# Formulation and Solution ofnth-Order Derivative Fuzzy Integrodifferential Equation Using New Iterative Method with a Reliable Algorithm

Authors: A. A. Hemeda
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Thenth-order derivative fuzzy integro-differential equation in
parametric form is converted to its crisp form, and then the new iterative method
with a reliable algorithm is used to obtain an approximate solution for this crisp
form. The analysis is accompanied by numerical examples which confirm efficiency and power of this method in solving fuzzy integro-differential equations.


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## Body

## 1. Introduction

In 1975 Zadeh and then Dubois and Prade [1] introduced fuzzy numbers and fuzzy arithmetic. This concept propagated widely to various problems, for example, fuzzy linear systems [2, 3], fuzzy differential equations [4, 5], fuzzy integral equations [6–8], and fuzzy integro-differential equations [9–12]. Additional topics can be found in [13, 14]. Recently, several numerical methods were suggested to solve integro-differential equations, for example, Sine-Cosine wavelet used by Tavassoli Kajani et al. to obtain a solution of linear integro-differential equations [15] and variational iteration method used by Abbasbandy and Hashemi to formulate and solve fuzzy integro-differential equation [12], by Saberi-Nadjafi and Tamamgar for solving system of integro-differential equations [16], and by Shang and Han for solving nth-order integro-differential equations [17]. Some other worthwhile works can be found in [18, 19]. Recently, Daftardar-Gejji and Jafari [20, 21] proposed the new iterative method. This method has proven useful for solving a variety of linear and nonlinear equations such as algebraic equations, integral equations, ordinary and partial differential equations of integer and fractional order, and system of equations as well. The new iterative method is simple to understand and easy to implement using computer packages and yields better results [21] than the existing Adomian decomposition method [22], homotopy perturbation method [23], or variational iteration method [24]. For more details, see [25–40].In the present work we apply the new iterative method with a reliable algorithm to solve thenth-order derivative fuzzy integro-differential equation in its crisp form.


## 2. Preliminaries

In this section we set up the basic definitions of fuzzy numbers and fuzzy functions.Definition 2.1 (see [2, 3, 12]).
A fuzzy number in parametric form is an ordered pair of functions(u_(r),u¯(r)), 0≤r≤1 which satisfy the following requirements:(1)
u_(r) is a bounded left continuous nondecreasing function over [0,1],(2)
u¯(r) is a bounded left continuous nonincreasing function over [0,1],(3)
u_(r)≤u¯(r), 0≤r≤1.Remark 2.2 (see [6, 12]).
Letu(r)=(u_(r),u¯(r)), 0≤r≤1 be a fuzzy number, and we can take
(2.1)uc(r)=12(u_(r)+u¯(r)),ud(r)=12(u¯(r)-u_(r)).
It is clear that ud(r)≥0, u_(r)=uc(r)-ud(r) and u¯(r)=uc(r)+ud(r), and also a fuzzy number u∈E is said symmetric if uc(r) is independent of r for all 0≤r≤1.
Letf:[a,b]→E for each partition P={t0,…,tn} of [a,b] and for arbitrary ξi∈ti-1-ti], 1≤i≤nsuppose that RP=∑i=1nf(ξi)(ti-ti-1) and Δ=max{|ti-ti-1|, i=1,…,n}. The definite integral of f(t) over [a,b] is ∫ab‍f(t)dt=limΔ→0RP provided that this limit exists in the metric D. If the fuzzy function f(t) is continuous in the metric D, its definite integral exists. Also we have (∫ab‍f(t,r)dt_)=∫ab‍f_(t,r)dt and (∫ab‍f(t,r)dt¯)=∫ab‍f¯(t,r)dt.Definition 2.3 (see [33]).
Letf:(a,b)→RF and t0∈(a,b). We say that f′ is Hukuhara differentiable at t0 if there exists an element f′(t0)∈RF, such that for all h>0 sufficiently small, ∃f(t0+h)⊝Hf(t0),f(t0)⊝Hf(t0-h) and
(2.2)limh→0f(t0+h)⊝Hf(t0)h=limh→0f(t0)⊝Hf(t0-h)h=f′(t0),
where ⊝H is Hukuhara difference, RF is the set of fuzzy numbers, and the limit is in the metric D.
In parametric form, iff=(f_,f¯), then f′=(f′_,f¯′), where f′_,f¯′ are Hukuhara differentiable of f_,f¯, respectively.
From (2.2), the nth-order Hukuhara differentiable, f(n), of f at t0 can be defined as in the following definition.Definition 2.4.
Letf(n-1):(a,b)→RF and t0∈(a,b) where f(n-1) is (n-1)th-orderHukuhara differentiable of f at t0 for all n>1. We say that f(n) is nth-order Hukuhara differentiable at t0 if there exists an element f(n)(t0)∈RF such that for all h>0 sufficiently small, ∃f(n-1)(t0+h)⊝Hf(n-1)(t0),f(n-1)(t0)⊝Hf(n-1)(t0-h) and(2.3a)limh→0f(n-1)(t0+h)⊝Hf(n-1)(t0)h=limh→0f(n-1)(t0)⊝Hf(n-1)(t0-h)h=f(n)(t0),
n
=
1,2
,
…. In parametric form, as above, if =(f_,f¯), then f(n)=(f_(n),f¯(n)), where f_(n),f¯(n) are nth-order Hukuhara differentiable of f_,f¯, respectively. From (2.3a), in case n=1, we obtain (2.2).


