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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
\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
{\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
\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
\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{\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{\overline f_\alpha}are differentiable functions and{\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{\overline f_\alpha}are differentiable functions and{\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
\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
\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
{\left[\lambda \right]^\alpha} = \left[{{{\underline \lambda}_\alpha},{{\overline \lambda}_\alpha}} \right]
,
\left( {{{\underline \lambda}_\alpha} > 0,\;\;{{\overline \lambda}_\alpha} > 0\;} \right)
,
{\left[\rho \right]^\alpha} = \left[{{{\underline \rho}_\alpha},{{\overline \rho}_\alpha}} \right]
,
{\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
{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
\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
{\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
\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'\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
\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}
3. (2,1)-solution: Let u be (2)-differentiable and u′ be (1)-differentiable. From the equation
\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
{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}.
In these figures, Blue is used to symbolize
\to {\overline y_\alpha}\left( t \right)
, Red is for → yα (t) and Green is used to explain
\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.