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On a fuzzy initial value problem

Hülya Gültekin Çitil
Hülya Gültekin Çitil
   | 02 jun 2024
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Article Category: Original Study
Publicado en línea: 02 jun 2024
Páginas: -
Recibido: 30 ago 2023
Aceptado: 01 ene 2024
DOI: https://doi.org/10.2478/ijmce-2025-0002
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© 2025 Hülya Gültekin Çitil, published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.
Introduction

Fuzzy number and fuzzy arithmetic were introduced by Zadeh [1]. Fuzzy differential equations are important topic in many fields. Thus, many researchers study fuzzy differential equation by different approach method [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The first approach is Hukuhara derivative [12, 13]. This approach has a drawback: the solution becomes fuzzier as time goes. Thus, the fuzzy solution behaves quite differently from the crisp solution. So, generalized Hukuhara derivative was studied [7, 8, 11, 14, 15, 16]. Generalized Hukuhara derivative allows us to resolve the above-mentioned shortcoming. The second approach is extension principle [17]. The third approach is differential inclusion [18]. Fuzzy Laplace transform method is useful to solve fuzzy differential equation. In many articles, solution of fuzzy differential equation was studied by fuzzy Laplace transform [19, 20, 21, 22, 23, 24].

The rest of this paper is organized as following sections. Preliminaries and basic definitions used in this paper are given in section 2. In section 3, some applications and main results are reported. In section 4, the conclusions of this paper are introduced.
Preliminaries

In this part of the paper, we present some important definitions and theorems which will be used in this paper.

Definition 1

[25] A fuzzy number is a mapping u:ℝ → [0, 1] satisfying the properties {x∈ℝ|u(x)>0}¯ \overline {\left\{{x \in {\rm{\mathbb R}} \left| {u\left( x \right) > 0} \right.} \right\}} is compact, u is normal, u is convex fuzzy set, u is upper semi-continuous on ℝ. Let denote the set of all fuzzy numbers with ℝF.
Definition 2

[11] Let be u ∈ ℝF. The α-level set of u is [u]α = [uα, ūα] = {x ∈ ℝ | u (x) ≥ α}, 0 < α ≤ 1.
Definition 3

[11] The parametric form [uα, ūα] of u fuzzy number satisfy the following requirements:

The lower part uα is bounded non-decreasing left-continuous on (0, 1], right-continuous for α = 0.

The upper part ūα is bounded non-increasing left-continuous on (0, 1], right-continuous for α = 0.

uα ≤ ūα, 0 ≤ α ≤ 1.
Definition 4

[25] If A is asymmietric triangular fuzzy number with support [a, ā], the α–level set of A is [A]α=[a_+(a¯−a_2)α,a¯−(a¯−a_2)α] {\left[A \right]^\alpha} = \left[{\underline a + \left( {{{\overline a - \underline a} \over 2}} \right)\alpha ,\overline a - \left( {{{\overline a - \underline a} \over 2}} \right)\alpha} \right] .
Definition 5

[26] Let u, v ∈ ℝF. The generalized Hukuhara difference (gH-difference) between u and v is the set w ∈ ℝF which u⊖gv = w if and only if u = v + w or v = u + (−1) w.
Definition 6

[11] Let f : [a, b] → ℝF and t0 [a, b]. We say that f is (1)-differentiable at t0, if there exists an element f′ (t0) ∈ ℝF such that for all h > 0 sufficiently small (near to 0), exist f (t0 + h) ⊖ f (t0), f (t0) ⊖ f (t0 − h) and the limits limh→0f(t0+h)⊖f(t0)h=limh→0f(t0)⊖f(t0−h)h=f′(t0), \mathop {\lim}\limits_{h \to 0} {{f\left( {{t_0} + h} \right) \ominus f\left( {{t_0}} \right)} \over h} = \mathop {\lim}\limits_{h \to 0} {{f\left( {{t_0}} \right) \ominus f\left( {{t_0} - h} \right)} \over h} = {f^{'}}\left( {{t_0}} \right), and f is (2)-differentiable if for all h > 0 sufficiently small (near to 0), exist f (t0) ⊖ f (t0 + h), f (t0 − h) ⊖ f (t0) and the limits limh→0f(t0)⊖f(t0+h)−h=limh→0f(t0−h)⊖f(t0)−h=f′(t0). \mathop {\lim}\limits_{h \to 0} {{f\left( {{t_0}} \right) \ominus f\left( {{t_0} + h} \right)} \over {- h}} = \mathop {\lim}\limits_{h \to 0} {{f\left( {{t_0} - h} \right) \ominus f\left( {{t_0}} \right)} \over {- h}} = {f^{'}}\left( {{t_0}} \right).

