### Abstract

This paper reveals a computational method based using a tau method with Jacobi polynomials
for the solution of fuzzy linear fractional differential equations of order

**PACS Codes: **
02, 02.30.Jr, 02.60.-x, 45.10.Hj.

##### Keywords:

fuzzy fractional differential equation; Caputo-type fuzzy fractional derivative; single-term Caputo fractional differential equation; Jacobi polynomials; operational matrix### 1 Introduction

Recently, the enormous number of applications in the field of fractional calculus and fractional differential equations has been visualized. Fractional differential equations provide an outstanding instrument to describe the complex phenomena in fields of viscoelasticity, electromagnetic waves, diffusion equations and so on [1-5]. Moreover, the fractional order models of real systems are more sufficient in comparison with the integer order cases. Therefore, the field of fractional calculus has motivated the interest of researchers in various fields like physics, chemistry, engineering and even finance [6-10].

Finding a high accurate and efficient numerical method has become a significant research due to except for a few number of these equations, there exists difficulty to find the exact solution of fractional differential equations (FDEs). Consequently, various numerical methods have appeared to approximate reasonably the analytical solutions. These methods are such as the predictor corrector method [11], Adomian decomposition method (ADM) [12-15], variational iteration method (VIM) [16,17] and homotopy analysis method (HAM) [18,19].

Orthogonal functions have received noticeable consideration in dealing with various
problems. The main advantage behind the approach using this method is that it reduces
these problems to those of solving a system of algebraic equations leading to simplify
the original problem clearly. Saadatmandi and Dehghan [20] presented a shifted Legendre tau method with an operational matrix for the numerical
solution of a multilinear and nonlinear fractional differential equation. Esmaeili
*et al.*[21] introduced a direct method using the collocation method and Müntz polynomials for
the solution of FDEs. Consequently, the operational matrix of the other orthogonal
polynomials has been derived for solving FDEs with boundary conditions and initial
conditions, like Chebyshev polynomials [22,23], Laguerre series [24], fractional Legendre polynomials [25], generalized hat basis functions [26] and Jacobi polynomials [27,28].

The study of fuzzy differential equations (*e.g.*, in this contribution, we consider fuzzy fractional differential equation) creates
a suitable setting for mathematical modeling of real-world problems in which uncertainties
or vagueness penetrate. A comprehensive approach to this kind of equations has been
considered by Seikkala [29] and Kaleva [30]. Despite the vast applications of the H-derivative introduced by them, due to an
important drawback in this kind of derivative, Bede and Gal [31] introduced strongly generalized differentiability and followed up by the authors
in [32,33]. Actually, strongly generalized differentiability can be applied for a more enormous
class of fuzzy differential equations than Hukuhara differentiability.

Recently, some attempts have been made for solving fuzzy fractional differential equations
(FFDEs) that Agarwal *et al.* was a pioneer [34]. They considered the solution of FFEDs under Riemann-Liouville’s differentiability.
Also, Salahshour *et al.*[35] studied the existence, uniqueness and approximate solutions of (FFDEs) under Caputo’s
H-differentiability. Afterward, Mazandarani, Vahidian Kamyad [36] applied the fractional Euler method for FFDEs under Caputo-type differentiability
and Salahshour *et al.*[37] extended fuzzy Laplace transforms for solving FFDEs under the Riemann-Liouville H-derivative.

Our main motivation for preparing this paper is to generalize shifted Jacobi function
operational matrix for solving fuzzy fractional differential equations of order
*α* and *β*. Therefore, instead of using with particular indexes, the solution can be derived
generally to extend for other requests.

The paper organized as follows. In Section 2, we present some relevant properties of fuzzy sets, fuzzy differential equations and Jacobi polynomials with its error bound for approximate function accompanied by some details of JOM based on shifted Jacobi polynomials in crisp concept. Also, Caputo type derivative definition and its properties in the crisp sense is considered in this section. Some basic concepts of fuzzy fractional derivatives are explained in Section 3. Section 4 is devoted to the fuzzy approximation function using shifted Jacobi polynomials. Additionally, the Jacobi operational matrix (JOM) based on shifted Jacobi polynomials is extended for solving FFDEs in this section. Several examples are experienced to depict the effectiveness of the proposed method in Section 5. Finally, some conclusions are drawn in Section 6.

### 2 Preliminaries

Let us denote by
*u* of the real axis ℝ (*i.e.*

(i) *u* is upper semicontinuous,

(ii) *u* is fuzzy convex, *i.e.*,

(iii) *u* is normal, *i.e.*,

(iv)
*u*, and its closure

Then
*e.g.*, [38]).