## 3.  Fuzzy Integro-Differential Equation

Consider thenth-order derivative integro-differential equation [12] as follows:(3.1a)u(n)(t)+u(t)+λ∫0T‍k(s,t)u(m)(s)ds=g(t),
where m, n∈ℕ, m<n, t∈[0,T], λ∈ℝ, g(t) is a known function and the kernel k(s,t)≥0 with the initial conditions as follows:
(3.1b)u(k)(0)=hk,k=0,1,…,n-1,hk∈ℝ,in the fuzzy case; that is,u and g be fuzzy functions. Let
(3.2)u(t,r)=(u_(t,r),u¯(t,r)),g(t,r)=(g_(t,r),g¯(t,r)),u(n)(t,r)=(u_(n)(t,r),u¯(n)(t,r)),g(n)(t,r)=(g_(n)(t,r),g¯(n)(t,r)),
where all derivatives are, with respect to t, fuzzy functions. Therefore, related fuzzy integro-differential equation of (3.1a) can be written as follows:(3.3a)u_(n)(t,r)+u_(t,r)+λ∫0T‍k(s,t)u_(m)(s,r)ds=g_(t,r),(3.3b)u¯(n)(t,r)+u¯(t,r)+λ∫0T‍k(s,t)u¯(m)(s,r)ds=g¯(t,r),and its two crisp equations can be written as follows:(3.4a)uc(n)(t,r)+uc(t,r)+λ∫0T‍k(s,t)uc(m)(s,r)ds=gc(t,r),(3.4b)ud(n)(t,r)+ud(t,r)+λ∫0T‍k(s,t)ud(m)(s,r)ds=gd(t,r).


## 4. New Iterative Method

For simplicity, we present a review of the new iterative method [20, 21, 34–39, 41], and then we introduce a suitable algorithm of this method for solving the nth-order derivative fuzzy integro-differential equations.Consider the following general functional equation:(4.1)u=f+N(u),
where N is a nonlinear operator from a Banach space B→B, and f is a known function.We are looking for a solutionu of (4.1) having the series form:
(4.2)u=∑i=0∞ui.
The nonlinear operator N can be decomposed as follows:
(4.3)N(∑i=0∞ui)=N(u0)+∑i=0∞{N(∑j=0iuj)-N(∑j=0i-1uj)}.
From (4.2) and (4.3), (4.1) is equivalent to
(4.4)∑i=0∞ui=f+N(u0)+∑i=1∞{N(∑j=0iuj)-N(∑j=0i-1uj)}.
We define the recurrence relation:
(4.5)u0=f,u1=N(u0),un+1=N(u0+u1+⋯+un)-N(u0+u1+⋯+un-1),n=1,2,….
Then
(4.6)(u1+⋯+un+1)=N(u0+⋯+un),n=1,2,…,∑i=0∞ui=f+∑i=1∞ui.
The n-term approximate solution of (4.1) is given by u=u0+u1+⋯+un-1.