Theorem 1

[27] Let f : [a, b] → ℝF be fuzzy function, where [f(t)]α=[f_α(t),f¯α(t)] {\left[{f\left( t \right)} \right]^\alpha} = \left[{{{\underline f}_\alpha}\left( t \right),{{\overline f}_\alpha}\left( t \right)} \right] , for each α ∈ [0, 1].

If f is (1)-differentiable then fα and f¯α {\overline f_\alpha} are differentiable functions and [f′(t)]α=[f_α′(t),f¯α′(t)] {\left[{{f^{'}}\left( t \right)} \right]^\alpha} = \left[{\underline f_\alpha^{'}\left( t \right),\overline f_\alpha^{'}\left( t \right)} \right] ,

If f is (2)-differentiable then fα and f¯α {\overline f_\alpha} are differentiable functions and [f′(t)]α=[f¯α′(t),f_α′(t)] {\left[{{f^{'}}\left( t \right)} \right]^\alpha} = \left[{\overline f_\alpha^{'}\left( t \right),\underline f_\alpha^{'}\left( t \right)} \right] .
Definition 7

[28] The fuzzy Laplace transform of fuzzy function f is F(s)=L(f(t))=∫0∞e−stf(t)dt=[limρ→∞∫0ρe−stf_(t)dt ,limρ→∞∫0ρe−stf¯(t)dt ],F(s,α)=L([f(t)]α)=[L(f_α(t)),L(f¯α(t))].L(f_α(t))=∫0∞e−stf_α(t)dt=limρ→∞∫0ρe−stf_α(t)dt ,L(f¯α(t))=∫0∞e−stf¯α(t)dt=limρ→∞∫0ρe−stf¯α(t)dt.  \matrix{{F\left( s \right) = L\left( {f\left( t \right)} \right) = \int_0^\infty {e^{- st}}f\left( t \right)dt = \left[{\mathop {\lim}\limits_{\rho \to \infty} \int_0^\rho {e^{- st}}\underline f \left( t \right)dt\;,\mathop {\lim}\limits_{\rho \to \infty} \int_0^\rho {e^{- st}}\overline f \left( t \right)dt\;} \right],} \cr {F\left( {s,\alpha} \right) = L\left( {{{\left[{f\left( t \right)} \right]}^\alpha}} \right) = \left[{L\left( {{{\underline f}_\alpha}\left( t \right)} \right),L\left( {{{\overline f}_\alpha}\left( t \right)} \right)} \right].} \cr {L\left( {{{\underline f}_\alpha}\left( t \right)} \right) = \int_0^\infty {e^{- st}}{{\underline f}_\alpha}\left( t \right)dt = \mathop {\lim}\limits_{\rho \to \infty} \int_0^\rho {e^{- st}}{{\underline f}_\alpha}\left( t \right)dt\;,} \cr {L\left( {{{\overline f}_\alpha}\left( t \right)} \right) = \int_0^\infty {e^{- st}}{{\overline f}_\alpha}\left( t \right)dt = \mathop {\lim}\limits_{\rho \to \infty} \int_0^\rho {e^{- st}}{{\overline f}_\alpha}\left( t \right)dt.\;} \cr}

Theorem 2

[19] Let f′ (t) be an integrable fuzzy function and f (t) is primitive of f′ (t) on (0, ∞].

If f is (1)-differentiable, L (f′ (t)) = sL ( f (t)) ⊖ f (0).

If f is (2)-differentiable, L (f′ (t)) = (− f (0)) ⊖ (−sL (f (t))).
Theorem 3

[28] Let f″ (t) be an integrable fuzzy function and f (t), f′ (t) are primitive of f′ (t), f″ (t) on (0, ∞].

If f and f′ are (1)-differentiable, L (f″ (t)) = s2L (f (t)) ⊖ s f (0) ⊖ f′ (0).

If f and f′ are (2)-differentiable, L (f″ (t)) = s2L(f (t)) ⊖ s f (0) − f′ (0).

If f is (1)-differentiable and f′ is (2)-differentiable, L (f″ (t)) = ⊖ (−s2) L(f (t)) − s f (0) − f′ (0).

