A Jacobi operational matrix for solving a fuzzy linear fractional differential equation
© Ahmadian et al.; licensee Springer 2013
Received: 31 January 2013
Accepted: 3 April 2013
Published: 16 April 2013
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 . A suitable representation of the fuzzy solution via Jacobi polynomials diminishes its numerical results to the solution of a system of algebraic equations. The main advantage of this method is its high robustness and accuracy gained by a small number of Jacobi functions. The efficiency and applicability of the proposed method are proved by several test examples.
PACS Codes:02, 02.30.Jr, 02.60.-x, 45.10.Hj.
Keywordsfuzzy fractional differential equation Caputo-type fuzzy fractional derivative single-term Caputo fractional differential equation Jacobi polynomials operational matrix
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 , 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  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.  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 , fractional Legendre polynomials , generalized hat basis functions  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  and Kaleva . Despite the vast applications of the H-derivative introduced by them, due to an important drawback in this kind of derivative, Bede and Gal  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 . They considered the solution of FFEDs under Riemann-Liouville’s differentiability. Also, Salahshour et al.  studied the existence, uniqueness and approximate solutions of (FFDEs) under Caputo’s H-differentiability. Afterward, Mazandarani, Vahidian Kamyad  applied the fractional Euler method for FFDEs under Caputo-type differentiability and Salahshour et al.  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 under Caputo’s H-differentiability. We introduce a suitable way to approximate fuzzy solution of linear fuzzy fractional differential equations by means of shifted Jacobi functions based on the fuzzy residual of the problem in which the Jacobi operational matrix is introduced to be applied in the derivation of the proposed method. Another motivation is that the approximate solution based on the shifted Jacobi polynomials, (), can be obtained in terms of the Jacobi parameters 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.
u is upper semicontinuous,
u is fuzzy convex, i.e., for all , ,
u is normal, i.e., for which ,
is the support of the u, and its closure is compact.
Then is called the space of fuzzy real numbers and any is called fuzzy real number (see, e.g., ).
ℝ can be embedded in .
, , ,
is a complete metric space.
Definition 1 ()
Remark 1 ()
Definition 2 ()
Let . If there exists such that , then 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 . Also, throughout of paper is assumed the Hukuhara difference and generalized Hukuhara differentiability are existed.
Definition 3 ()
Proposition 1 ()
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 .
Definition 4 ()
- (i)for all sufficiently small, , and the limits (in the metric d)
- (ii)for all sufficiently small, , and the limits (in the metric d)
- (iii)for all sufficiently small, , and the limits (in the metric d)
- (iv)for all sufficiently small, , and the limits (in the metric d)
Remark 2 f is so-called (1)-differentiable on , if f is differentiable in the sense (i) of Definition 4 and also f is (2)-differentiable on , if f is differentiable in the sense (ii) of Definition 4.
The following theorem was proved by Chalco-Cano and Román-Flores  based on Definition 4.
Theorem 1 (see )
- (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 )
Definition 5 ()
We also call an f as above -integrable.
Definition 6 ()
where the coefficient matrix , is a crisp matrix and , . This system is called a fuzzy linear system (FLS).
Definition 7 ()
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 , 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 , which are functions of 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 ()
Definition 9 ()
where is the classical differential operator of order m.
The ceiling function is used to denote the smallest integer greater than or equal to v, and the floor function to denote the largest integer less than or equal to v. Also and .
Definition 10 ()
where λ and μ are constants.
Definition 11 ()
A classical (crisp) set is normally defined as a collection of elements or objects which can be finite, accountable, or overcountable. Each single element can either belong to or not belong to a set A, while in a fuzzy set (subset) elements of the set have a degree of membership in the set.
Remark 4 Throughout the paper, we use the crisp context frequently, regarding to Definition 11.
2.1 Jacobi polynomials
Also, the shifted Jacobi polynomial can be stated by the following concise form.
Lemma 1 ()
Lemma 2 ()
So, the following lemma provides the upper bound of approximate function using shifted Jacobi polynomials. This error bound proves that the approximate function converges to based on shifted Jacobi polynomials.
Lemma 3 ()
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 ()
Note that in , the first rows, are all zeros.
Proof Analogously to the demonstration of Lemma 5 in , we can prove this lemma. □
3 Fuzzy Caputo-type fractional differentiability
The fuzzy fractional differentiability of order , particularly Caputo type, is considered in this section. Some basic definitions and theorems are given and introduced the necessary notation, which will be used in the rest of paper. See, for example, [34, 35, 37].
