FUZZY COLLOCATION METHODS FOR SECOND- ORDER FUZZY ABEL

19 Pages·2014·0.54 MB·English

IranianJournalofFuzzySystems Vol. 11, No. 2,(2014)pp. 71-88 71
FUZZY COLLOCATION METHODS FOR SECOND- ORDER
FUZZY ABEL-VOLTERRA INTEGRO-DIFFERENTIAL
EQUATIONS
S.S.BEHZADI,T.ALLAHVIRANLOOANDS.ABBASBANDY
Abstract. In this paper we intend to offer new numerical methods toDsolve
the second-order fuzzy Abel-Volterra integro-differential equations under the
generalized H-differentiability. The existence and uniqueness of the solution
and convergence of the proposed methods are proved in details and the effi-
I
ciencyofthemethodsisillustratedthroughanumericalexample.
S
1. Introduction
Thefuzzyintegralequationsandthefuzzyinteg ro-differentialequationsarevery
useful for solving many problems in several apfplied fields like mathematical eco-
nomics, electrical engineering, medicine and biology and optimal control theory.
o
Sincetheseequationsusuallycannotbesolvedexplicitly,soitisrequiredtoobtain
approximate solutions. There are numerous numerical methods which have been

focusing on the solution of these equations [16, 14, 28, 7, 2, 3, 1, 20, 4, 5, 21, 23,
e
24, 18, 25, 26, 6, 27].
In this work, we develope vthe Jacobi polynomials and the Airfoil polynomials
fuzzy collocation methods to solve the second- order fuzzy Abel-Volterra integro-
differential equation of thie second kind as follows:
u(cid:101)(cid:48)(cid:48)h(x)=f(cid:101)(x)+µ(cid:90) xk(x,t)√ 1 u(cid:101)(t) dt, (1)
x−t
a
with fuzzy initial conditions:
c
u(cid:101)r(a)=b(cid:101)r, r =0,1. (2)
r
Where a and µ are crisp constant values and k(x,t) is function that has deriva-
tivesoAnanintervala≤t≤x≤bandbr arefuzzyconstantvaluesandf(cid:101)(x)isfuzzy
function. The second- order Abel-Volterra integro-differential equations occur in
various physical applications including heat transfer and viscoelasticity [10] .
Hereisanoutlineofthepaper. Insection2,thebasicnotationsanddefinitionsin
fuzzycalculusarebrieflypresented. Section3describeshowtofindanapproximate
solutionofthegivensecond-orderfuzzyAbel-Volterraintegro-differentialequations
by using proposed methods. The existence and uniqueness of the solution and
convergence of the proposed methods are proved in Section 4 respectively. Finally
Received: January2013;Revised: June2013;Accepted: January2014
Key words and phrases: Jacobi polynomials, Airfoil polynomials, Collocation method, Fuzzy
integro-differentialequations,AbelandVolterraintegralequations,Generalizeddifferentiability.
www.SID.ir
72 S.S.Behzadi,T.AllahviranlooandS.Abbasbandy
in section 5, we apply the proposed methods by an example to show the simplicity
and efficiency of the methods, and a brief conclusion is given in Section 6.
2. Preliminaries
Herebasicdefinitionsofafuzzynumberaregivenasfollows, [15,22,17,29,12,
8, 11]
Definition2.1. Anarbitraryfuzzynumberuintheparametricformisrepresented
(cid:101)
by an ordered pair of functions (u,u) which satisfy the following requirements:
(i) u:r →u− ∈R is a bounded left-continuous non-decreasing function over [0,1],
r
(ii)u:r →u+r ∈Risaboundedleft-continuousnon-increasingfunctiDonover[0,1],
(iii) u≤u, 0≤r ≤1.
Definition 2.2. Forarbitraryfuzzynumbersu˜,v˜∈E ,weusethedistance(Haus-
I
dorff metric) [17]
D(u(r),v(r))=max{sup |u(r)−v(r)|,sup |u(r)−v(r)|},
r∈[0,1] S
and it is shown [8] that (E , D) is a complete metric space and the following
properties are well known:

f
D(u+w,v+w)=D(u,v),∀ u,v ∈E,
(cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101)
D(ku,kv)=|k |D(u,v),∀ ko∈R,u,v ∈E,
(cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101)
D(u+v,w+e)≤D(u,w)+D(v,e),∀ u,v,w,e∈E.
(cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101) (cid:101)

Definition 2.3. Consider x, y ∈E. If there exists z ∈E such that x=y+z then
(cid:101) (cid:101) e (cid:101) (cid:101) (cid:101) (cid:101)
z is called the H- difference of x and y, and is denoted by x(cid:9)y, [8].
(cid:101) (cid:101) (cid:101) (cid:101) (cid:101)
v
Definition 2.4. Let f(cid:101): (a,b) → E and x0 ∈ (a,b). We say that f is generalized
differentiable at x0 ( Bedie-Gal differentiability), if there exists an element f(cid:101)(cid:48)(x0)∈
E, such that [8]:
h
i) for all h>0 sufficiently small, ∃f(cid:101)(x0+h)(cid:9)f(cid:101)(x0),∃f(cid:101)(x0)(cid:9)f(cid:101)(x0−h) and the
following limits hold:
c
rhli→m0f(cid:101)(x0+hh)(cid:9)f(cid:101)(x0) =hli→m0f(cid:101)(x0)(cid:9)hf(cid:101)(x0−h) =f(cid:101)(cid:48)(x0)
or
A
ii) for all h > 0 sufficiently small, ∃f(cid:101)(x0)(cid:9)f(cid:101)(x0 +h),∃f(cid:101)(x0 −h)(cid:9)f(cid:101)(x0) and
the following limits hold:
hli→m0f(cid:101)(x0)(cid:9)−f(cid:101)h(x0+h) =hli→m0f(cid:101)(x0−−h)h(cid:9)f(cid:101)(x0) =f(cid:101)(cid:48)(x0)
or
iii) for all h > 0 sufficiently small, ∃f(cid:101)(x0+h)(cid:9)f(cid:101)(x0),∃f(cid:101)(x0−h)(cid:9)f(cid:101)(x0) and
the following limits hold:
hli→m0f(cid:101)(x0+hh)(cid:9)f(cid:101)(x0) =hli→m0f(cid:101)(x0−h−)h(cid:9)f(cid:101)(x0) =f(cid:101)(cid:48)(x0)
or
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FuzzyCollocationMethodsforSecond-orderFuzzyAbel-VolterraIntegro-differentialEquations73
iv) for all h > 0 sufficiently small, ∃f(cid:101)(x0)(cid:9)f(cid:101)(x0+h),∃f(cid:101)(x0)(cid:9)f(cid:101)(x0−h) and
the following limits hold:
hli→m0f(cid:101)(x0)(cid:9)−f(cid:101)h(x0+h) =hli→m0f(cid:101)(x0)(cid:9)fh(cid:101)(x0−h) =f(cid:101)(cid:48)(x0).
Definition 2.5. Let f(cid:101): (a,b) → E. We say f(cid:101)is (i)-differentiable on (a,b) if f(cid:101)is
differentiable in the sense (i) of Definition (2.4) and similarly for (ii), (iii) and (iv)
differentiability [11].
Definition 2.6. A fuzzy number A˜ is of LR-type if there exist shape functions
L(for left), R(for right) and scalar α≥0,β ≥0 with
D
(cid:26) L(a−x), x≤a,
µ˜ (x)= α
A R(x−b), x≥a,
β
I
the mean value of A˜,a is a real number, and α,β are called the left and right
spreads, respectively. A˜ is denoted by (a,α,β). S
Definition 2.7. Let M˜ =(m,α,β) and N˜ =(n,γ,δ) and λ∈R+. Then,
LR LR
(1):λM˜ =(λm,λα,λβ)
LR
(2):−λM˜ =(−λm,λβ,λα) f
LR
(3):M˜ ⊕N˜ =(m+n,α+γ,β+δ) o
LR
 (mn,mγ+ nα,mδ+nβ)LR, M˜,N˜ >0,
(4):M˜ (cid:12)N˜ (cid:39) (mn,nαe−mδ,nβ−mγ)LR, M˜ >0,N˜ <0,
 (mn,−nβ−mδ,−nα−mγ) , M˜,N˜ <0.