If f is (2)-differentiable and f′ is (1)-differentiable L (f″ (t)) = ⊖ (−s2) L(f (t)) − s f (0) ⊖ f′ (0).
Applications

We investigate solutions of the problem u″(t)=[λ]αu(t),  t>0u(0)=[ρ]α, u′(0)=[ν]α \matrix{{u''(t) = {{\left[\lambda \right]}^\alpha}u(t),\;\;t > 0} \cr {u\left( 0 \right) = {{\left[\rho \right]}^\alpha},\;u'\left( 0 \right) = {{\left[\nu \right]}^\alpha}} \cr} by the fuzzy Laplace transform and the generalized Hukuhara differentiability, where [λ]α=[λ_α,λ¯α] {\left[\lambda \right]^\alpha} = \left[{{{\underline \lambda}_\alpha},{{\overline \lambda}_\alpha}} \right] , (λ_α>0,  λ¯α>0 ) \left( {{{\underline \lambda}_\alpha} > 0,\;\;{{\overline \lambda}_\alpha} > 0\;} \right) , [ρ]α=[ρ_α,ρ¯α] {\left[\rho \right]^\alpha} = \left[{{{\underline \rho}_\alpha},{{\overline \rho}_\alpha}} \right] , [ν]α= [ν_α,ν¯α] {\left[\nu \right]^\alpha} = \;\left[{{{\underline \nu}_\alpha},{{\overline \nu}_\alpha}} \right] are symmetric triangular fuzzy numbers, u (t) is positive fuzzy function, the fuzzy Laplace transform of fuzzy function u (t) is L(u (t)) = U(s) and (i, j)-solution means that u is (i)-differentiable, u′ is (j)-differentiable, i, j = 1, 2.

1. (1,1)-solution: Let u, u′ be (1)-differentiable. Then, the equation s2U(s)⊖su(0)⊖u′(0)=[λ]αU(s) {s^2}U\left( s \right) \ominus su\left( 0 \right) \ominus u'\left( 0 \right) = {\left[\lambda \right]^\alpha}U\left( s \right) is obtained by using the fuzzy Laplace transform. From this, we have the equations s2U_α(s)−su_α(0)−u_α′(0)=λ_αU_α(s),s2U¯α(s)−su¯α(0)−u¯α′(0)=λ¯αU¯α(s). \matrix{{{s^2}{{\underline U}_\alpha}\left( s \right) - s{{\underline u}_\alpha}\left( 0 \right) - \underline u_\alpha^{'}\left( 0 \right) = {{\underline \lambda}_\alpha}{{\underline U}_\alpha}\left( s \right),} \cr {{s^2}{{\overline U}_\alpha}\left( s \right) - s{{\overline u}_\alpha}\left( 0 \right) - \overline u_\alpha^{'}\left( 0 \right) = {{\overline \lambda}_\alpha}{{\overline U}_\alpha}\left( s \right).} \cr}

Using the initial conditions, we obtain U_α(s)=sρ_αs2−λ_α+ν_αs2−λ_α, U¯α(s)=sρ¯αs2−λ¯α+ν¯αs2−λ¯α. {\underline U_\alpha}\left( s \right) = {{s{{\underline \rho}_\alpha}} \over {{s^2} - {{\underline \lambda}_\alpha}}} + {{{{\underline \nu}_\alpha}} \over {{s^2} - {{\underline \lambda}_\alpha}}},\;{\overline U_\alpha}\left( s \right) = {{s{{\overline \rho}_\alpha}} \over {{s^2} - {{\overline \lambda}_\alpha}}} + {{{{\overline \nu}_\alpha}} \over {{s^2} - {{\overline \lambda}_\alpha}}}.

Taking the inverse Laplace transform of these equations, (1, 1)-solution is obtained as u_α(t)=ρ_αcosh(λ_αt)+ν_αλ_αsinh(λ_αt),u¯α(t)=ρ¯αcosh(λ¯αt)+ν¯αλ¯αsinh(λ¯αt),[u(t)]α=[u_α(t),u¯α(t)]. \matrix{{{{\underline u}_\alpha}\left( t \right) = {{\underline \rho}_\alpha}\cosh \left( {\sqrt {{{\underline \lambda}_\alpha}} t} \right) + {{{{\underline \nu}_\alpha}} \over {\sqrt {{{\underline \lambda}_\alpha}}}}\sinh \left( {\sqrt {{{\underline \lambda}_\alpha}} t} \right),{{\overline u}_\alpha}\left( t \right) = {{\overline \rho}_\alpha}\cosh \left( {\sqrt {{{\overline \lambda}_\alpha}} t} \right) + {{{{\overline \nu}_\alpha}} \over {\sqrt {{{\overline \lambda}_\alpha}}}}\sinh \left( {\sqrt {{{\overline \lambda}_\alpha}} t} \right),} \cr {{{\left[{u\left( t \right)} \right]}^\alpha} = \left[{{{\underline u}_\alpha}\left( t \right),{{\overline u}_\alpha}\left( t \right)} \right].} \cr}

