# Applications of the Fuzzy Sumudu Transform for the Solution of First Order Fuzzy Differential Equations

^{*}

## Abstract

**:**

## 1. Introduction

_{0}”, “about x

_{0}” or “more than x

_{0}”. If this is the case, the classical differential equations cannot be used to handle this situation. Therefore, it is necessary to study other theories in order to overcome this problem. One of the most popular theories for describing this situation is the fuzzy set theory [22]. By incorporating fuzziness into classical mathematics, many authors studied fuzzy derivatives [23–28], fuzzy differential equations (FDEs) [29–31] and fuzzy fractional differential equations (FFDEs) [32–34].

## 2. Preliminaries

**Definition 1.**[44] By ℝ, we denote the set of all real numbers. A fuzzy number is a mapping U : ℝ → [0, 1] with the following properties:

- U is upper semi-continuous,
- U is fuzzy convex, i.e., U(λx + (1 − λ)y) ≥ min{U(x), U(y)} for all x, y ∈ ℝ, λ ∈ [0, 1],
- U is normal, i.e., ∃x
_{0}∈ ℝ for which U(x_{0}) = 1, - supp U = {x ∈ ℝ|U(x) > 0} is the support of the U, and its closure, i.e. cl(supp U) is compact.

**Definition 2.**[44] Let $\mathcal{F}$(ℝ) be the set of all fuzzy numbers on ℝ. The α-level set of a fuzzy number U ∈ $\mathcal{F}$(ℝ), α ∈ [0, 1], denoted by U

_{α}, is defined as:

**Remark.**[45] Let X be the Cartesian product of universes X = X

_{1}× … × X

_{n}and A

_{1}, …, A

_{n}be n fuzzy numbers in X

_{1}, …, X

_{n}, respectively. A fuzzy function f maps from X to a universe Y, y = f(x

_{1}, …, x

_{n}). Then, the extension principle allows us to define a fuzzy set B in Y by:

^{−}

^{1}is the inverse of f.

_{α}represents the α-level set of a fuzzy number.

_{α}. This leads to other representation of a fuzzy number, which will be defined by two endpoint functions ${\underset{\xaf}{u}}_{\alpha}$ and ${\overline{u}}_{\alpha}$. Friedman et al. [46] and Ma et al. [47] defined the representation as:

**Definition 3.**A fuzzy number U in parametric form is a pair [ ${\underset{\xaf}{u}}_{\alpha}$, ${\overline{u}}_{\alpha}$] of functions${\underset{\xaf}{u}}_{\alpha}$ and${\overline{u}}_{\alpha}$, α ∈ [0, 1], which satisfy the following requirements:

- ${\underset{\xaf}{u}}_{\alpha}$ is a bounded non-decreasing left continuous function in (0, 1] and right continuous at zero,
- ${\overline{u}}_{\alpha}$ is a bounded non-increasing left continuous function in (0, 1] and right continuous at zero,
- ${\underset{\xaf}{u}}_{\alpha}$.≤ ${\overline{u}}_{\alpha}$.

**Definition 4.**Let U ∈ $\mathcal{F}$(ℝ). U is called a triangular fuzzy number if its membership function has the following form:

_{α}= [a + (b − c)α, c − (c − b)α], for any α ∈ [0, 1].

**Definition 5.**[46] For arbitrary${U}_{\alpha}=[{\underset{\xaf}{u}}_{\alpha},{\overline{u}}_{\alpha}]$, ${V}_{\alpha}=[{\underset{\xaf}{v}}_{\alpha},{\overline{v}}_{\alpha}]$ and k > 0, we define addition, subtraction and multiplication by k for U

_{α}and V

_{α}as:

- addition,$${U}_{\alpha}\oplus {V}_{\alpha}=[{\underset{\xaf}{u}}_{\alpha}+{\underset{\xaf}{v}}_{\alpha},{\overline{u}}_{\alpha}+{\overline{v}}_{\alpha}],$$
- subtraction,$${U}_{\alpha}\ominus {V}_{\alpha}=[{\underset{\xaf}{u}}_{\alpha}-{\overline{v}}_{\alpha},{\overline{u}}_{\alpha}-{\underset{\xaf}{v}}_{\alpha}],$$
- scalar multiplication,$$k\odot {U}_{\alpha}=\{\begin{array}{ll}[k{\underset{\xaf}{u}}_{\alpha},k{\overline{u}}_{\alpha}],& k\ge 0,\\ [k{\overline{u}}_{\alpha},k{\underset{\xaf}{u}}_{\alpha}],& k<0.\end{array}$$

_{α}= −U

_{α}.

**Definition 6.**[48] The distance D(U, V) between two fuzzy intervals U and V is defined as:

_{α}and V

_{α}.

- D(U ⊕ W, V ⊕ W) = D(U, V), ∀U, V, W ∈ $\mathcal{F}$(ℝ),
- D(k ⊙ U, k ⊙ V) = |k|D(U, V), ∀k ∈ ℝ, U, V ∈ $\mathcal{F}$(ℝ),
- D(U ⊕ V, W ⊕ E) ≤ D(U, W) + D(V, E), ∀U, V, W, E ∈ $\mathcal{F}$(ℝ),
- (D, $\mathcal{F}$(ℝ)) is a complete metric space.

