View of Solving a Circuit System Using Fuzzy Aboodh Transform

(1)

Research Article

3317

Solving a Circuit System Using Fuzzy Aboodh Transform

Samer Thaaban Abaas Alshibley(1,2)_{, Ameera N. Alkiffai}(2)_{, Athraa Neamah Albukhuttar}(2)

(1) Department of Mathematics, Faculty of Education, University of Kufa, Najaf, Iraq.

(2) Department of Mathematics, Faculty of Education for girls, University of Kufa, Najaf, Iraq. Corresponding Author Email samir.alshebly@uokufa.edu.iq

Article History: Received: 11 January 2021; Revised: 12 February 2021; Accepted: 27 March 2021; Published online: 23 May 2021

Abstract:

This paper proposed a new fuzzy transform that based on Aboodh transform and using this new fuzzy transform to calculate the exact solutions of first order fuzzy differential equations. To explain this approach, a related theorems and properties are proved in detail associated with some examples. In addition, this approach has been applied on physical application of an electrical circuit system.

Keywords: fuzzy number; fuzzy differential equation; strongly generalized differentiable; fuzzy Aboodh transforms; fuzzy first-order differential equation.

Introduction:

In the last decades, fuzzy differential equations have been used in many fields due to their numerous and important applications in a wide range of fields. In order to keep pace with the rapid development and progress in the field of fuzzy differential equations, we presented this paper that contains a new method for solving this type of equations, and our research will be limited to solving fuzzy differential equations of the first order. Fuzzy derivative was first introduced by Chang and Zadeh (1972) [1], then the concept of fuzzy differential equations was introduced by Kandel and Byatt (1978-1980) [2], after a while over time, the method of numerical solution was introduced for solving fuzzy differential equations by Abbasbandy and Allahviranloo (2002) [3]. Seikkala (1987) defined the fuzzy derivative which is a generalization of the Hukuhara derivative [4]. Generalized differential is strongly introduced in Bede and Gal (2005) [5] and studied by Bede et al (2006) [6]. This paper will construct a new fuzzy transform based on Aboodh transform, which for solving this type of equations and a circuit system is showed as an application for this technique.

Definition 1 [7]

A fuzzy number in parametric form is an ordered pair(s , s ) of functions s (κ), s (κ) , κ ∈ [0,1],which satisfies the following requirements:

• s (κ)is a bounded non-decreasing, right Continuous at 0 and left continuous function in (0,1]. • s (κ) is a bounded non- increasing, right Continuous at 0 and left continuous function in (0,1]. • s (κ) ≤ s (κ) , κ ∈ [0,1] .

Let η and ζ are fuzzy numbers, where η = (η (κ), η (κ)), ζ = (ζ (κ), ζ (κ)), 0 ≤ κ ≤ 1 and α > 0 we define • Addition η⨁ζ = (η (κ) + ζ (κ), η (κ) + ζ (κ)). • Subtraction η ⊝ ζ = (η (κ) − ζ (κ), η (κ) − ζ (κ)). • Scalar multiplication α⨀η = { (αη, αη , α ≥ 0 (αη, αη) , α < 0 }. Definition 3 [9]

(2)

Research Article

3318 Let η and ζ are fuzzy numbers then the distance between fuzzy numbers in the Hausdorff is given by Γ: ℜf× ℜf→ [0, +∞],where ℜf be the set of all fuzzy numbers on ℜ:

Γ(η, ζ) = supκ∈[0,1] max {|η (κ) − ζ (κ)| , |η (κ) − ζ (κ)|},where η = (η (κ), η (κ)),ζ = (ζ (κ), ζ (κ))

and (ℜf, Γ ) is a complete metric space and following properties are well known:

• Γ( η⨁ω, ζ⨁ω) = Γ(η, ζ), ∀η, ζ, ω ∈ ℜf .

• Γ(κ⨀η, κ⨀ ζ ) = |κ| Γ(η, ζ), ∀η, ζ, ∈ ℜf, κ ∈ ℜ .

• Γ( η⨁ω, ζ⨁ν) ≤ Γ(η, ζ) + Γ(ω, ν) , ∀η, ζ, ω, ν ∈ ℜ_f .

