¯(r), 0≤r≤1, which satisfies the following requirements:(1)
u_(r) is a bounded monotonic increasing left continuous function;(2)
u¯(r) is a bounded monotonic decreasing left continuous function;(3)
u_(r)≤u¯(r), 0≤r≤1.
LetI be a real interval. A mapping y:I→E1 is called a fuzzy process and its α-level set is denoted by
(2)[y(t)]α=[y_α(t),y¯α(t)],t∈I,α∈(0,1].Definition 3.
A triangular fuzzy number is a fuzzy setu in E1 that is characterized by an ordered triple (xl,xc,xr)∈R3 with xl≤xc≤xr such that [u]0=[xl,xr] and [u]1={xc}.
Theα-level set of a triangular fuzzy number u is given by
(3)[u]α=[xc-(1-α)(xc-xl),xc+(1-α)(xr-xc)],
for any α∈I.Definition 4 (see [9]).
The supremum metricd∞ on E1 is defined by
(4)d∞(u,v)=sup{dH{[u]α,[v]α}:α∈I}.
With the spermium metric, the space(E1,d∞) is a complete metric space.Definition 5 (see [9]).
A mappingF:T→E1 is Hukuhara differentiable at t0∈T⊂R if for some h0>0 the Hukuhara difference
(5)F(t0+Δt)~hF(t0),F(t0)~hF(t0-Δt)
exists in E1, for all 0<Δt<h0, and if there exists an F′(t0)∈E1 such that
(6)limh→0+d∞(F(t0+Δt)~hF(t0)Δt,F′(t0))=0,limh→0+d∞(F(t0)~hF(t0-Δt)Δt,F′(t0))=0.
The fuzzy set F′(t0) is called the Hukuhara derivative of F at t0.Recall thatu~hv=w∈E1 is defined on α-level sets, where [u]α~h[v]α=[w]α, for all α∈I. By the definition of the metric d∞, all the α-level set mappings [F(·)]α are Hukuhara differentiable at t0 with Hukuhara derivative [F′(t0)]α for each α∈I when F:T→E1 is Hukuhara differentiable at t0 with Hukuhara derivative F′(t0).Remark 6.
IfF:T→E1 is Hukuhara differentiable and its Hukuhara derivative F′ is integrable over [0,1], then
(7)F(t)=F(t0)+∫t0t‍F′(s)ds,
for all 0≤t0≤t≤1.Definition 7.
A mappingy:I→E1 is called a fuzzy process. We designate
(8)[y(t)]α=[y1α(t),y2α(t)],t∈I,0≤α≤1.
The Seikkala derivative y′(t) of a fuzzy process y is defined by
(9)[y′(t)]α=[(y1α)′(t),(y2α)′(t)],0≤α≤1,
provided that this equation in fact defines a fuzzy number y′(t)∈E1.Remark 8.
Ify:I→E1 is Seikkala differentiable and its Seikkala derivative y′ is integrable over [0,1], then
(10)y(t)=y(t0)+∫t0t‍y′(s)ds,
for all t0,t∈I.


## 2.2. A Fuzzy Cauchy Problem

Consider the first-order fuzzy differential equationy′=f(t,y), where y is a fuzzy function of t, f(t,y) is a a fuzzy function of the crisp variable t and the fuzzy variable y, and y′ is the Hukuhara or Seikkala fuzzy derivative of y. Given an initial value y(t0)=α0, we can define a first-order fuzzy Cauchy problem as follows:
(11)y′(t)=f(t,y),t0≤t≤T,y(t0)=y0.
The existence theorem is obtained for the Cauchy problem (11).Let[y(t)]α=[y_α(t),y¯α(t)]; if y(t) is Hukuhara differentiable then [y′(t)]α=[(y_α(t))′,(y¯α(t))′]. So (11) translates into the following system of ODEs:
(12)(y_α(t))′=f_α(t,y_α(t),y¯α(t)),(y¯α(t))′=f¯α(t,y_α(t),y¯α(t)),y_α(t0)=y_0α,y¯α(t0)=y¯0α.Theorem 9 (see [33]).
Let one consider the FCP (11) where f:[t0,T]×E1→E1 is such that(i)
[f(t,y)]α=[f_α(t,y_α(t),y¯α(t)),f¯α(t,y_α(t),y¯α(t))];(ii)
f_α and f¯α are equicontinuous and uniformly bounded on any bounded set;(iii)
f_α and f¯α satisfy the Lipschitz conditions.
Then the FCP (11) and the system of ODEs (12) are equivalent.


## 2.3. Interpolation for Fuzzy Numbers

The problem of interpolation for fuzzy sets is as follows.Suppose that at various time instantt information f(t) is presented as fuzzy set. The aim is to approximate the function f(t), for all t in the domain of f. Let t0<t1<⋯<tn be n+1 distinct points in R and let u0,u1,…,un be n+1 fuzzy sets in E1. A fuzzy polynomial interpolation of the data is a fuzzy-value function F:R→E1 satisfying the following conditions:(1)
f(ti)=ui, for all i=0,1,…,n;(2)
f is continuous;(3)
if the data is crisp, then the interpolationf is a crisp polynomial.A functionf fulfilling these conditions may be constructed as follows. Let Cαi=[ui]α, for any α∈[0,1], i=0,1,…,n. For each x=(x0,x1,…,xn)∈Rn+1, the unique polynomial of degree ≤n is denoted by PX such that
(13)PX(ti)=xi,i=0,1,…,n,PX(t)=∑i=0n‍xi(∏i≠j‍t-tjti-tj).Finally, for allt, ξ∈R are defined by f(t)∈E1 such that
(14)f(t)(ξ)=sup{α∈[0,1]:∃X∈Cα0211111×Cα1×⋯×Cαn,PX(t)=ξ}.The interpolation polynomial can be written level setwise as(15)[f(t)]α={y∈R:y=PX(t),x∈Cαi,i=0,1,…,n},
for 0≤α≤1.Theorem 10 (see [34]).
Let(ti,ui), i=0,1,…,n be the observed data and suppose that each of the ui=(uil,uic,uir) is an element of E1. Then for each t∈[t0,tn], f(t)=(fl(t),fc(t),fr(t))∈E1,
(16)fl(t)=∑li(t)≥0‍li(t)uil+∑li(t)<0‍li(t)uir,fc(t)=∑i=0n‍li(t)uic,fr(t)=∑li(t)≥0‍li(t)uir+∑li(t)<0‍li(t)uil,
where li(t)=∏i≠j‍((t-tj)/(ti-tj)), i=0,1,…,n.


