Abstract and Applied Analysis

Volume 2013 (2013), Article ID 505903, 29 pages

http://dx.doi.org/10.1155/2013/505903

Research Article

An Operational Matrix Based on Legendre Polynomials for Solving Fuzzy Fractional-Order Differential Equations

Ali Ahmadian,1,2 Mohamed Suleiman,¹ and Soheil Salahshour³

¹Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

²Department of Mathematics, Science Faculty, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

³Young Researchers and Elite Club, Islamic Azad University, Mobarakeh Branch, P.O. Box 9189945113, Mobarakeh, Iran

Received 20 April 2013; Accepted 16 May 2013

Academic Editor: Dumitru Baleanu

Copyright © 2013 Ali Ahmadian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper deals with the numerical solutions of fuzzy fractional differential equations under Caputo-type fuzzy fractional derivatives of order . We derived the shifted Legendre operational matrix (LOM) of fuzzy fractional derivatives for the numerical solutions of fuzzy fractional differential equations (FFDEs). Our main purpose is to generalize the Legendre operational matrix to the fuzzy fractional calculus. The main characteristic behind this approach is that it reduces such problems to the degree of solving a system of algebraic equations which greatly simplifies the problem. Several illustrative examples are included to demonstrate the validity and applicability of the presented technique.

1. Introduction

The subject of fractional calculus has gained considerable popularity and importance during the past three decades. Some of the most recent works on this topic, that is, the theory of derivatives and integrals of fractional (noninteger) order, are such as the book of Podlubny [1], Baleanu et al. [2], Diethelm [3], Baleanu et al. [4], and Sabatier et al. [5]. Only in the last few years, the various applications of fractional calculus have been extended in the area of physics and engineering such as the modeling of nonlinear oscillation of earthquake [6], the fluid-dynamic models [7], continuum and statistical mechanics [8], and solid mechanics [9]. In a notably enormous number of recent works, one can find the robustness upon the usefulness of fractional calculus to derive particular solutions of different kinds of classical differential equations like Bessel differential equation of general order [10, 11]. Also, the most significant advantage of applying FDEs is their nonlocal property, which interprets that the next state of a system relies not only pon its current phase but also pon all of its past records of phases [12]. For example, with the fractional differentiability, the fluid dynamic traffic model can get rid of the shortage arising from the hypothesis of continuum traffic flow [12, 13].

On the other hand, the modeling of natural phenomena is stated using mathematical tools (mathematical arithmetic, mathematical logics, etc.). However, obtaining a deterministic model of such problems is not easy, even does not occur and always has some errors and vagueness. So, investigating a popular way to interpret such vagueness is important. Since 1965 with Zadeh’s well-known paper on introducing fuzzy sets, applications of fuzzy concept to the structure of any modeling has appeared more and more, instead of deterministic case. So the topic of fuzzy differential equations (FDEs) of integer order has been rapidly growing in recent years [14–20]. Additionally, the application of various techniques has been expanded by means of the interpolations and polynomials for approximating the fuzzy solutions of fuzzy integral equations vastly, like Bernstein polynomials [21, 22], Lagrange interpolation [23, 24], Chebyshev interpolation [25], Legendre wavelets [26], and Galerkin-type technique [27].

Recently, Agarwal et al. [28] proposed the concept of solutions for the fractional differential equations with uncertainty. They have considered the Riemann-Liouville’s differentiability with a fuzzy initial condition to solve FFDEs. In [29, 30], the authors considered the generalization of H-differentiability for the fractional case. Discovering a suitable approximate or exact solution for FFDEs is a significant task which has been aroused simultaneously with the emerging of FFDES, except for a few number of these equations, and we have hardship in finding their analytical solutions. Consequently, there have been limited efforts to develop new methods for gaining approximate solutions which reasonably estimate the exact solutions. Salahshour et al. [31] considered fuzzy laplace transforms for solving FFDEs under Riemann-Liouville H-differentiability. Also Mazandarani and Kamyad [32] generalized the fractional Euler method for solving FFDEs under Caputo-type derivative.

From another point of view, several methods have been exploited to solve fractional differential equations, and fractional partial differential equations, fractional integrodifferential equations such as Adomian’s decomposition method [7], He’s variational iteration method [33], homotopy perturbation method [34], and spectral methods [35, 36]. In this way, orthogonal functions have received considerable attention in dealing with the various kinds of fractional differential equations. The main characteristic behind the approach using this technique is that it reduces these problems to those of solving a system of algebraic equations thus greatly simplifying the problem. Saadatmandi and Dehghan [36] presented the shifted Legendre operational matrix for fractional derivatives and applied it with tau method for the numerical solution of fractional differential equations subject to initial conditions. Also in [37–39], the authors derived new formulas using shifted Chebyshev polynomials and shifted Jacobi polynomials of any degree, respectively and applied them together with tau and collocation spectral methods for solving multiterm linear and nonlinear fractional differential equations.

The essential target of this paper is to recommend a suitable way to approximate FFDEs using a shifted Legendre tau approach. This strategy demands a formula for fuzzy fractional-order Caputo derivatives of shifted Legendre polynomials of any degree which is provided and applied together with the tau method for solving FFDEs with initial conditions. Up till now, and to the best of our knowledge, few methods corresponding to those mentioned previously have been devoted to solve FFDEs and are traceless in the literature for FFDEs under Caputo differentiability. This partially motivates our interest in the operational matrix of fuzzy fractional derivative of shifted Legendre polynomials. Also another motivation is based on the reality that only a few terms of expansion of the shifted Legendre function is needed to reach to a high accuracy, therefore, it does not need to implement the method frequently for finding the approximate results in each particular point.

For finding the fuzzy solution, the shifted Legendre operational matrix is generalized for the fuzzy fractional derivative () which is based on the Legendre tau method for solving numerically FFDEs with the fuzzy initial conditions. It is worthy to note here that the method based on using the operational matrix of the Legendre orthogonal function for solving FFDEs is computer oriented.