### 4.1. Reliable Algorithm

After the above presentation of the new iterative method, we introduce a reliable algorithm of this method for solving anynth-order derivative fuzzy integro-differential equation. Consider the nth-order derivative integro-differential equation (3.1a) and (3.1b) defined as4.7au(n)(t)+u(t)+λ∫0T‍k(s,t)u(m)(s)ds=g(t),
with the following initial conditions:
4.7bu(k)(0)=hk,k=0,1,…,n-1,hk∈ℝ.The initial value problem (4.7a) and (4.7b) is equivalent to the following integral equation:
(4.8)u(t)=u0(t)-Itn[u(t)+λ∫0T‍k(s,t)u(m)(s)ds]=u0(t)-N(u),
where u0(t) is the solution of the nth-order differential equation:(4.9a)dnu0dtn=g(t),u0(k)(0)=hk,k=0,1,2,…,n-1,hk∈ℝ,(4.9b)N(u)=Itn[u(t)+λ∫0T‍k(s,t)u(m)(s)ds],andItn is an integral operator of order n. Also, the two crisp equations (3.4a) and (3.4b) are equivalent to the two integral equations:(4.10a)uc(t,r)=u0c(t,r)-Itn[uc(t,r)+λ∫0T‍k(s,t)uc(m)(s,r)ds]=u0c(t,r)-N(uc),(4.10b)ud(t,r)=u0d(t,r)-Itn[ud(t,r)+λ∫0T‍k(s,t)ud(m)(s,r)ds]=u0d(t,r)-N(ud),
where u0c(t,r) and u0d(t,r) are the solutions of the nth-order differential equations:
(4.10c)dnu0cdtn=gc(t),u0c(k)(0)=hkc,k=0,1,2,…,n-1,hkc∈ℝ,(4.10d)dnu0ddtn=gd(t),u0d(k)(0)=hkd,k=0,1,2,…,n-1,hkd∈ℝ.N(uc) and N(ud) are the following two integral equations:
(4.10e)N(uc)=Itn[uc(t,r)+λ∫0T‍k(s,t)uc(m)(s,r)ds],(4.10f)N(uc)=Itn[ud(t,r)+λ∫0T‍k(s,t)uc(m)(s,r)ds].We get the solution of (4.8) or the two (4.10a), (4.10b) by employing the recurrence relation (4.5).


### 4.2. Convergence of the New Iterative Method

Now, we introduce the condition of convergence of the new iterative method, which is proposed by Daftardar-Gejji and Jafari in (2006) [20], also called (DJM) [39]. From (4.3), the nonlinear operator N is decomposed as follows [39]:
(4.11)N(u)=N(u0)+[N(u0+u1)-N(u0)]+[N(u0+u1+u2)-N(u0+u1)]+⋯.
Let G0=N(u0) and
(4.12)Gn=N(∑i=0n‍ui)-N(∑i=0n-1‍ui),n=1,2,….
Then N(u)=∑i=0∞‍Gi.Set(4.13)u0=f,(4.14)un=Gn-1,n=1,2,….
Then
(4.15)u=∑i=0∞‍ui
is a solution of the general functional equation (4.1). Also, the recurrence relation (4.5) becomes
(4.16)u0=f,un=Gn-1,n=1,2,….
Using Taylor series expansion for Gi, i=1,2,…,n, we have
(4.17)G1=N(u0+u1)-N(u0)=N(u0)+N′(u0)u1+N′′(u0)u122!+⋯-N(u0)=∑k=1∞‍N(k)(u0)u1kk!,G2=N(u0+u1+u2)-N(u0+u1)=N′(u0+u1)u2+N′′(u0+u1)u222!+⋯=∑j=1∞‍[∑i=0∞‍N(i+j)(u0)u1ii!]u2jj!,G3=∑i3=1∞‍∑i2=0∞‍∑i1=0∞‍N(i1+i2+i3)(u0)u3i3i3!u2i2i2!u1i1i1!.
In general:
(4.18)Gn=∑in=1∞‍∑in-1=0∞‍⋯∑i1=0∞‍[N(∑k=1n‍ik)(u0)(∏j=1nujijij!)].
In the following theorem we state and prove the condition of convergence of the method.Theorem 4.1.
If N isC(∞) in a neighborhood of u0 and(4.19a)∥N(n)(u0)∥=sup{N(n)(u0)(h1,…,hn):∥hi∥≤1,1≤i≤n}≤L,
for any n and for some real L>0 and ∥ui∥≤M<1/e, i=1,2,…, then the series ∑n=0∞‍Gn is absolutely convergent, and, moreover,
(4.19b)∥Gn∥≤LMnen-1(e-1),n=1,2,….Proof.
In view of (4.18),
(4.19c)∥Gn∥≤LMn∑in=1∞‍∑in-1=0∞‍⋯∑i1=0∞‍(∏j=1nujijij!)=LMnen-1(e-1).
Thus, the series ∑n=1∞‍∥Gn∥ is dominated by the convergent series LM(e-1)∑n=1∞‍(Me)n-1, where M<1/e. Hence, ∑n=0∞‍Gn is absolutely convergent, due to the comparison test.
For more details, see [39].