LR
v
Definition 2.8. A triangular fuzzy number is defined as a fuzzy set in E, that is
specified by an ordered triple u = (a,b,c) ∈ R3 with a ≤ b ≤ c such that [u]r =
i
[ur ,ur] are the endpoints of r-level sets for all r ∈[0,1], where ur =a+(b−a)r
− + h −
and ur =c−(c−b)r. Here, u0 =a,u0 =c,u1 =u1 =b, which is denoted by u1.
+ − + − +
The set of triangular fuzzy numbers will be denoted by E.
c
Definition 2.9. The mapping f(cid:101): T → E for some interval T is called a fuzzy
process. Trherefore, its r-level set can be written as follows:
A
[f(t)]r =[fr(t),fr(t)], t∈T, r ∈[0,1].
− +
Definition 2.10. Let f(cid:101) : T → E be Hukuhara differentiable and denote by
[f(t)]r = [fr,fr]. Then, the boundary function fr and fr are differentiable (
− + − +
or Seikkala differentiable) and
[f(cid:48)(t)]r =[(fr)(cid:48)(t),(fr)(cid:48)(t)], t∈T, r ∈[0,1].
− +
3. Description of the Methods
In this section we are going to solve the equation (1) by using the Jacobi poly-
nomials and the Airfoil polynomials fuzzy collocation methods.
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74 S.S.Behzadi,T.AllahviranlooandS.Abbasbandy
To obtain the approximation solution of equation (1), based on Definition (2.4)
there are two possible cases for each derivation, so there are four possible cases for
the second order derivation. The conditions for the existence of solution in each
case can now be started.
Theorem 3.1. Let x0 ∈ [a,b] and assume that f(cid:101)is continuous. A mapping u :
[a,b]→E is a solution to the initial value equation (1) if and if only u; and u(cid:48)(cid:48) are
continuous and satisfy on the following conditions
(a)
(cid:82)x (cid:82)s
µu(cid:101)((cid:82)xxx)0=(cid:82)xs0x(cid:82)0axkx0(sf(cid:101),(ts))√ds1−stdus(cid:101)+(t) dt ds ds+(cid:101)b1(x−x0)+(cid:101)b0,D (3)
where u(cid:48) and u(cid:48)(cid:48) are (i)-differentials, or
(cid:101) (cid:101)
(b)
I
(cid:82)x (cid:82)s
uµ(cid:101)((cid:82)xxx)0=(cid:82)xs0(cid:9)(cid:82)(a−xk1)([s(cid:101)b,1t()x√−s1−xt0u(cid:101))((cid:9)t)(d−t1d)s[ xd0s]]x+0f(cid:101)(cid:101)bS(0s,) ds ds+ (4)
where u(cid:101)(cid:48) and u(cid:101)(cid:48)(cid:48) are (ii)-differentials, or
(c)
f
(cid:82)x (cid:82)s
uµ(cid:101)((cid:82)xxx)0=(cid:82)xs0(cid:9)(cid:82)(a−xk1)([s(cid:101)b,1t()x√−s1−xt0u(cid:101))(+ot) dxt0dxs0df(cid:101)s(]s+) d(cid:101)b0s,ds+ (5)
where u(cid:48) is the (i)-differential and u (cid:48)(cid:48) is the (ii)-differential, or
(cid:101) (cid:101)
e
(d)
(cid:82)x (cid:82)s
uµ(cid:101)((cid:82)xxx)0=(cid:82)ixs0[(cid:101)b(cid:82)1a(xxvk−(s,xt0))√(cid:9)s1−(t−u(cid:101)1()t[)xd0t xd0sf(cid:101)d(ss])]+ds(cid:101)b0ds+ (6)
where u(cid:48) is the (ii)-differential and u(cid:48)(cid:48) is the (i)-differential.