The *α*-level set of a fuzzy number

It is clear that the *r*-level set of a fuzzy number is a closed and bounded interval

ℝ can be embedded in

The addition and scaler multiplication of fuzzy number in

We can define a matrix *D* on

Then the following properties are known (see [38,39]):

(i)

(ii)

(iii)

(iv)

(v)

**Definition 1** ([40])

Let *f* and *g* be the two fuzzy-number-valued functions on the interval
*i.e.*,

**Remark 1** ([39])

Let

**Definition 2** ([30])

Let
*z* is called the H-difference of *x* and *y*, and it is denoted by

In this paper, the sign ‘⊖’ always stands for H-difference and note that

**Definition 3** ([41])

The generalized difference (*g*-difference for short) of two fuzzy numbers

**Proposition 1** ([41])

*For any fuzzy numbers*
*the**g*-*difference*
*exists and it is a fuzzy number*.

In this paper, we consider the definition of fuzzy differentiability presented by Bede and Gal in [32].

**Definition 4** ([32])

Let
*f* is strongly generalized differential at

(i) for all
*d*)

(ii) for all
*d*)

(iii) for all
*d*)

(iv) for all
*d*)

**Remark 2***f* is so-called (1)-differentiable on
*f* is differentiable in the sense (i) of Definition 4 and also *f* is (2)-differentiable on
*f* is differentiable in the sense (ii) of Definition 4.

The following theorem was proved by Chalco-Cano and Román-Flores [42] based on Definition 4.

**Theorem 1** (see [42])

*Let*
*be a function and denote*
*for each*
*Then*:

(1) *If**F**is* (1)-*differentiable*, *then*
*and*
*are differentiable functions and*

(2) *If**F**is* (1)-*differentiable*, *then*
*and*
*are differentiable functions and*

**Theorem 2** (see [43])

*Let*
*be a fuzzy*-*valued function on*
*and it is represented by*
*For any fixed*
*assume* (
*and*
*are Riemann*-*integrable on*
*for every*
*and assume there are two positive*
*and*
*such that*
*and*
*for every*
*Then*
*is improper fuzzy Riemann*-*integrable on*
*and the improper fuzzy Riemann*-*integral is a fuzzy number*. *Furthermore*, *we have*

**Definition 5** ([39])

*f**fuzzy-Riemann integrable* to

where ∑^{∗} means addition with respect to ⊕ in

We also call an *f* as above

**Definition 6** ([44])

Consider the

The matrix form of the above equations is

where the coefficient matrix

**Definition 7** ([44])

A fuzzy number vector

If for a particular *k*,

To solve fuzzy linear systems, one can refer to [44,45].

Now, we review some basic definitions of fractional integral and derivative, especially Caputo type, with their properties presented in crisp context [6,46].

**Remark 3** The fuzzy fractional derivative, in this paper, is assumed in the Caputo sense. The
reason for adopting the Caputo definition, as pointed by Momani and Noor [47], is as follows: to solve differential equations (both classical and fractional),
we need to specify additional conditions in order to produce a unique solution. Therefore,
for the case of the fuzzy Caputo fractional differential equations, these additional
conditions are just the traditional conditions, which are akin to those of classical
fuzzy differential equations, and are therefore familiar to us. In contrast, for the
fuzzy Riemann-Liouville fractional differential equations, these additional conditions
constitute certain fuzzy fractional derivatives (and/or integrals) of the unknown
solution at the initial point
*x*. These fuzzy initial conditions are not physical like in the crisp concept; furthermore,
it is not clear how such quantities are to be measured from experiment, say, so that
they can be appropriately assigned in an analysis. See more details in [35,37,48].

**Definition 8** ([46])

The Riemann-Liouville fractional integral operator of order *v*,

**Definition 9** ([6])

The Caputo fractional derivatives of order *v* is defined as

where
*m*.

For the Caputo derivative, we have:

The ceiling function
*v*, and the floor function
*v*. Also

**Definition 10** ([48])

Similar to the differential equation of integer order, the Caputo’s fractional differentiation
is a linear operation, *i.e.*,

where *λ* and *μ* are constants.

**Definition 11** ([49])

A classical (crisp) set is normally defined as a collection of elements or objects
*A*,

**Remark 4** Throughout the paper, we use the crisp context frequently, regarding to Definition 11.

#### 2.1 Jacobi polynomials

The Jacobi polynomials, denoted by

where

Also, the Jacobi polynomials can be created by means of the following recurrence formula:

for

at which
*i* is acquired by

that

Also, the shifted Jacobi polynomial can be stated by the following concise form.