The aim of this paper is to introduce the shifted Legendre operational matrix of fuzzy fractional derivative which is based on Legendre tau method for solving FFDEs under generalized differentiability. Also, we introduce a suitable way to estimate the nonlinear fuzzy fractional initial problems on the interval , by spectral shifted Legendre collocation method based on Legendre operational matrix, to find the approximate fuzzy solution. Finally, the accuracy of the proposed algorithms is demonstrated by several test problems. We note that the two shifted Legendre and shifted Jacobi operational matrices have been introduced by Saadatmandi and Dehghan [36] and Doha et al. [39], respectively, in the crisp concept. We, therefore, motivated our interest in the shifted Legendre operational matrix in the fuzzy settings.

This paper is organized as follows: In Section 2, we begin by introducing some necessary definitions and mathematical preliminaries of the fuzzy calculus and fractional calculus. Some basic concepts, properties and theorems of fuzzy fractional calculus are presented in Section 3. Section 4 is devoted to the fuzzy Legendre functions and their properties. The shifted Legendre operational matrix of fuzzy fractional derivative for solving fuzzy fractional differential equation is obtained in Section 5. Section 6 illustrates the effectiveness of the proposed method through solving several examples which some of them are modelled based on the real phenomena. Finally, a conclusion is given in the last section.

2. Preliminaries

We give some definitions and introduce the necessary notation which will be used throughout the paper, see, for example, [40, 41]. Also for some definitions related to generalized fuzzy difference, one can find more in [42, 43].

We denote the set of all real numbers by . A fuzzy number is a mapping with the following properties:

(a) is upper semicontinuous,(b) is fuzzy convex, that is, for all ,(c) is normal, that is, for which ,(d) is the support of the , and its closure cl (supp ) is compact.

Let be the set of all fuzzy number on . The -level set of a fuzzy number , denoted by , is defined as

It is clear that the -level set of a fuzzy number is a closed and bounded interval , where denotes the left-hand endpoint of and denotes the right-hand endpoint of . Since each can be regarded as a fuzzy number defined by

can be embedded in .

The addition and scalar multiplication of fuzzy number in are defined as follows:

The metric structure is given by the Hausdorff distance ,

It is easy to see that is a metric in and has the following properties

(i), for all ,(ii), for all ,(iii), for all ,(iv), for all ,(v) is a complete metric space.

Definition 1. The property (iv) in the properties of the above metric space suggests the definition of a function that that has the properties of usual norms. In [44], the properties of this function are presented as follows:

(i), for all and if and only if ,(ii) and , for all .(iii) and .

Definition 2 (see [45]). Let and be the two fuzzy-number-valued functions on the interval , that is, . The uniform distance between fuzzy-number-valued functions is defined by

Remark 3 (see [46]). Let be fuzzy continuous. Then from property (iv) of Hausdorff distance, we can define

Definition 4 (see [42]). Let be the space of nonempty compact and convex sets of . The generalized Hukuhara difference of two sets (gH-difference for short) is defined as follows:

In case (a) of the above equation, the gH-difference is coincident with the H-difference. Thus the gH-difference is a generalization of the H-difference.

Definition 5 (see [47]). Let . If there exists such that , and then is called the H-difference of and , and it is denoted by .

In this paper, the sign “” always stands for H-difference and note that . Also throughout the paper is assumed that the Hukuhara difference and Hukuhara generalized differentiability existed.

Definition 6 (see [42]). The generalized difference (g-difference for short) of two fuzzy numbers is given by its level sets as

where the gH-difference is with interval operands and .

Proposition 7. The g-difference in Definition 6 is given by the expression

Proof. See  [42].

The next proposition gives simplified notation for and .

Proposition 8. For any two fuzzy numbers the two g-difference and exist and, for any , one with

where

and the sets are

Proof.  See [42].

The following proposition prove that the g-difference is well-defined.

Proposition 9 (see [14]). For any fuzzy numbers the g-difference exists and it is a fuzzy number.

Proof. See [42].

The following property holds for g-derivative.

Proposition 10. Let be two fuzzy numbers, and then

(i) whenever the expressions on the right exist, in particular, , (ii), (iii), (iv) if and only if . Furthermore, if and only if .

In this paper, we consider the following definition which was introduced by Bede and Gal in [14].

Definition 11 (see [14]). Let and . One says that is strongly generalized differentiable at , if there exists an element , such that

(i)for all sufficiently small,  , and the limits (in the metric )

(ii)for all sufficiently small,  , and the limits (in the metric )

(iii)for all sufficiently small,  , and the limits (in the metric )

(iv)for all sufficiently small,  , and the limits (in the metric )

Remark 12. Throughout this paper, we say that is -differentiable on , if is differentiable in the sense (i) of Definition 11 and also is -differentiable on , if is differentiable in the sense (ii) of Definition 11.

Theorem 13 (see [17]). Let be a function and denote , for each . Then

(1)if is -differentiable, then and are differentiable functions and

(2)if is -differentiable, then and are differentiable functions and

Definition 14 (see [42]). Let and . We say that is g-differentiable at , if there exists an element such that

Next we review one of the main results from Bede [15] for fuzzy initial value problem (FIVP) under -differentiability which Nieto et al. [48] generalized this results for FIVP under -differentiability (let denote the usual Euclidean norm).

Theorem 15 (see [15], characterization theorem). Let one consider the fuzzy initial value problem

where is such that

(i), (ii) and are equicontinuous (i.e., for any there is a such that and for all , whenever and and uniformly bounded on any bounded set,(iii)there exists an such that for all , for all .

Then the FIVP (20) and system of ODEs

are equivalent.

Corollary 16 (see [48]). If we consider FIVP (20) under -differentiability then the FIVP (20) and the following system of ODEs are equivalent:

Theorem 17 (see [49]). Let be a fuzzy-valued function on and it is represented by . For any fixed , assume that and are Riemann-integrable on for every , and assume that there are two positive and such that:

Then is improper fuzzy Riemann-integrable on and the improper fuzzy Riemann-integral is a fuzzy number. Further more, we have

Definition 18 (see [45]). . We say that Fuzzy-Riemann integrable to , if for any , there exists such that for any division of with the norms , we have

where means addition with respect to in ,

We also call an as above, -integrable.