(cid:101) h (cid:101)
Proof. Since f(cid:101)is continuous, it must be integrable [9]. So,
cu(cid:101)(cid:48)(cid:48)(x)=f(cid:101)(x)+µ(cid:82)axk1(x,t)√s1−t u(cid:101)(t) dt, (7)
r u(cid:101)(x0)=(cid:101)b0, u(cid:101)(cid:48)(x0)=(cid:101)b1,
can be written in each as follows:
A
(a) Let u(cid:48) and u(cid:48)(cid:48) be (i)-differentiable. So, for equation (3) we have equivalently
(cid:101) (cid:101)
(cid:90) x (cid:90) x(cid:90) x 1
u(cid:101)(cid:48)(x)= f(cid:101)(s) ds+µ k(s,t)√ u(cid:101)(t) dt ds+(cid:101)b1,
s−t
x0 x0 a
and thus,
u(cid:101)(x)=(cid:82)xx0(cid:82)xs0f(cid:101)(s) ds ds+µ(cid:82)xx0(cid:82)xs0(cid:82)axk(s,t)√s1−t u(cid:101)(t) dt ds ds
+(cid:101)b1(x−x0)+(cid:101)b0.
(b) Let u(cid:48) and u(cid:48)(cid:48) be (ii)-differentiable. So, for equation(3) we have
(cid:101) (cid:101)
(cid:90) x (cid:90) x(cid:90) x 1
u(cid:101)(cid:48)(x)=(cid:9)(−1)[ f(cid:101)(s) ds+µ k(s,t)√ u(cid:101)(t) dt ds]+(cid:101)b1,
s−t
x0 x0 a
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FuzzyCollocationMethodsforSecond-orderFuzzyAbel-VolterraIntegro-differentialEquations75
and equivalently,
(cid:82)x (cid:82)s
(c) Let u(cid:48) beuµ(cid:101)(((cid:82)xixix))0-=(cid:82)dxis0ff(cid:9)e(cid:82)(ra−xenk1t)(i[sa(cid:101)b,b1t(l)ex√−as1−nxdt0u(cid:101))u((cid:48)(cid:9)(cid:48)t)b(d−et1(d)i)s[-dxd0isff]]exr+0efn(cid:101)(cid:101)bt(0is.a)bdles.dTs+hen for equation
(cid:101) (cid:101)
(3), we have equivalently
(cid:90) x (cid:90) x(cid:90) x 1
u(cid:101)(cid:48)(x)= f(cid:101)(s) ds+µ k(s,t)√ u(cid:101)(t) dt ds+(cid:101)b1,
s−t
x0 x0 a
so,
(cid:82)x (cid:82)s
u+(cid:101)(µx)(cid:82)x=x0(cid:82)(cid:9)xs0(−(cid:82)a1x)k[(cid:101)b(1s(,xt)−√sx1−0t)+u(cid:101)(t)x0dtxd0sf(cid:101)d(ss)]+ds(cid:101)b0d.s D
(d) Let u(cid:48) be (i)-differentiable and u(cid:48)(cid:48) be (ii)-differentiable. Then for equation
(cid:101) (cid:101) I
(3) we have equivalently
S
(cid:90) x (cid:90) x(cid:90) x 1
u(cid:101)(cid:48)(x)=(cid:9)(−1)[ f(cid:101)(s) ds+µ k(s,t)√ u(cid:101)(t) dt ds]+(cid:101)b1,
s−t
so, x0 x0 a
f
(cid:82)x (cid:82)s
u+(cid:101)(µx)(cid:82)x=x0(cid:82)[(cid:101)bxs10((cid:82)xax−k(xs0,)t)(cid:9)√(s1−−t1o)u(cid:101)[(xt)0 dxt0df(cid:101)s(sd)s]d]s+d(cid:101)bs0,
which completes the proof.
e (cid:3)
