# On Analytical Solutions of the Fractional Differential Equation with Uncertainty: Application to the Basset Problem

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- u is upper semi-continuous,
- u is fuzzy convex, i.e., for all x, y ∈ R, λ ∈ [0, 1]: u(λx + (1 – λ)y) ≥ min{u(x), u(y)},
- 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 cl(supp u) is compact.

**Definition 1.**A fuzzy number u in parametric form is a pair ( $\underset{\xaf}{u}$, $\overline{u}$) of functions$\underset{\xaf}{u}$(r), $\overline{u}$(r), 0 ≤ r ≤ 1, where the following requirements hold:

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

**Definition 2.**Let E be the set of all fuzzy numbers on R. The r-level set of a fuzzy number u ∈ E, 0 ≤ r ≤ 1, denoted by [u]

^{r}, is defined as:

^{r}and$\overline{u}$(r) denotes the right-hand endpoint of [u]

^{r}.

- d(u ⊕ w, v ⊕ w) = d(u, v), ∀u, v, w ∈ E,
- d(k ⊙ u, k ⊙ v) = |k|d(u, v), ∀k ∈ R, u, v ∈ E,
- d(u ⊕ v, w ⊕ e) ≤ d(u, w) + d(v, e), ∀u, v, w, e ∈ E,
- (E, d) is a complete metric space.

**Definition 3.**A mapping f : R × E → E is continuous at point (x

_{0}, y

_{0}) ∈ R × E, provided that for any fixed r ∈ [0, 1] and arbitrary ϵ > 0, there exists an δ(ϵ, r), such that:

_{0}| < δ(ϵ, r) and d([y(x)]

^{r}, [y

_{0}]

^{r}) < δ(ϵ, r) for all x ∈ R and y ∈ E.

**Theorem 1.**(see [46]) Let f(x) be a fuzzy-valued function on [a, ∞), and it is represented by$(\underset{\xaf}{f}(x;r),\overline{f}(x;r))$. For any fixed r ∈ [0, 1], assume$\underset{\xaf}{f}(x;r)$ and$\overline{f}(x;r)$ are Riemann-integrable on [a, b] for every b ≥ a, and assume that there are two positive functions$\underset{\xaf}{M}(r)$ and$\overline{M}(r)$, such that${\int}_{a}^{b}|\underset{\xaf}{f}}(x;r)|dx\le \underset{\xaf}{M}(r)$ and${\int}_{a}^{b}|\overline{f}}(x;r)|dx\le \overline{M}(r)$ for every b ≥ a; then, f(x) is improper fuzzy Riemann-integrable on [a; ∞). The improper fuzzy Riemann-integral is a fuzzy number, and we have:

**Definition 4.**Let x, y ∈ E, such that E is the set of all fuzzy numbers on R; if there exists z ∈ E, such that x = y ⊕ z, then z is called the H-difference of x and y and is denoted as x ⊖ y. In this paper, the sign “⊖” always refers to the H-difference, unless specified otherwise, and also, note that x ⊖ y ≠ x+(−1)y.

**Definition 5.**Let f : (a, b) → E and x

_{0}∈ (a, b). f be called strongly-generalized differentiable at x

_{0}, if an element f′(x

_{0}) ∈ E exists, such that:

- for all h > 0 sufficiently small, ∃f(x
_{0}+ h) ⊝ f(x_{0}), ∃f(x_{0}) ⊝ f(x_{0}− h) and the limits (in the metric d):${\mathrm{lim}}_{h\searrow 0}\frac{f({x}_{0}+h)\ominus f({x}_{0})}{h}={\mathrm{lim}}_{h\searrow 0}\frac{f({x}_{0})\ominus f({x}_{0}-h)}{h}={f}^{\prime}({x}_{0}),$

- for all h > 0 sufficiently small, ∃f(x
_{0}) ⊖ f(x_{0}+ h), ∃f(x_{0}h) ⊖ f(x_{0}) and the limits (in the metric d):${\mathrm{lim}}_{h\searrow 0}\frac{f({x}_{0})\ominus f({x}_{0}+h)}{-h}={\mathrm{lim}}_{h\searrow 0}\frac{f({x}_{0}-h)\ominus f({x}_{0})}{-h}={f}^{\prime}({x}_{0}),$

