Representation of solution of initial value problem for fuzzy linear multi-term fractional differential equation with continuous variable coefficient

1.   Introduction

Nowadays the fractional differential equations (FDEs) are powerful tools representing many problems in various fields such as viscoelasticity, control engineering, diffusion processes, signal processing and electromagnetism.

Recently many researchers are interested in research on fuzzy fractional differential equations(FFDEs). In one of the earliest papers, Agarwal et al. ^[1] considered a fractional differential equation of order $\alpha\in$ with uncertainty and introduced the concept of solution for proposed equations. Arshad and Lupulescu ^[2] presented the existence and uniqueness of solution for fractional differential equations with uncertainty using the concept of solution in means of ^[1]. Next, Khastan et al.^[3] proved the existence result of solution for nonlinear fuzzy fractional differential equations with the Riemann-Liouville derivative by Schauder fixed-point theorem. Allahviranloo et al. ^[4] proved that uncertain fractional differential equation is equivalent to integral equation under Riemann-Liouville H-differentiability and obtained the explicit solution in case of the fuzzy linear fractional differential equations. Salahshour et al. ^[5] studied the analytical methods to solve the FFDE under the fuzzy Caputo fractional differentiability.

Abdollahi et al. introduced the linear fuzzy Caputo-Fabrizio fractional differential equation with initial value problems and presented the general form of their solutions under generalized Hukuhara differentiability in ^[6] and H. V. Ngo et al. proved that the fractional fuzzy differential equation is not equal to the fractional fuzzy integral equation in general in ^[7]. Salahshour et al. ^[8] introduced that the solutions of FFDEs of order $0 < \beta < 1$ are obtained by fuzzy Laplace transforms under Riemann-Liouville H-differentiability. Chehlabi and Allahviranloo ^[9] investigated the concreted solutions for fuzzy linear fractional differential equations of order $0 < \alpha < 1$ under Riemann-Liouville H-differentiability by using fractional hyperbolic functions and extended the results obtained in ^[8]. Also Arqub et al. introduced the exact and the numerical solutions of various fuzzy differential equations based on the reproducing kernel Hilbert space method under strongly generalized differentiability ^[10,11,12].

Generally, the most of the FFDEs have no known exact solution. Thus the researches of approximate and numerical solutions of the FFDEs are important. Mazandarani and Vahidian Kamyad ^[13] obtained a fuzzy approximate solution to solve FFDEs with the Caputo-type fuzzy fractional derivative based on Hukuhara difference and strongly generalized fuzzy differentiability by using modified fractional Euler method. Also the operational matrix methods based on orthogonal polynomials have been proposed to solve FFDEs ^{[14,15,16,17,18]}.

Based on above considerations, we are going to obtain the representation of solutions of the initial value problems of fuzzy linear multi-term in-homogeneous fractional differential equations with continuous variable coefficients.

The paper is organized as follows. In Section 2, we introduced some definitions and properties for fuzzy calculus and proposed basic results of our paper. The representation of solution of proposed problem is described in Section 3. The examples are presented to illustrate our result in Section 4. Finally, the conclusion is summarized in Section 5.

2.   Preliminaries and basic results

We denote the set of all fuzzy numbers on ${\bf{R}}$ by ${{\bf{R}}_{\rm{F}}}$. A fuzzy number is a mapping $u :{\bf{R}} \to [0, 1]$ with the following properties:

(ⅰ) $u$ is normal, i.e., $\exists {x_0} \in {\bf{R}} \; \; u ({x_0}) = 1$,

(ⅱ) $u$ is a convex fuzzy subset, i.e., $u (\lambda x + (1 - \lambda)y) \ge \min \{u (x), u (y)\}, \forall x, y \in {\bf{R}} \; \; \forall \lambda \in [0, 1]$,

(ⅲ) $u$ is upper semi-continuous on ${\bf{R}}$,

(ⅳ) The set $\overline {{\rm{supp(}}u {\rm{)}}}$ is compact in ${\bf{R}}$ (where ${\rm{supp}}(u): = \{ x \in {\bf{R}}|u (x) > 0\}$).

Then ${{\bf{R}}_{ {\rm F} }}$ is called the space of fuzzy numbers.

We denote the $r$-level form of the fuzzy number $u \in {{\bf{R}}_{ {\rm F} }}$ by ${[u]^r}: = [{u_1}(r), \; \; {u_2}(r)]$, $0 \leq r \leq 1$.

Let's $u, {\rm{ }}v \in {{\bf{R}}_{\rm{F}}}$. The distance $d:{{\bf{R}}_{\rm{F}}} \times {{\bf{R}}_{\rm{F}}} \to {{\bf{R}}_ + }$ is defined by

We introduce following notations to understand our paper.

${C}(I)$ is the set of all continuous real functions on $I$.

${C^F}(I)$ is the set of all continuous fuzzy-valued functions on $I$.

$A{C^F}(I)$ is the set of all absolutely continuous fuzzy-valued functions on $I$.