## 3. Adams Method

### 3.1. Adams-Bashforth Method

Now we are going to solve fuzzy initial problemy′(t)=f(t,y) by Adams-Bashforth four-step method. Let the fuzzy initial values be y~(ti-3), y~(ti-2), y~(ti-1), y~(ti), that is,
(17)f~(ti-3,y(ti-3)),f~(ti-2,y(ti-2)),f~(ti-1,y(ti-1)),f~(ti,y(ti)),
which are triangular fuzzy numbers and are shown by
(18){fl(ti-3,y(ti-3)),fc(ti-3,y(ti-3)),fr(ti-3,y(ti-3))},{fl(ti-2,y(ti-2)),fc(ti-2,y(ti-2)),fr(ti-2,y(ti-2))},{fl(ti-1,y(ti-1)),fc(ti-1,y(ti-1)),fr(ti-1,y(ti-1))},{fl(ti,y(ti)),fc(ti,y(ti)),fr(ti,y(ti))},
also
(19)y~(ti+1)=y~(ti)+∫titi+1‍f(t,y(t))dt.
By fuzzy interpolation of f~(ti-3,y(ti-3)), f~(ti-2,y(ti-2)), f~(ti-1,y(ti-1)), f~(ti,y(ti)), we have
(20)fl(t,y(t))=∑lj(t)≥0‍lj(t)fl(tj,y(tj))+∑lj(t)<0‍lj(t)fr(tj,y(tj)),fc(t)=∑j=i-3i‍lj(t)fc(tj,y(tj)),fr(t,y(t))=∑lj(t)≥0‍lj(t)fr(tj,y(tj))+∑lj(t)<0‍lj(t)fl(tj,y(tj)),
for ti-1≤t≤ti,
(21)li-3(t)=(t-ti-2)(t-ti-1)(t-ti)(ti-3-ti-1)(ti-3-ti-1)(ti-3-ti)≥0,li-2(t)=(t-ti-3)(t-ti-1)(t-ti)(ti-2-ti-3)(ti-2-ti-1)(ti-2-ti)≤0,li-1(t)=(t-ti-3)(t-ti-2)(t-ti)(ti-1-ti-3)(ti-1-ti-2)(ti-1-ti)≥0,li(t)=(t-ti-3)(t-ti-2)(t-ti-1)(ti-ti-3)(ti-ti-2)(ti-ti-1)≥0,
therefore the following results will be obtained:
(22)fl(t,y(t))=li-3(t)fl(ti-3,y(ti-3))+li-2(t)fr(ti-2,y(ti-2))+li-1(t)fl(ti-1,y(ti-1))+li(t)fl(ti,y(ti)),fc(t,y(t))=li-3(t)fc(ti-3,y(ti-3))+li-2(t)fc(ti-2,y(ti-2))+li-1(t)fc(ti-1,y(ti-1))+li(t)fc(ti,y(ti)),fr(t,y(t))=li-3(t)fr(ti-3,y(ti-3))+li-2(t)fl(ti-2,y(ti-2))+li-1(t)fr(ti-1,y(ti-1))+li(t)fr(ti,y(ti)).From (3) and (19) it follows that
(23)y~α(ti+1)=[y_α(ti+1),y¯α(ti+1)],
where
(24)y_α(ti+1)=y_α(ti)+∫titi+1‍{fl(t,y(t))αfc(t,y(t))2003222222222222+(1-α)fl(t,y(t))}dt,y¯α(ti+1)=y¯α(ti)+∫titi+1‍{αfc(t,y(t))2003222222222222+(1-α)fr(t,y(t))}dt.If (22) are situated in (24), we have
(25)y_α(ti+1)=y_α(ti)+∫titi+1‍{+li(t)fl(ti,y(ti)))α(+li(t)fl(ti,y(ti)))li-3(t)fc(ti-3,y(ti-3))20032212222+li-2(t)fc(ti-2,y(ti-2))20032212222+li-1(t)fc(ti-1,y(ti-1))20032212222+li(t)fc(ti,y(ti))(+li(t)fl(ti,y(ti)))li-3(t)fl(ti-3,y(ti-3)))200322222+(1-α)(+li(t)fl(ti,y(ti)))li-3(t)fl(ti-3,y(ti-3))200322222222222222+li-2(t)fr(ti-2,y(ti-2))200322222222222222+li-1(t)fl(ti-1,y(ti-1))200322222222222222+li(t)fl(ti,y(ti)))}dt,y¯α(ti+1)=y¯α(ti)+∫titi+1‍{+li(t)fr(ti,y(ti)))α(+li(t)fc(ti,y(ti)))li-3(t)fc(ti-3,y(ti-3))20032221222+li-2(t)fc(ti-2,y(ti-2))20032221222+li-1(t)fc(ti-1,y(ti-1))20032221222+li(t)fc(ti,y(ti)))200322222+(1-α)(+li(t)fr(ti,y(ti)))li-3(t)fr(ti-3,y(ti-3))200322222222222222+li-2(t)fl(ti-2,y(ti-2))200322222222222222+li-1(t)fr(ti-1,y(ti-1))200322222222222222+li(t)fr(ti,y(ti)))}dt.The following results will be obtained by integration:(26)y_α(ti+1)=y_α(ti)+h24×{-9[αfc(ti-3,y(ti-3))2003212+(1-α)fr(ti-3,y(ti-3))]+37[(ti-2,y(ti-2))αfc(ti-2,y(ti-2))2003212+(1-α)fl(ti-2,y(ti-2))]-59[αfc(ti-1,y(ti-1))2003212+(1-α)fr(ti-1,y(ti-1))]+55[(ti,y(ti))αfc(ti,y(ti))2003212+(1-α)fl(ti,y(ti))]},y¯α(ti+1)=y¯α(ti)+h24×{-9[(ti-3,y(ti-3))αfc(ti-3,y(ti-3))20032+(1-α)fl(ti-3,y(ti-3))]+37[αfc(ti-2,y(ti-2))200312+(1-α)fr(ti-2,y(ti-2))]-59[(ti-3,y(ti-3))αfc(ti-1,y(ti-1))200312+(1-α)fl(ti-1,y(ti-1))]+55[αfc(ti,y(ti))200312+(1-α)fr(ti,y(ti))]}.Thus(27)y_α(ti+1)=y_α(ti)+h24[-9f¯α(ti-3,y(ti-3))+37f_α(ti-2,y(ti-2))2032222222222-59f¯α(ti-1,y(ti-1))+55f_α(ti,y(ti))],(28)y¯α(ti+1)=y¯α(ti)+h24[f_α(ti-3,y(ti-3))+37f¯α(ti-2,y(ti-2))2003222222222-59f_α(ti-1,y(ti-1))+55f¯α(ti,y(ti))].Therefore the four-step Adams-Bashforth method for solving fuzzy initial problems is obtained as follows:(29)y_α(ti+1)=y_α(ti)+h24[-9f¯α(ti-3,y(ti-3))+37f_α(ti-2,y(ti-2))2003222222221,-59f¯α(ti-1,y(ti-1))+55f_α(ti,y(ti))],y¯α(ti+1)=y¯α(ti)+h24[-9f_α(ti-3,y(ti-3))+37f¯α(ti-2,y(ti-2))2003222222221,-59f_α(ti-1,y(ti-1))+55f¯α(ti,y(ti))],y_α(ti-3)=α0,y_α(ti-2)=α1,y_α(ti-1)=α2,y_α(ti)=α3,y¯α(ti-3)=α4,y¯α(ti-2)=α5,y¯α(ti-1)=α6,y¯α(ti)=α720032222222222222222222222222221,,i=3,4,…,N-1.