- for all h > 0 sufficiently small, ∃f(x
_{0}+ h) ⊖ f(x_{0}), ∃f(x_{0}−h) ⊖ f(x_{0}) and the limits (in the metric d):${\mathrm{lim}}_{h\searrow 0}\frac{f({x}_{0}+h)\ominus f({x}_{0})}{h}={\mathrm{lim}}_{h\searrow 0}\frac{f({x}_{0}-h)\ominus f({x}_{0})}{-h}={f}^{\prime}({x}_{0}),$

- for all h > 0 sufficiently small, ∃f(x
_{0}) ⊖ f(x_{0}+ h), ∃f(x_{0}) ⊖ f(x_{0}− h) and the limits (in the metric d):${\mathrm{lim}}_{h\searrow 0}\frac{f({x}_{0})\ominus f({x}_{0}+h)}{-h}={\mathrm{lim}}_{h\searrow 0}\frac{f({x}_{0})\ominus f({x}_{0}-h)}{h}={f}^{\prime}({x}_{0})$ (h and −h at denominators mean$\frac{1}{h}$ and$\frac{-1}{h}$, respectively).

**Theorem 2.**(see [47]) Let f : R → E be a function, and denote$f(x;r)=[\underset{\xaf}{f}(x;r),\overline{f}(x;r)]$, for each r ∈ [0,1].

- If f is a (1)-differentiable function, then$\underset{\xaf}{f}(x;r)$ and$\overline{f}(x;r)$ are differentiable functions and${[{f}^{\prime}(x)]}^{r}=\left[{\underset{\xaf}{f}}^{\prime}(x;r),{\overline{f}}^{\prime}(x;r)\right]$,

- If f is a (2)-differentiable function, then$\underset{\xaf}{f}(x;r)$ and$\overline{f}(x;r)$ are differentiable functions and${[{f}^{\prime}(x)]}^{r}=\left[{\overline{f}}^{\prime}(x;r),{\underset{\xaf}{f}}^{\prime}(x;r)\right]$.

## 3. Caputo’s H-Differentiability

**Definition 6.**Let f : [a, b] → E, for 0 < β ≤ 1; the fuzzy Riemann–Liouville integral of fuzzy-valued function f is defined as:

**Theorem 3.**(see [37]) Let f : [a, b] → E, for 0 ≤ r ≤ 1 and 0 < β ≤ 1; the fuzzy Riemann–Liouville integral of fuzzy-valued function f is defined as:

**Definition 7.**Let f : (a,b) → E, x

_{0}∈ (a,b) and$\Phi (x)=\frac{1}{\mathrm{\Gamma}(1-\beta )}{\displaystyle {\int}_{a}^{x}\frac{f(t)dt}{{(x-t)}^{\beta}}}$. For all 0 ≤ r ≤ 1, h > 0. f(x) is called fuzzy Riemann–Liouville fractional differentiable of order 0 < β < 1 at x0, if there exists an element$\left({}^{RL}{D}_{a+}^{\beta}f\right)\phantom{\rule{0.2em}{0ex}}({x}_{0})\in E$, such that:

- $$\left({}^{RL}{D}_{a+}^{\beta}f\right)({x}_{0})=\underset{h\to 0}{\mathrm{lim}}\frac{\Phi ({x}_{0}+h)\ominus \Phi ({x}_{0})}{h}=\underset{h\to 0}{\mathrm{lim}}\frac{\Phi ({x}_{0})\ominus \Phi ({x}_{0}-h)}{h},$$

- $$\left({}^{RL}{D}_{a+}^{\beta}f\right)({x}_{0})=\underset{h\to 0}{\mathrm{lim}}\frac{\Phi ({x}_{0})\ominus \Phi ({x}_{0}+h)}{-h}=\underset{h\to 0}{\mathrm{lim}}\frac{\Phi ({x}_{0}-h)\ominus \Phi ({x}_{0})}{-h}.$$

^{RL}[(1)−β] -differentiable if it is differentiable, as in Definition 7, Case (1), and is

^{RL}[(2) − β]-differentiable if it is differentiable as in Definition 7, Case (2).