Suppose that x, y ∈ ℜf. If there exists z ∈ ℜf such that x = y + z then z is called the H-differential of

x and y and it is denoted by x ⊝ y.

Note that in this work, the sign ⊝ always meant the H-difference as well as x ⊝ y ≠ x + (−1)y . Definition 5 [10]

Let continuous fuzzy - valued function υ: (a, b): ⟶ ℜf and x0∈ (a, b) . We say that a mapping υ is

strongly generalized differentiable at x0 if there exists an element υ`(x0) ∈ ℜf , such that :

i. For all τ > 0 sufficiently small, ∃υ(x0 + τ) ⊖ υ(x0 ), υ(x0 ) ⊖ υ(x0 − τ),

where lim τ⟶0 υ(x₀ +τ)⊖υ(x₀ ) τ = limτ⟶0 υ(x₀ )⊖υ(x₀ −τ) τ = υ `_(x 0 ) , or

ii. For all τ > 0 sufficiently small, ∃υ(x0 ) ⊖ υ(x0 + τ), υ(x0 − τ) ⊖ υ(x0 )

where lim τ⟶0 υ(x0 )⊖υ(x0 +τ) −τ = limτ⟶0 υ(x0 −τ)⊖υ(x0 ) −τ = υ `_(x 0 ), or

iii. For all τ > 0 sufficiently small, ∃υ(x0+ τ) ⊖ υ(x0), υ(x0− τ) ⊖ υ(x0)

where lim τ⟶0 υ(x₀+τ)⊖υ(x₀) τ = limτ⟶0 υ(x₀−τ)⊖υ(x₀) −τ = υ `_(x 0), or

iv. For all τ > 0 sufficiently small, ∃υ(x0) ⊖ υ(x0+ τ), υ(x0) ⊖ υ(x0− τ)

where lim τ⟶0 υ(x0)⊖υ(x0+τ) −τ = limτ⟶0 υ(x0)⊖υ(x0−τ) τ = υ `_(x 0). Theorem 1 [11]

Assume that Γ: [𝑎, 𝑏] → [0,1], be a function such that [Γ(𝛾)]ω = [υω(𝛾),πω(𝛾)] for each ω∈ [0,1]. Then:

(i) If Γ is differentiable of the first form (i), then υω and πω are differentiable functions and

[Γ′(𝛾)]ω = [υ′(𝛾), π′(π)].

(ii) If F is differentiable of the second form (ii), then fα and gα are differentiable functions and

[Γ′_(𝛾)]

ω = [π′(π), υ′(𝛾)].

Theorem 2 [12]

Let υ(x) be a fuzzy valued function on [a, ∞) represented by((υ(x, κ),υ(x, κ)) .For any fixed κ ∈ [0,1] , let (υ(x, κ),υ(x, κ) are Riemann-integrals on [a, b]. For every b ≥a, if there exists two positive functions Μ(κ) and M(κ) such that∫ | υ(x , κ)|dx ≤_ab M(κ) and ∫ | υ(x, κ) |dx ≤_ab M(κ) for every b ≥a, then υ(x) is said to be improper fuzzy Riemann-Liouville integrals function on [a,∞), i.e.

∫ υ(x) dx_a∞ = [∫ (υ(x, κ)dx_a∞ , ∫ υ(x, κ)dx_a∞ ].

Let υ(x) be a fuzzy valued function on [a, b]. Suppose that υ(x, κ) and υ(x, κ) are improper Riemman-integrable on [a, b], then υ(x) is an improper on [a, b] and (∫ υ(τ, κ)𝑑τ)_𝑎𝑏 = (∫ υ(τ, κ)𝑑τ),_𝑎𝑏

(3)

Research Article

3319 (∫ υ(τ, κ)𝑑τ_𝑎𝑏 ) = (∫ υ(τ, κ)𝑑τ_𝑎𝑏 ).