### 3.2. Adams-Moulton Method

From [30], the Adams-Moulton three-step method to solve fuzzy initial problem y′(t)=f(t,y) is as follows:
(30)y_α(ti+1)=y_α(ti)20032221+h24[f_α(ti-2,y(ti-2))2003222222221,-5f¯α(ti-1,y(ti-1))2003222222221,+19f_α(ti,y(ti))+9f_α(ti+1,y(ti+1))],y¯α(ti+1)=y¯α(ti-1)20032221+h24[f¯α(ti-2,y(ti-2))2003222222221,-5f_α(ti-1,y(ti-1))2003222222221,+19f¯α(ti,y(ti))+9f¯α(ti+1,y(ti+1))],y_α(ti-2)=α0,y_α(ti-1)=α1,y_α(ti)=α2,y¯α(ti-2)=α3,y¯α(ti-1)=α4,y¯α(ti)=α520032222222222222222222222222221i=2,3,…,N-1.


## 3.1. Adams-Bashforth Method

Now we are going to solve fuzzy initial problemy′(t)=f(t,y) by Adams-Bashforth four-step method. Let the fuzzy initial values be y~(ti-3), y~(ti-2), y~(ti-1), y~(ti), that is,
(17)f~(ti-3,y(ti-3)),f~(ti-2,y(ti-2)),f~(ti-1,y(ti-1)),f~(ti,y(ti)),
which are triangular fuzzy numbers and are shown by
(18){fl(ti-3,y(ti-3)),fc(ti-3,y(ti-3)),fr(ti-3,y(ti-3))},{fl(ti-2,y(ti-2)),fc(ti-2,y(ti-2)),fr(ti-2,y(ti-2))},{fl(ti-1,y(ti-1)),fc(ti-1,y(ti-1)),fr(ti-1,y(ti-1))},{fl(ti,y(ti)),fc(ti,y(ti)),fr(ti,y(ti))},
also
(19)y~(ti+1)=y~(ti)+∫titi+1‍f(t,y(t))dt.
By fuzzy interpolation of f~(ti-3,y(ti-3)), f~(ti-2,y(ti-2)), f~(ti-1,y(ti-1)), f~(ti,y(ti)), we have
(20)fl(t,y(t))=∑lj(t)≥0‍lj(t)fl(tj,y(tj))+∑lj(t)<0‍lj(t)fr(tj,y(tj)),fc(t)=∑j=i-3i‍lj(t)fc(tj,y(tj)),fr(t,y(t))=∑lj(t)≥0‍lj(t)fr(tj,y(tj))+∑lj(t)<0‍lj(t)fl(tj,y(tj)),
for ti-1≤t≤ti,
(21)li-3(t)=(t-ti-2)(t-ti-1)(t-ti)(ti-3-ti-1)(ti-3-ti-1)(ti-3-ti)≥0,li-2(t)=(t-ti-3)(t-ti-1)(t-ti)(ti-2-ti-3)(ti-2-ti-1)(ti-2-ti)≤0,li-1(t)=(t-ti-3)(t-ti-2)(t-ti)(ti-1-ti-3)(ti-1-ti-2)(ti-1-ti)≥0,li(t)=(t-ti-3)(t-ti-2)(t-ti-1)(ti-ti-3)(ti-ti-2)(ti-ti-1)≥0,
therefore the following results will be obtained:
(22)fl(t,y(t))=li-3(t)fl(ti-3,y(ti-3))+li-2(t)fr(ti-2,y(ti-2))+li-1(t)fl(ti-1,y(ti-1))+li(t)fl(ti,y(ti)),fc(t,y(t))=li-3(t)fc(ti-3,y(ti-3))+li-2(t)fc(ti-2,y(ti-2))+li-1(t)fc(ti-1,y(ti-1))+li(t)fc(ti,y(ti)),fr(t,y(t))=li-3(t)fr(ti-3,y(ti-3))+li-2(t)fl(ti-2,y(ti-2))+li-1(t)fr(ti-1,y(ti-1))+li(t)fr(ti,y(ti)).From (3) and (19) it follows that
(23)y~α(ti+1)=[y_α(ti+1),y¯α(ti+1)],
where
(24)y_α(ti+1)=y_α(ti)+∫titi+1‍{fl(t,y(t))αfc(t,y(t))2003222222222222+(1-α)fl(t,y(t))}dt,y¯α(ti+1)=y¯α(ti)+∫titi+1‍{αfc(t,y(t))2003222222222222+(1-α)fr(t,y(t))}dt.If (22) are situated in (24), we have