**Theorem 4.**(see [37]) Let f : (a, b) → E and x

_{0}∈ (a, b), 0 < β < 1; then, for all 0 ≤ r ≤ 1, we have the following relations:

- If f is a
^{RL}[(1) − β]-differentiable fuzzy-valued function, then:$$\left({}^{RL}{D}_{a+}^{\beta}f\right)({x}_{0};r)=\left[\left({}^{RL}{D}_{a+}^{\beta}\underset{\xaf}{f}\right)({x}_{0};r),\left({}^{RL}{D}_{a+}^{\beta}\overline{f}\right)({x}_{0};r)\right]$$

- If f(x) is a
^{RL}[(2) − β]-differentiable fuzzy-valued function, then:$$\left({}^{RL}{D}_{a+}^{\beta}f\right)({x}_{0};r)=\left[\left({}^{RL}{D}_{a+}^{\beta}\overline{f}\right)({x}_{0};r),\phantom{\rule{0.2em}{0ex}}\left({}^{RL}{D}_{a+}^{\beta}\underset{\xaf}{f}\right)({x}_{0};r)\right]$$

**Proposition 1.**(see [37,48]) Let b > 0 and J = (a, b]; we denote C(J, E) as the space of all continuous fuzzy functions defined on J. Furthermore, let f ∈ C(J, E); we say that$f\in {L}^{1}(J,E)\phantom{\rule{0.2em}{0ex}}i\phantom{\rule{0.2em}{0ex}}f\phantom{\rule{0.2em}{0ex}}f\phantom{\rule{0.2em}{0ex}}d({\displaystyle {\int}_{a}^{b}f(s)\phantom{\rule{0.2em}{0ex}}ds,}\phantom{\rule{0.2em}{0ex}}\tilde{0})<\infty $, where d is the fuzzy metric defined in Section 2.

**Definition 8.**(see [28,37]) Let f ∈ C(J, E) ∩ L

^{1}(J, E) be a fuzzy set-value function; then f is Caputo fuzzy H-differentiable at x when:

^{C}[(1) − β]-differentiable if Equation (7) holds, while f is (1)-differentiable, and f is

^{C}[(2) − β]-differentiable if Equation (7) holds, while f is (2)-differentiable.

**Theorem 5.**(see [37]) Let 0 < β < 1 and f ∈ C(J, E); then the fuzzy Caputo fractional derivative exists on (a, b), and for all 0 ≤ r ≤ 1, we have:

## 4. The Fuzzy Laplace Transforms

**Definition 9.**(see[30]) Let f(x) be a continuous fuzzy-value 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 the fuzzy Laplace transforms and can be denoted as:

**Theorem 6.**(see [30]) Let f(x), g(x) be continuous-fuzzy-valued functions; suppose that c

_{1},c

_{2}are constants, then:

**Lemma 1.**(see [30]) Let f(x) be a continuous fuzzy-value function on [0, ∞) and λ ∈ R, then:

**Lemma 2.**(see [30]) Let f be a continuous fuzzy-value function and g(x) ≥ 0 a real value function; suppose that (f(x) ⊙ g(x)) ⊙ e

^{−px}is improper fuzzy Riemann-integrable on [0, ∞); then, for fixed r ∈ [0, 1]:

**Theorem 7.**(see [30]) Let f be a continuous fuzzy value function and

**L**{f(x)} = F (p); then:

^{ax}is a real value function.

#### 4.1. Proposed Solution

**L**{f(x)}:

**Theorem 8.**Derivative theorem: Suppose that f is a continuous fuzzy-valued function on [0, ∞). Then, we have:

^{C}[(1) − β]-differentiable and:

^{C}[(2) − β]-differentiable.

