Fractional and Time-Scales Differential EquationsView this Special Issue
Research Article | Open Access
Ali Ahmadian, Mohamed Suleiman, Soheil Salahshour, "An Operational Matrix Based on Legendre Polynomials for Solving Fuzzy Fractional-Order Differential Equations", Abstract and Applied Analysis, vol. 2013, Article ID 505903, 29 pages, 2013. https://doi.org/10.1155/2013/505903
An Operational Matrix Based on Legendre Polynomials for Solving Fuzzy Fractional-Order Differential Equations
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.
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 , Baleanu et al. , Diethelm , Baleanu et al. , and Sabatier et al. . 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 , the fluid-dynamic models , continuum and statistical mechanics , and solid mechanics . 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 . 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 , Legendre wavelets , and Galerkin-type technique .
Recently, Agarwal et al.  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.  considered fuzzy laplace transforms for solving FFDEs under Riemann-Liouville H-differentiability. Also Mazandarani and Kamyad  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 , He’s variational iteration method , homotopy perturbation method , 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  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  and Doha et al. , 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.
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 , 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 ). 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 ). Let be fuzzy continuous. Then from property (iv) of Hausdorff distance, we can define
Definition 4 (see ). 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 ). 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 ). 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 .
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 .
The following proposition prove that the g-difference is well-defined.
Proposition 9 (see ). For any fuzzy numbers the g-difference exists and it is a fuzzy number.
Proof. See .
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 .
Definition 11 (see ). 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 ). 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 ). 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  for fuzzy initial value problem (FIVP) under -differentiability which Nieto et al.  generalized this results for FIVP under -differentiability (let denote the usual Euclidean norm).
Theorem 15 (see , 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 ODEsare equivalent.
Theorem 17 (see ). 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 ). . 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 ). 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 ). 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.
Definition 21 (see ). The Riemann-Liouville fractional integral operator of order , , of a function is defined as
For , and ,
Definition 22 (see ). The Riemann-Liouville fractional derivatives of order of a function are expressed by
Definition 23 (see ). 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 ). 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 ). 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
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
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 .
Theorem 30. Let one assume that , and then one has the following: when is -differentiable and when is -differentiable.
Proof. See .
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 .
Theorem 32. Let and suppose that for where , and . Then we have where .
Proof. See .
Now, characterization theorem (Theorem 15), which was introduced by Bede in  and established by Pederson and Sambandham in  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  and Theorem 3.1 in .
4. Proposed Method for Solving FFDEs
Saadatmandi and Dehghan  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 , 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.
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 ). 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 .
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 .
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