Definition 19 (see [50]). Consider the linear system of equations

The matrix form of the above equations is

where the coefficient matrix , , is a crisp matrix and . This system is called a fuzzy linear system (FLS).

Definition 20 (see [50]). A fuzzy number vector given by is called a solution of the fuzzy linear system (27) if

If for a particular , , we simply get

To solve fuzzy linear systems, one can refer to [51–53].

Now we define some notations which are used for the fuzzy fractional calculus throughout the paper.

(i) is the set of all fuzzy-valued measurable functions on where .(ii) is a space of fuzzy-valued functions which are continuous on .(iii) denotes the set of all fuzzy-valued functions which are absolutely continuous.

One can easily find this definition in the crisp sense in [1, 54].

Definition 21 (see [54]). The Riemann-Liouville fractional integral operator of order , , of a function is defined as

Properties of the operator can be found in [1, 54, 55]. we refer to only the following

For ,   and ,

Definition 22 (see [54]). The Riemann-Liouville fractional derivatives of order of a function are expressed by

Definition 23 (see [55]). The fractional Caputo derivatives and on for are defined via the above Riemann-Liouville fractional derivatives by

which can be simplified as

Also, the fractional Caputo derivative can be defined in a sense of integral form described in Definition 24.

Definition 24 (see [56]). The Caputo definition of the fractional-order derivative is defined as

where is the order of the derivative and is the smallest integer greater than . For the Caputo derivative, we have

The ceiling function is used to denote the smallest integer greater than or equal to , and the floor function to denote the largest integer less than or equal to . Also and .

Definition 25 (see [1]). Similar to the differential equation of integer order, the Caputo’s fractional differentiation is a linear operation, that is,

where and are constants.

Lemma 26. Let and . Then the Caputo fractional derivatives are bounded for any and as

Proof. See [54, 57].

3. Fuzzy Caputo Fractional Derivatives

In this section, some definitions and theorems related to the fuzzy Caputo fractional derivatives are presented which are an extension of the fractional derivative in the crisp sense. The generalized differentiability should be considered to expand the concept of Caputo fractional derivatives for the fuzzy space. For more details, see [14, 30].

Definition 27. Let be a fuzzy set-value function and , and then is said to be g-Caputo fuzzy fractional differentiable at , when

where

Remark 28. A fuzzy-valued function is -differentiable, if it is differentiable as in Definition 27, Case (i), and it is -differentiable, if it is differentiable as in Definition 27, case (ii).

Theorem 29. Let and , then Caputo fuzzy fractional derivative exists almost everywhere on and for all we have

when is -differentiable, and

when is -differentiable, in which for .

Proof. It is straightforward by applying Definitions 20 and 24.

Theorem 30. Let one assume that , and then one has the following:

when is -differentiable and

when is -differentiable.

Proof. See [30].

Lemma 31. Let and , then the fuzzy Caputo derivative can be expressed by means of the fuzzy Riemann-Liouville integral as follows:

when is -differentiable, and

when is -differentiable.

Now we consider the generalization of Taylor’s formula for the fuzzy Caputo fractional derivative which was introduced in [32, 45]. It should be mentioned that this theorem is the extension of Taylor’s formula for the Caputo fractional derivative in the crisp context [58].

Theorem 32. Let and suppose that for where , and . Then we have

where .

Proof. See [32].

Now, characterization theorem (Theorem 15), which was introduced by Bede in [15] and established by Pederson and Sambandham in [59] for hybrid fuzzy differential equations, is extended for fuzzy Caputo-type fractional differential equations. To this end, we first consider the FFDEs under Caputo’s H-differentiability for as follows:

Theorem 33 (characterization theorem). Let one consider the fuzzy fractional differential equation under Caputo’s H-differentiability (50) where and such that:

(i), (ii) and are equicontinuous (i.e., for any there is a such that and for all , whenever and and uniformly bounded on any bounded set,(iii) there exists an such that , .

Then, (50) and the following system of FDEs are equivalent when is -differentiable

also (50) and the following system of FDEs is equivalent when is -differentiable

Proof. In the papers [15, 59], the authors proved for fuzzy ordinary differential equations and hybrid fuzzy differential equations. The result for FFDEs is obtained analogously by using Theorem 2 in [15] and Theorem  3.1 in [59].

4. Proposed Method for Solving FFDEs

Saadatmandi and Dehghan [36] introduced the shifted Legendre operational matrix for derivative with fractional order using a spectral method which has been followed by Doha et al. [37–39]. They presented the shifted Chebyshev polynomials and Jacobi polynomials for solving fractional differential equations by tau method. In this section, we try to approximate fuzzy solution using shifted Legendre polynomials under H-differentiability as follows.

4.1. Properties of Shifted Legendre Polynomials

The Legendre polynomials, denoted by , are orthogonal with Legendre weight function: over , namely [60],

where is the Kronecker function and can be specified with the help of following recurrence formula:

In order to use these polynomials on the interval , we define the so-called shifted Legendre polynomials by introducing the change of variable . Let the shifted Legendre polynomials be denoted by . The shifted Legendre polynomials are orthogonal with respect to the weight function in the interval with the following orthogonality property:

The shifted Legendre polynomials are generated from the following three-term recurrence relation:

The analytic form of the shifted Legendre polynomial of degree is given by

in which

where

A function of independent variable defined for may be expanded in terms of shifted Legendre polynomials as

where the shifted Legendre coefficients matrix and the shifted Legendre vector are given by

Also, the derivative of can be expressed by

where is the operational matrix of derivative given by

The graph of some shifted Legendre polynomials (for ) shown in Figure 1 to depict their behaviors.

Figure 1: The shifted Legendre functions for different .

Now, we use the shifted Legender functions due to approximate a fuzzy function.