2. (1,2)-solution: Let u be (1)-differentiable and u′ be (2)-differentiable. Then, since −u′(0)⊖(−s2U(s))−su(0)=[λ]αU(s), - u'\left( 0 \right) \ominus \left( {- {s^2}U\left( s \right)} \right) - su\left( 0 \right) = {\left[\lambda \right]^\alpha}U\left( s \right), we have the equations −u¯α′(0)+s2U¯α(s)−su¯α(0)=λ_αU_α(s),−u_α′(0)+s2U_α(s)−su_α(0)=λ¯αU¯α(s). \matrix{{- \overline u_\alpha^{'}\left( 0 \right) + {s^2}{{\overline U}_\alpha}\left( s \right) - s{{\overline u}_\alpha}\left( 0 \right) = {{\underline \lambda}_\alpha}{{\underline U}_\alpha}\left( s \right),} \cr {- \underline u_\alpha^{'}\left( 0 \right) + {s^2}{{\underline U}_\alpha}\left( s \right) - s{{\underline u}_\alpha}\left( 0 \right) = {{\overline \lambda}_\alpha}{{\overline U}_\alpha}\left( s \right).} \cr}

Using the initial conditions and making the necessary operations, we obtain the equations U_α(s)=λ¯αν¯αs4−λ_αλ¯α+sλ¯αρ¯αs4−λ_αλ¯α+s2ν_αs4−λ_αλ¯α+s3ρ_αs4−λ_αλ¯α,U¯α(s)=λ_αν_αs4−λ_αλ¯α+sλ_αρ_αs4−λ_αλ¯α+s2ν¯αs4−λ_αλ¯α+s3ρ¯αs4−λ_αλ¯α. \matrix{{{{\underline U}_\alpha}\left( s \right) = {{{{\overline \lambda}_\alpha}{{\overline \nu}_\alpha}} \over {{s^4} - {{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}}} + {{s{{\overline \lambda}_\alpha}{{\overline \rho}_\alpha}} \over {{s^4} - {{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}}} + {{{s^2}{{\underline \nu}_\alpha}} \over {{s^4} - {{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}}} + {{{s^3}{{\underline \rho}_\alpha}} \over {{s^4} - {{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}}},} \cr {{{\overline U}_\alpha}\left( s \right) = {{{{\underline \lambda}_\alpha}{{\underline \nu}_\alpha}} \over {{s^4} - {{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}}} + {{s{{\underline \lambda}_\alpha}{{\underline \rho}_\alpha}} \over {{s^4} - {{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}}} + {{{s^2}{{\overline \nu}_\alpha}} \over {{s^4} - {{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}}} + {{{s^3}{{\overline \rho}_\alpha}} \over {{s^4} - {{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}}}.} \cr}

