The relationship between the solutions according to the noniterative method and the generalized differentiability of the fuzzy boundary value problem

https://doi.org/10.26637/MJM0604/0012

The relationship between the solutions according to

the noniterative method and the generalized

differentiability of the fuzzy boundary value problem

Hülya Gültekin Çitil

*

Abstract

In this paper is studied the fuzzy boundary value problem according to the noniterative method and according to the generalized differentiability. Comparison result and relationship between them are given. It is seen on the example.

Keywords

Two-point fuzzy boundary value problems, Fuzzy differential equation.

AMS Subject Classification 03E72, 34B05

1_{Department of Mathematics, Faculty of Arts and Sciences, Giresun University, Giresun-28100, Turkey.}

*Corresponding author:1_{[email protected]}

Article History: Received21July2018; Accepted26October2018 c2018 MJM.

Contents

1 Introduction. . . 781

2 Preliminaries. . . 781

3 Findings and Main Results . . . 782

References. . . 786

1. Introduction

Fuzzy differential equations is investigated by many re-searchers [1–3,5–7,14,15]. Because fuzzy differential equa-tions are a suitable tool for modeling science and engineering problems in which uncertainties. Fuzzy differential equations can be studied by different approach. The first approach is Hukuhara differentiability. The second approach is general-ized differentiability [17]. Two-point boundary value prob-lems have four different solution using the generalized differ-entiabilty. The third approach is to generate the fuzzy solution from the crips solution [7,8,11,13].

In this paper is studied the fuzzy boundary value problem according to the noniterative method and according to the gen-eralized differentiability. Comparison result and relationship between them are given. It is seen on the example.

2. Preliminaries

Definition 2.1. [18] A fuzzy number is a function u:R→

[0,1]satisfying the properties normal, convex fuzzy set, up-per semi-continuous onRand cl{xεR|u(x)>0}is com-pact,where cl denotes the closure of a subset.

LetRFdenote the space of fuzzy numbers.

Definition 2.2. [17] Let uεRF. Theα-level set of u, denoted ,[u]α_,0<

α≤1,is

[u]α_={x

εR|u(x)≥0}.

The notation, denotes explicitly theα-level set of u. The notation,[u]α_{= [}

u_α,uα]denotes explicitly theα-level set of u.We refer to u and u as the lower and upper branches of u, respectively.

Remark 2.3. [10,17] The sufficient and necessary conditions for[u_α,uα]to define the parametric form of a fuzzy number as follows:

u_αis bounded monotonic increasing (nondecreasing) left-continuous function on(0,1]and right-continuous forα=0, uαis bounded monotonic decreasing (nonincreasing) left-continuous function on(0,1]and right-continuous forα=0,

u_α≤uα,0≤α≤1.

Randλ[u]α

means the usual product between a scalar and a subset ofR.

The metric structure is given by the Hausdorff distance

D:RF×RF→R+∪ {0},

by

D(u,v) = sup α∈[0,1]

max{|u_α−v_α|,|uα−vα|}.

Definition 2.5. [18] If A is a symmetric triangular numbers with supports[a,a], theα−level sets of [A]αis

[A]α=

a+

a−a 2

α,a−

a−a

2

α

.

Definition 2.6. [12, 17, 19] Let u,v∈RF. If there exists w∈RF such that u=v+w,then w is called the Hukuhara difference of fuzzy numbers u and v, and it is denoted by w=u v.

Definition 2.7. [4,12,17] Let f :[a,b]→_RFand t0∈[a,b].We say that f is Hukuhara differentiable at t0,if there exists an element f0(t0)∈RF such that for all h>0sufficiently small,

∃f(t0+h) f(t0), f(t0) f(t0−h)and the limits

lim

h→0

f(t0+h) f(t0) h =hlim→0

f(t0) f(t0−h)

h =f

0 (t0).

Definition 2.8. [17] Let f :[a,b]→RF and t0∈[a,b].We say that f is (1)-differentiable at t0,if there exists an element f0(t0)∈RFsuch that for all h>0sufficiently small near to 0, exist f(t0+h) f(t0), f(t0) f(t0−h)and the limits

lim

h→0

f(t0+h) f(t0) h =hlim→0

f(t0) f(t0−h)

h =f

0 (t0),

and f is (2)-differentiable if for all h>0sufficiently small near to 0, exist f(t0) f(t0+h), f(t0−h) f(t0)and the limits

lim

h→0

f(t0) f(t0+h)

−h =hlim→0

f(t0−h) f(t0)

−h =f

0 (t0).