4.2. The Approximation of Fuzzy Function

In this section, we propose a shifted Legendre approximation for the fuzzy-valued functions. To this end, we use Legendre’s nodes and fuzzy shifted Legendre polynomials to calculate the fuzzy best approximation. For more details, see [61–67].

Definition 34. For and Legendre polynomial a real valued function over , the fuzzy function is approximated by

where the fuzzy coefficients are obtained by

in which is the same as in (57), and means addition with respect to in .

Remark 35. In actuality, only the first -terms shifted Legendre polynomials are considered. So we have

hence

that the fuzzy shifted Legendre coefficient vector and shifted Legendre function vector are defined as

Definition 36 (see [68]). A fuzzy-valued polynomial is the best approximation to fuzzy function on , if

in which is the set of all fuzzy valued polynomials.

The problem is referred to as the best shifted Legendre approximation, as we use Legendre’s nodes.

Theorem 37. The best approximation of a fuzzy function based on the Legendre nodes exists and is unique.

Proof. The proof is an instantaneous outcome of Theorem   in [68].

Now, we want to show that the fuzzy approximation converges of Legendre functions to function .

Lemma 38. Suppose that and , , , and , and also assume that is -differentiable. Therefore using Theorem 32, we have

in which and .

Proof. From Theorem 32, we have

and the following relation can be obtained:

that and .

Remark 39. If we consider Lemma 38 and define , then (71) can be stated in the following form, regarding to Section 2:

Theorem 40. Let , and . Also consider a sequence of finite dimensional fuzzy space , , in which have dimension . Additionally, have a basis . If one assumes that is the best fuzzy approximation for fuzzy function from , then the error estimation is as follows:

Proof. Let be a fuzzy valued function such that . Also consider the fuzzy Taylor’s formula in Theorem 32, and . From Lemma 38 we have

From the assumption, is the best fuzzy approximation to from , and , . So one has

and thus, taking into account Theorem  1 in [69] and above relations, we have

From Remark 39 and Lemma 38, we have

and if , we get , . Therefore, from Remark 39 and the definition of Hausdorff distance in Section 2, it can be implied that

which completes the proof.

Remark 41. The same result can be obtained for under -differentiability.

4.3. Operational Matrix of Caputo Fractional Derivative

Lemma 42. The fuzzy Caputo fractional derivative of order over the shifted Legendre functions can be gained in the form of

where when and for , and so one has .

Proof. Employing the analytic form of shifted Legendre polynomials explained in Section 4.1 and Definition 25, we have:

Now, by exploiting Definition 24, the lemma can be proved.

Lemma 43. Let , and the integral of the product of the fuzzy Caputo fractional derivative with order over the shifted legendre functions can be obtained by

Proof. Using Lemma 42 and the analytic form of shifted Legendre polynomials explained in Section 4.1, we can acquire:

The operational matrix of different orthogonal functions for solving differential equations was introduced in the crisp concept [36, 37, 39]. Here, the Legendre operational matrix (LOM) in [36] is applied to the FFDEs using Caputo-type derivative.

The Caputo fractional derivatives operator of order of the vector defined in (69) can be stated by

where is the -square operational matrix of fractional Caputo’s derivative of Legendre functions. Regarding the following theorem, the LOM elements are determined under Caputo fractional derivative. This theorem is generalizing the operational matrix of derivatives of shifted Legendre given in Section 4.1 to the fractional calculus.

Theorem 44. Let be Legendre functions vector. is the -square operational matrix of fractional Caputo’s derivative of order . Then the elements of are achieved as

in which are acquired by

Consider that in , the first rows, are all zero.

Proof. Employing the relation (85) and the orthogonal properties of shifted Legendre functions (56), we have

in which and are -square matrixes defined as

Hence, applying Lemma 43 and inserting the above matrixes in the product , the theorem be proved.

Remark 45. If , then Theorem 44 gives the mentioned result as in Section 4.1.

The following property of the product of two shifted Legendre polynomials fuzzy vectors will also be applied which is the extension of the crisp case introduced in [36],

where is an -square product operational matrix for the fuzzy vector . Using the above equation and by the orthogonal property equation (56), the elements can be computed from

where denotes the fuzzy summation and indicates fuzzy multiplication. Moreover, are obtained by

that in the simpler form, it is given by

in which .

Remark 46. As a matter of fact, from (93), it is obvious that the elements of the product of shifted Legendre operational matrix are independent from the fuzzy vector.

5. Application of the LOM for Solving FFDEs

In this section, Legendre operational matrix of fractional Caputo’s derivative is considered to present its significance for solving fuzzy fractional differential equation which is alike to the concept of the Caputo-type derivative in crisp case [36]. The existence and uniqueness of the solutions under fuzzy Caputo’s type derivative to this problem are discussed in [30].

5.1. General Linear FFDEs

Let us consider the general linear fuzzy fractional differential equation

in which for are fuzzy constant coefficients, is a continuous fuzzy-valued function, and represents the fuzzy Caputo fractional derivative of order .

Remark 47. In this section, considering more simplicity, it is assumed that for the fuzzy set-valued function , we present the functions by , which are in the same previous manner, called the left and right -cut functions of .

Let and . Then from (94), we have that

Therefore,

and thus

Hence, we obtain

We can rewrite (94) in the operator form

in which the fuzzy operator is supposed to be compact on a fuzzy Banach space to [45, 70] and is the notation for g-difference.

For solving fuzzy fractional differential equation (94), we attempt to find a fuzzy function ; therefore, let , and be approximated using Definition 34 as:

that

where is obtained as

Also, using relations (85) and (100), we obtain

Substituting (100)–(103) in problem (94), the coefficients are specified by imposing the equation to be almost fuzzy exact in the Legendre operational matrix form. Now, we establish fuzzy residual for the approximation of (94), when .

Theorem 48. Let and , and then

Proof. Let . We have , and regarding to what was conversed previously,

so

Using g-difference, we have

or in the sense of fuzzy operator,

It is anticipated that the deriving fuzzy function will be a suitable approximation of the exact solution . To this end, let , and let indicate the fuzzy inner product for . It is demanded that satisfy

in which . The left side is the shifted legendre coefficients associated with . If are the main members of shifted Legendre family which is complete in , then (109) needs the main terms to be zero in the Fourier extension of with respect to which is called tau method in the crisp context and it is in a similar manner with the meaning of fuzzy (for more details, see [71–73]).