(25)y_α(ti+1)=y_α(ti)+∫titi+1‍{+li(t)fl(ti,y(ti)))α(+li(t)fl(ti,y(ti)))li-3(t)fc(ti-3,y(ti-3))20032212222+li-2(t)fc(ti-2,y(ti-2))20032212222+li-1(t)fc(ti-1,y(ti-1))20032212222+li(t)fc(ti,y(ti))(+li(t)fl(ti,y(ti)))li-3(t)fl(ti-3,y(ti-3)))200322222+(1-α)(+li(t)fl(ti,y(ti)))li-3(t)fl(ti-3,y(ti-3))200322222222222222+li-2(t)fr(ti-2,y(ti-2))200322222222222222+li-1(t)fl(ti-1,y(ti-1))200322222222222222+li(t)fl(ti,y(ti)))}dt,y¯α(ti+1)=y¯α(ti)+∫titi+1‍{+li(t)fr(ti,y(ti)))α(+li(t)fc(ti,y(ti)))li-3(t)fc(ti-3,y(ti-3))20032221222+li-2(t)fc(ti-2,y(ti-2))20032221222+li-1(t)fc(ti-1,y(ti-1))20032221222+li(t)fc(ti,y(ti)))200322222+(1-α)(+li(t)fr(ti,y(ti)))li-3(t)fr(ti-3,y(ti-3))200322222222222222+li-2(t)fl(ti-2,y(ti-2))200322222222222222+li-1(t)fr(ti-1,y(ti-1))200322222222222222+li(t)fr(ti,y(ti)))}dt.The following results will be obtained by integration:(26)y_α(ti+1)=y_α(ti)+h24×{-9[αfc(ti-3,y(ti-3))2003212+(1-α)fr(ti-3,y(ti-3))]+37[(ti-2,y(ti-2))αfc(ti-2,y(ti-2))2003212+(1-α)fl(ti-2,y(ti-2))]-59[αfc(ti-1,y(ti-1))2003212+(1-α)fr(ti-1,y(ti-1))]+55[(ti,y(ti))αfc(ti,y(ti))2003212+(1-α)fl(ti,y(ti))]},y¯α(ti+1)=y¯α(ti)+h24×{-9[(ti-3,y(ti-3))αfc(ti-3,y(ti-3))20032+(1-α)fl(ti-3,y(ti-3))]+37[αfc(ti-2,y(ti-2))200312+(1-α)fr(ti-2,y(ti-2))]-59[(ti-3,y(ti-3))αfc(ti-1,y(ti-1))200312+(1-α)fl(ti-1,y(ti-1))]+55[αfc(ti,y(ti))200312+(1-α)fr(ti,y(ti))]}.Thus(27)y_α(ti+1)=y_α(ti)+h24[-9f¯α(ti-3,y(ti-3))+37f_α(ti-2,y(ti-2))2032222222222-59f¯α(ti-1,y(ti-1))+55f_α(ti,y(ti))],(28)y¯α(ti+1)=y¯α(ti)+h24[f_α(ti-3,y(ti-3))+37f¯α(ti-2,y(ti-2))2003222222222-59f_α(ti-1,y(ti-1))+55f¯α(ti,y(ti))].Therefore the four-step Adams-Bashforth method for solving fuzzy initial problems is obtained as follows:(29)y_α(ti+1)=y_α(ti)+h24[-9f¯α(ti-3,y(ti-3))+37f_α(ti-2,y(ti-2))2003222222221,-59f¯α(ti-1,y(ti-1))+55f_α(ti,y(ti))],y¯α(ti+1)=y¯α(ti)+h24[-9f_α(ti-3,y(ti-3))+37f¯α(ti-2,y(ti-2))2003222222221,-59f_α(ti-1,y(ti-1))+55f¯α(ti,y(ti))],y_α(ti-3)=α0,y_α(ti-2)=α1,y_α(ti-1)=α2,y_α(ti)=α3,y¯α(ti-3)=α4,y¯α(ti-2)=α5,y¯α(ti-1)=α6,y¯α(ti)=α720032222222222222222222222222221,,i=3,4,…,N-1.


## 3.2. Adams-Moulton Method

From [30], the Adams-Moulton three-step method to solve fuzzy initial problem y′(t)=f(t,y) is as follows:
(30)y_α(ti+1)=y_α(ti)20032221+h24[f_α(ti-2,y(ti-2))2003222222221,-5f¯α(ti-1,y(ti-1))2003222222221,+19f_α(ti,y(ti))+9f_α(ti+1,y(ti+1))],y¯α(ti+1)=y¯α(ti-1)20032221+h24[f¯α(ti-2,y(ti-2))2003222222221,-5f_α(ti-1,y(ti-1))2003222222221,+19f¯α(ti,y(ti))+9f¯α(ti+1,y(ti+1))],y_α(ti-2)=α0,y_α(ti-1)=α1,y_α(ti)=α2,y¯α(ti-2)=α3,y¯α(ti-1)=α4,y¯α(ti)=α520032222222222222222222222222221i=2,3,…,N-1.