If υ :( a, b) → ℜf is a continuous fuzzy valued function, then π(x) = ∫ υ(τ)𝑑τ 𝑥

𝑎 is differentiable with

derivative π′(x) = υ(x) . Theorem 3 [15]

Let υ(x): [a, b] ⟶ ℜf be a function and denote υ(x) = ((υ(x, κ),υ(x, κ)) for each κ ∈ [0,1] . Then:

• If υ is differentiable of the first form (i), then (υ(x, κ) and υ(x, κ) are differentiable functions and

υ`(x)= ( υ`(x, κ), υ`_{(x, κ)) .}

• If υ is differentiable of the second form (ii), then υ(x, κ) and υ(x, κ) are differentiable functions and υ`_{(x) = ( υ}`_{(x, κ), υ}`_{(x, κ)) .}

Consider functions in the set Η defined by Η = {υ(x): ∃ Μ, , κ1, κ2> 0, |υ(x)| < Μe−sx}, for a given

function in the set Η, the constant Μ must be finite number, κ1, κ2 may be finite or infinite. The Aboodh

transform denoted by the operator A and defined by the integral equations

A[υ(x)] = Η(s) =1_s∫ υ(x)e₀∞ −sxdx , x ≥ 0, κ1 ≤ s ≤ κ2, the variable s in this transform is used to

factor the variable x in the argument of the function υ . Theorem 4 [17]

Let υ(x) is a continuous function in [0, k] and A[υ(x)] = Η(s),then • A[υ(ax)] = 1

𝑎2Η (

s

𝑎) , for any constant a.

• For any functions υ(x) and π(x) and any constants a, b then: A[𝑎 υ(x) + 𝑏 π(x)] = 𝑎(A[ υ(x)]) + 𝑏(A[ π(x)])

• A [υ(𝑛)_{(x)] = s}𝑛_{Η(s) − ∑} υ(𝑖)(0) s2−𝑛−𝑖

𝑛−1 𝑖 = 1

• If A[υ(x)] = Η(s) and L[υ(x)] = F(s), then Η(s) =1 s F(s)

Where F(s)is Laplace transformation of υ(x). Definition 9

Let υ(x) be a continuous fuzzy-valued function. Suppose that 1

sυ(x)e

−sx_{is an improper fuzzy}

Rimann-integrable on [0, ∞), then 1

s∫ υ(x)e −sx ∞

0 dx is called fuzzy Aboodh transform and it is denoted by

Â[υ(x)] =1

0 dx , (s > 0 and integer). For theorem (2),we have. 1 s∫ υ(x)e −sx ∞ 0 dx = ( 1 s∫ υ(x, κ)e −sx ∞ 0 dx, 1 s∫ υ(x, κ)e −sx ∞ 0 dx) .

Using the definition of classical Aboodh transform, we have. A[υ(x, κ)] =1 s∫ υ(x, κ)e −sx ∞ 0 dx and A[ υ(x, κ)] = 1 s∫ υ(x, κ)e −sx ∞ 0 dx ,then Â[υ(x)] = (A[υ(x, κ)], A[ υ(x, κ)]) . Definition 10

(4)

Research Article

3320 The integral transform Â[υ(x)] =1

0 dx is said to be absolutely convergent integral if 1

sx⟶∞lim ∫ |υ(x)e −sx_dx| ∞

0 exists, that is mean:

1 sx⟶∞lim ∫ |υ(x, κ)e −sx_dx| ∞ 0 and 1 sx⟶∞lim ∫ |υ(x, κ)e −sx_dx| ∞ 0 are exist. Theorem 5

Let υ(x), π(x) be continuous fuzzy-valued functions assume that c1 and c2 are constants then