**Proof.**For arbitrary fixed r ∈ [0, 1], we have:

^{C}[(1) − β],

**L**,

^{C}[(2) − β]-differentiable; then for arbitrary fixed r ∈ [0, 1], we have:

^{C}[(2) − β]-differentiable, we get:

## 5. FFDEs under Caputo’s H-Differentiability

**Case I:**We consider y(x) a

^{C}[(1) − β]-differentiable function; then Equation (18) is extended based on its lower and upper functions for 0 < β < 1 as:

_{1}(p; r) and K

_{1}(p; r) are solutions of System (19). By using the inverse Laplace transform, $\underset{\xaf}{y}(x;r)$ and $\overline{y}(x;r)$ are calculated as follows:

**L**

^{−1}as the inverse of fuzzy Laplace transform, and it is equivalent to:

**Case II**: Let us consider y(x) as

^{C}[(2) − β]-differentiable; then for 0 < β < 1, Equation (18) can be written as:

_{2}(p; r) and K

_{2}(p; r) are solutions of the system (Equation (23)). By using the inverse Laplace transform, $\underset{\xaf}{y}(x;r)$ and $\overline{y}(x;r)$ are computed as follows:

## 6. Application

**Example 1**. We analyze the following FFDE:

**Example 2**. In the following, we analyze the following FFDE,

**Case I**: We make the assumption that λ ∈ R

^{+}= (0, +∞), and after that, we use the Laplace transform and finally conclude:

^{C}[(1) − β]-differentiability, we get:

_{β,}

_{1}is the Mittag–Leffler function (see [11]).

**Case II**: Suppose that λ ∈ R

^{−}= (−∞, 0); then, using

^{C}[(2) − β]-differentiability and Theorem 8, the obtained solution will be similar to Equation (34). For a special case, let us consider β = 0.5, λ = 1 and y(0; r) = [1 + r, 3 − r]; then, the solution for Case I is derived as follows:

**Example 3**. The Basset problem: The dynamics of a sphere immersed in an incompressible viscous fluid is a classical problem with huge applications in material sciences, as well as in the study of geophysical flows. A particularly important problem is the study of a sphere subjected to gravity, which was first presented by Basset in 1888 [51] and followed in 1910 by [52], who then introduced a special hydraulic force, known as “Basset’s force”.

_{0}:

_{0}, the fuzzy-valued function g(x) and the concept of Caputo’s H-differentiability for the fractional derivative of y(x), $\left({}^{C}{D}_{{0}^{+}}^{\beta}y\right)$ and the generalized H-differentiability [32] for the first order derivative of y(x), y′(x). Let us consider the fuzzy version of the dynamics of a sphere immersed in an incompressible viscous fluid (Basset’s problem) as follows:

^{C}[(1)−β]-differentiability, we apply the fuzzy Laplace transform to both sides of Equation (37), which leads to:

^{C}[(1) − β]-differentiability, we get:

## 7. Conclusion and Future Works

^{C}[(1) − β] Caputo H-differentiability, Equations (24) and (25) under

^{C}[(2) − β] Caputo H-differentiability). Experimental results using some real-world problems (nuclear decay Equation (30) and Basset problem (38)) illustrated the effectiveness and applicability of the proposed method.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Salahshour, S.; Ahmadian, A.; Senu, N.; Baleanu, D.; Agarwal, P.
On Analytical Solutions of the Fractional Differential Equation with Uncertainty: Application to the Basset Problem. *Entropy* **2015**, *17*, 885-902.
https://doi.org/10.3390/e17020885

**AMA Style**

Salahshour S, Ahmadian A, Senu N, Baleanu D, Agarwal P.
On Analytical Solutions of the Fractional Differential Equation with Uncertainty: Application to the Basset Problem. *Entropy*. 2015; 17(2):885-902.
https://doi.org/10.3390/e17020885

**Chicago/Turabian Style**

Salahshour, Soheil, Ali Ahmadian, Norazak Senu, Dumitru Baleanu, and Praveen Agarwal.
2015. "On Analytical Solutions of the Fractional Differential Equation with Uncertainty: Application to the Basset Problem" *Entropy* 17, no. 2: 885-902.
https://doi.org/10.3390/e17020885