## 4.1. Reliable Algorithm

After the above presentation of the new iterative method, we introduce a reliable algorithm of this method for solving anynth-order derivative fuzzy integro-differential equation. Consider the nth-order derivative integro-differential equation (3.1a) and (3.1b) defined as4.7au(n)(t)+u(t)+λ∫0T‍k(s,t)u(m)(s)ds=g(t),
with the following initial conditions:
4.7bu(k)(0)=hk,k=0,1,…,n-1,hk∈ℝ.The initial value problem (4.7a) and (4.7b) is equivalent to the following integral equation:
(4.8)u(t)=u0(t)-Itn[u(t)+λ∫0T‍k(s,t)u(m)(s)ds]=u0(t)-N(u),
where u0(t) is the solution of the nth-order differential equation:(4.9a)dnu0dtn=g(t),u0(k)(0)=hk,k=0,1,2,…,n-1,hk∈ℝ,(4.9b)N(u)=Itn[u(t)+λ∫0T‍k(s,t)u(m)(s)ds],andItn is an integral operator of order n. Also, the two crisp equations (3.4a) and (3.4b) are equivalent to the two integral equations:(4.10a)uc(t,r)=u0c(t,r)-Itn[uc(t,r)+λ∫0T‍k(s,t)uc(m)(s,r)ds]=u0c(t,r)-N(uc),(4.10b)ud(t,r)=u0d(t,r)-Itn[ud(t,r)+λ∫0T‍k(s,t)ud(m)(s,r)ds]=u0d(t,r)-N(ud),
where u0c(t,r) and u0d(t,r) are the solutions of the nth-order differential equations:
(4.10c)dnu0cdtn=gc(t),u0c(k)(0)=hkc,k=0,1,2,…,n-1,hkc∈ℝ,(4.10d)dnu0ddtn=gd(t),u0d(k)(0)=hkd,k=0,1,2,…,n-1,hkd∈ℝ.N(uc) and N(ud) are the following two integral equations:
(4.10e)N(uc)=Itn[uc(t,r)+λ∫0T‍k(s,t)uc(m)(s,r)ds],(4.10f)N(uc)=Itn[ud(t,r)+λ∫0T‍k(s,t)uc(m)(s,r)ds].We get the solution of (4.8) or the two (4.10a), (4.10b) by employing the recurrence relation (4.5).


## 4.2. Convergence of the New Iterative Method

Now, we introduce the condition of convergence of the new iterative method, which is proposed by Daftardar-Gejji and Jafari in (2006) [20], also called (DJM) [39]. From (4.3), the nonlinear operator N is decomposed as follows [39]:
(4.11)N(u)=N(u0)+[N(u0+u1)-N(u0)]+[N(u0+u1+u2)-N(u0+u1)]+⋯.
Let G0=N(u0) and
(4.12)Gn=N(∑i=0n‍ui)-N(∑i=0n-1‍ui),n=1,2,….
Then N(u)=∑i=0∞‍Gi.Set(4.13)u0=f,(4.14)un=Gn-1,n=1,2,….
Then
(4.15)u=∑i=0∞‍ui
is a solution of the general functional equation (4.1). Also, the recurrence relation (4.5) becomes
(4.16)u0=f,un=Gn-1,n=1,2,….
Using Taylor series expansion for Gi, i=1,2,…,n, we have
(4.17)G1=N(u0+u1)-N(u0)=N(u0)+N′(u0)u1+N′′(u0)u122!+⋯-N(u0)=∑k=1∞‍N(k)(u0)u1kk!,G2=N(u0+u1+u2)-N(u0+u1)=N′(u0+u1)u2+N′′(u0+u1)u222!+⋯=∑j=1∞‍[∑i=0∞‍N(i+j)(u0)u1ii!]u2jj!,G3=∑i3=1∞‍∑i2=0∞‍∑i1=0∞‍N(i1+i2+i3)(u0)u3i3i3!u2i2i2!u1i1i1!.
In general:
(4.18)Gn=∑in=1∞‍∑in-1=0∞‍⋯∑i1=0∞‍[N(∑k=1n‍ik)(u0)(∏j=1nujijij!)].
In the following theorem we state and prove the condition of convergence of the method.Theorem 4.1.
If N isC(∞) in a neighborhood of u0 and(4.19a)∥N(n)(u0)∥=sup{N(n)(u0)(h1,…,hn):∥hi∥≤1,1≤i≤n}≤L,
for any n and for some real L>0 and ∥ui∥≤M<1/e, i=1,2,…, then the series ∑n=0∞‍Gn is absolutely convergent, and, moreover,
(4.19b)∥Gn∥≤LMnen-1(e-1),n=1,2,….Proof.
In view of (4.18),
(4.19c)∥Gn∥≤LMn∑in=1∞‍∑in-1=0∞‍⋯∑i1=0∞‍(∏j=1nujijij!)=LMnen-1(e-1).
Thus, the series ∑n=1∞‍∥Gn∥ is dominated by the convergent series LM(e-1)∑n=1∞‍(Me)n-1, where M<1/e. Hence, ∑n=0∞‍Gn is absolutely convergent, due to the comparison test.
For more details, see [39].