(a) Â[c1υ(x)] = c1Â[υ(x)] . (b) Â[c1(υ(x))⨁c2( π(x))] = c1Â[υ(x)]⨁c2Â[π(x)] . Proof (a) Â[c1υ(x)] = (A[c1 υ(x, κ)], A[c1υ(x, κ)]) = ( 1 s∫ c1 υ(x, κ)e −sx ∞ 0 dx,1 s∫ c1 υ(x, κ)e −sx ∞ 0 dx) = ( c1 s ∫ υ(x, κ)e −sx ∞ 0 dx,c1 s ∫ υ(x, κ)e −sx ∞ 0 dx) = c1( 1 s∫ υ(x, κ)e −sx ∞ 0 dx,1 s∫ υ(x, κ)e −sx ∞ 0 dx) = c1Â[υ(x)] (b) Suppose υ(x) = (υ(x, κ), υ(x, κ)) , π(x) = (π(x, κ), π(x, κ) Â[c1(υ(x))⨁c2( π(x))] = (A[ c1(υ(x, κ) + c2(π(x, κ)], A[c1 υ(x, κ) + c2π(x, κ)]) = (1 s∫ e −sx ∞ 0 ( c1υ(x, κ) + c2π(x, κ)) , 1 s∫ e −sx ∞ 0 (c1 υ(x, κ) + c2π(x, κ))) = (1 s∫ e −sx_c 1υ(x, κ) ∞ 0 +1 s∫ e −sx ∞ 0 c2π(x, κ)), ( 1 s∫ e −sx ∞ 0 c1 υ(x, κ) + 1 s∫ e −sx ∞ 0 c2π(x, κ)) = ( 1 s∫ e −sx ∞ 0 c1υ(x, κ), 1 s∫ e −sx ∞ 0 c1 υ(x, κ)) + ( 1 s∫ e −sx ∞ 0 c2π(x, κ), 1 s∫ e −sx_c 2π(x, κ) ∞ 0 ) = c1( 1 s∫ e −sx ∞ 0 υ(x, κ), 1 s∫ e −sx ∞ 0 υ(x, κ)) + c2( 1 s∫ e −sx ∞ 0 π(x, κ), 1 s∫ e −sx_{π(x, κ)} ∞ 0 ) = c1A (υ(x, κ), υ(x, κ)) + c2A (π(x, κ), π(x, κ)) = c1Â [υ(x)]⨁ c2Â [π(x)] . Theorem 6

Let μ(x) is the primitive of μ`_{(x) on [0, ∞) and μ(x) be an integrable fuzzy-valued function then}

a) if μ is (i)-differentiable then Â[μ`(x) ] = sÂ[μ(x)] ⊝ 1_sμ(0) . b) if μ is (ii)-differentiable then Â[μ`(x) ] = (−1_sμ(0)) ⊝ (−sÂ[μ(x)] . Proof (a)

For arbitrary fixed κ ∈ [0,1] sÂ[μ(x)] ⊝1 sμ(0) = (sA [ μ (x, κ)] − 1 s μ(0, κ), sA[ μ (x, κ)] − 1 s μ(0, κ)) Since

(5)

Research Article

3321 sA [ μ (x, κ)] −1 s μ(0, κ) = A [ μ `_{(x, κ) ] and sA[ μ (x, κ)] −}1 s μ(0, κ) = A [μ `_{(x, κ)]} Â[μ`_{(x) ] = (A [ μ}`_{(x, κ)] , A [μ}`_{(x, κ)])} sÂ[μ(x)] ⊝1 sμ(0) = (A [ μ `_{(x, κ)] , A [μ}`_{(x, κ)])} Â[μ`_{(x) ] = sA}_{̂[μ(x)] ⊝}1 sμ(0) . Proof (b)

(−1_sμ(0)) ⊝ (−sÂ[μ(x)] = (−1_s μ(0, κ) + (−sA[μ(x, κ)] , −1_sμ(0, κ) + (−sA [μ(x, κ)]) Since (−1 sμ(0, κ)) + (−sA[μ(x, κ)] = A [μ `_{(x, κ)] and (−}1 sμ(0, κ)) + (−sA [μ(x, κ)] = A [ μ `_{(x, κ) ]} Â[μ`(x) ] = (A [μ`_{(x, κ)] , A [ μ}`_{(x, κ)] )} (−1 sμ(0)) ⊝ (−sÂ[μ(x)] = (A [μ `_{(x, κ)] , A [ μ}`_{(x, κ)] )} Â[μ`(x) ] = (−1_sμ(0)) ⊝ (−sÂ[μ(x)] . Example 1

Consider a fuzzy initial value problem υ`(x) = υ(x), υ(0, κ) = (κ − 1,1 − κ), 0 ≤ κ ≤ 1 . Solution:

Using fuzzy Aboodh transform on both sides, to get Â[υ`(x) ] = Â[υ(x)] .