${L^F}(I)$ is the space of all Lebesque integrable fuzzy-valued functions on $I$, where $I = [0, L]$ and without losing generality, we promise that $L = 1$.

Definition 2.1. ^[4] Let $x, y \in {{\bf{R}}_{\rm{F}}}$. If there exists $z \in {{\bf{R}}_{\rm{F}}}$ such that $x = y \oplus z$, then $z$ is called the H-difference of $x$ and $y$, and it is denoted by $z = x {\ominus_H}y$.

Definition 2.2. ^[8] Let $f:I \to {{\bf{R}}_{\rm F}}$ and fix ${x_0} \in (0, 1)$.

We say that $f$ is (1)-differentiable at ${x_0}$, if for all $h > 0$ sufficiently near to $0$, $f({x_0} + h){ \ominus _H}f({x_0})$, $f({x_0}){ \ominus _H}f({x_0} - h)$ and the relation

exists. Then the limit is defined by $D^{(1)}f({x_0})$.

Definition 2.3. ^[4] Let $f \in {L^F}(I)$. The fuzzy Riemann-Liouville integral of fuzzy-valued function $f$ is defined as follow

where $0 < \nu \leq 1$.

Definition 2.4. ^[4,7] Let $f:I \to {{\bf{R}}_{\rm F}}, 0 < \beta < 1$. We say that $f$ is fuzzy Riemann-Liouville H-differentiable of order $\beta$ if

is (1)-differentiable. Then fuzzy Riemann-Liouville H-derivative of order $\beta$ of function $f$ is denoted ${}^{RL}D_{0 + }^\beta f(x): = D_{}^{(1)}I_{0 + }^{1 - \beta }f(x)$.

Definition 2.5. Let $f:I \to {{\bf{R}}_{\rm F}}, 0 < \beta < 1$. We say that $f$ is fuzzy Caputo-type differentiable of order $\beta$ if H-difference $f(x){ \ominus _H}f(0)$ exists and $I_{0 + }^{1 - \beta }({f(x){ \ominus _H}f(0)}) \in A{C^F}(I)$ satisfies. Then fuzzy Caputo-type derivative of order $\beta$ of function $f$ is denoted by

We consider initial value problem of fuzzy multi-term fractional differential equation as

Definition 2.6. Let $y:I \to {{\bf{R}}_{\rm{F}}}$. $y$ is called the solution of Eq. (2.1) if $y$ is to be ${}^cD_{{\rm{0}} + }^\alpha y(x), {}^cD_{{\rm{0}} + }^\beta y(x) \in {C^F}(I)$ and satisfies Eq. (2.1).

Now let consider as follows

Theorem 2.1. Let $f$ of Eq. (2.1) be a continuous function with respect to every variable. If $y(x)$ is the solution of Eq. (2.1), the fuzzy-valued function $z(x)$ which is constructed by $z(x): = {}^cD_{0 + }^\alpha y(x)$ is the solution in ${C^F}(I)$ of fuzzy integral equation as follows

Conversely if $z(x)$ is the solution in ${C^F}(I)$ of fuzzy integral equation (2.3), $y(x)$ which is constructed by

is the solution of Eq. (2.1).

Let use the following distance structure in ${C^F}(I)$

$({{C^F}(I), {d^*}})$ is a complete metric space. For any $k$, let consider distance as

Then the distance $d_k^*$ is equal to the distance $d_{}^*$. Namely

Theorem 2.2. Assume that the function in Eq. (2.3) is continuous in its all variables and especially, for $\forall {y_1}, {y_2}, {z_1}, {z_2} \in {{\bf{R}}_{\rm F}}$ and $f$ satisfies the following condition

Then the fuzzy integral equation (2.3) has the unique solution.

3.   Main result

We consider the representation of solution of initial value problem for fuzzy linear multi-term fractional differential equation with continuous variable coefficient as

where $a, b \in C(I), g \in {C^F}(I)$.

First we will obtain the representation of solution for corresponding fuzzy integral equation (2.3).

If $z(x): = {}^cD_{0 + }^\alpha y(x)$, the following relations satisfy

Then we have

Now let define the operator $L$ by

From $a, b \in C(I)$, it is obvious that $L:{C^F}(I) \to {C^F}(I)$. By using the operator $L$, the integral equation (3.2) can be expressed as

Let $\hat g(x): = g(x) \oplus b(x){y_0}$. Then the integral equation (3.2) is denoted by

Also let denote the operator $\mathcal{I} { \ominus _H}L$ as

where the operator $\mathcal{I}$ is identity operator. From Eq. (3.3), we have

Lemma 3.1. Let $a, b, c, d \in {{\bf{R}}_{\rm{F}}}$. The following relations are satisfied

(1) If $(a \oplus b){ \ominus _H}b$ exists, then $(a \oplus b){ \ominus _H}b = a$.