## 5. Numerical Examples

Example 5.1.
Consider the following fuzzy integro-differential equation:(5.1)u′(t,r)+λ∫01‍k(s,t)u(s,r)ds=g(t,r),0≤t,r≤1,
where u(t,r)=(u_(t,r),u¯(t,r))=(-2rt,4+rt) is the exact solution for (5.1), g(t,r)=(g_(t,r),g¯(t,r))=(-2r-2rt/3, r+2t+rt/3, k(s,t)=st, and λ=1, with the initial conditions u_(0,r)=0, u¯(0,r)=4. Taking gc(t,r)=(1/2)(g_(t,r)+g¯(t,r))=-r/2+t-rt/6, gd(t,r)=(1/2)(g¯(t,r)-g_(t,r))=3r/2+t+rt/2. The exact solution for related crisp equations is uc(t,r)=2-rt/2, ud(t,r)=2+3rt/2. At first, we identify uc(t,r) and ud(t,r). From (3.4a) and (3.4b), (5.1), where n=1, we have(5.2a)uc′(t,r)+∫01‍stuc(s,r)ds=gc(t,r),uc(0,r)=2,(5.2b)ud′(t,r)+∫01‍stud(s,r)ds=gd(t,r),ud(0,r)=2.
From (4.10c), (4.10d), we obtain
(5.3)u0c(t,r)=2-rt2+t22-rt212,u0d(t,r)=2+3rt2+t22+rt24.
Therefore, from (4.10a), (4.10b) the fuzzy integro-differential equations (5.2a) and (5.2b) are equivalent to the following integral equations:
(5.4)uc(t,r)=2-rt2+t22-rt212-It[∫01‍stuc(s,r)ds],ud(t,r)=2+3rt2+t22+rt24-It[∫01‍stud(s,r)ds].
Let N(uc)=-It[∫01‍stuc(s,r)ds],N(ud)=-It[∫01‍stud(s,r)ds]. Therefore, from (4.5), we can obtain easily the following first few components of the new iterative solution for (5.1):
(5.5)u0c(t,r)=2-rt2+t22-rt212,u0d(t,r)=2+3rt2+t22+rt24,u1c(t,r)=N(u0c)=-9t216+9rt296,u1d(t,r)=N(u0d)=-9t216-9rt232,u2c(t,r)=9t2128-9rt2768,u2d(t,r)=9t2128+9rt2256,⋮u5c(t,r)=-9t265536+9rt2393216,u5d(t,r)=-9t265536-9rt2131072,
and so on. The 6-term approximate solution is
(5.6)uc(t,r)=∑n=05unc=2-rt2-t265536+rt2393216,ud(t,r)=∑n=05und=2+3rt2-t265536-rt2131072.
It is clear that the iterations converge to the exact solution of the two crisp equations uc,ud as the number of iteration converges to ∞, that  is, limn→∞unc(t,r)=2-rt/2=uExactc, and limn→∞und(t,r)=2+3rt/2=uExactd. From the relations between u_,u¯,uc,ud, and u in Section 2, it follows immediately that this solution is the solution of the fuzzy integro-differential equation (5.1).Example 5.2.
Consider the following fuzzy integro-differential equation:(5.7)u′′(t,r)+λ∫01‍k(s,t)u(s,r)ds=g(t,r),0≤t,r≤1.
In this example u(t,r)=(u_(t,r),u¯(t,r))=(2rt,4rt) is the exact solution for (5.7), g(t,r)=(g_(t,r),g¯(t,r))=2rt/3,4rt/3, k(s,t)=st, λ=1, and the initial conditions are u_(0,r)=0, u′_(0,r)=2r, u¯(0,r)=0, u¯′(0,r)=4r. Also, the exact solution for related crisp equations is uc(t,r)=3rt, ud(t,r)=rt with gc(t,r)=rt, gd(t,r)=rt/3. At first, we identify uc(t,r), ud(t,r). From (3.4a) and (3.4b), (5.7), where n=2, we have(5.8a)uc′′(t,r)+∫01‍stuc(s,r)ds=gc(t,r),uc(0,r)=0,uc′(0,r)=3r,(5.8b)ud′′(t,r)+∫01‍stud(s,r)ds=gd(t,r),ud(0,r)=0,ud′(0,r)=r.
From (4.10c), (4.10d), we obtain
(5.9)u0c(t,r)=3rt+rt36,u0d(t,r)=rt+rt318.