To discover , implementing (109) to (108), regarding the (100)–(103), we generate fuzzy linear equations as

for . Equation (110) generate () set of fuzzy linear equations. These fuzzy linear equations can be acquired as the following system of equations:

or

Afterwards, substitution of (100) in the initial condition of (94) yields

that this equation be coupled with the previous fuzzy linear equations and constructed fuzzy linear equations. Clearly, after solving this fuzzy system, the coefficients will be gained.

5.1.1. Error Analysis

The aim of this section is to acquire the error bound for the Legendre approximation using shifted Legendre polynomials. We consider the best shifted Legendre approximation of a smooth fuzzy function under Caputo derivative for to reach the result. It should be considered that the results of this section are the extension of this concept in the crisp context (see more in [69, 74–76]). Initially, we state the following lemma which can offer an upper bound for approximating the error of Caputo fractional derivative. So we define the error vector as

where

Now, we can propose the next lemma by using Theorem  1 in [69] to depict the error bound of Caputo fractional derivative operator for the shifted Legendre polynomials.

Lemma 49. Let the error function of Caputo fractional derivative operator for Legendre polynomials be times continuously differentiable for , . Also and and then the error bound is presented as follows:

Proof. Firstly, we acquire a bound for , regarding Definition 24, as

Then using Lemma  1 in [69], we have:

in which

Now, utilizing Theorem  1 in [69], the lemma can be proved.

Therefore, the maximum norm of error vector is attained as

Now we extend these results for the fuzzy case. Hence, the error of fuzzy Caputo fractional derivative is defined as

regarding Lemmas 38 and 49 and Theorem 40, and we have:

where ,  .

Lemma 50. Consider the error function of the fuzzy Caputo fractional derivative operator for shifted Legendre polynomials which is continuously differentiable and . Then the error bound is given as follows:

that .

Proof. It is straightforward from the definition of , Definition 1, and Remark 3.

For an error assessment of the approximation solution of (94), we assume that and reveal the fuzzy approximate and exact solutions of the fuzzy fractional differential equations, respectively. Then we rewrite (94) as

that is the fuzzy perturbation function that depends only on . By subtracting (94) from above equation and using Lemmas 38 and 50, Theorem 40, and Definition 2, one can obtain

If we substitute the error bound from Lemmas 38 and 50 and the proof of Theorem 40, then we have

or

at which , for , . Therefore, is bounded.

5.2. Nonlinear FFDEs

Consider the nonlinear FFDE

where can be nonlinear in general, is a continuous fuzzy-valued function, and indicates the fuzzy Caputo fractional derivative of order .

In order to use LOM for this problem, we first approximate and as (100) and (103), respectively. By replacing these equations in (128), we have

We intend to find the fuzzy coefficients . Also by substituting (100) in the initial condition of nonlinear FFDE (128), we have

To find the approximate fuzzy solution , we first collocate (129) at points. For appropriate collocation points we use the first shifted Legendre polynomials roots. These equations together with (130) generate nonlinear fuzzy equations which can be solved using Newton’s iterative method presented in [77].

Remark 51. The solvability of system (129) and (130) is a complicated problem and we cannot prove the existence and uniqueness of such a fuzzy solution. But in our accomplishment, we have solved this system, regarding the method in [77], using MATLAB functions. In the assumed example, these functions have prospered to gain an accurate fuzzy approximate solution of the system, even starting with a zero initial guess.

6. Numerical Examples

In this section, to demonstrate the effectiveness of the proposed method in the present paper, two different test examples are carried out. Also, the obtained numerical solutions be compared with exact solutions.

Example 52. Let us consider the following FFDE:

in which is a continuous fuzzy-valued function and denotes the fuzzy Caputo fractional derivative of order . We solve this example according to two following cases for .

Case 1. Suppose that , and then using -differentiability and Theorem 33, we have the following:

in which .

The exact solution of FFDE is as follows:

where is the classical Mittag-Leffler function

Note that for this representation is still valid since and .

Remark 53. If , then the fuzzy fractional differential equations (132) are converted to fuzzy integer-order differential equations. So the exact solution of problem under -differentiability using Theorem 15 is as follows:

By utilizing the technique explained in Section 5, (131) with becomes:

or we can rewrite it in the matrix form as

in which and for . As it was described in Section 5, we produce fuzzy algebraic equations multiplied above fuzzy residual system by for using orthogonal property, and so we obtain

Now, using the initial condition (131), we have

The above three equations generate a set of fuzzy linear algebraic equations. As a result, for and , one can gain

the unknown coefficients can be gained by substituting matrix in the previous mentioned systems.

Case 2. Suppose that , then using -differentiability and Theorem 33, the FDEs are obtained as same as (132). Also the exact solution is given by

By applying the technique described in Sections 4 and 5, namely , we may write the approximate solution for Cases 1 and 2 in the forms

So regarding the (100)–(110) and (142), the unknown parameters, , are achieved for both cases.

Remark 54. If , then the fuzzy fractional differential equation (131) with is converted to fuzzy integer-order differential equation. So the exact solution of problem under -differentiability using Corollary 16 is as follows:

The approximate solution, exact solution and absolute errors are depicted for Case 1 of Example 52 in Table 1 for with . It can be seen that a few terms of the shifted Legendre functions are required to achieve a suitable approximation which demonstrate the applicability of the proposed method for this problem. Also the absolute errors for with are plotted in Figure 2 which show the decreasing of the error with the increasing of the number of Legendre functions. The fuzzy approximate solution is shown in Figure 3 for .

Table 1: The result of the proposed method for Case 1 of Example 52 with and .