Theorem 2.9. [16] Let f :[a,b]→RF be fuzzy function, where[f(t)]α₌h

f

α(t),fα(t) i

,for eachα∈[0,1].

(i)If f is (1)-differentiable then f

α and fα are differen-tiable functions andhf0(t)iα=hf0

α(t),f 0

α(t) i

,

(ii)If f is (2)-differentiable then f

αand fαare differen-tiable functions and

h

f0(t)iα=hf 0

α(t),f 0

α(t) i

.

Theorem 2.10. [16] Let f0 :[a,b]→_RF be fuzzy function, where[f(t)]α=hf

α(t),fα(t) i

,for eachα∈[0,1],f is (1)-differentiable or (2)-(1)-differentiable.

(i)If f and f0 are (1)-differentiable then f0 α and f

0

α are

differentiable functions andhf00(t)iα=hf00 α(t),f

00

α(t) i

,

(ii) If f is (1)-differentiable and f0 is (2)-differentiable then f0

αand f 0

α are differentiable functions and h

f 00

(t)iα=

f 00

α(t),f 00

α(t)

,

(iii)If f is (2)-differentiable and f0 is (1)-differentiable then f0

αand f 0

α are differentiable functions and h

f 00

(t)iα=

f 00

α(t),f 00

α(t)

,

(iv)If f and f0 are (2)-differentiable then f0 α and f

0

α are

differentiable functions andhf00(t)iα=hf00 α(t),f

00

α(t) i

.

3. Findings and Main Results

Consider the solutions of the following problem

L:= d2 dt2,

Lu+u= [g(t)]α, (3.1)

u(0) = [ϕ]α_,

u(1) = [ψ]α_{, (3.2)}

where

[g(t)]α₌h_g

α(t),gα(t) i

= [g(t)−1+α,g(t) +1−α]

is fuzzy function,

[ϕ]α₌h ϕ

α,ϕα i

and[ψ]α₌h ψ

α,ψα i

are symmetric triangular fuzzy numbers.

The Solution According to the Noniterative Method

[9] Let look for the solution of the problem (3.1)-(3.2) as

u(t) =u1(t) +λu2(t), (3.3)

whereλ is a constant to be determined.

Substituting the equation (3.3) in (3.1), we have

u00₁(t) +u1(t) = [g(t)]α,

u00₂(t) +u2(t) =0.

u1(0) +λu2(0) = [ϕ]α,

u1(0) = [ϕ]α,u2(0) =0.

If

u0₁(0) =0,u0₂(0) =1,

the unknown constantλ is identified. Then, from the second

boundary condition,

u1(1) +λu2(1) = [ψ]α

λ =[ψ] α

−u1(1) u2(1)

is obtained. Then, fuzzy initial value problems

 



u00₁(t) +u1(t) = [g(t)]α u1(0) = [ϕ]α

u0₁(0) =0

(3.4)

and

 



u00₂(t) +u2(t) =0 u2(0) =0 u0₂(0) =1

(3.5)

is solved. The solution of the problem (3.4) is

u_1α(t) =c1(α)cost+c2(α)sint+upα(t),

u1α(t) =c3(α)cost+c4(α)sint+upα(t),

[u1(t)]α= [u1α(t),u1α(t)],

where

[up(t)]α=

u_pα(t),upα(t)

is the solution nonhomogeneous fuzzy differential equation in (3.4). Using the initial conditions, the coefficientsc1(α), c2(α),c3(α),c4(α)are obtained as

c1(α) =ϕ

α−upα(0),c2(α) =−u 0 pα(0),

c3(α) =ϕ_α−upα(0),c4(α) =−u 0 pα(0). The solution of the problem (3.5) is

u_2α(t) =a1(α)cost+a2(α)sint,

u2α(t) =a3(α)cost+a4(α)sint,

[u2(t)]α= [u2α(t),u2α(t)].

Using the initial conditions, the coefficientsa1(α),a2(α), a3(α),a4(α)are obtained as

a1(α) =a3(α) =0,

a2(α) =a4(α) =1.

Then,

[u2(t)]α= [sint,sint].