Therefore, from (4.10a), (4.10b), the fuzzy integro-differential equations (5.8a) and (5.8b) are equivalent to the following integral equations:
(5.10)uc(t,r)=3rt+rt36-It2[∫01‍stuc(s,r)ds],ud(t,r)=rt+rt318-It2[∫01‍stud(s,r)ds].
Let N(uc)=-It2[∫01‍stuc(s,r)ds], N(ud)=-It2[∫01‍stud(s,r)ds]. Therefore, from (4.5), we can obtain easily the following first few components of the new iterative solution for (5.7):
(5.11)u0c(t,r)=3rt+rt36,u0d(t,r)=rt+rt318,u1c(t,r)=N(u0c)=-31rt3180,u1d(t,r)=N(u0d)=-31rt3540,u2c(t,r)=31rt35400,u2d(t,r)=31rt216200,⋮u5c(t,r)=-31rt3145800000,u5d(t,r)=-29rt2437400000,
and so on. The 6-term approximate solution is
(5.12)uc(t,r)=∑n=05unc=3rt-rt3145800000,ud(t,r)=∑n=05und=rt-rt3437400000.
It is clear that the iterations converge to the exact solution of uc,ud as n→∞, that is, limn→∞unc(t,r)=3rt=uExactc, limn→∞und(t,r)=rt=uExactd. As the above example the solution of (5.7) follows immediately.Example 5.3.
The third example is the fuzzy integro-differential equation:(5.13)u′′(t,r)+λ∫01‍k(s,t)u′(s,r)ds=g(t,r),0≤t,r≤1,
where u(t,r)=(u_(t,r),u¯(t,r))=(rt,4-3rt) is the exact solution for (5.13), g(t,r)=(g_(t,r),g¯(t,r))=(-rt/2,3rt/2), k(s,t)=st, λ=-1, and the initial conditions are u_(0,r)=0, u′_(0,r)=r, u¯(0,r)=4, u¯′(0,r)=-3r. The exact solution for related crisp equations is uc(t,r)=2-rt, ud(t,r)=2-2rt with gc(t,r)=rt/2, gd(t,r)=rt. At first, we identify uc(t,r), ud(t,r), and from (3.4a) and (3.4b), (5.13), where n=2, we have(5.14a)uc′′(t,r)-∫01‍stuc′(s,r)ds=gc(t,r),uc(0,r)=2,uc′(0,r)=-r,(5.14b)ud′′(t,r)-∫01‍stud′(s,r)ds=gd(t,r),ud(0,r)=2,ud′(0,r)=-2r.
From (4.10c), (4.10d), we obtain
(5.15)u0c(t,r)=2-rt+rt312,u0d(t,r)=2-2rt+rt36.
Therefore, from (4.10a) and (4.10b), the fuzzy integro-differential equations (5.14a) and (5.14b) are equivalent to the integral equations:
(5.16)uc(t,r)=2-rt+rt312+It2[∫01‍stuc′(s,r)ds],ud(t,r)=2-2rt+rt36+It2[∫01‍stud′(s,r)ds].
Let N(uc)=It2[∫01‍stuc′(s,r)ds], N(ud)=It2[∫01‍stud′(s,r)ds]. Therefore, from (4.5), we can obtain easily the following first few components of the new iterative solution for (5.13):
(5.17)u0c(t,r)=2-rt+rt312,u0d(t,r)=2-2rt+rt36,u1c(t,r)=N(u0c)=-7rt396,u1d(t,r)=N(u0d)=-7rt348,u2c(t,r)=-7rt3768,u2d(t,r)=-7rt3384,⋮u5c(t,r)=-7rt3393216,u5d(t,r)=-7rt3196608,
and so on. The 6-term approximate solution is
(5.18)uc(t,r)=∑n=05unc=2-rt+rt3393216,ud(t,r)=∑n=05und=2-2rt+rt3196608.
It is clear that the iterations converge to the exact solution of uc,ud as n→∞, that is, limn→∞unc(t,r)=2-rt=uExactc, limn→∞und(t,r)=2-2rt=uExactd. Therefore the solution of (5.13) follows immediately.Example 5.4.
Let us consider the following fuzzy integro-differential equation:(5.19)u′′(t,r)+u(t,r)+λ∫01‍k(s,t)u(s,r)ds=g(t,r),0≤t,r≤1,