## 4. Predictor-Corrector Method

### 4.1. Adams Predictor-Corrector Method

The following algorithm is based on Adams-Bashforth four-step method as a predictor and also an iteration of Adams-Moulton three-step method as a corrector.Algorithm 11 (predictor-corrector four-step method).
To approximate the solution of the following fuzzy initial value problem:(31)y′(t)=f(t,y),t0≤t≤T,y_α(t0)=α0,y_α(t1)=α1,y_α(t2)=α2,y_α(t3)=α3,y¯α(t0)=α4,y¯α(t1)=α5,y¯α(t2)=α6,y¯α(t3)=α7,
positive integer N is chosen.Step 1.
Leth=(T-t0)/N,
(32)w_α(t0)=α0,w_α(t1)=α1,w_α(t2)=α2,w_α(t3)=α3,w¯α(t0)=α4,w¯α(t1)=α5,w¯α(t2)=α6,w¯α(t3)=α7.Step 2.
Leti=3.Step 3.
Let(33)w_(0)α(ti+1)=w_α(ti)2003222211+h24[-9f¯α(ti-3,w(ti-3))200322221111111+37f_α(ti-2,w(ti-2))200322221111111-59f¯α(ti-1,w(ti-1))200322221111111+55f_α(ti,w(ti))],w¯(0)α(ti+1)=w¯α(ti)2003222211+h24[-9f_α(ti-3,w(ti-3))200322221111111+37f¯α(ti-2,w(ti-2))200322221111111-59f_α(ti-1,w(ti-1))200322221111111+55f¯α(ti,w(ti))].Step 4.
Letti+1=t0+ih.Step 5.
Let(34)w_α(ti+1)=w_α(ti)200322111+h24[f_α(ti-2,w(ti-2))20032222222211-5f¯α(ti-1,w(ti-1))20032222222211+19f_α(ti,w(ti))20032222222211+9f_α(ti+1,w_(0)(ti+1))],w¯α(ti+1)=w¯α(ti-1)200322111+h24[f¯α(ti-2,w(ti-2))20032222222211-5f_α(ti-1,w(ti-1))20032222222211+19f¯α(ti,w(ti))20032222222211+9f¯α(ti+1,w¯(0)(ti+1))].Step 6.
i
=
i
+
1.Step 7.
Ifi≤N-1 goto Step 3.Step 8.
Algorithm is completed and(w_α(ti),w¯α(ti)) approximates real value of (Y_α(ti),Y¯α(ti)) to the original differential equations (i=4,5,…,N).


### 4.2. Improved Adams Predictor-Corrector Method

In the above section, the predicted valuesy_i+10,y¯i+10 and the corrected values y_i+1,y¯i+1 have the local truncation errors as follows:
(35)y_(ti+1)-y_i+10≈251720h5y_(5)(x_i),y_(ti+1)-y_i+10≈251720h5y_(5)(x_i),y¯(ti+1)-y¯i+1≈-19720h5y¯(5)(xi),y¯(ti+1)-y¯i+1≈-19720h5y¯(5)(x¯i).
Thus there exists the error estimations
(36)y_(ti+1)-y_i+10≈-251270(y_i+10-y_i+1),y¯(ti+1)-y¯i+10≈-251270(y¯i+10-y¯i+1),y¯(ti+1)-y¯i+1≈-19720(y_i+10-y_i+1),y¯(ti+1)-y¯i+1≈-19720(y¯i+10-y¯i+1).Based on above results, we improve Adams predictor-corrector four-step method into the following iterative computation algorithm.Algorithm 12 (improved predictor-corrector systems).
To approximate the solution of the following fuzzy initial value problem:(37)y′(t)=f(t,y),t0≤t≤T,y_α(t0)=α0,y_α(t1)=α1,y_α(t2)=α2,y_α(t3)=α3,y¯α(t0)=α4,y¯α(t1)=α5,y¯α(t2)=α6,y¯α(t3)=α7,
positive integer N is chosen.Step 1.
Leth=(T-t0)/N,
(38)w_α(t0)=α0,w_α(t1)=α1,w_α(t2)=α2,w_α(t3)=α3,w¯α(t0)=α4,w¯α(t1)=α5,w¯α(t2)=α6,w¯α(t3)=α7.Step 2.
Leti=3.Step 3.
Let(39)p_i+1(0)α=w_α(ti)200311+h24[-9f¯α(ti-3,w(ti-3))20032222211+37f_α(ti-2,w(ti-2))20032222211-59f¯α(ti-1,w(ti-1))20032222211+55f_α(ti,w(ti))],p¯i+1(0)α=w¯α(ti)200311+h24[-9f_α(ti-3,w(ti-3))20032222211+37f¯α(ti-2,w(ti-2))20032222211-59f_α(ti-1,w(ti-1))20032222211+55f¯α(ti,w(ti))].Step 4.
Letm_i+1=f_(ti+1,p_i+1)+(251/270)(c_i-p_i), m¯i+1=f¯(ti+1,p¯i+1)+(251/270)(c¯i-p¯i).Step 5.
Let(40)c_i+1α=y_α(ti)20031+h24[f_α(ti-2,w(ti-2))2003222222-5f_α(ti-1,w(ti-1))2003222222+19f_α(ti,w(ti))2003222222+9f_α(ti+1,w_(0)(ti+1))],c¯i+1α=y¯α(ti-1)20031+h24[f¯α(ti-2,w(ti-2))2003222222-5f_α(ti-1,w(ti-1))2003222222+19f¯α(ti,w(ti))2003222222+9f¯α(ti+1,w¯(0)(ti+1))].Step 6.
Let(41)y_i+1=c_i+1-19720(c_i+1-p_i+1),y¯i+1=c¯i+1-19720(c¯i+1-p¯i+1).Step 7.
i
=
i
+
1.Step 8.
Ifi≤N-1 goto Step 3.Step 9.
Algorithm is completed and(w_α(ti),w¯α(ti)) approximates real value of (Y_α(ti),Y¯α(ti)) to the original differential equations (i=4,5,…,N).