**Definition 7.**[49] Let f : ℝ → $\mathcal{F}$(ℝ). The function f is called continuous if for every x

_{0}∈ ℝ and every ϵ > 0, there exists δ > 0, such that if |x − x

_{0}| < δ, then D(f(x), f(x

_{0})) < ϵ.

**Theorem 1.**[50] Let f : ℝ → $\mathcal{F}$(ℝ), and it is represented by$[{\underset{\xaf}{f}}_{\alpha}(x),{\overline{f}}_{\alpha}(x)]$. For any fixed α ∈ [0, 1], assume that${\underset{\xaf}{f}}_{\alpha}(x)$ and${\overline{f}}_{\alpha}(x)$ are Riemann-integrable on [a, b] for every b ≥ a, and assume that there are two positive${\underset{\xaf}{M}}_{\alpha}$ and${\overline{M}}_{\alpha}$, such that${\int}_{a}^{b}|{\underset{\xaf}{f}}_{\alpha}(x)|dx\le {\underset{\xaf}{M}}_{\alpha}$ and${f}_{a}^{b}|{\overline{f}}_{\alpha}(x)|dx\le {\overline{M}}_{\alpha}$ for every b ≥ a. Then, f(x) is improper fuzzy Riemann-integrable on [a, ∞), and the improper fuzzy Riemann-integrable is a fuzzy number. Furthermore, we have:

**Proposition 1.**[51] If each of f(x) and g(x) is a fuzzy-valued function and fuzzy Riemann-integrable on [a, ∞), then f(x) ⊕ g(x) is fuzzy Riemann-integrable on [a, ∞). Moreover, we have:

**Definition 8.**[27] Let x, y ∈ $\mathcal{F}$(ℝ). If there exists z ∈ $\mathcal{F}$(ℝ), such that x = y ⊕ z, then z is called the H-difference of x and y, and it is denoted by x −

^{H}y.

**Definition 9.**[52,53] Let f : (a, b) → $\mathcal{F}$(ℝ) and x

_{0}∈ (a, b). We say that f is strongly generalized differentiable at x

_{0}, if there exists an element f′(x

_{0}) ∈ $\mathcal{F}$(ℝ), such that:

- for all h > 0 sufficiently small, there exist f(x
_{0}+ h) −^{H}f(x_{0}), f(x_{0}) −^{H}f(x_{0}− h) and the limits (in the metric D):$$\underset{h\to 0}{\mathrm{lim}}\frac{f({x}_{0}+h){-}^{H}f({x}_{0})}{h}=\underset{h\to 0}{\mathrm{lim}}\frac{f({x}_{0}){-}^{H}f({x}_{0}-h)}{h}={f}^{\prime}({x}_{0}),$$ - for all h > 0 sufficiently small, there exist f(x
_{0})−^{H}f(x_{0}+ h), f(x_{0}− h)−^{H}f(x_{0}) and the limits (in the metric D):$$\underset{h\to 0}{\mathrm{lim}}\frac{f({x}_{0}){-}^{H}f({x}_{0}+h)}{-h}=\underset{h\to 0}{\mathrm{lim}}\frac{f({x}_{0}-h){-}^{H}f({x}_{0})}{-h}={f}^{\prime}({x}_{0}).$$

**Note.**In this paper, we only consider Cases (1) and (2) in the strongly generalized differentiability proposed by Bede and Gal [52]. Chalco-Cano and Roman-Florés [54] stated that Cases (1) and (2) are more important, since Cases (3) and (4) occur only on a discrete set of points.

**Theorem 2.**[54] Let f : ℝ → $\mathcal{F}$(ℝ) be a continuous fuzzy-valued function and denote$f(x)=[{\underset{\xaf}{f}}_{\alpha}(x),{\overline{f}}_{\alpha}(x)]$, for each α ∈ [0, 1]. Then:

- if f is (1)-differentiable, then${\underset{\xaf}{f}}_{\alpha}(x)$ and${\overline{f}}_{\alpha}(x)$ are differentiable functions and${f}^{\prime}(x)=[{{\underset{\xaf}{f}}^{\prime}}_{\alpha}(x),{{\overline{f}}^{\prime}}_{\alpha}(x)]$,
- if f is (2)-differentiable, then${\underset{\xaf}{f}}_{\alpha}(x)$ and${\overline{f}}_{\alpha}(x)$ are differentiable functions and${f}^{\prime}(x)=[{{\overline{f}}^{\prime}}_{\alpha}(x),{{\underset{\xaf}{f}}^{\prime}}_{\alpha}(x)]$.