⧫ , is the set of all fuzzy-valued measurable functions f on where .
⧫ is a space of fuzzy-valued functions, which are continuous on .
⧫ indicates the set of all fuzzy-valued functions, which are continuous up to order n.
⧫ denotes the set of all fuzzy-valued functions, which are absolutely continuous.
Definition 12 ()
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 ()
in which and .
Definition 14 ()
Definition 15 ()
Remark 5 A fuzzy-valued function f is -differentiable, if it is differentiable as in the Definition 15, case (i), and it is -differentiable, if it is differentiable as in the Definition 15, case (ii).
Theorem 3 ()
when f is -differentiable.
Lemma 6 ()
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 . It should be mentioned that this method is the extension of the researches implemented in the crisp sense by Doha et al.  and Kazem .
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 . Firstly, we define the approximate fuzzy function using shifted Jacobi polynomials.
in which , is as the same as the shifted Jacobi polynomials described in Section 2.1, and ∑∗ means addition with respect to ⊕ in .
Since , for all , then we can point out the fuzzy approximation function , according to the lower and upper functions as follows.
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 . □
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 .
that and .
in which and . This completes the proof of theorem. □
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.
where for and for , we have .
Proof It is straightforward from Section 2.1 and the Caputo derivative 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 based on the shifted Jacobi polynomials. The 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.
in which is a continuous fuzzy-valued function, denotes the fuzzy Caputo fractional derivative of order v and .
In the following theorem, we clarify the way to find the fuzzy unknown coefficient of the fuzzy approximate function , using the fuzzy residual function of the problem easily.
from the above equation with Eq. (22), -fuzzy linear algebraic equations are generated. It is obvious that the unknown fuzzy coefficients are obtained with solving this fuzzy system using the method presented for example in .
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.
The absolute error of the proposed method for Example 1 with different values of α , β and
in which is a continuous fuzzy function and indicates the fuzzy Caputo fractional derivative of order v.
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 . Actually, it is related to the oscillatory behavior of fuzzy exact solution. Although the approximate solution using the proposed method has a smooth behavior, it can not respond appropriately to match the exact solution, especially when the 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.
as it can be seen, the approximate solution for lower bound has the negotiable coefficients for x, , terms and constant value. Therefore, in this condition, the approximate solution, with initial value , approaches the analytical fuzzy solution rapidly with the increasing the number of Jacobi functions.
where is a continuous fuzzy set-value function and points out the fuzzy fractional derivative order of Caputo type.
in which .
The absolute error of the proposed method for Example 3 with different values of α , β and
CPU time (in seconds) on MATLAB-7.6 (R2008a) for all of the examples, with different m and
This article adopted the operational Jacobi operational matrix based on the fuzzy Caputo fractional derivative using shifted Jacobi polynomials. The clear advantage of the usage of this method is that the matrix operators have the main role to find the approximate fuzzy solution of FFDEs instead of considering the methods required the complicated fractional derivatives and their calculations, which consume more time and cost in comparison with this method.
Theorems 5 and 6 were proved to demonstrate the error bound of the fuzzy approximate solution and fuzzy fractional Caputo derivative of order . Also, various kinds of problems are solved to illustrate the effectiveness and strength of the method, which can be reached to the suitable accuracy with a lower number of Jacobi functions. In addition, the problems were tested for different values of α and β to show that the method is adaptable in dealing with the various issues.
For future researches, we will try to extend this method for solving multilinear and nonlinear problems as well as solving FFDEs of the order . Furthermore, the generalization of the other orthogonal polynomials for solving FFDEs is the another scope of our attempts. Finally, we will attempt to expand the proposed method under other kinds of fuzzy derivatives like Riemann-Liouville differentiability.
The authors thank the referees for careful reading and helpful suggestions on the improvement of the manuscript. Also, the research of the first and second authors was partially supported by the Ministry of Higher Education (MOHE), Malaysia, Project FRGS 02-01-12-1142FR.