**Lemma 1** ([28])

*The shifted Jacobi polynomial*
*can be obtained in the form of*

*in which*
*are*

**Lemma 2** ([28])

*For*

*in which*
*is the Beta function and stated as*

Let
*f* belonging to

where the coefficients

Realistically, only the first

that

Regarding to
*f* has a unique best approximation from

So, the following lemma provides the upper bound of approximate function

**Lemma 3** ([28])

*Let the function*
*be*
*times continuously differentiable for*
*and*
*If*
*is the best approximation to**f**from*
*then the error bound is presented as follows*:

*that*
*and*

#### 2.2 Operational matrix of Caputo’s derivative of order *v*

In this section, the Jacobi operational matrix method based on the Caputo-type fractional derivative with using shifted Jacobi polynomials is explained. Afterward, an upper bound for the absolute error between the exact and approximate values of Caputo fractional derivative operator is provided (for more details, see [27,28]).

**Lemma 4** ([27])

*Let*
*be shifted Jacobi vector defined in Eq*. (4) *and also let*
*Then*

*where*
*is*
*operational matrix of derivatives of order**v**in the Caputo sense and is defined by*:

*where*

*and*
*is given by*

*Note that in*
*the first*
*rows*, *are all zeros*.

Now, the following lemma gives us an upper bound for error estimation of Caputo derivative
operator mentioned in Lemma 4. But initially, we define the error vector

where

**Lemma 5***If the error function of Caputo fractional derivative operator for Jacobi polynomials*
*is*
*times continuously differentiable for*
*Also*
*and*
*then the error bound is gained as follows*:

*Proof* Analogously to the demonstration of Lemma 5 in [28], we can prove this lemma. □

Therefore, the maximum norm of error vector

### 3 Fuzzy Caputo-type fractional differentiability

The fuzzy fractional differentiability of order

At first, some notations are presented which are put to use throughout the remaining sections. It is easy to find these notations in the crisp sense. See [46,48].

⧫
*f* on

⧫

⧫
*n*.

⧫

**Definition 12** ([37])

Let
*f* is as follows:

To specify the fuzzy Riemann-Liouville integral of fuzzy-valued function *f* based on the lower and upper functions, the following definition is determined.

**Definition 13** ([37])

Let
*f* can be expressed by

in which

**Definition 14** ([35])

Let
*f* is said to be Caputo’s fuzzy differentiable at *x* when

where

**Definition 15** ([35])

Let

or

or

or

**Remark 5** A fuzzy-valued function *f* is

**Theorem 3** ([35])

*Let us assume that*
*then we have the following*:

*when**f**is*
*differentiable and*

*when**f**is*
*differentiable*.

**Lemma 6** ([35])

*Let*
*and*
*then the fuzzy Caputo derivative can be stated using the fuzzy fractional Riemann*-*Liouville integral operator as follows*:

*when**f**is*
*differentiable*, *and*

*when**f**is*
*differentiable*.

### 4 Extension of JOM method for FFEDs

In this section, fuzzy approximation function by means of shifted Jacobi polynomials
is derived. Moreover, the Jacobi operational matrix based of fuzzy shifted Jacobi
polynomials is introduced with details and provided the application of the method
for solving fuzzy linear fractional differential equations of order
*et al.*[27] and Kazem [28].

In [52-54], the authors established the concepts of the best approximation of fuzzy function and as an application, Lowen introduced fuzzy approximation of fuzzy function by means of Lagrange interpolation [55]. Firstly, we define the approximate fuzzy function using shifted Jacobi polynomials.

**Definition 16** For

where the fuzzy coefficients

in which
^{∗} means addition with respect to ⊕ in

**Remark 6** In practice, only the first

that the fuzzy shifted Jacobi coefficient vector

Since

**Definition 17** Let

**Theorem 4***The best approximation of a fuzzy function based on the Jacobi points exists and is
unique*.

*Proof* The proof is an immediate result of Theorem 4.2.1 in [54]. □

Now, in the following theorem, we will achieve the error bound for the fuzzy approximate
function based on shifted Jacobi polynomials. Actually, this error bound depicts that
the approximation converges to the fuzzy function

**Theorem 5***Consider the function*
*is*
*times continuously fuzzy differentiable for*
*and*
*If*
*is the best fuzzy approximation to*
*from*
*then the error bound is presented as follows*:

*that*
*and*

*Proof* It follows from Definition 1 of

in which

#### 4.1 Jacobi operational matrix

This part is devoted to the operational matrix of shifted Jacobi polynomials regarding to fuzzy Caputo’s derivative. The operational matrix play an important role in solving fractional differential equations by means of orthogonal functions [22,23,25,27,28]. Our aim in this section is to generalize this method for solving fuzzy linear fractional differential equations.