Then, (1, 2)-solution is obtained as u_α(t)=(λ¯αν¯α2(λ_αλ¯α)34+ν_α2(λ_αλ¯α)4)(sin((λ_αλ¯α)4t)+sinh((λ_αλ¯α)4t))+(λ¯αρ¯α2(λ_αλ¯α)+ρ_α2)(cos((λ_αλ¯α)4t)+cosh((λ_αλ¯α)4t)),u¯α(t)=(λ_αν_α2(λ_αλ¯α)34+ν¯α2(λ_αλ¯α)4)(sin((λ_αλ¯α)4t)+sinh((λ_αλ¯α)4t))+(λ_αρ_α2(λ_αλ¯α)+ρ¯α2)(cos((λ_αλ¯α)4t)+cosh((λ_αλ¯α)4t)),[u(t)]α=[u_α(t),u¯α(t)]. \matrix{\hfill {{{\underline u}_\alpha}\left( t \right) = \left( {{{{{\overline \lambda}_\alpha}{{\overline \nu}_\alpha}} \over {2\root 4 \of {{{\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)}^3}}}} + {{{{\underline \nu}_\alpha}} \over {2\root 4 \of {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)}}}} \right)\left( {\sin \left( {\root 4 \of {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)} t} \right) + \sinh \left( {\root 4 \of {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)} t} \right)} \right)} \cr \hfill {+ \left( {{{{{\overline \lambda}_\alpha}{{\overline \rho}_\alpha}} \over {2\sqrt {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)}}} + {{{{\underline \rho}_\alpha}} \over 2}} \right)\left( {\cos \left( {\root 4 \of {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)} t} \right) + \cosh \left( {\root 4 \of {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)} t} \right)} \right),} \cr \hfill {{{\overline u}_\alpha}\left( t \right) = \left( {{{{{\underline \lambda}_\alpha}{{\underline \nu}_\alpha}} \over {2\root 4 \of {{{\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)}^3}}}} + {{{{\overline \nu}_\alpha}} \over {2\root 4 \of {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)}}}} \right)\left( {\sin \left( {\root 4 \of {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)} t} \right) + \sinh \left( {\root 4 \of {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)} t} \right)} \right)} \cr \hfill {+ \left( {{{{{\underline \lambda}_\alpha}{{\underline \rho}_\alpha}} \over {2\sqrt {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)}}} + {{{{\overline \rho}_\alpha}} \over 2}} \right)\left( {\cos \left( {\root 4 \of {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)} t} \right) + \cosh \left( {\root 4 \of {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)} t} \right)} \right),} \cr \hfill {{{\left[{u\left( t \right)} \right]}^\alpha} = \left[{{{\underline u}_\alpha}\left( t \right),{{\overline u}_\alpha}\left( t \right)} \right].} \cr}

3. (2,1)-solution: Let u be (2)-differentiable and u′ be (1)-differentiable. From the equation ⊖(−s2U(s))−su(0)⊖u′(0)=[λ]αU(s), \ominus \left( {- {s^2}U\left( s \right)} \right) - su\left( 0 \right) \ominus {u^{'}}\left( 0 \right) = {\left[\lambda \right]^\alpha}U\left( s \right), we have the equations s2U¯α(s)−λ_αU_α(s)=sρ¯α+ν_α,s2U_α(s)−λ¯αU¯α(s)=sρ_α+ν¯α. {s^2}{\overline U_\alpha}\left( s \right) - {\underline \lambda_\alpha}{\underline U_\alpha}\left( s \right) = s{\overline \rho_\alpha} + {\underline \nu_\alpha},{s^2}{\underline U_\alpha}\left( s \right) - {\overline \lambda_\alpha}{\overline U_\alpha}\left( s \right) = s{\underline \rho_\alpha} + {\overline \nu_\alpha}.

From this, making the necessary operations, (2, 1)-solution is obtained as u_α(t)=(λ¯αν_α2(λ_αλ¯α)34+ν¯α2(λ_αλ¯α)4)(sin((λ_αλ¯α)4t)+sinh((λ_αλ¯α)4t))+(λ¯αρ¯α2(λ_αλ¯α)+ρ_α2)(cos((λ_αλ¯α)4t)+cosh((λ_αλ¯α)4t)),u¯α(t)=(λ_αν¯α2(λ_αλ¯α)34+ν_α2(λ_αλ¯α)4)(sin((λ_αλ¯α)4t)+sinh((λ_αλ¯α)4t))+(λ_αρ_α2(λ_αλ¯α)+ρ¯α2)(cos((λ_αλ¯α)4t)+cosh((λ_αλ¯α)4t)),[u(t)]α=[u_α(t),u¯α(t)]. \matrix{\hfill {{{\underline u}_\alpha}\left( t \right) = \left( {{{{{\overline \lambda}_\alpha}{{\underline \nu}_\alpha}} \over {2\root 4 \of {{{\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)}^3}}}} + {{{{\overline \nu}_\alpha}} \over {2\root 4 \of {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)}}}} \right)\left( {\sin \left( {\root 4 \of {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)} t} \right) + \sinh \left( {\root 4 \of {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)} t} \right)} \right)} \cr \hfill {+ \left( {{{{{\overline \lambda}_\alpha}{{\overline \rho}_\alpha}} \over {2\sqrt {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)}}} + {{{{\underline \rho}_\alpha}} \over 2}} \right)\left( {\cos \left( {\root 4 \of {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)} t} \right) + \cosh \left( {\root 4 \of {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)} t} \right)} \right),} \cr \hfill {{{\overline u}_\alpha}\left( t \right) = \left( {{{{{\underline \lambda}_\alpha}{{\overline \nu}_\alpha}} \over {2\root 4 \of {{{\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)}^3}}}} + {{{{\underline \nu}_\alpha}} \over {2\root 4 \of {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)}}}} \right)\left( {\sin \left( {\root 4 \of {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)} t} \right) + \sinh \left( {\root 4 \of {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)} t} \right)} \right)} \cr \hfill {+ \left( {{{{{\underline \lambda}_\alpha}{{\underline \rho}_\alpha}} \over {2\sqrt {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)}}} + {{{{\overline \rho}_\alpha}} \over 2}} \right)\left( {\cos \left( {\root 4 \of {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)} t} \right) + \cosh \left( {\root 4 \of {\left( {{{\underline \lambda}_\alpha}{{\overline \lambda}_\alpha}} \right)} t} \right)} \right),} \cr \hfill {{{\left[{u\left( t \right)} \right]}^\alpha} = \left[{{{\underline u}_\alpha}\left( t \right),{{\overline u}_\alpha}\left( t \right)} \right].} \cr}