- Caputo M: Linear models of dissipation whose Q is almost frequency. Part II. Geophys. J. R. Astron. Soc. 1967, 13: 529–539. 10.1111/j.1365-246X.1967.tb02303.xView ArticleGoogle Scholar
- de Pablo A, Quirós F, Rodríguez A, Vázquez JL: A fractional porous medium equation. Adv. Math. 2011, 226: 1378–1409. 10.1016/j.aim.2010.07.017MathSciNetView ArticleMATHGoogle Scholar
- Hilfer R: Applications of Fractional Calculus in Physics. Word Scientific, Singapore; 2000.View ArticleMATHGoogle Scholar
- Oldham KB, Spanier J: The Fractional Calculus. Academic Press, New York; 1974.MATHGoogle Scholar
- Vázquez L, Mendes RV: Fractionally coupled solutions of the diffusion equation. Appl. Math. Comput. 2003, 141: 125–130. 10.1016/S0096-3003(02)00326-0MathSciNetView ArticleMATHGoogle Scholar
- Diethelm K Lecture Notes in Mathematics. In The Analysis of Fractional Differential Equations. Springer, Berlin; 2010.View ArticleGoogle Scholar
- Machado JT, Kiryakova V, Mainardi F: Recent history of fractional calculus. Commun. Nonlinear Sci. Numer. Simul. 2011, 3: 1140–1153.MathSciNetView ArticleGoogle Scholar
- Rivero M, Trujillo JJ, Vázquez L, Velasco MP: Fractional dynamics of populations. Appl. Math. Comput. 2011, 218: 1089–1095. 10.1016/j.amc.2011.03.017MathSciNetView ArticleMATHGoogle Scholar
- Rossikhin YA, Shitikova MV: Application of fractional calculus for dynamic problems of solid mechanics: novel trends and recent result. Appl. Mech. Rev. 2010., 63: Article ID 010801Google Scholar
- Mehdinejadiani B, Naseri AA, Jafari H, Ghanbarzadehd A, Baleanu D: A mathematical model for simulation of a water table profile between two parallel subsurface drains using fractional derivatives. Comput. Math. Appl. 2013. doi:10.1016/j.camwa.2013.01.002Google Scholar
- Diethelm K, Ford NJ, Freed AD: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 2002, 29: 3–22. 10.1023/A:1016592219341MathSciNetView ArticleMATHGoogle Scholar
- Daftardar-Gejji V, Jafari H: Adomian decomposition: a tool for solving a system of fractional differential equations. J. Math. Anal. Appl. 2005, 301: 508–518. 10.1016/j.jmaa.2004.07.039MathSciNetView ArticleMATHGoogle Scholar
- Jafari H, Daftardar-Gejji V: Solving a system of nonlinear fractional differential equations using Adomian decomposition. J. Comput. Appl. Math. 2006, 196: 644–651. 10.1016/j.cam.2005.10.017MathSciNetView ArticleMATHGoogle Scholar
- Odibat Z, Momani S: Numerical methods for nonlinear partial differential equations of fractional order. Appl. Math. Model. 2008, 32: 28–39. 10.1016/j.apm.2006.10.025View ArticleMATHGoogle Scholar
- Daftardar-Gejji V, Jafari H: Solving a multi-order fractional differential equation using Adomian decomposition. Appl. Math. Comput. 2007, 189: 541–548. 10.1016/j.amc.2006.11.129MathSciNetView ArticleMATHGoogle Scholar
- Das S: Analytical solution of a fractional diffusion equation by variational iteration method. Comput. Math. Appl. 2009, 57: 483–487. 10.1016/j.camwa.2008.09.045MathSciNetView ArticleMATHGoogle Scholar
- Sweilam NH, Khadar MM, Al-Bar RF: Numerical studies for a multi-order fractional differential equation. Phys. Lett. A 2007, 371: 26–33. 10.1016/j.physleta.2007.06.016MathSciNetView ArticleMATHGoogle Scholar
- Jafari H, Momani S: Solving fractional diffusion and wave equations by modified homotopy perturbation method. Phys. Lett. A 2007, 370: 388–396. 10.1016/j.physleta.2007.05.118MathSciNetView ArticleMATHGoogle Scholar
- Pandey RK, Singh OP, Baranwal VK: An analytic algorithm for the space-time fractional advection-dispersion equation. Comput. Phys. Commun. 2011, 182: 1134–1144. 10.1016/j.cpc.2011.01.015MathSciNetView ArticleMATHGoogle Scholar