Case (1)

υ(x) be (i)-differentiable ,

Â[υ`(x) ] = sÂ[υ(x)] ⊝ 1_sυ(0) , Â[υ(x)] = sÂ[υ(x)] ⊝ 1_sυ(0) Using upper and lower functions, to have

A[ υ (x, κ)] = sA[ υ (x, κ)] −1_s υ(0, κ) , A[ υ (x, κ)] = sA[ υ (x, κ)] −1_s υ(0, κ) sA[ υ (x, κ)] − A[ υ (x, κ)] =1 s υ(0, κ) , sA[ υ (x, κ)] − A[ υ (x, κ)] = 1 s υ(0, κ) A[ υ (x, κ)](s − 1) =1 s υ(0, κ) , A[ υ (x, κ)](s − 1) = 1 s υ(0, κ) A[ υ (x, κ)] = 1 s2_−s υ(0, κ) , A[ υ (x, κ)] = 1 s2_−s υ(0, κ) [ υ (x, κ)] = A−1[ 1 s2_−s ] υ(0, κ) , [ υ (x, κ)] = A −1_[ 1 s2_−s] υ(0, κ) υ (x, κ) = (κ − 1)ex_{, υ (x, κ) = (1 − κ)e}x_. Case (2) υ(x) be (ii)-differentiable, Â[υ`(x) ] = (−1 sυ(0)) ⊝ (−sÂ[υ(x)], Â[υ(x)] = (− 1 sυ(0)) ⊝ (−sÂ[υ(x)] Using upper and lower functions, to have

A[ υ (x, κ)] = −1 s υ(0, κ) + sA[ υ (x, κ)], A[ υ (x, κ)] = − 1 s υ(0, κ) + sA[ υ (x, κ)] A[ υ (x, κ)] = −1 s (κ − 1) + sA[ υ (x, κ)], A[ υ (x, κ)] = − 1 s (1 − κ) + sA[ υ (x, κ)] Using Cramer's rule to get

A[ υ (x, κ)] = (1 − κ) 1 s3_{− 𝑠}+ (κ − 1) 1 s2_{− 1} , A[ υ (x, κ)] = (κ − 1) 1 s3_{− 𝑠}+ (1 − κ) 1 s2_{− 1}

(6)

Research Article

3322 υ (x, κ) = (1 − κ)A−1_[ 1 s3_{− 𝑠}] + (κ − 1)A−1[ 1 s2_{− 1}] , υ (x, κ) = (κ − 1)A−1[ 1 s3_{− 𝑠}] + (1 − κ)A−1[ 1 s2_{− 1}]

υ (x, κ) = (1 − κ)( sinh x − cosh x), υ (x, κ) = (κ − 1) (sinh x − cosh x) . Example 2 (Application of Â-transform of first-order differential equation)

Consider an RL circuit with R = 10, L = 2, E0= 0 Suppose that the initial charge on the capacitor is

I(0, κ) = (κ − 1,1 − κ) , 0 ≤ κ ≤ 1 . Find the charge I(t) for t ≥ 0 . Solution:

LI`(t) + 𝑅I(t) = E₀ , 2I`(t) + 10I(t) = 0

Using fuzzy Aboodh transform on both sides, to get Â[2I`(t) + 10I(t) ] = Â[0], 2Â[I`_{(t) ] + 10A}_{̂[I(t) ] = 0 .}

Case (1)

I(t) be (i)-differentiable Â[I`(t) ] = sÂ[I(t)] ⊝ 1

sI(0)

Using upper and lower functions, to have sA[I (t, κ)] −1

sI(0, κ) + 5A[I (t, κ)] = 0 , sA[ I (t, κ)] − 1 sI(0, κ) + 5A[I (t, κ)] = 0 A[I (t, κ)](𝑠 + 5) =1 s(κ − 1), A[I (t, κ)](𝑠 + 5) = 1 s(1 − κ) A[I (t, κ)] = 1 s(𝑠 + 5)(κ − 1), A[I (t, κ)] = 1 s(𝑠 + 5)(1 − κ) A[I (t, κ)] = 1 (𝑠2_{+ 5s)}(κ − 1), A[I (t, κ)] = 1 (𝑠2_{+ 5s)}(1 − κ) I (t, κ) = (κ − 1)e−5t, I (t, κ) = (1 − κ)e−5t . Case (2) I(t) be (ii)-differentiable: Â[I`(t) ] = (−1

sI(0)) ⊝ (−sÂ[I(t)]