Error

Error

0
1.5625
1.5627

4.6875
4.6882

0.1
1.7188
1.7190

4.5313
4.5320

0.2
1.8750
1.8753

4.3750
4.3757

0.3
2.0313
2.0316

4.2188
4.2194

0.4
2.1875
2.1878

4.0625
4.0631

0.5
2.3438
2.3441

3.9063
3.9069

0.6
2.5000
2.5004

3.7500
3.7506

0.7
2.6563
2.6567

3.5938
3.5943

0.8
2.8125
2.8129

3.4375
3.4380

0.9
2.9688
2.9692

3.2813
3.2818

1
3.1250
3.1255

3.1250
3.1255

Figure 2: The absolute error for different of Example 52, Case 1. .

Figure 3: The Fuzzy approximate solution of Example 52, Case 1. .

Remark 55. Figure 2 depicts the absolute errors for of Example 52, Case 1. In the same way, if we consider , then analogously to the demonstration of Figure 2, we can obtain the absolute errors.

In the Table 2, the fuzzy approximate solution for Case 2 of Example 52 is considered. The fractional Caputo derivative is and the number of Legendre functions are . The result are computed for with different -cut. Again we can see that Table 2 demonstrates the validity of the method for this kind of problems. Furthermore, the absolute errors of Example 52, Case 2, are plotted in Figure 4 with and the fuzzy approximate solution is shown in Figure 5 for .

Table 2: The result of the proposed method for Case 2 of Example 52 with and .

Error

Error

0
0.1962
0.1966

0.5887
0.5897

0.1
0.2159
0.2162

0.5691
0.5700

0.2
0.2355
0.2359

0.5495
0.5504

0.3
0.2551
0.2555

0.5298
0.5307

0.4
0.2747
0.2752

0.5102
0.5110

0.5
0.2944
0.2948

0.4906
0.4914

0.6
0.3140
0.3145

0.4710
0.4717

0.7
0.3336
0.3341

0.4514
0.4521

0.8
0.3532
0.3538

0.4317
0.4324

0.9
0.3729
0.3735

0.4121
0.4128

1
0.3925
0.3931

0.3925
0.3931

Figure 4: The Absolute Error for different of Example 52, Case 2. .

Figure 5: The fuzzy approximate solution of Example 52, Case 2. .

Remark 56. Figure 4 shows the absolute errors for of Example 52, Case 2. In the same way, if we consider , then analogously to the demonstration of Figure 4, we can obtain the absolute errors.

Example 57. Let us consider the following FFDE:

Here, suppose that , then using -differentiability and Theorem 33 we have the following:

where . Also, regarding (145), the exact solution of FFDE (144) be determined as

Applying the shifted Legendre method with LOM technique explained in Sections 4 and 5, we have

or it can be written in the matrix form as

in which and for . As was illustrated in Section 5, we make fuzzy algebraic equations by the result of the inner product of fuzzy residual with ,  , so we gain

Also for the initial condition (144), we have

The above equations produce a set of fuzzy linear algebraic equations. As a result, namely, with , the approximate fuzzy solution should be written in the form

Finally, the corresponding fuzzy approximate solution from (151) can be acquired.

For case and , with , we obtain the fuzzy approximate solution in a series expansion as

Remark 58. If , then the fuzzy fractional differential equation (144) with is converted to fuzzy integer-order differential equation. Therefore, the exact solution of problem under -differentiability using Corollary 16 is as follows:

In order to evaluate the advantages and the accuracy of using the presented method for the fuzzy fractional differential equations, Example 57 is considered. The results are illustrated in Table 3 are the approximate solutions are compared with exact solutions and the absolute errors are derived for with at . It is obvious that the fuzzy approximate solutions are in high agreement with the fuzzy exact solutions. Moreover, different numbers of Legendre polynomials are applied to obtain the absolute errors for this problem which can be seen in Figure 6. This graph shows that the method has a good convergence rate. Also, the fuzzy approximate solution is shown for in Figure 7. The absolute error of different order of fractional differentiability can be considered for in Figure 8 and the approximate solutions are shown in Figure 9 that shows that this approach can apply to solve the problem effectively with different fuzzy fractional order of derivatives with suitable errors. It can bee seen that the approaches an integer order, and the error has a tendency to decrease, as expected.

Table 3: The result of the proposed method for Example 57 with and .

Error

Error

0
1.191161
1.191169

1.562753
1.562743

0.1
1.209740
1.209748

1.544173
1.544164

0.2
1.228320
1.228327

1.525594
1.525586

0.3
1.246899
1.246905

1.507014
1.507007

0.4
1.265479
1.265484

1.488435
1.488428

0.5
1.284059
1.284063

1.469855
1.469850

0.6
1.302638
1.302642

1.451275
1.451271

0.7
1.321218
1.321220

1.432696
1.432692

0.8
1.339798
1.339799

1.414116
1.414114

0.9
1.358377
1.358378

1.395536
1.395535

1
1.376957
1.376956

1.376957
1.376956

Figure 6: The absolute error for different of Example 57, .

Figure 7: The fuzzy approximate solution of Example 57, .

Figure 8: The absolute error for different of Example 57, .

Figure 9: The approximate solution for of Example 57, .

Remark 59. Figure 6 illustrates the absolute errors for of Example 57. In the same way, if we consider for , then analogously to the demonstration of Figure 6, we can obtain the absolute errors.

Example 60. For our third example, consider the inhomogeneous linear equation in [78] with fuzzy initial values, and so we have:

in which is a continuous fuzzy-valued function and denotes the fuzzy Caputo fractional derivative of order .