## 4.1. Adams Predictor-Corrector Method

The following algorithm is based on Adams-Bashforth four-step method as a predictor and also an iteration of Adams-Moulton three-step method as a corrector.Algorithm 11 (predictor-corrector four-step method).
To approximate the solution of the following fuzzy initial value problem:(31)y′(t)=f(t,y),t0≤t≤T,y_α(t0)=α0,y_α(t1)=α1,y_α(t2)=α2,y_α(t3)=α3,y¯α(t0)=α4,y¯α(t1)=α5,y¯α(t2)=α6,y¯α(t3)=α7,
positive integer N is chosen.Step 1.
Leth=(T-t0)/N,
(32)w_α(t0)=α0,w_α(t1)=α1,w_α(t2)=α2,w_α(t3)=α3,w¯α(t0)=α4,w¯α(t1)=α5,w¯α(t2)=α6,w¯α(t3)=α7.Step 2.
Leti=3.Step 3.
Let(33)w_(0)α(ti+1)=w_α(ti)2003222211+h24[-9f¯α(ti-3,w(ti-3))200322221111111+37f_α(ti-2,w(ti-2))200322221111111-59f¯α(ti-1,w(ti-1))200322221111111+55f_α(ti,w(ti))],w¯(0)α(ti+1)=w¯α(ti)2003222211+h24[-9f_α(ti-3,w(ti-3))200322221111111+37f¯α(ti-2,w(ti-2))200322221111111-59f_α(ti-1,w(ti-1))200322221111111+55f¯α(ti,w(ti))].Step 4.
Letti+1=t0+ih.Step 5.
Let(34)w_α(ti+1)=w_α(ti)200322111+h24[f_α(ti-2,w(ti-2))20032222222211-5f¯α(ti-1,w(ti-1))20032222222211+19f_α(ti,w(ti))20032222222211+9f_α(ti+1,w_(0)(ti+1))],w¯α(ti+1)=w¯α(ti-1)200322111+h24[f¯α(ti-2,w(ti-2))20032222222211-5f_α(ti-1,w(ti-1))20032222222211+19f¯α(ti,w(ti))20032222222211+9f¯α(ti+1,w¯(0)(ti+1))].Step 6.
i
=
i
+
1.Step 7.
Ifi≤N-1 goto Step 3.Step 8.
Algorithm is completed and(w_α(ti),w¯α(ti)) approximates real value of (Y_α(ti),Y¯α(ti)) to the original differential equations (i=4,5,…,N).


## 4.2. Improved Adams Predictor-Corrector Method

In the above section, the predicted valuesy_i+10,y¯i+10 and the corrected values y_i+1,y¯i+1 have the local truncation errors as follows:
(35)y_(ti+1)-y_i+10≈251720h5y_(5)(x_i),y_(ti+1)-y_i+10≈251720h5y_(5)(x_i),y¯(ti+1)-y¯i+1≈-19720h5y¯(5)(xi),y¯(ti+1)-y¯i+1≈-19720h5y¯(5)(x¯i).
Thus there exists the error estimations
(36)y_(ti+1)-y_i+10≈-251270(y_i+10-y_i+1),y¯(ti+1)-y¯i+10≈-251270(y¯i+10-y¯i+1),y¯(ti+1)-y¯i+1≈-19720(y_i+10-y_i+1),y¯(ti+1)-y¯i+1≈-19720(y¯i+10-y¯i+1).Based on above results, we improve Adams predictor-corrector four-step method into the following iterative computation algorithm.Algorithm 12 (improved predictor-corrector systems).
To approximate the solution of the following fuzzy initial value problem:(37)y′(t)=f(t,y),t0≤t≤T,y_α(t0)=α0,y_α(t1)=α1,y_α(t2)=α2,y_α(t3)=α3,y¯α(t0)=α4,y¯α(t1)=α5,y¯α(t2)=α6,y¯α(t3)=α7,
positive integer N is chosen.Step 1.
Leth=(T-t0)/N,
(38)w_α(t0)=α0,w_α(t1)=α1,w_α(t2)=α2,w_α(t3)=α3,w¯α(t0)=α4,w¯α(t1)=α5,w¯α(t2)=α6,w¯α(t3)=α7.Step 2.
Leti=3.Step 3.
Let(39)p_i+1(0)α=w_α(ti)200311+h24[-9f¯α(ti-3,w(ti-3))20032222211+37f_α(ti-2,w(ti-2))20032222211-59f¯α(ti-1,w(ti-1))20032222211+55f_α(ti,w(ti))],p¯i+1(0)α=w¯α(ti)200311+h24[-9f_α(ti-3,w(ti-3))20032222211+37f¯α(ti-2,w(ti-2))20032222211-59f_α(ti-1,w(ti-1))20032222211+55f¯α(ti,w(ti))].Step 4.
Letm_i+1=f_(ti+1,p_i+1)+(251/270)(c_i-p_i), m¯i+1=f¯(ti+1,p¯i+1)+(251/270)(c¯i-p¯i).Step 5.
Let(40)c_i+1α=y_α(ti)20031+h24[f_α(ti-2,w(ti-2))2003222222-5f_α(ti-1,w(ti-1))2003222222+19f_α(ti,w(ti))2003222222+9f_α(ti+1,w_(0)(ti+1))],c¯i+1α=y¯α(ti-1)20031+h24[f¯α(ti-2,w(ti-2))2003222222-5f_α(ti-1,w(ti-1))2003222222+19f¯α(ti,w(ti))2003222222+9f¯α(ti+1,w¯(0)(ti+1))].Step 6.
Let(41)y_i+1=c_i+1-19720(c_i+1-p_i+1),y¯i+1=c¯i+1-19720(c¯i+1-p¯i+1).Step 7.
i
=
i
+
1.Step 8.
Ifi≤N-1 goto Step 3.Step 9.
Algorithm is completed and(w_α(ti),w¯α(ti)) approximates real value of (Y_α(ti),Y¯α(ti)) to the original differential equations (i=4,5,…,N).