## 3. Fuzzy Sumudu Transform

**Definition 10.**Let f : ℝ → $\mathcal{F}$(ℝ) be a continuous fuzzy-valued function. Suppose that f(ux) ⊙ e

^{−x}is improper fuzzy Riemann-integrable on [0, ∞), then${\int}_{0}^{\infty}f(ux)}\odot {e}^{-x}dx$ is called the fuzzy Sumudu transform and is denoted by:

#### 3.1. Duality Properties of the Fuzzy Laplace and Fuzzy Sumudu Transform

**Definition 11.**[35] Let f(x) be a continuous fuzzy-valued function. Suppose that f(x)⊙e

^{−px}is improper fuzzy Riemann-integrable on [0, ∞), then${\int}_{0}^{\infty}f(x)\odot {e}^{-px}dx$ is called the fuzzy Laplace transform and is denoted by:

**Theorem 3.**Let f(x) be a continuous fuzzy-valued function. If F is the fuzzy Laplace transform of f(x) and G is the fuzzy Sumudu transform of f(x), then:

**Proof.**Let f(x) ∈ $\mathcal{F}$(ℝ), then for −τ

_{1}< u < τ

_{2},

**Corollary 1.**Let f(x) ∈ $\mathcal{F}$(ℝ), having F and G for the fuzzy Laplace transform and fuzzy Sumudu transform, respectively. Then:

#### 3.2. Fundamental Theorems and Properties of the Fuzzy Sumudu Transform

**Theorem 4.**Let f, g : ℝ → $\mathcal{F}$(ℝ) be two continuous fuzzy-valued functions. Suppose that c

_{1}and c

_{2}are arbitrary constants, then:

**Proof.**Assume that $f(x)=[{\underset{\xaf}{f}}_{\alpha}(x),{\overline{f}}_{\alpha}(x)]$ and $g(x)=[{\underset{\xaf}{g}}_{\alpha}(x),{\overline{g}}_{\alpha}(x)]$. First, we proof for the lower bound of f(x) and g(x).

**Theorem 5.**Let f : ℝ → $\mathcal{F}$(ℝ) be a continuous fuzzy-valued function and a an arbitrary constant, then:

**Proof.**From Definition 10,

**Theorem 6.**Let f : ℝ → $\mathcal{F}$(ℝ) be a continuous fuzzy-valued function, then:

**Proof.**From the definition of FST, $G(u)={\displaystyle {\int}_{0}^{\infty}f(ux)\odot {e}^{-x}dx}$. Then, for Case 1 in Theorem 2,

**Theorem 7.**Let f : ℝ → $\mathcal{F}$(ℝ) be a continuous fuzzy-valued function and f the primitive of f′ on [0, ∞). Then:

**Proof.**First, we assume f is (1)-differentiable. Therefore,

**Theorem 8.**Let f : ℝ → $\mathcal{F}$(ℝ) be a continuous fuzzy-valued function and a an arbitrary constant, then:

**Proof.**From Definition 10,

**Theorem 9.**Let f, g : ℝ → $\mathcal{F}$(ℝ) be two continuous fuzzy-valued functions. Let F(p) and G(p) be fuzzy Laplace transforms, and let M(u) and N(u) be fuzzy Sumudu transforms for f and g, respectively. Then, the Sumudu transform of the convolution of f and g,

**Proof.**The FLT for (f ∗ g) as in [55] is given by:

## 4. Procedure for Solving Fuzzy Differential Equations

_{0}, T ] × ℝ → ℝ. Suppose that the initial value in Equation (5) is not precisely known and modeled with a fuzzy number, we have the following fuzzy initial value problem [56]:

_{0}, T] × $\mathcal{F}$(ℝ) → $\mathcal{F}$(ℝ) is a continuous fuzzy mapping. By referring to Kaleva [57], we observe that Theorem 2 provides a procedure to solve the Equation (6). As a matter of fact, ${Y}_{\alpha}(t)=[{\underset{\xaf}{y}}_{\alpha}(t),{\overline{y}}_{\alpha}(t)]$.

## 5. A Numerical Example

**Example 1.**Consider the following initial value problem:

**Remark.**From the example, we notice that the solutions depend on the differential equation we chose. For Case 1, the solution has the property that the diameter, i.e. diam(supp y(t)) = 2ae

^{t}, which is unbounded as t approaches infinity. Comparing to Case 2, the diam(supp y(t)) = 2ae

^{−t}→ 0 as t approaches infinity, which leads to much more intuitive results.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Rahman, N.A.A.; Ahmad, M.Z.
Applications of the Fuzzy Sumudu Transform for the Solution of First Order Fuzzy Differential Equations. *Entropy* **2015**, *17*, 4582-4601.
https://doi.org/10.3390/e17074582

**AMA Style**

Rahman NAA, Ahmad MZ.
Applications of the Fuzzy Sumudu Transform for the Solution of First Order Fuzzy Differential Equations. *Entropy*. 2015; 17(7):4582-4601.
https://doi.org/10.3390/e17074582

**Chicago/Turabian Style**

Rahman, Norazrizal Aswad Abdul, and Muhammad Zaini Ahmad.
2015. "Applications of the Fuzzy Sumudu Transform for the Solution of First Order Fuzzy Differential Equations" *Entropy* 17, no. 7: 4582-4601.
https://doi.org/10.3390/e17074582