4. (2,2)-solution: Let u and u′ be (2)-differentiable. Since s2U(s)⊖su(0)−u′(0)=[λ]αU(s), {s^2}U\left( s \right) \ominus su\left( 0 \right) - {u^{^{'}}}\left( 0 \right) = {\left[\lambda \right]^\alpha}U\left( s \right), (2, 2)-solution is obtained as u_α(t)=ρ_αcosh(λ_αt)+ν¯αλ_αsinh(λ_αt),u¯α(t)=ρ¯αcosh(λ¯αt)+ν_αλ¯αsinh(λ¯αt),[u(t)]α=[u_α(t),u¯α(t)]. \matrix{{{{\underline u}_\alpha}\left( t \right) = {{\underline \rho}_\alpha}\cosh \left( {\sqrt {{{\underline \lambda}_\alpha}} t} \right) + {{{{\overline \nu}_\alpha}} \over {\sqrt {{{\underline \lambda}_\alpha}}}}\sinh \left( {\sqrt {{{\underline \lambda}_\alpha}} t} \right),{{\overline u}_\alpha}\left( t \right) = {{\overline \rho}_\alpha}\cosh \left( {\sqrt {{{\overline \lambda}_\alpha}} t} \right) + {{{{\underline \nu}_\alpha}} \over {\sqrt {{{\overline \lambda}_\alpha}}}}\sinh \left( {\sqrt {{{\overline \lambda}_\alpha}} t} \right),} \cr {{{\left[{u\left( t \right)} \right]}^\alpha} = \left[{{{\underline u}_\alpha}\left( t \right),{{\overline u}_\alpha}\left( t \right)} \right].} \cr}

These solutions must be valid fuzzy functions.

Example 3.1

Consider the solutions of the fuzzy problem u′′(t)=[1]αu(t),u(0)=[1]α,  u′(0)=[2]α, \matrix{{{u^{''}}\left( t \right) = {{\left[1 \right]}^\alpha}u\left( t \right),} \cr {u\left( 0 \right) = {{\left[1 \right]}^\alpha},\;\;{u^{'}}\left( 0 \right) = {{\left[2 \right]}^\alpha},} \cr} where [1]α = [α, 2 − α], [2]α = [1 + α, 3 − α].

(1, 1)-solution is u_α(t)=αcosh(αt)+1+ααsinh(αt),u¯α(t)=(2−α)cosh((2−α)t)+3−α(2−α)sinh((2−α)t),[u(t)]α=[u_α(t),u¯α(t)], \matrix{{{{\underline u}_\alpha}\left( t \right) = \alpha \cosh \left( {\sqrt \alpha t} \right) + {{1 + \alpha} \over {\sqrt \alpha}}\sinh \left( {\sqrt \alpha t} \right),} \cr {{{\overline u}_\alpha}\left( t \right) = \left( {2 - \alpha} \right)\cosh \left( {\sqrt {\left( {2 - \alpha} \right)} t} \right) + {{3 - \alpha} \over {\sqrt {\left( {2 - \alpha} \right)}}}\sinh \left( {\sqrt {\left( {2 - \alpha} \right)} t} \right),} \cr {{{\left[{u\left( t \right)} \right]}^\alpha} = \left[{{{\underline u}_\alpha}\left( t \right),{{\overline u}_\alpha}\left( t \right)} \right],} \cr}