- Saadatmandi A, Dehghan M: A new operational matrix for solving fractional-order differential equations. Comput. Math. Appl. 2010, 59: 1326–1336. 10.1016/j.camwa.2009.07.006MathSciNetView ArticleMATHGoogle Scholar
- Esmaeili S, Shamsi M, Luchko Y: Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials. Comput. Math. Appl. 2011, 62: 918–929. 10.1016/j.camwa.2011.04.023MathSciNetView ArticleMATHGoogle Scholar
- Bhrawy AH, Alofi AS: The operational matrix of fractional integration for shifted Chebyshev polynomials. Appl. Math. Lett. 2013, 26: 25–31. 10.1016/j.aml.2012.01.027MathSciNetView ArticleMATHGoogle Scholar
- Doha EH, Bhrawy AH, Ezz-Eldien SS: A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order. Comput. Math. Appl. 2011, 62: 2364–2373. 10.1016/j.camwa.2011.07.024MathSciNetView ArticleMATHGoogle Scholar
- Hwang C, Shih YP: Parameter identification via Laguerre polynomials. Int. J. Syst. Sci. 1982, 13: 209–217. 10.1080/00207728208926341MathSciNetView ArticleMATHGoogle Scholar
- Kazem S, Abbasbandy S, Kumar S: Fractional-order Legendre functions for solving fractional-order differential equations. Appl. Math. Model. 2013, 37: 5498–5510. 10.1016/j.apm.2012.10.026MathSciNetView ArticleGoogle Scholar
- Tripathi MP, Baranwal VK, Pandey RK, Singh OP: A new numerical algorithm to solve fractional differential equations based on operational matrix of generalized hat functions. Commun. Nonlinear Sci. Numer. Simul. 2013, 18: 1327–1340. 10.1016/j.cnsns.2012.10.014MathSciNetView ArticleMATHGoogle Scholar
- Doha EH, Bhrawy AH, Ezz-Eldien SS: A new Jacobi operational matrix: an application for solving fractional differential equations. Appl. Math. Model. 2012, 36: 4931–4943. 10.1016/j.apm.2011.12.031MathSciNetView ArticleMATHGoogle Scholar
- Kazem S: An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations. Appl. Math. Model. 2013, 37: 1126–1136. 10.1016/j.apm.2012.03.033MathSciNetView ArticleGoogle Scholar
- Seikkala S: On the fuzzy initial value problem. Fuzzy Sets Syst. 1987, 24: 319–330. 10.1016/0165-0114(87)90030-3MathSciNetView ArticleMATHGoogle Scholar
- Kaleva O: Fuzzy differential equations. Fuzzy Sets Syst. 1987, 24: 301–317. 10.1016/0165-0114(87)90029-7MathSciNetView ArticleMATHGoogle Scholar
- Bede B, Gal SG: Almost periodic fuzzy-number-valued functions. Fuzzy Sets Syst. 2004, 147: 385–403. 10.1016/j.fss.2003.08.004MathSciNetView ArticleMATHGoogle Scholar
- Bede B, Gal SG: Generalizations of the differentiability of fuzzy number value functions with applications to fuzzy differential equations. Fuzzy Sets Syst. 2005, 151: 581–599. 10.1016/j.fss.2004.08.001MathSciNetView ArticleMATHGoogle Scholar
- Bede B, Rudas IJ, Bencsik AL: First order linear fuzzy differential equations under generalized differentiability. Inf. Sci. 2007, 177: 1648–1662. 10.1016/j.ins.2006.08.021MathSciNetView ArticleMATHGoogle Scholar
- Agarwal RP, Lakshmikantham V, Nieto JJ: On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. 2010, 72: 2859–2862. 10.1016/j.na.2009.11.029MathSciNetView ArticleMATHGoogle Scholar
- Salahshour S, Allahviranloo T, Abbasbandy S, Baleanu D: Existence and uniqueness results for fractional differential equations with uncertainty. Adv. Differ. Equ. 2012., 2012: Article ID 112Google Scholar
- Mazandarani M, Vahidian Kamyad A: Modified fractional Euler method for solving fuzzy fractional initial value problem. Commun. Nonlinear Sci. Numer. Simul. 2013, 18: 12–21. 10.1016/j.cnsns.2012.06.008MathSciNetView ArticleMATHGoogle Scholar