3.1. Description of the Jacvobli Polynomials Fuzzy Collocation Method.
To obtain the approximation solution of equation (1), according to the Jacobli
polynomials method [19]iwe can write:
h n
(cid:88)
u(cid:101)n(x)=w(x) (cid:101)aipαi,β(x), α,β >−1, (8)
where,
i=0
c
w(x)= (1−x)α,
(1+x)β
rpα,β(x)= (1−x)−α(1+x)−β dn [(1−x)n+α(1+x)n+β],
i (−2)n n! dxn
A (pα,β)(cid:48)(s)= 1(n+α+β+1)p(α+1,β+1)(s). (9)
i 2 n−1
Now from this method, we have four cases as follows:
Case (a):
wµ((cid:82)xxx0)(cid:80)(cid:82)xs0ni=(cid:82)0ax(cid:101)akip(sαi,,βt)(x√)s1−=t (cid:82)(xwx0((cid:82)tx)s0(cid:80)f(cid:101)ni(=s)0(cid:101)adispαid,sβ+(t)) dt ds ds+(cid:101)b1(x−x0)+(cid:101)b0. (10)
We can write equation (10) as follows:
(cid:82)xx0(cid:82)xs0f(cid:101)(s) ds ds+µ(cid:80)ni=0(cid:101)ai(cid:82)xx0(cid:82)xs0(cid:82)axw(t)k(s,t)√s1−tpαi,β(t) dt ds ds+
(cid:101)b1(x−x0)+(cid:101)b0 =w(x)(cid:80)ni=0(cid:101)aipαi,β(x)+R(cid:101)n(x). (11)
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76 S.S.Behzadi,T.AllahviranlooandS.Abbasbandy
So, we have
(cid:82)xx0(cid:82)xs0f(cid:101)(s) ds ds+µ(cid:80)ni=0(cid:101)ai(cid:82)xx0(cid:82)xs0(cid:82)axw(t)k(s,t)√s1−tpαi,β(t) dt ds ds+
(cid:101)b1(x−x0)+(cid:101)b0(cid:9)w(x)(cid:80)ni=0(cid:101)aipαi,β(x)=R(cid:101)n(x). (12)
R(cid:101)n(xj)=(cid:101)0. (13)
It means,
Rr(x )=0, Rr(x )=0, ∀r ∈[0,1],
n j n j
where x (j =1,...,n) are collocation points.
j
Therefore we can write,
D
(cid:82)xx0j(cid:82)xs0f(cid:101)(s) ds ds+µ(cid:80)ni=0(cid:101)ai(cid:82)xx0j(cid:82)xs0(cid:82)axjw(t)k(s,t)√s1−tpαi,β(t) dt ds ds+
(cid:101)b1(xj −x0)+(cid:101)b0(cid:9)w(xj)(cid:80)ni=0(cid:101)aipαi,β(xj)=(cid:101)0. I (14)
Equation (14) can be written in the following operator form
S
F(cid:101)⊕A(cid:101)a(cid:9)B(cid:101)a=(cid:101)0. (15)
(cid:88) (cid:88) (cid:88) (cid:88)
F(cid:101)j ⊕ aij(cid:101)aj ⊕ aij (cid:9)( bfij(cid:101)aj ⊕ bij(cid:101)aj)=(cid:101)0.
aij≥0 aij<0 bij≥0 bij<0
Where, o
((FA(cid:101)))ijj==(cid:82)µxx0(cid:82)jxx(cid:82)0jxs0(cid:82)xfs(cid:101)0(s(cid:82))axjdesw(d ts)k+(s(cid:101)b,1t()x√js1−−txp0αi),β+(t(cid:101)b)0,dt ds ds,
(B) =w(x )pα,β(x ). (16)
ij j i j
Case (b): v
wµ((cid:82)xxx0)(cid:80)(cid:82)xs0ni=(cid:82)0ax(cid:101)akip(sαih,,βt)(ix√)s1−=t (cid:9)(w(−(t1))(cid:80)[(cid:101)b1ni=(x0(cid:101)a−ipxαi0,)β(cid:9)(t)()−d1t)[d(cid:82)sxx0d(cid:82)sx]s]0+f(cid:101)((cid:101)bs0).ds ds+ (17)
We can write equation (17) as follows:
c
(cid:82)x (cid:82)s
rµ(cid:9)(cid:80)(−ni1=)0[(cid:101)b(cid:101)a1i((cid:82)xxx−0(cid:82)xxs00)(cid:82)(cid:9)axk(−(s1,)t[)√x0s1−xt0(fw(cid:101)((st))pdαis,βd(st+)) dt ds ds]]+(cid:101)b0 =
Aw(x)(cid:80)ni=0(cid:101)aipαi,β(x)+R(cid:101)n(x). (18)
So, we have
(cid:82)x (cid:82)s
µ(cid:9)(cid:80)(−ni1=)0[(cid:101)b(cid:101)a1i((cid:82)xxx−0(cid:82)xxs00)(cid:82)(cid:9)axk(−(s1,)t[)√x0s1−xt0(fw(cid:101)((st))(cid:80)dsni=ds0+(cid:101)aipαi,β(t)) dt ds ds]]+(cid:101)b0
(cid:9)w(x)(cid:80)ni=0(cid:101)aipαi,β(x)=R(cid:101)n(x). (19)
Therefore we can write,
µ(cid:9)(cid:80)(−ni1=)0[(cid:101)b(cid:101)a1i((cid:82)xxjx0j−(cid:82)xxs00(cid:82))a(cid:9)xj(k−(s1,)t[)(cid:82)x√x0js1−(cid:82)txs0(fw(cid:101)((st))(cid:80)dsni=ds0+(cid:101)aipαi,β(t)) dt ds ds]]+(cid:101)b0
(cid:9)w(xj)(cid:80)ni=0(cid:101)aipαi,β(xj)=(cid:101)0. (20)
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FuzzyCollocationMethodsforSecond-orderFuzzyAbel-VolterraIntegro-differentialEquations77
Equation (20) can be written in the following operator form
F(cid:101)⊕A(cid:101)a(cid:9)B(cid:101)a=(cid:101)0. (21)
(cid:88) (cid:88) (cid:88) (cid:88)
F(cid:101)j ⊕ aij(cid:101)aj ⊕ aij (cid:9)( bij(cid:101)aj ⊕ bij(cid:101)aj)=(cid:101)0.