Using upper and lower functions, to have sA[I (t, κ)] −1

sI(0, κ) + 5A[I (t, κ)] = 0 , sA[ I (t, κ)] − 1 sI(0, κ) + 5A[I (t, κ)] = 0 sA[I (t, κ)] + 5A[I (t, κ)] =1 s(κ − 1), sA[ I (t, κ)] + 5A[I (t, κ)] = 1 s(1 − κ)

(7)

Research Article

3323 Using Cramer's rule to get

A[ I (t, κ)] = (κ − 1) 1 s2_{− 25}− (1 − κ) 5 s3_{− 25𝑠} , A[I (t, κ)] = (1 − κ) 1 s2_{− 25}− (κ − 1) 5 s3_{− 25𝑠}

I (t, κ) = (κ − 1)( cosh5t + sinh5t), I (t, κ) = (1 − κ)(cosh5t + sinh5t) . Conclusion:

The main aim of the paper is to solve first -order linear fuzzy differential equations using proposed fuzzy Aboodh transform. Two numerical examples are given to illustrate the efficiency of the proposed method. References:

1. Chang S S L, Zadeh L (1972) “On fuzzy mapping and control”. IEEE Trans Syst Cybern 2:30– 34.

2. Kandel A, Byatt WJ (1978) “Fuzzy differential equations”. In: Proceedings of the interational conference on cybernetics and society. Tokyo, 1213–12160.

3. Abbasbandy S, Allahviranloo T (2002) “Numerical solution of fuzzy differential equation by

Tailor method”. J Comput Method Appl Math 2:113–124.

4. Kaleva O (1987) “Fuzzy differential equations”. Fuzzy Set Syst 24:301– 317.

5. Bede B, Gal SG (2005) “Generalizations of the differentiability of fuzzy-number-valued

functions with applications to fuzzy differential equations”. Fuzzy Set Syst 151:581–599.

6. Bede B, Gal SG (2006) “Remark on the new solutions of fuzzy differential equations”. Chaos Solitons Fractals.

7. Friedman M, Ma M, Kandel A (1999) “Numerical solution of fuzzy differential and integral

equations”. Fuzzy Set Syst 106:35–48.

8. Khudair R A, Alkiffai N A, ALbukhuttar A N (2020) “Solving the Vibrating Spring Equation

Using Fuzzy Elzaki Transform”. Mathematical Modelling of Engineering Problems

7(4):549-555.

9. Kaleva O (1987) “Fuzzy differential equations”. Fuzzy sets and systems”. 24(3) 301-317. 10. Sleibe A S, Alkiffai A N (2020). “Solving Ordinary Differential Equations Using Fuzzy

Transformation”. M.Sc. Thesis University of Kufa, College of Education for Girls, Department

of Mathematics.

11. Hooshangian L, Allahviranloo T (2014 ) “A New Method to Find Fuzzy Nth Order Derivation

and Applications to Fuzzy Nth Order Arithmetic Based on Generalized H-Derivation,”.

IJOCTA,4(2):105-121.

12. Wu HC (1999).The improper fuzzy Riemann integral and its numerical integration. Inform Sci 111:109–137.

13. Wu HC (2000).The fuzzy Riemann integral and its numerical integration. Fuzzy Set Syst 110:1– 25.

14. Salahshour S (2011) "Nth-order fuzzy differential equations under generalized

differentiability." Journal of fuzzy set Valued Analysis ,2011:1-14.

15. Abbasbandy.S,Allahviranloo.T and Darabi P (2011) “Numerical solution of N-order fuzzy

differential equations by Runge-Kutta method”. Mathematical and Computational

Applications,16(4): 935-946.

16. Aboodh K.S(2013)”The New Integral Transform Aboodh Transform”. Global Journal of Pure and Applied Mathematics 9(1): 35-43.

17. Chaudhary R, Sharma S D, Kumar N, Aggarwal S (2019) “Connections between Aboodh