Now, using -differentiability and Theorem 33, we have the following:

Solving (155) leads to determining the solution of FFDE (154) as follows:

By exploiting the method proposed in Section 5, the equations are acquired by

or we can rewrite it in the matrix form as

in which and for . As it was described in Section 5, we create fuzzy algebraic equations multiplied in above system by for and implemented in the inner product using orthogonal property, and so we gain:

Also for the initial condition (154), we have

Equations (159) and (160) produce a set of fuzzy linear algebraic equations. As a result, for and the unknown coefficients can be achieved as and . Also with these assumptions, one has

The approximate and exact solution of FFDEs (154) is illustrated in Table 4 at with the absolute errors. In this table, three shifted Legendre polynomials are considered to derive the approximate solution. It is clear that with the lower number of shifted Legendre functions, using proposed method, one can reach a resealable approximate solution. Also Figure 10 depicts the absolute error with and various values of . This figure exhibits the applicability and validity of the proposed technique for this example. Furthermore, the fuzzy approximate solution for with is shown in Figure 11 at . Additionally, in Figure 12, the fuzzy approximate solution is compared with different values of . It can be seen that as tends to 1, the solution of the fuzzy fractional differential equations tends to that of the fuzzy integer-order differential equations. Finally, Figure 13 displays the absolute error for with different values of which is obvious that with increasing the number of shifted Legendre functions, the absolute error of the problem has been decreased gently.

Table 4: The result of the proposed method for Example 60 with and .

Error

Error

0
−0.371581
−0.371573

0.371581
0.371573

0.1
−0.334423
−0.334416

0.334423
0.334416

0.2
−0.297265
−0.297258

0.297265
0.297258

0.3
−0.260107
−0.260101

0.260107
0.260101

0.4
−0.222948
−0.222944

0.222948
0.222944

0.5
−0.185790
−0.185786

0.185790
0.185786

0.6
−0.148632
−0.148629

0.148632
0.148629

0.7
−0.111474
−0.111472

0.111474
0.111472

0.8
−0.074316
−0.074314

0.074316
0.074314

0.9
−0.037158
−0.037157

0.037158
0.037157

1
0.000000
0.000000

0.000000
0.000000

Figure 10: The absolute error for different of Example 60, .

Figure 11: The fuzzy approximate solution of Example 60, .

Figure 12: The fuzzy approximate solution for different : (1: star, 0.95: square, 0.85: , 0.75: , 0.65: pentagram) of Example 60, .

Figure 13: The absolute error for different of Example 60, .

Remark 61. Figures 10 and 13 illustrate the absolute errors for of Example 60. In the same way, if we consider , then analogously to the demonstration of Figures 10 and 13, we can obtain the absolute errors.

Now, we consider two applicable examples which are new under uncertainty represented by fuzzy-valued functions. Firstly, a brief history is given in the deterministic case of Example 62. For more details, see [1, 79].

There are two kinds of electrical circuits which are related to the fractional calculus. Circuits of the first types are supposed to consist of capacitors and resistors, which are described by conventional (integer-order) models. A circuit expressing fractional order behavior is called fractance. Circuits of the second type may consist of resistors, capacitors, and fractances. The term fractance was suggested initially by Mehaute and Crepy [80]. Now, in order to scrutinize the mentioned problem in a real case, we use the fuzzy initial value and the concept of Caputo’s H-differentiability for fractional derivative of . So we have the following fuzzy fractional oscillation differential equation.

Example 62. Consider an electrical circuit (LR circuit) with an AC source. The current equation of this circuit can be written as follows:

in which is the circuit resistance, and is a coefficient, corresponding to the solenoid and . Suppose that , and , so (162) can be rewritten as

Now, using -differentiability and Theorem 33, we have the following:

The exact solution of (163) under -differentiability for integer order is given by

By applying the technique described in Section 5, we approximate fuzzy solution as

where vector is obtained as (102). With , , and , we have

and subsequently, we generate fuzzy linear equations using (110) and also from substituting (100) in the initial conditions of (164), we can obtain the unknown fuzzy coefficients as

and finally, the fuzzy approximate solution is given by

In Table 5, the numerical results of (162) are depicted at for and . The absolute error confirms that the proposed method approximates the fuzzy solution with a suitable accuracy. Figure 14 shows the absolute error of the proposed method with a different value of with which is clear as tends to 1, and the error is decreasing dramatically, also with the increasing of the number of Legendre functions, the absolute error is dereasing that is, shown in Figure 16. Finally the fuzzy approximate solution of (162) is demonstrated in Figure 15 in the interval .

Table 5: The result of the proposed method for Example 62 with and .

Error

Error

0
0.7208
0.7206

0.7398
0.7397

0.1
0.7223
0.7221

0.7394
0.7393

0.2
0.7238
0.7237

0.7391
0.7389

0.3
0.7253
0.7252

0.7387
0.7385

0.4
0.7269
0.7267

0.7383
0.7381

0.5
0.7284
0.7282

0.7379
0.7378

0.6
0.7299
0.7298

0.7375
0.7374

0.7
0.7314
0.7313

0.7372
0.7370

0.8
0.7330
0.7328

0.7368
0.7366

0.9
0.7345
0.7343

0.7364
0.7362

1
0.7360
0.7359

0.7360
0.7359

Figure 14: The absolute error for different of Example 62, .

Figure 15: The fuzzy approximate solution of Example 62, .

Figure 16: The absolute error for different of Example 62, .

Now we consider the second application of FFDES in the real world. But before we consider the problem of fuzzy model, We glimpse the vision of a non-fuzzy case.

A pharmacodynamic model is usually separated in two parts: a link model and a transduction model. The link model depicts the distribution of drug from an observed compartment (e.g., plasma) into a biophase (the effect compartment). Drug in the biophase induces a pharmacodynamic response (the effect), which is represented by a transduction model. For more details, see [81, 82].

The model describing the transfer of drug to the effect compartment, from the mechanistic interpretation, it can be considered as the pharmacokinetics of the drug. However, practically, and because drug condensation in the effect compartment does not appears, it is not often clear if the link model is actually linked with a site of action, or if it exhibits some pharmacodynamic related delay, or a combination of the two. It is begun by representing drug concentration in the effect compartment by the (Caputo) fractional differential equation as follows:

in which describes the drug input into the central compartment and is the elimination rate constant from the effect compartment. Also the action of drug is described by a nonlinear memory-less function of .

Now, we consider the action of drug delivery by a fuzzy function under the fuzzy Caputo’s H-differentiability which is modelled by fuzzy fractional differential equation.