aij≥0 aij<0 bij≥0 bij<0
Where,
(cid:82)x (cid:82)s
((FA(cid:101)))ijj==(cid:9)µ((cid:82)−xx10j)(cid:82)[(cid:101)bxs10((cid:82)xax−jwx(0t))(cid:9)k(s(−,t1))√[s1x−0tpxαi0,βf(cid:101)((ts))ddtsddss]d+s,(cid:101)b0,
(B) =w(x )pα,β(x ). (22)
ij j i j
D
Case (c)
wµ((cid:82)xxx0)(cid:80)(cid:82)xs0ni=(cid:82)0ax(cid:101)akip(sαi,,βt)(x√)s1−=t (cid:9)(w(−(t1))(cid:80)[(cid:101)b1ni=(x0(cid:101)a−ipxαi0,)β+(t)(cid:82))xxd0t(cid:82)xds0sfI(cid:101)d(ss])+ds(cid:101)b0d.s+ (23)
We can write equation (23) as follows: S
(cid:82)x (cid:82)s
µ(cid:9)(cid:82)(−xx01(cid:82))x[s(cid:101)b01(cid:82)(axxk−(sx,0t))+√s1−xt0(wx0(tf(cid:101))((cid:80)s)nid=s0(cid:101)adisfp+αi, β(t)) dt ds ds]+(cid:101)b0 =
w(x)(cid:80)ni=0(cid:101)aipαi,β(x)+R(cid:101)n(x). (24)
o
So, we have
(cid:82)x (cid:82)s
µ(cid:9)(cid:82)(−xx01(cid:82))x[s(cid:101)b01(cid:82)(axxk−(sx,0t))+√s1−xt0e(wx0 (tf(cid:101))((cid:80)s)nid=s0(cid:101)adisp+αi,β(t)) dt ds ds]+(cid:101)b0
(cid:9)w(x)(cid:80)ni=0(cid:101)aipαi,βv(x)=R(cid:101)n(x). (25)
Therefore we can write,
i
(cid:9)µ(cid:82)(−xx0j1)(cid:82)[x(cid:101)bs01((cid:82)xahxjj−k(xs,0t))+√s(cid:82)1−xxt0j((cid:82)wxs0(tf(cid:101))((cid:80)s)ni=ds0(cid:101)adisp+αi,β(t)) dt ds ds]+(cid:101)b0
(cid:9)w(cxj)(cid:80)ni=0(cid:101)aipαi,β(xj)=(cid:101)0. (26)
Equation (26) can be written in the following operator form
r
F(cid:101)(cid:9)A(cid:101)a(cid:9)B(cid:101)a=(cid:101)0. (27)
A
(cid:88) (cid:88) (cid:88) (cid:88)
F(cid:101)j (cid:9)( aij(cid:101)aj ⊕ aij)(cid:9)( bij(cid:101)aj ⊕ bij(cid:101)aj)=(cid:101)0.