with the exact solution u(t,r)=(u_(t,r),u¯(t,r))=(3rt,8-rt), g(t,r)=(g_,g¯)=(2rt,8-4t-(2rt/3)), k(s,t)=st, λ=-1, and the initial conditions u_(0,r)=0, u′_(0,r)=3r, u¯(0,r)=8, u¯′(0,r)=-r. The exact solution for related crisp equations is uc(t,r)=4+rt, ud(t,r)=4-2rt with gc(t,r)=4-2t+2rt/3, gd(t,r)=4-2t-4rt/3. From (3.4a), (3.4b), and (5.19), where n=2, the two crisp equations are(5.20a)uc′′(t,r)+uc(t,r)-∫01‍stuc(s,r)ds=gc(t,r),uc(0,r)=4,uc′(0,r)=r,(5.20b)ud′′(t,r)+ud(t,r)-∫01‍stud(s,r)ds=gd(t,r),ud(0,r)=4,ud′(0,r)=-2r.
From (4.10c), (4.10d) we obtain
(5.21)u0c(t,r)=4+rt+2t2-t33+rt39,u0d(t,r)=4-2rt+2t2-t33-2rt39.
Therefore, from (4.10a), (4.10b), the fuzzy integro-differential equations (5.20a) and (5.20b) are equivalent to the following integral equations:
(5.22)uc(t,r)=4+rt+2t2-t33+rt39+It2[-uc(t,r)+∫01‍stuc(s,r)ds],ud(t,r)=4-2rt+2t2-t33-2rt39+It2[-ud(t,r)+∫01‍stud(s,r)ds].
Let
(5.23)N(uc)=It2[-uc(t,r)+∫01‍stuc(s,r)ds],N(ud)=It2[-ud(t,r)+∫01‍stud(s,r)ds].
Therefore, from (4.5), we can obtain the following first approximate solutions for (5.19):
(5.24)u0c(t,r)=4+rt+2t2-t33+rt39,u0d(t,r)=4-2rt+2t2-t33-2rt39,u1c(t,r)=-2t2+73t3180-29rt3270-t46+t560-rt5180,u1d(t,r)=-2t2+73t3180+29rt3135-t46+t560+rt590,u2c(t,r)=-311t34200-421rt3113400+t46-73t53600+29rt55400+t6180-t72520+rt77560,u2d(t,r)=-311t34200+421rt356700+t46-73t53600-29rt52700+t6180-t72520-rt73780,⋮
and so on. In the same manner the rest of components can be obtained. The 5-term approximate solution is(5.25)uc(t,r)=∑n=04unc=4+rt+2.8739t33143448-0.1019rt39430344+0.0523t527216+0.0023rt520412-0.023t710584-0.001rt795256-0.13t9108864-0.01rt9163296+0.01t109072-0.01t11199584+0.01rt11598752,ud(t,r)=∑n=04und=4-2rt+2.8739t33143448+0.1019rt34715172+0.0523t527216-0.0023rt510206-0.023t710584+0.001rt747628-0.13t9108864+0.01rt981648+0.01t109072-0.01t11199584-0.01rt11299376.
Now, let us define the absolute value of thenth-term error by |en(t,r)|=|u*(t,r)-un(t,r)|, where u*(t,r) is the exact solution and un(t,r) is the nth-term approximate solution respectively. It is clear from previous above solutions that the smallest value of the absolute nth-term error is at t=r=0, that is, |en(0,0)|=(|enc(0,0)|,|end(0,0)|)=(0,0), n=0,1,…,4 and the largest value is at t=r=1 as shown in Table 1. Therefore the iterations converge to the exact solution of uc,ud, that is, limn→∞unc(t,r)=4+rt=uExactc, limn→∞und(t,r)=4-2rt=uExactd, and the solution of (5.19) follows immediately.Table 1
n
0
1
2
3
4
|
e
n
c
|
1.777778
7.9629629−2
3.395065−4
4.588−5
5.67−7
|
e
n
d
|
1.444444
7.4074074−2
1.543214−4
4.3601−5
3.96−7Example 5.5.
The final example is the following fuzzy integro-differential equation:(5.26)u′′(t,r)+u(t,r)+λ∫01‍k(s,t)u′(s,r)ds=g(t,r),0≤t,r≤1,