## 5. Convergence

To integrate the system given in (11) from t0 to a prefixed T>t0, the interval [t0,T] is replaced by a set of discrete equally spaced grid points t0<t1<⋯<tn=T, and the exact solution (Y_(t,α),Y¯(t,α)) is approximated by some (y_(t,α),y¯(t,α)). The exact and approximate solutions at ti,0≤i≤N, are denoted by Yi(α)=(Y_i(α),Y¯i(α)) and yi(α)=(y_i(α),y¯i(α)), respectively. The grid points at which the solution is calculated are ti=t0+ih, h=(T-t0)/N, 0≤i≤N.From (30), the polygon curves
(42)y_(t,h,α)={[t0,y_0(α)],[t1,y_1(α)],…,[tN,y_N(α)]},y¯(t,h,α)={[t0,y¯0(α)],[t1,y¯1(α)],…,[tN,y¯N(α)]}
are the implicit three-step approximation to Y_(t,α) and Y¯(t,α), respectively, over the interval [t0,T]. The following lemma will be applied to show the convergence of these approximations; that is,
(43)limh→0y_(t,h,α)=Y_(t,α),limh→0y¯(t,h,α)=Y¯(t,α).Lemma 13.
Let a sequence of numbers{wn}n=0N satisfy
(44)|wn+1|≤A|wn|+B|wn-1|+C,0≤n≤N-1,
for some given positive A and B, C. Then
(45)|wn|≤(An-1+β1An-3B+β2An-5B2+⋯+βsBn/2)|w1|+(An-2B+γ1An-4B2+⋯+γtABn/2)|w0|+(An-2+An-3+⋯+1)C+(δ1An-4+δ2An-5+⋯+δmA+1)BC+(ζ1An-6+ζ2An-7+⋯+ζlA+1)B2C+(λ1An-8+λ2An-9+⋯+λpA+1)B3C+⋯,
when n is odd and
(46)|wn|≤(An-1+β1An-3B+β2An-5B2+⋯+βsB(n/2)-1)|w1|+(An-2B+γ1An-4B2+⋯+γtABn/2)|w0|+(An-2+An-3+⋯+1)C+(δ1An-4+δ2An-5+⋯+δmA+1)BC+(ζ1An-6+ζ2An-7+⋯+ζlA+1)B2C+(λ1An-8+λ2An-9+⋯+λpA+1)B3C+⋯,
when n is even, where βs, γt, δm, ζl, λp are constants for all s, t, l, m, and p.Theorem 14.
For any arbitrary fixedr:0≤r≤1, the implicit three-step Simpson approximations of (30) converge to the exact solutions Y_(t,α), Y¯(t,α) for Y_, Y¯∈C3[t0,T].Proof.
It is sufficient to show(47)limh→0y_N(α)=Y_(T,α),limh→0y¯N(α)=Y¯(T,α).
By Taylor’s theorem, we have(48)Y_n+1(α)=Y_n-1(α)+h3[f(tn-1,Y_n-1(α))+4f(tn,Y_n(α))20032222222222+f(tn+1,Y_n+1(α))]-h590Y_(5)(ξ_n),Y¯n+1(α)=Y¯n-1(α)+h3[f(tn-1,Y¯n-1(α))+4f(tn,Y¯n(α))20032222222222+f(tn+1,Y¯n+1(α))]-h590Y¯(5)(ξ¯n),
where tn-1<ξ_n, ξ¯n<tn+1. Consequently
(49)Y_n+1(α)-y_n+1(α)=Y_n-1(α)-y_n-1(α)+h3{(tn-1,y_n-1(α))f(tn-1,Y_n-1(α))20032222222222222222222-f(tn-1,y_n-1(α))}+4h3{f(tn,Y_n(α))-f(tn,y_n(α))}+h3{f(tn+1,Y_n+1(α))-f(tn+1,y_n+1(α))}-h590Y_(5)(ξ_n),Y¯n+1(α)-y¯n+1(α)=Y¯n-1(α)-y¯n-1(α)+h3{f(tn-1,Y¯n-1(α))20032222222222222222222-f(tn-1,y¯n-1(α))f(tn-1,Y¯n-1(α))}+4h3{f(tn,Y¯n(α)){-f(tn,y¯n(α))}×h3{f(tn+1,Y¯n+1(α))-f(tn+1,y¯n+1(α))f(tn+1,Y¯n+1(α))}-h590Y¯(5)(ξ¯n).
Denotewn=Y_n(α)-y_n(α), vn=Y¯n+1(α)-y¯n+1(α). Then
(50)|wn+1|≤4hL13|wn+1|+(1+hL23)|wn-1|+hL33|wn+1|+h590M_,|vn+1|≤4hL43|vn+1|+(1+hL53)|vn-1|+hL63|vn+1|+h590M¯,
where M_=maxt0≤t≤T|Y_(5)(t,α)| and M¯=maxt0≤t≤T|Y¯(5)(t,α)|.
Set(51)L=max{L1,L2,L3,L4,L5,L6}<3h,
then
(52)|wn+1|≤4hL3-hL|wn|+3+hL3-hL|wn-1|+130(3-hL)h5M_,|vn+1|≤4hL3-hL|vn|+3+hL3-hL|vn-1|+130(3-hL)h5M¯
are obtained, where |un|=|wn|+|vn|, so by Lemma 13, since w0=v0=0 and w1=v1=0 we have
(53)|un|≤((4hL)/(3-hL))n-1-1(30(hL-3))/(3-hL)×130(3-hL)h5(M_+M¯)+{δ1(4hL3-hL)n-4+δ2(4hL3-hL)n-520032+⋯+δm(4hL3-hL)+1δ1(4hL3-hL)n-4}×(3+hL3-hL)(130(3-hL)h5(M_+M¯))+{ζ1(4hL3-hL)n-6+ζ2(4hL3-hL)n-720032+⋯+ζl(4hL3-hL)+1ζ1(4hL3-hL)n-6}×(3+hL3-hL)2(130(3-hL)h5(M_+M¯))+{λ1(4hL3-hL)n-8+λ2(4hL3-hL)n-920032+⋯+λp(4hL3-hL)+1λ1(4hL3-hL)n-8}×(3+hL3-hL)3(130(3-hL)h5(M_+M¯))+⋯.
If h→0 then wn→0,vn→0, which concludes the proof.Theorem 15.
For any arbitrary fixedr:0≤r≤1, the explicit four-step Milne approximations of (29) converge to the exact solutions Y_(t,α), Y¯(t,α) for Y_,Y¯∈C3[t0,T].Proof.
Like Theorem14, the conclusion can be obtained easily.