(1, 2)-solution is u_α(t)=((2−α)(3−α)2(α(2−α))34+1+α2(α(2−α))4)(sin((α(2−α))4t)+sinh((α(2−α))4t))+((2−α)22(α(2−α))+α2)(cos((α(2−α))4t)+cosh((α(2−α))4t)),u¯α(t)=(α(1+α)2(α(2−α))34+3−α2(α(2−α))4)(sin((α(2−α))4t)+sinh((α(2−α))4t))+(α22(α(2−α))+2−α2)(cos((α(2−α))4t)+cosh((α(2−α))4t)),[u(t)]α=[u_α(t),u¯α(t)], \matrix{{{{\underline u}_\alpha}\left( t \right) = \left( {{{\left( {2 - \alpha} \right)\left( {3 - \alpha} \right)} \over {2\root 4 \of {{{\left( {\alpha \left( {2 - \alpha} \right)} \right)}^3}}}} + {{1 + \alpha} \over {2\root 4 \of {\left( {\alpha \left( {2 - \alpha} \right)} \right)}}}} \right)\left( {\sin \left( {\root 4 \of {\left( {\alpha \left( {2 - \alpha} \right)} \right)} t} \right) + \sinh \left( {\root 4 \of {\left( {\alpha \left( {2 - \alpha} \right)} \right)} t} \right)} \right)} \cr {+ \left( {{{{{\left( {2 - \alpha} \right)}^2}} \over {2\sqrt {\left( {\alpha \left( {2 - \alpha} \right)} \right)}}} + {\alpha \over 2}} \right)\left( {\cos \left( {\root 4 \of {\left( {\alpha \left( {2 - \alpha} \right)} \right)} t} \right) + \cosh \left( {\root 4 \of {\left( {\alpha \left( {2 - \alpha} \right)} \right)} t} \right)} \right),} \cr {{{\overline u}_\alpha}\left( t \right) = \left( {{{\alpha \left( {1 + \alpha} \right)} \over {2\root 4 \of {{{\left( {\alpha \left( {2 - \alpha} \right)} \right)}^3}}}} + {{3 - \alpha} \over {2\root 4 \of {\left( {\alpha \left( {2 - \alpha} \right)} \right)}}}} \right)\left( {\sin \left( {\root 4 \of {\left( {\alpha \left( {2 - \alpha} \right)} \right)} t} \right) + \sinh \left( {\root 4 \of {\left( {\alpha \left( {2 - \alpha} \right)} \right)} t} \right)} \right)} \cr {+ \left( {{{{\alpha^2}} \over {2\sqrt {\left( {\alpha \left( {2 - \alpha} \right)} \right)}}} + {{2 - \alpha} \over 2}} \right)\left( {\cos \left( {\root 4 \of {\left( {\alpha \left( {2 - \alpha} \right)} \right)} t} \right) + \cosh \left( {\root 4 \of {\left( {\alpha \left( {2 - \alpha} \right)} \right)} t} \right)} \right),} \cr {{{\left[{u\left( t \right)} \right]}^\alpha} = \left[{{{\underline u}_\alpha}\left( t \right),{{\overline u}_\alpha}\left( t \right)} \right],} \cr}