aij≥0 aij<0 bij≥0 bij<0
Where,
(cid:82)x (cid:82)s
((AF(cid:101)))ijj==(cid:9)µ((cid:82)−xx10j)(cid:82)[(cid:101)bxs10((cid:82)xax−jwx(0t))+k(s,xt0)√xs01−f(cid:101)t(psαi),βd(st)dsd]t+d(cid:101)bs0,ds,
(B) =w(x )pα,β(x ). (28)
ij j i j
Case (d)
wµ((cid:82)xxx0)(cid:80)(cid:82)xs0ni=(cid:82)0ax(cid:101)akip(sαi,,βt)(x√)s1−=t [((cid:101)bw1((xt)−(cid:80)xni0=)0(cid:9)(cid:101)ai(p−αi,1β)([t(cid:82))x)x0d(cid:82)txs0dfs(cid:101)(dss)]]d+s d(cid:101)bs0.+ (29)
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We can write equation (29)
(cid:82)x (cid:82)s
µ[(cid:101)b1(cid:82)(xxx0(cid:82)−xs0x(cid:82)0a)x(cid:9)k((s−,t1))√[s1x−0t x(w0f((cid:101)t()s(cid:80)) dnis=0d(cid:101)asi+pαi,β(t)) dt ds ds]]+(cid:101)b0 =
w(x)(cid:80)ni=0(cid:101)aipαi,β(x)+R(cid:101)n(x). (30)
So, we have
(cid:82)x (cid:82)s
µ[(cid:101)b1(cid:82)(xxx0(cid:82)−xs0x(cid:82)0a)x(cid:9)k((s−,t1))√[s1x−0t x(w0f((cid:101)t()s(cid:80)) dnis=0d(cid:101)asi+pαi,β(t)) dt ds ds]]+(cid:101)b0
(cid:9)w(x)(cid:80)ni=0(cid:101)aipαi,β(x)=R(cid:101)n(x). (31)
D
Therefore we can write,
[µ(cid:101)b1(cid:82)(xxx0jj(cid:82)−xs0x(cid:82)0ax)j(cid:9)k((s−,1t))[√(cid:82)sx1x−0jt(cid:82)(xsw0f((cid:101)t()s(cid:80)) dni=s0d(cid:101)asi+pαi,β(t)) dt dIs ds]]+(cid:101)b0
(cid:9)w(xj)(cid:80)ni=0(cid:101)aipαi,β(xj)=(cid:101)0. S (32)
Equation (32) can be written in the following operator form

F(cid:101)⊕A(cid:101)a(cid:9)B(cid:101)a=f(cid:101)0. (33)
(cid:88) (cid:88) (cid:88)o (cid:88)
F(cid:101)j ⊕ aij(cid:101)aj ⊕ aij (cid:9)( bij(cid:101)aj ⊕ bij(cid:101)aj)=(cid:101)0.
aij≥0 aij<0 bij≥0 bij<0
Where,
e
(cid:82)x (cid:82)s
((AF(cid:101)))ijj==(cid:101)bµ1((cid:82)xxx0−j(cid:82)xxvs00)(cid:82)(cid:9)axj(w−(1t))kx(s0,tx)0√f(cid:101)s1(−st)pdαis,β(dts)+d(cid:101)bt0d,s ds,
(B)ij =w(ixj)pαi,β(xj). (34)
h
3.2. Description of the Airfoil polynomials fuzzy collocation method. To
obtain the approximation solution of equation (1), according to the airfoil polyno-
mials method ocf the first kind [13] we can write:
n
r x(cid:101)n(x)=w(x)(cid:88)(cid:101)aiti(x), (35)
wheAre i=0
(cid:113)
w(x)= 1+x,
1−x
cos[(1)arccosx]
ti(x)= cos(21arccosx) ,
2
sin[(1)arcsinx]
ui(x)= cos(21arcsinx) ,
(1+x)t(cid:48)(x)=2 (i+ 1)u (x)− 1t (x). (36)
i 2 i 2 i
Now from this method, we have four cases as follows:
Case (a):
µw((cid:82)xxx0)(cid:80)(cid:82)xs0ni=(cid:82)0ax(cid:101)akit(is(,xt))√=s1−(cid:82)xtx0((cid:82)wxs(0tf)(cid:101)((cid:80)s)ni=d0s(cid:101)adisti+(t)) dt ds ds+(cid:101)b1(x−x0)+(cid:101)b0. (37)