## 6. Numerical Examples

Example 16 (see [33]).
Consider the following fuzzy differential equation:(54)y~′(t)=-y~(t)+t+1,t≥0,y~(0)=(0.96,1,1.01).
The exact solution of equation is(55)Y_(t)=t-0.025et+0.985e-t,Y1(t)=t+1.0e-t,Y¯(t)=t+0.025et+0.985e-t.
By using the Adams predictor-corrector four-step method withN=10 for some t∈[0,1], the results shown in Tables 1, 2, and 3 are obtained.
And by using the improved Adams predictor-corrector systems withN=10 for some t∈[0,1], the results shown in Tables 4, 5, and 6 are obtained.
The results of Example16 are shown by Figures 1, 2, and 3.Table 1
Comparisons between the exact solution and the numerical solution.
t
Y
_
(
t
,
r
)
y
_
(
t
,
r
)
Error
0.1
0.963635583813
0.963082998354
0.000552585459
0.2
0.975914722827
0.975304021448
0.000610701379
0.3
0.995959477182
0.995284547778
0.000674929403
0.4
1.022969627904
1.022223715555
0.000745912348
0.5
1.056214668049
1.055390307414
0.000824360635
0.6
1.095026491542
1.094115432142
0.000911059400
0.7
1.138792706547
1.137785830194
0.001006876353
0.8
1.186950506443
1.185837735978
0.001112770464
0.9
1.238981037065
1.237751235509
0.001229801555
1
1.294404203842
1.293045062928
0.001359140914Table 2
Comparisons between the exact solution and the numerical solution.
t
Y
1
(
t
,
r
)
y
1
(
t
,
r
)
Error
0.1
1.004837418035
1.006647092872
0.001809674836
0.2
1.018730753077
1.020368214584
0.001637461506
0.3
1.040818220681
1.042299857123
0.001481636441
0.4
1.070320046035
1.071660686127
0.001340640092
0.5
1.106530659712
1.107743721032
0.001213061319
0.6
1.148811636094
1.149909259366
0.001097623272
0.7
1.196585303791
1.197578474398
0.000993170607
0.8
1.249328964117
1.250227622045
0.000898657928
0.9
1.306569659740
1.307382799060
0.000813139319
1
1.367879441171
1.368615200053
0.000735758882Table 3
Comparisons between the exact solution and the numerical solution.
t
Y
¯
(
t
,
r
)
y
¯
(
t
,
r
)
Error
0.1
1.018894129717
1.019446715176
0.000552585459
0.2
1.036984860735
1.037595562114
0.000610701379
0.3
1.063452417560
1.064127346964
0.000674929403
0.4
1.097560862786
1.098306775134
0.000745912348
0.5
1.138650731584
1.139475092219
0.000824360635
0.6
1.186132431562
1.187043490962
0.000911059400
0.7
1.239480341921
1.240487218275
0.001006876353
0.8
1.298227552867
1.299340323332
0.001112770464
0.9
1.361961192623
1.363190994178
0.001229801555
1
1.430318295265
1.431677436179
0.001359140914Table 4
Comparisons between the exact solution and the numerical solution.
t
Y
_
(
t
,
r
)
y
_
(
t
,
r
)
Error
0.1
0.963635583813
0.963525066721
0.110517091807
e
-
003
0.2
0.975914722827
0.975792582551
0.122140275815
e
-
003
0.3
0.995959477182
0.995824491301
0.134985880757
e
-
003
0.4
1.022969627904
1.022820445434
0.149182469764
e
-
003
0.5
1.056214668049
1.056049795922
0.164872127069
e
-
003
0.6
1.095026491542
1.094844279662
0.182211880038
e
-
003
0.7
1.138792706547
1.138591331277
0.201375270747
e
-
003
0.8
1.186950506443
1.186727952350
0.222554092849
e
-
003
0.9
1.238981037065
1.238735076754
0.245960311115
e
-
003
1
1.294404203842
1.294132375659
0.271828182845
e
-
003Table 5
Comparisons between the exact solution and the numerical solution.
t
Y
1
(
t
,
r
)
y
1
(
t
,
r
)
Error
0.1
1.004837418035
1.005742255454
0.904837418035
e
-
003
0.2
1.018730753077
1.019549483831
0.818730753077
e
-
003
0.3
1.040818220681
1.041559038902
0.740818220682
e
-
003
0.4
1.070320046035
1.070990366081
0.670320046035
e
-
003
0.5
1.106530659712
1.107137190372
0.606530659712
e
-
003
0.6
1.148811636094
1.149360447730
0.548811636093
e
-
003
0.7
1.196585303791
1.197081889095
0.496585303791
e
-
003
0.8
1.249328964117
1.249778293081
0.449328964117
e
-
003
0.9
1.306569659740
1.306976229400
0.406569659740
e
-
003
1
1.367879441171
1.368247320612
0.367879441171
e
-
003Table 6
Comparisons between the exact solution and the numerical solution.
t
Y
¯
(
t
,
r
)
y
¯
(
t
,
r
)
Error
0.1
1.018894129717
1.019004646809
0.110517091807
e
-
003
0.2
1.036984860735
1.037107001011
0.122140275816
e
-
003
0.3
1.063452417560
1.063587403441
0.134985880757
e
-
003
0.4
1.097560862786
1.097710045255
0.149182469763
e
-
003
0.5
1.138650731584
1.138815603711
0.164872127070
e
-
003
0.6
1.186132431562
1.186314643442
0.182211880038
e
-
003
0.7
1.239480341921
1.239681717192
0.201375270747
e
-
003
0.8
1.298227552867
1.298450106960
0.222554092849
e
-
003
0.9
1.361961192623
1.362207152934
0.245960311115
e
-
003
1
1.430318295265
1.430590123448
0.271828182845
e
-
003Figure 1
Comparisons between the exact solutionY_ and the numerical solution y_.Figure 2
Comparisons between the exact solutionY1 and the numerical solution y1.Figure 3
Comparisons between the exact solutionY¯ and the numerical solution y¯.


## 7. Conclusion

In this paper two numerical methods with higher order of convergence and not much amounts of computation for solving fuzzy differential equations were discussed in detail. The proposed algorithms were generated by updating the Adams-Bashforth four-step method and Adams-Moulton three-step method. An example showed that the proposed methods is more efficient and practical than some methods appeared in the literature before.



---

*Source: 312328-2013-06-24.xml*