(2, 1)-solution is u_α(t)=((2−α)(1+α)2(α(2−α))34+(3−α)2(α(2−α))4)(sin((α(2−α))4t)+sinh((α(2−α))4t))+((2−α)22(α(2−α))+α2)(cos((α(2−α))4t)+cosh((α(2−α))4t)),u¯α(t)=(α(3−α)2(α(2−α))34+(1+α)2(α(2−α))4)(sin((α(2−α))4t)+sinh((α(2−α))4t))+(α22(α(2−α))+2−α2)(cos((α(2−α))4t)+cosh((α(2−α))4t)),[u(t)]α=[u_α(t),u¯α(t)], \matrix{{{{\underline u}_\alpha}\left( t \right) = \left( {{{\left( {2 - \alpha} \right)\left( {1 + \alpha} \right)} \over {2\root 4 \of {{{\left( {\alpha \left( {2 - \alpha} \right)} \right)}^3}}}} + {{\left( {3 - \alpha} \right)} \over {2\root 4 \of {\left( {\alpha \left( {2 - \alpha} \right)} \right)}}}} \right)\left( {\sin \left( {\root 4 \of {\left( {\alpha \left( {2 - \alpha} \right)} \right)} t} \right) + \sinh \left( {\root 4 \of {\left( {\alpha \left( {2 - \alpha} \right)} \right)} t} \right)} \right)} \cr {+ \left( {{{{{\left( {2 - \alpha} \right)}^2}} \over {2\sqrt {\left( {\alpha \left( {2 - \alpha} \right)} \right)}}} + {\alpha \over 2}} \right)\left( {\cos \left( {\root 4 \of {\left( {\alpha \left( {2 - \alpha} \right)} \right)} t} \right) + \cosh \left( {\root 4 \of {\left( {\alpha \left( {2 - \alpha} \right)} \right)} t} \right)} \right),} \cr {{{\overline u}_\alpha}\left( t \right) = \left( {{{\alpha \left( {3 - \alpha} \right)} \over {2\root 4 \of {{{\left( {\alpha \left( {2 - \alpha} \right)} \right)}^3}}}} + {{\left( {1 + \alpha} \right)} \over {2\root 4 \of {\left( {\alpha \left( {2 - \alpha} \right)} \right)}}}} \right)\left( {\sin \left( {\root 4 \of {\left( {\alpha \left( {2 - \alpha} \right)} \right)} t} \right) + \sinh \left( {\root 4 \of {\left( {\alpha \left( {2 - \alpha} \right)} \right)} t} \right)} \right)} \cr {+ \left( {{{{\alpha^2}} \over {2\sqrt {\left( {\alpha \left( {2 - \alpha} \right)} \right)}}} + {{2 - \alpha} \over 2}} \right)\left( {\cos \left( {\root 4 \of {\left( {\alpha \left( {2 - \alpha} \right)} \right)} t} \right) + \cosh \left( {\root 4 \of {\left( {\alpha \left( {2 - \alpha} \right)} \right)} t} \right)} \right),} \cr {{{\left[{u\left( t \right)} \right]}^\alpha} = \left[{{{\underline u}_\alpha}\left( t \right),{{\overline u}_\alpha}\left( t \right)} \right],} \cr} and (2, 2)-solution is u_α(t)=αcosh(αt)+3−ααsinh(αt),u¯α(t)=(2−α)cosh((2−α)t)+1+α(2−α)sinh((2−α)t),[u(t)]α=[u_α(t),u¯α(t)]. \matrix{{{{\underline u}_\alpha}\left( t \right) = \alpha \cosh \left( {\sqrt \alpha t} \right) + {{3 - \alpha} \over {\sqrt \alpha}}\sinh \left( {\sqrt \alpha t} \right),} \cr {{{\overline u}_\alpha}\left( t \right) = \left( {2 - \alpha} \right)\cosh \left( {\sqrt {\left( {2 - \alpha} \right)} t} \right) + {{1 + \alpha} \over {\sqrt {\left( {2 - \alpha} \right)}}}\sinh \left( {\sqrt {\left( {2 - \alpha} \right)} t} \right),} \cr {{{\left[{u\left( t \right)} \right]}^\alpha} = \left[{{{\underline u}_\alpha}\left( t \right),{{\overline u}_\alpha}\left( t \right)} \right].} \cr}

From Definition 3, according to Figure (1a) and Figure (2b), (1, 1) and (2, 2) solutions are valid fuzzy functions. But, according to Figure (1b) and Figure (2a), (1, 2) and (2, 1) solutions are not valid fuzzy functions.

Fig. 1

a) Graphic of (1,1)-solution for α = 0.5. b) Graphic of (1,2)-solution for α = 0.5.

Fig. 2

a) Graphic of (2,1)-solution for α = 0.5. b) Graphic of (2,2)-solution for α = 0.5.

In these figures, Blue is used to symbolize →y¯α(t) \to {\overline y_\alpha}\left( t \right) , Red is for → yα (t) and Green is used to explain →y¯1(t)=y_1(t) \to {\overline y_1}\left( t \right) = {\underline y_1}\left( t \right) .
Conclusion

In this work, we studied a fuzzy initial problem with fuzzy coefficient. We used the generalized differentiability and the fuzzy Laplace transform. We solved an application on the problem and draw the figures of the solutions. Finally, it is shown that whether the solutions are valid fuzzy functions or not.
Declarations
Conflict of interest 

The authors hereby declare that there is no conflict of interests regarding the publication of this paper.
Funding

Not applicable.
Author's contribution

H.G.Ç. - Methodology, Writing-Original Draft, Validation, Conceptualization, Formal analysis, Investigation, Writing-Review and Editing. All authors read and approved the final submitted version of this manuscript.
Acknowledgement

The authors deeply appreciate the anonymous reviewers for their helpful and constructive suggestions, which can help improve this paper further.
Data availability statement

All data that support the findings of this study are included within the article.
Using of AI tools

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.
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