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FuzzyCollocationMethodsforSecond-orderFuzzyAbel-VolterraIntegro-differentialEquations79
We can write equation (37) as follows:
(cid:82)xx0(cid:82)xs0f(cid:101)(s) ds ds+µ(cid:80)ni=0(cid:101)ai(cid:82)xx0(cid:82)xs0(cid:82)axw(t)k(s,t)√s1−tti(t) dt ds ds+
(cid:101)b1(x−x0)+(cid:101)b0 =w(x)(cid:80)ni=0(cid:101)aipαi,β(x)+R(cid:101)n(x). (38)
So, we have
(cid:82)xx0(cid:82)xs0f(cid:101)(s) ds ds+µ(cid:80)ni=0(cid:101)ai(cid:82)xx0(cid:82)xs0(cid:82)axw(t)k(s,t)√s1−tti(t) dt ds ds+
(cid:101)b1(x−x0)+(cid:101)b0(cid:9)w(x)(cid:80)ni=0(cid:101)aipαi,β(x)=R(cid:101)n(x), (39)
R(cid:101)n(xj)=(cid:101)0. (40)
It means, D
Rr(x )=0, Rr(x )=0, ∀r ∈[0,1],
n j n j
where xj (j =1,...,n) are collocation points. I
2i−1
x =−cos positiveimplicative, j =S0,1,...,n.
j 2n+3
Therefore we can write,

(cid:82)xx0j(cid:82)xs0f(cid:101)(s) ds ds+µ(cid:80)ni=0(cid:101)ai(cid:82)xx0j(cid:82)xs0(cid:82)axjwf(t)k(s,t)√s1−tti(t) dt ds ds+
(cid:101)b1(xj −x0)+(cid:101)b0(cid:9)w(xj)(cid:80)ni=0(cid:101)aiti(xj)o=(cid:101)0. (41)
Equation (41) can be written in the following operator form

F(cid:101)⊕eA(cid:101)a(cid:9)B(cid:101)a=(cid:101)0. (42)
(cid:88) (cid:88) (cid:88) (cid:88)
F(cid:101)j ⊕ aij(cid:101)aj ⊕vaij (cid:9)( bij(cid:101)aj ⊕ bij(cid:101)aj)=(cid:101)0.
aij≥0 aij<0 bij≥0 bij<0
Where,
i
((FA(cid:101)))ijj==h(cid:82)µxx0(cid:82)jxx(cid:82)0jxs0(cid:82)xfs(cid:101)0(s(cid:82))axjdsw(dts)k+(s(cid:101)b,1t()x√js1−−txt0i()t+) (cid:101)bd0t,ds ds,
c(B) =w(x )t (x ). (43)
ij j i j
Case (b):
r
Awµ((cid:82)xxx0)(cid:80)(cid:82)xs0ni=(cid:82)0ax(cid:101)akit(is(,xt))√=s1−(cid:9)t(−(w1()t[(cid:101)b)1(cid:80)(xni=−0x(cid:101)a0it)i(cid:9)(t)()−d1t)[d(cid:82)sxx0d(cid:82)sx]s]0+f(cid:101)((cid:101)bs0).ds ds+ (44)
We can write equation (44) as follows:
(cid:82)x (cid:82)s
(cid:9)µ(cid:80)(−ni1=)0[(cid:101)b(cid:101)a1i((cid:82)xxx−0(cid:82)xxs00)(cid:82)(cid:9)axk(−(s1,)t[)√x0s1−xt0(fw(cid:101)((st))tdi(st)d)s+dt ds ds]]+(cid:101)b0 =
w(x)(cid:80)ni=0(cid:101)aiti(x)+R(cid:101)n(x). (45)
So, we have
(cid:82)x (cid:82)s
µ(cid:9)(cid:80)(−ni1=)0[(cid:101)b(cid:101)a1i((cid:82)xxx−0(cid:82)xxs00)(cid:82)(cid:9)axk(−(s1,)t[)√x0s1−xt0(fw(cid:101)((st))(cid:80)dsni=ds0+(cid:101)aiti(t)) dt ds ds]]+(cid:101)b0
(cid:9)w(x)(cid:80)ni=0(cid:101)aiti(x)=R(cid:101)n(x). (46)
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