Example 63. Consider the following fuzzy pharmacodynamic model

in which represents drug concentration in the central compartment following, for example, an oral administration of a dose, in this example and parameters values are , and .

The exact solution of (171) with under generalized differentiability is given by

We apply the method presented in Section 5 and solve this problem for ,  , and . Hence, we have

and thereafter the fuzzy unknown coefficients for the fuzzy approximation of (171) can be acquired easily by replacing the above results in (110) and solving this algebraic fuzzy linear equations system. So we have

and ultimately the fuzzy approximate solution is given by

In Table 6, the approximate fuzzy solutions (171) are compared with the exact solution at for and . The absolute error shows that the proposed method approximates the fuzzy solution with a high accuracy. Moreover, Figure 17 describes the absolute error of the proposed method with a different value of with . It is obvious that the method reaches a good approximation as approaches 1; also again with the increasing of the number of Legendre functions, the absolute error is decreasing that is, shown in Figure 19. The fuzzy approximate solution of the problem is displayed in Figure 18 in the interval .

Table 6: The result of the proposed method for Example 63 with and .

Error

Error

0
−0.243121
−0.243123

0.128454
0.128450

0.1
−0.224543
−0.224544

0.109875
0.109871

0.2
−0.205964
−0.205965

0.091296
0.091293

0.3
−0.187385
−0.187387

0.072717
0.072714

0.4
−0.168806
−0.168808

0.054139
0.054135

0.5
−0.150227
−0.150229

0.035560
0.035557

0.6
−0.131648
−0.131651

0.016981
0.016978

0.7
−0.113070
−0.113072

−0.001597
−0.001600

0.8
−0.094491
−0.094493

−0.020176
−0.020178

0.9
−0.075912
−0.075914

−0.038754
−0.038757

1
−0.057333
−0.057336

−0.057333
−0.057336

Figure 17: The absolute error for different of Example 63, .

Figure 18: The fuzzy approximate solution of Example 63, .

Figure 19: The absolute error for different of Example 63, .

Example 64. Consider the following initial value problem of nonlinear FFDE:

in which is a continuous fuzzy-valued function and denotes the fuzzy Caputo fractional derivative of order . Also is a constant fuzzy number.

For approximating the fuzzy solution of (162), using the method described in Sections 4 and 5, if we approximate the solution by shifted Legendre functions, then it needs to consider the first roots of the shifted Legendre polynomial . Initially using (162), we have

or we can rewrite it in the matrix form as

in which and for . As it was described in Section 5, we produce fuzzy algebraic equations by collecting (162) in the first root of . Also using the initial condition (162), we have

These roots can be put in the (178) and deriving three nonlinear fuzzy equation, then these equations be coupled with (179). Finally with solving the fuzzy nonlinear equations system, the unknown coefficients are obtained.

Now, as an explanation, assume that , then the first three roots of are as follows:

which for and are

Finally, using the above results and (100), the approximate solution is computed in each particular point easily.

The approximate solution for Example 64 with different fractional Caputo order is derived in Table 7. The solution has been estimated using 3 shifted Legendre functions. The results in this table are at . It is clear that the method is applicable and valid for different fuzzy fractional order . Furthermore, by changing the number of shifted Legendre functions, the approximate solution for at is revealed in Figure 20 which shows that with only a few number of shifted Legendre polynomials, desired results are available. Ultimately, the fuzzy approximate solution is described in Figure 21 with and .

Table 7: The approximate solution of the proposed method for Example 64 at with .

0
0.1402
0.1019
0.1040
0.8224
0.8856
0.9381

0.1
0.1434
0.1053
0.1080
0.3061
0.3250
0.3456

0.2
0.1468
0.1091
0.1123
0.2769
0.2917
0.3111

0.3
0.1503
0.1131
0.1170
0.2539
0.2646
0.2826

0.4
0.1539
0.1175
0.1222
0.2356
0.2418
0.2586

0.5
0.1578
0.1224
0.1278
0.2209
0.2223
0.2380

0.6
0.1618
0.1278
0.1340
0.2087
0.2056
0.2202

0.7
0.1662
0.1328
0.1409
0.1986
0.1912
0.2046

0.8
0.1706
0.1388
0.1485
0.1908
0.1785
0.1909

0.9
0.1752
0.1455
0.1571
0.1838
0.1674
0.1788

1
0.1778
0.1576
0.1680
0.1778
0.1576
0.1680

Figure 20: The approximate solution for different of Example 64, .

Figure 21: The fuzzy approximate solution of Example 64, .

Remark 65. It is consequential to note that, for a crisp differential equation of integer or fractional order, the problem

is the same as

in which is is nonlinear operator, and in the case of space , both problems are not equivalent [83].

7. Conclusion

In this paper, a cluster of orthogonal functions named shifted Legendre function is used to solve fuzzy fractional differential equations under Caputo type. The benefit of the shifted Legendre operational matrices method over other existing orthogonal polynomials is its simplicity of execution besides some other advantages. A general formulation of the proposed method is provided in details. Also a complete error analysis is considered in Section 5. In this paper, we tried to answer some questions:

(i)how to derive a shifted Legendre operational matrice method over the fuzzy fractional integration and differentiation(ii)how to solve the fuzzy fractional order differential equations via the shifted Legendre operational matrices of the fuzzy fractional derivative.

Moreover, it should be mentioned that another advantage of this technique is that it decreases these problems to the degree of solving a system of fuzzy algebraic equations thus extremely making the problems easier. Several examples was carried out to depict the effectiveness and the absence of complexity of the proposed method. The achieved solutions have a satisfactory results obtained with only a small number of Legendre polynomials.

For future research, we will consider this method for solving FFDEs with order . Also we will apply it under Riemann-Liouville’s H-differentiability. Apart from this, the other orthogonal functions like Jacobi polynomials will be extended for solving FFDEs.

Acknowledgments

The authors would like to thank Professor Dumitru Baleanu, the Leader Guest Editor, for his consideration about this paper and the anonymous reviewers for their helpful suggestions and comments.

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