with the exact solution u(t,r)=(u_(t,r), u¯(t,r))=(3rt,4+rt), g(t,r)=(g_,g¯)=(2rt,4+2rt/3, k(s,t)=s2t, λ=-1 and the initial values u_(0,r)=0, u′_(0,r)=3r, u¯(0,r)=4, u¯′(0,r)=r. The exact solution for related crisp equations is uc(t,r)=2+2rt, ud(t,r)=2-rt with gc(t,r)=2+4rt/3, gd(t,r)=2-2rt/3. As above, where n=2, the two crisp equations are(5.27a)uc′′(t,r)+uc(t,r)-∫01‍s2tuc′(s,r)ds=gc(t,r),uc(0,r)=2,uc′(0,r)=2r,(5.27b)ud′′(t,r)+ud(t,r)-∫01‍s2tud′(s,r)ds=gd(t,r),ud(0,r)=2,ud′(0,r)=-r.
As the above examples, we obtain(5.28)u0c(t,r)=2+2rt+t2+2rt39,u0d(t,r)=2-rt+t2-rt39.
Therefore, the fuzzy integro-differential equations (5.27a) and (5.27b) are equivalent to the following integral equations:
(5.29)uc(t,r)=2+2rt+t2+2rt39+It2[-uc(t,r)+∫01‍s2tuc′(s,r)ds],ud(t,r)=2-rt+t2-rt39+It2[-ud(t,r)+∫01‍s2tud′(s,r)ds].
Let
(5.30)N(uc)=It2[-uc(t,r)+∫01‍s2tuc′(s,r)ds],N(ud)=It2[-ud(t,r)+∫01‍s2tud′(s,r)ds].
Therefore, we obtain
(5.31)u0c(t,r)=2+2rt+t2+2rt39,u0d(t,r)=2-rt+t2-rt39,u1c(t,r)=-t2+t312-rt35-t412-rt590,u1d(t,r)=-t2+t312+rt310-t412+rt5180,u2c(t,r)=-91t31080-403rt318900+t412-t5240+rt5100+t6360+rt73780,u2d(t,r)=-91t31080+403rt337800+t412-t5240-rt5200+t6360-rt77560,⋮
and so on. The 5-term approximate solution is(5.32)uc(t,r)=∑n=04unc=2+2rt-0.337t327216-5.0753rt31571724+0.073t56048+0.0041rt510206-0.01t79072+0.017rt715876-0.1t972576-0.01rt927216+0.01t1018144+0.01t11299376,ud(t,r)=∑n=04und=2-rt-0.337t327216+5.0753rt33143448+0.073t56048-0.0041rt520412-0.01t79072-0.017rt731752-0.1t972576+0.01rt954432+0.01t1018144-0.01t11598752.
As the previous example, it is clear from the obtained results that the smallest value of the absolutenth-term error is |en(0,0)|=(|enc(0,0)|,|end(0,0)|)=(0,0), n=0,1,…,4, and the largest value is at t=r=1 as shown in Table 2. Therefore, the iterations converge to the exact solution of uc,ud, that is, limn→∞unc(t,r)=2+2rt=uExactc, limn→∞und(t,r)=2-rt=uExactd. Therefore, the solution of (5.26) follows immediately.Table 2
n
0
1
2
3
4
|
e
n
c
|
1.2222222
1.1111111−2
2.2222222−3
1.757303−4
4.3324−6
|
e
n
d
|
8.888889−1
5.5555556−3
2.23412699−3
1.3038902−4
1.19612−6


## 6. Conclusion

In this work, the new iterative method used with a reliable algorithm to solve thenth-order derivative fuzzy integro-differential equations in crisp form and the cases of first- and second-order derivatives are taken into account. The obtained results concluded that the approximate solutions are in high agreement with corresponding exact solutions, which means that this method is suitable and effective to solve fuzzy integro-differential equations. Moreover, the solutions of the higher order fuzzy integro-differential equations can be calculated, as a future prospects, in a similar manner.



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*Source: 325473-2012-10-03.xml*