(2) If $(a \oplus b){ \ominus _H}(c \oplus d), (a{ \ominus _H}c)$ and $(b{ \ominus _H}d)$ exist, then

Proof. First we prove (1). Now let $E: = (a \oplus b){ \ominus _H}b$. Then we get

Also for $\forall r \in [0, 1]$, we have ${[E]^r} = [E_-^r, E_ + ^r], {[a]^r} = [a_-^r, a_ + ^r]$ and ${[b]^r} = [b_-^r, b_ + ^r]$. Therefore following relations holds

Namely $E = a$.

We prove (2). Let $E: = (a \oplus b){ \ominus _H}(c \oplus d)$, $F: = (a{ \ominus _H}c) \oplus (b{ \ominus _H}d)$. As $E \oplus (c \oplus d) = a \oplus b$ stands, for $\forall r \in [0, 1]$, we get

Therefore

In the one hand, let $F: = (a{ \ominus _H}c) \oplus (b{ \ominus _H}d), {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} G: = a{ \ominus _H}c$. Because $G \oplus c = a$ stands, we have

Therefore the proof is completed.

If $z \in {C^F}(I)$ is the solution of the fuzzy integral equation (3.3) and $D(\mathcal{I}{ \ominus _H}L)$ is the domain of the operator $\mathcal{I}{ \ominus _H}L$, we can think the following Lemma.

Lemma 3.2. If $z \in D(L)$ is the solution of the integral equation (3.3) and

Then the following relations are satisfied

(1) $\forall k \in N, {L^k}z \in D(\mathcal{I}{ \ominus _H}L)$,

(2) $\sum\limits_{k = 0}^n {{L^k}} z \in D({\mathcal{I}{ \ominus _H}L})$,

where $D(L)$ is the domain of the operator $L$.

Proof. First we prove (1). Because $z \in D(L)$ is the solution of the integral equation (3.3), the following relations are satisfied

Consequently we have $Lz \in D(\mathcal{I}{ \ominus _H}L)$. Similarity we can prove

Next let prove (2). By the assumption of $z$, we get

Also the following relation is satisfied

Therefore we have

In the same way, we can prove $\sum\limits_{k = 0}^n {{L^k}} z \in D(\mathcal{I}{ \ominus _H}L)$.

For $z \in D(\mathcal{I}{ \ominus _H}L)$, we can know that

where the symbol $\circ$ means that the operator $(\mathcal{I}{ \ominus _H}L)$ is applied to $\sum\limits_{k = 0}^n {{L^k}} z(x)$.

Because $z \in D(\mathcal{I}{ \ominus _H}L)$, $z(x){ \ominus _H}Lz(x)$ exists. Also applying the operator $L$ to the both of Eq. (3.4), we get

Similarity, we can obtain

On the other hand, from (2) of Lemma 3.1, we get

Namely

Let estimate $d_k^*({\hat 0, Lz})$ for $\forall k > 0$. Since

and

we have

If there is a positive number ${k_*}$ satisfying

then we have

Therefore

Now let denote as

Then we get

Therefore

From Eq. (3.7), we have

Namely

Consequently from Eq. (3.6), we hold

Similarity from Eq. (3.4), Since

we get

The following Theorem gives the representation of solution for the proposed problem (3.1).

Theorem 3.1. If $\forall k \in \{ 0, {\rm{ }}1, {\rm{ }} \cdots \}, {L^k}\hat g \in D(L)$ holds, the initial value problem (3.1) has the representation of the solution as follows

Proof. Eq. (3.8) can be rewritten as

Therefore we obtain

The proof is completed.

4.   Examples

We consider the analytical representation of solution for the fuzzy fractional differential equation as following

By using Theorem 3.3, we get $\hat g(x): = \lambda {y_0}$, $(Lz)(x): = \lambda I_{0 + }^\alpha z(x), \; \; x \in I$. Therefore the conditions of Theorem 3.1 are satisfied.

Corollary 4.1. The initial value problem (4.1) have the solution as follows

in case of $\lambda > 0$,

in case of $\lambda < 0$,

Proof. In case of $\lambda > 0$, we get

Also in case of $\lambda < 0$, we have

Next we consider the representation of solution for initial value problem of the non-homogeneous fuzzy fractional differential equation with constant coefficient as following

Let's denote $\hat g(x): = g(x) \oplus \lambda {y_0}$, $(Lz)(x): = \lambda I_{0 + }^\alpha z(x), x \in I$ in order to use Theorem 3.1.

Corollary 4.2. If $\forall k \in \{ 0, 1, \cdots \}, {L^k}g \in D(L)$, then initial value problem (4.2) have the solution as follows

in case of $\lambda > 0$,

in case of $\lambda < 0$,

Proof. In case of $\lambda > 0$, we get

And in case of $\lambda < 0$, we obtain

5.   Conclusion

In this manuscript, we have studied the representation of the solution for initial value problem for fuzzy linear multi-term fractional differential equations with continuous variable coefficients. We obtained the representation of solutions for proposed problem by using the representation of solution of the corresponding fuzzy integral equations.

Acknowledgments

The authors thank the referees for careful reading and helpful suggestions on the improvement of the manuscript.

Conflict of interest

The authors declare no conflict of interest.