Since

λ=[ψ] α

−u1(1) u2(1)

,

λ_α= ψ

α− h

ϕ

α−upα(0)

cos(1) +u_pα(1)i

sin(1) −u

0 pα(0)

λα=

ψ_α−[(ϕ_α−upα(0))cos(1) +upα(1)]

sin(1) −u

0 pα(0)

[λ]α₌h λ_α,λα

i

is found. Since the solution of the problem (3.1)-(3.2)

the lower and the upper solutions the fuzzy problem (3.1)-(3.2) are

u_α(t) = ϕ

α−upα(0)

cos(t) +−u0_pα(0)sin(t) +u_pα(t) +λ_αsin(t),

uα(t) = (ϕα−upα(0))cos(t) +

−u0_pα(0)sin(t)

+upα(t) +λαsin(t).

Substitutingλ_α andλα are in the lower and the upper solutions, the solution problem (3.1)-(3.2) is obtained as

u_α(t) = ϕ

α−upα(0)

cos(t) (3.6)

+ψα−(ϕα−upα(0))cos(1)−upα(1)

sin(1) sin(t)

+u_pα(t),

uα(t) = (ϕα−upα(0))cos(t) (3.7)

+ψα−(ϕα−upα(0))cos(1)−upα(1)

sin(1) sin(t)

+upα(t),

[u(t)]α_{= [u}

α(t),uα(t)] (3.8)

The Solution According to the Generalized Differen-tiability

Let find the solution of the problem (3.1)-(3.2) according to the generalized differentiability, where (i,j) solution means that y is (i)-differentiable, y0 is (j) differentiable. (i,j=1,2)

For the (1,1) and (2,2) solutions

Using the generalized differentiability for (1,1) and (2,2) solu-tions, we have

(

u00_α(t) +u_α(t) =g α(t) u_α(0) =ϕ

α,uα(1) =ψα

, (3.9)

u00_α(t) +uα(t) =g_α(t)

uα(0) =ϕ_α,uα(1) =ψ_α . (3.10)

From (3.9) and (3.10), the lower and the upper solution is found as

u_α(t) =b1(α)cos(t) +b2(α)sin(t) +upα(t),

uα(t) =b3(α)cos(t) +b4(α)sin(t) +upα(t),

[u(t)]α_{= [}

u_α(t),uα(t)],

whereu_pα(t)andupα(t)are the nonhomogeneos solutions of the differential equations in (3.9) and (3.10). Using the boundary conditions, the coefficients are obtained as

b1(α) =ϕ

α−upα(0),

b2(α) = ψ

α−

ϕ

α−upα(0)

cos(1)−u_pα(1)

sin(1) ,

b3(α) =ϕα−upα(0),

b4(α) =

ψ_α−(ϕ_α−upα(0))cos(1)−upα(1)

sin(1) .

Then for (1,1) and (2,2) solutions, the solution of the problem (3.1)-(3.2) is

u_α(t) = ϕ

α−upα(0)

cos(t) (3.11)

+ψα−(ϕα−upα(0))cos(1)−upα(1)

sin(1) sin(t)

+u_pα(t),

uα(t) = (ϕ_α−upα(0))cos(t) (3.12)

+ψα−(ϕα−upα(0))cos(1)−upα(1)

sin(1) sin(t)

+upα(t),

[u(t)]α_{= [}

u_α(t),uα(t)]. (3.13)

For the (1,2) and (2,1) solutions

Using the generalized differentiability for (1,2) and (2,1) solu-tions, we have

u00_α(t) +u_α(t) =g α(t),

From here,

u(_α4)(t)−u_α(t) =g00_α(t)−g α(t),

u(_α4)(t)−uα(t) =g00

α(t)−gα(t) are obtained. That is, we have

(

u(_α4)(t)−u_α(t) =g00(t)−g(t) +1−α u_α(0) =ϕ

α,uα(1) =ψα

, (3.14)

u(_α4)(t)−uα(t) =g 00

(t)−g(t)−1+α uα(0) =ϕα,uα(1) =ψα

. (3.15)

From (3.14) and (3.15), the lower and the upper solution is found as

u_α(t) = c1(α)et+c2(α)e−t−c3(α)sin(t)(3.16)

−c4(α)cos(t) +upα(t),

uα(t) = c1(α)et+c2(α)e−t+c3(α)sin(t)(3.17) +c4(α)cos(t) +upα(t),

[u(t)]α_{= [}

u_α(t),uα(t)], (3.18)

whereu_pα(t)andupα(t)are the nonhomogeneos solutions of the differential equations in (3.14) and (3.15). Using the boundary conditions, the coefficients are obtained as

c1(α) =

−e−1ϕ

α+ϕα−upα(0)−upα(0)

2(e−e−1₎ ,

c2(α) =

−e

ϕ

α+ϕα−upα(0)−upα(0)

2(e−1_−e) ,

c3(α) = ϕ_α−ϕ

α+upα(0)−upα(0)

2 ,

c4(α) =

ψ_α−ψ

α+upα(1)−upα(1)

2 sin(1)

−

ϕ_α−ϕ

α+upα(0)−upα(0)

cos(1)

2 sin(1) .