- Salahshour S, Allahviranloo T, Abbasbandy S: Solving fuzzy fractional differential equations by fuzzy Laplace transforms. Commun. Nonlinear Sci. Numer. Simul. 2012, 17: 1372–1381. 10.1016/j.cnsns.2011.07.005MathSciNetView ArticleMATHGoogle Scholar
- Dubois D, Prade H: Fuzzy numbers: an overview. 1. In Analysis of Fuzzy Information. CRC Press, Boca Raton; 1987:3–39.Google Scholar
- Anastassiou GA, Gal SG: On a fuzzy trigonometric approximation theorem of Weierstrass-type. J. Fuzzy Math. 2001, 9: 701–708.MathSciNetMATHGoogle Scholar
- Anastassiou GA: Fuzzy Mathematics: Approximation Theory. Springer, Heidelberg; 2010.View ArticleGoogle Scholar
- Bede B, Stefanini L: Generalized differentiability of fuzzy-valued functions. Fuzzy Sets Syst. 2012. doi:10.1016/j.fss.2012.10.003Google Scholar
- Chalco-Cano Y, Román-Flores H: On new solutions of fuzzy differential equations. Chaos Solitons Fractals 2008, 38: 112–119. 10.1016/j.chaos.2006.10.043MathSciNetView ArticleMATHGoogle Scholar
- Wu HC: The improper fuzzy Riemann integral and its numerical integration. Inf. Sci. 1999, 111: 109–137.View ArticleGoogle Scholar
- Allahviranloo T, Afshar Kermani M: Solution of a fuzzy system of linear equation. Appl. Math. Comput. 2006, 175: 519–531. 10.1016/j.amc.2005.07.048MathSciNetView ArticleMATHGoogle Scholar
- Friedman M, Ming M, Kandel A: Fuzzy linear systems. Fuzzy Sets Syst. 1998, 96: 201–209. 10.1016/S0165-0114(96)00270-9MathSciNetView ArticleMATHGoogle Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.MATHGoogle Scholar
- Momani S, Noor MA: Numerical methods for fourth-order fractional integro-differential equations. Appl. Math. Comput. 2006, 182: 754–760. 10.1016/j.amc.2006.04.041MathSciNetView ArticleMATHGoogle Scholar
- Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.MATHGoogle Scholar
- Zimmermann HJ: Fuzzy Set Theory and Its Applications. Kluwer Academic, Dordrecht; 1991.View ArticleMATHGoogle Scholar
- Szegö G Am. Math. Soc. Colloq. Pub. 23. Orthogonal Polynomials 1985.Google Scholar
- Shen J, Tang T: High Order Numerical Methods and Algorithms. Chinese Science Press, Beijing; 2005.Google Scholar
- Abbasbandy S, Amirfakhrian M: Numerical approximation of fuzzy functions by fuzzy polynomials. Appl. Math. Comput. 2006, 174: 1001–1006. 10.1016/j.amc.2005.05.028MathSciNetView ArticleMATHGoogle Scholar
- Abbasbandy S, Asady B: The nearest trapezoidal fuzzy number to a fuzzy quantity. Appl. Math. Comput. 2004, 156: 381–386. 10.1016/j.amc.2003.07.025MathSciNetView ArticleMATHGoogle Scholar
- Abbasbandy S, Amirfakhrian M: Best approximation of fuzzy functions. Nonlinear Stud. 2007, 14: 88–103.MathSciNetGoogle Scholar
- Lowen R: A fuzzy Lagrange interpolation theorem. Fuzzy Sets Syst. 1990, 34: 33–38. 10.1016/0165-0114(90)90124-OMathSciNetView ArticleMATHGoogle Scholar
- Canuto C, Hussaini MY, Quarteroni A, Zang TA: Spectral Methods in Fluid Dynamics. Springer, New York; 1989.Google Scholar
- Freilich JH, Ortiz EL: Numerical solution of systems of ordinary differential equations with the Tau method: an error analysis. Math. Comput. 1982, 39: 467–479. 10.1090/S0025-5718-1982-0669640-6MathSciNetView ArticleMATHGoogle Scholar
- Diethelm K, Ford NJ: Multi-order fractional differential equations and their numerical solution. Appl. Math. Comput. 2004, 154: 621–640. 10.1016/S0096-3003(03)00739-2MathSciNetView ArticleMATHGoogle Scholar
- Fakhr Kazemi B, Ghoreishi F: Error estimate in fractional differential equations using multiquadratic radial basis functions. Comput. Appl. Math. 2013, 245: 133–147.MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.