**Lemma 7***The fuzzy Caputo fractional derivative of order*
*over the shifted Jacobi functions can be acquired in the form of*

*where*
*for*
*and for*
*we have*

*Proof* It is straightforward from Section 2.1 and the Caputo derivative of

The fuzzy Caputo operational matrix based on the shifted Jacobi polynomials is expressed as well as relation (5). So, we have

where

The subsequent property of the product of two fuzzy Jacobi function vectors will also be utilized

that ^{∼}Θ is a

where

The error bound of fuzzy Caputo fractional differential operator is taken into consideration
in the next theorem for

where

**Theorem 6***Assume that the error function of fuzzy Caputo fractional derivative operator for
shifted Jacobi polynomials*
*be*
*times continuously fuzzy differentiable for*
*Additionally*,
*and*
*then the error bound is given by*

*Proof* Again using Definition 1 of

□

#### 4.2 Application of the JOM of the fractional Caputo derivative

In this section, the Jacobi operational matrix derived from the previous sections
is applied for solving linear FFDEs of order
*fuzzy residual* of the general single-term FFDEs is obtained and then using the orthogonal property
of the Jacobi polynomials, a fuzzy algebraic system is extracted, which is solved
easily to find the unknown fuzzy coefficient of the approximate solution of the problem.

Let us consider the general linear fuzzy fractional differential equation

in which
*v* and

In the following theorem, we clarify the way to find the fuzzy unknown coefficient
of the fuzzy approximate function

**Theorem 7***Let*
*and*
*then*

*Proof* Let

Then the fuzzy residual function of the problem is expressed by

hence,

□

Regarding to Definition 3 of *g*-difference, we have

or in the form of fuzzy operator, we can state

Let

where

From Eq. (21), we gain

for

where

Subsequently, replacing Eq. (10) in the initial condition of the problem (19)

from the above equation with Eq. (22),

### 5 Test problems

In this part, different examples are considered to depict the feasibility of the proposed method for solving FFDEs with a suitable accuracy.

**Example 1**

Consider the following FFDE:

Here, suppose that

and

where

By applying the technique explained in Section 4, the equation is gained in the matrix form as

where the values of vector

Also for

Finally, Eqs. (31) and (32) create

With

and with the assumption that

So, considering these two matrices and substituting them into Eqs. (30) and (32), we can obtain the fuzzy coefficients as

From Table 1, we can obtain a good approximation with the exact solution by making use of the
proposed method. In this table, the results are gained at
*α*, *β* which are depicted in Figure 1. The results is more accurate with
*m* grows. Finally, the approximate fuzzy solution is illustrated in Figure 3 for different values of *v* that shows this approach can solve FFDEs of different fractional order effectively.

**Figure 1.** **The absolute error for different values**
**of Example 1,**
**,**
**.**

**Figure 2.** **The absolute error for different values***m***of Example 1,**
**,**
**.**

**Figure 3.** **The fuzzy approximate solution of Example 1, for different fractional orders***v***,**
**,**
**.**

**Table 1.** **The absolute error of the proposed method for Example 1 with different values of***α***,***β***and**

**Example 2** Consider the inhomogeneous linear fractional relaxation equation in [58] in the sense of fuzzy context, so we have

in which
*v*.

Now, utilizing the definition of

and

Solving Eqs. (34)-(35) causes to specify the solution of FFDE (33) as follows:

Now, if we apply the technique explained in Section 4 in Eqs. (34) and (35) with

so if we consider

The absolute error for some various *α*, *β* at

**Figure 4.** **The absolute error for different values**
**of Example 2,**
**,**
**.**

**Figure 5.** **The absolute error for different values***m***of Example 2,**
**,**
**,**
**.**

**Figure 6.** **The fuzzy approximate solution of Example 2, for different fractional orders***v***,**
**,**
**,**
**.**

**Table 2.** **The absolute error of the proposed method for Example 2 with different values of***α***,***β***and**

**Remark 7** Figure 2 depicts that the approximate solution has a little bit oscillation when the number
of Jacobi functions assumed
*r*-cuts tend to 1. This has lead to the growing of the absolute error which is not significant.
This defect is removed with the increasing of the number of Jacobi functions which
is obvious according to Figures 1 and 2.

On the other hand, taking into account Figures 4 and 5. The approximate of lower fuzzy function using JOM method for

as it can be seen, the approximate solution for lower bound has the negotiable coefficients
for *x*,

**Example 3** Let us consider the fractional oscillation equation [59] with fuzzy initial conditions as

where

Again, regarding to the case (i) of Definition 15 and Theorem 1, one can determine the parametric form of (36) as

and

with the exact solution as

With exploiting of the presented method in Section 5, we can obtain following fuzzy equations system:

then multiplying this system by