Conclusion 3.1. The solution (3.6)-(3.8) according to the noniterative method is the same as the solution (3.11)-(3.13) according to the generalized differentiability for the solutions (1,1) and (2,2).

Example 3.2. Consider the following problem

u00(t) +u(t) =t2α, (3.19)

u(0) = [−1+α,1−α],u(1) = [α,2−α]. (3.20)

Firstly, let find the solution according to the noniterative method. Then, since

u(t) =u1(t) +λu2(t),

we have

u00₁(t) +u1(t) =t2 α

u1(0) = [−1+α,1−α],u 0 1(0) =0

, (3.21)

u00₂(t) +u2(t) =0 u2(0) =0,u

0 2(0) =1

, (3.22)

λ=[α,2−α]−u1(1) u2(1)

. (3.23)

The solution of the equation in (3.21) is

u_1α(t) =c1(α)cost+c2(α)sint+t2−3+α,

u1α(t) =c3(α)cost+c4(α)sint+t2−1−α,

[u1(t)]α= [u1α(t),u1α(t)].

Using the initial conditions, the coefficients are found as

c1(α) =c3(α) =2,

c2(α) =c4(α) =0.

The solution of the equation in (3.22) is

u_2α(t) =a1(α)cost+a2(α)sint,

[u2(t)]α= [u2α(t),u2α(t)].

Using the initial conditions, the coefficients are found as

a1(α) =a3(α) =0,

a2(α) =a4(α) =1.

Then, from (3.23)

λ =[α,2−α]−[2 cos(1)−2+α,2 cos(1)−α] [sin(1),sin(1)] ,

λ =λ=−2 cos(1) +2

sin(1) =

2(1−cos(1))

sin(1) .

Consequently, the solution of the problem (3.19)-(3.20) is

u_α(t) =2 cost+2(1−cos(1))

sin(1) sint+t

2_{−3+α, (3.24)}

uα(t) =2 cost+2(1−cos(1))

sin(1) sint+t 2₋₁₋

α, (3.25)

[u(t)]α_{= [}

u_α(t),uα(t)]. (3.26)

Secondly, let find the solution according to the generalized differentiability. For the solutions (1,1) and (2,2),

u00_α(t) +u_α(t) =t2−1+α, u_α(0) =−1+α,uα(1) =α

u00_α(t) +uα(t) =t2+1−α, uα(0) =1−α,uα(1) =2−α

are solved. Then for the solutions (1,1) and (2,2), the solution of the problem (3.19)-(3.20) is

u_α(t) =2 cost+2(1−cos(1))

sin(1) sint+t

2_{−3+α (3.27)}

uα(t) =2 cost+

2(1−cos(1))

sin(1) sint+t 2₋₁₋

α (3.28)

[u(t)]α_{= [u}

α(t),uα(t)]. (3.29)

For the solutions (1,2) and (2,1),

   

  

u00_α(t) +u_α(t) =t2−1+α, u00_α(t) +uα(t) =t2+1+α, u_α(0) =−1+α,u_α(1) =α, uα(0) =1−α,uα(`) =2−α

that is

u(_α4)(t)−u_α(t) =3−t2−α, u_α(0) =−1+α,u_α(1) =α,

u(4)α (t) +uα(t) =1−t2+α, uα(0) =1−α,uα(1) =2−α

are solved. Then for the solutions (1,2) and (2,1), the solution of the problem (3.19)-(3.20) is

u_α(t) =2 e

−1₋₁

e−1_−e e

t₊2(1−e) e−1_−ee

−t_+t2₋₃₊

α (3.30)

uα(t) =2 e

−1₋₁

e−1_−e e

t₊2(1−e) e−1_−ee

−t_+t2₋₁₋

α (3.31)

[u(t)]α= [u_α(t),uα(t)]. (3.32)

The solution (3.24)-(3.26) according to the noniterative method is the same as the solution (3.27)-(3.29) according to the generalized differentiability for the solutions (1,1) and (2,2). Also, all of the solutions are a validα−level set.

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