A Jacobi operational matrix for solving a fuzzy linear fractional differential equation

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 0<v<1. 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.

[1]  Tofigh Allahviranloo,et al.  SOLVING FUZZY FRACTIONAL DIFFERENTIAL EQUATIONS BY FUZZY LAPLACE TRANSFORMS , 2012 .

[2]  G. Szegő.  Zeros of orthogonal polynomials , 1939 .

[3]  Tofigh Allahviranloo,et al.  Existence and uniqueness results for fractional differential equations with uncertainty , 2012, Advances in Difference Equations.

[4]  S. Seikkala.  On the fuzzy initial value problem , 1987 .

[5]  Saeid Abbasbandy,et al.  Best Approximation of Fuzzy Functions , 2007 .

[6]  M. Caputo.  Linear models of dissipation whose Q is almost frequency independent , 1966 .

[7]  James C. Bezdek,et al.  Analysis of fuzzy information , 1987 .

[8]  Shahrokh Esmaeili,et al.  Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials , 2011, Comput. Math. Appl..

[9]  I. Podlubny.  Fractional differential equations , 1998 .

[10]  Bobby Schmidt,et al.  Fuzzy math , 2001 .

[11]  Hossein Jafari,et al.  Solving a system of nonlinear fractional differential equations using Adomian decomposition , 2006 .

[12]  Farideh Ghoreishi,et al.  Error estimate in fractional differential equations using multiquadratic radial basis functions , 2013, J. Comput. Appl. Math..

[13]  George A. Anastassiou.  A FUZZY TRIGONOMETRIC APPROXIMATION THEOREM OF WEIERSTRASS-TYPE , 2010 .

[14]  Rui Vilela Mendes,et al.  Fractionally coupled solutions of the diffusion equation , 2003, Appl. Math. Comput..

[15]  Robert Lowen,et al.  A fuzzy lagrange interpolation theorem , 1990 .

[16]  Saeed Kazem,et al.  An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations , 2013 .

[17]  S. Das,et al.  Analytical solution of a fractional diffusion equation by variational iteration method , 2009, Comput. Math. Appl..

[18]  Hossein Jafari,et al.  Adomian decomposition: a tool for solving a system of fractional differential equations , 2005 .

[19]  Saeid Abbasbandy,et al.  The nearest trapezoidal fuzzy number to a fuzzy quantity , 2004, Appl. Math. Comput..

[20]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[21]  E. Ortiz,et al.  Numerical solution of systems of ordinary differential equations with the Tau method: an error analysis , 1982 .

[22]  Barnabás Bede,et al.  Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations , 2005, Fuzzy Sets Syst..

[23]  K. Diethelm.  The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type , 2010 .

[24]  Osmo Kaleva.  Fuzzy differential equations , 1987 .

[25]  Imre J. Rudas,et al.  First order linear fuzzy differential equations under generalized differentiability , 2007, Inf. Sci..

[26]  Eid H. Doha,et al.  A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order , 2011, Comput. Math. Appl..

[27]  D. Dubois,et al.  FUZZY NUMBERS: AN OVERVIEW , 1993 .

[28]  Tofigh Allahviranloo,et al.  Solution of a fuzzy system of linear equation , 2006, Appl. Math. Comput..

[29]  O. P. Singh,et al.  A new numerical algorithm to solve fractional differential equations based on operational matrix of generalized hat functions , 2013, Commun. Nonlinear Sci. Numer. Simul..

[30]  E. H. Doha,et al.  A NEW JACOBI OPERATIONAL MATRIX: AN APPLICATION FOR SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS , 2012 .

[31]  Chyi Hwang,et al.  Parameter identification via Laguerre polynomials , 1982 .

[32]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[33]  Hsien-Chung Wu,et al.  The Improper Fuzzy Riemann Integral and its Numerical Integration , 1998, Inf. Sci..

[34]  Saeed Kazem,et al.  Fractional-order Legendre functions for solving fractional-order differential equations , 2013 .

[35]  R. Hilfer.  Applications Of Fractional Calculus In Physics , 2000 .

[36]  Ravi P. Agarwal,et al.  On the concept of solution for fractional differential equations with uncertainty , 2010 .

[37]  Hossein Jafari,et al.  SOLVING FRACTIONAL DIFFUSION AND WAVE EQUATIONS BY MODIFIED HOMOTOPY PERTURBATION METHOD , 2007 .

[38]  Mehdi Dehghan,et al.  A new operational matrix for solving fractional-order differential equations , 2010, Comput. Math. Appl..

[39]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

[40]  Saeid Abbasbandy,et al.  Numerical approximation of fuzzy functions by fuzzy polynomials , 2006, Appl. Math. Comput..

[41]  Ali H. Bhrawy,et al.  The operational matrix of fractional integration for shifted Chebyshev polynomials , 2013, Appl. Math. Lett..

[42]  F. Quirós,et al.  A P ] 1 4 Ja n 20 10 A fractional porous medium equation by , 2010 .

[43]  Muhammad Aslam Noor,et al.  Numerical methods for fourth-order fractional integro-differential equations , 2006, Appl. Math. Comput..

[44]  Hossein Jafari,et al.  Solving a multi-order fractional differential equation using adomian decomposition , 2007, Appl. Math. Comput..

[45]  Abraham Kandel,et al.  Fuzzy linear systems , 1998, Fuzzy Sets Syst..

[46]  F. Mainardi,et al.  Recent history of fractional calculus , 2011 .

[47]  Barnabás Bede,et al.  Generalized differentiability of fuzzy-valued functions , 2013, Fuzzy Sets Syst..

[48]  N. Sweilam,et al.  Numerical studies for a multi-order fractional differential equation , 2007 .

[49]  George A. Anastassiou,et al.  Fuzzy Mathematics: Approximation Theory , 2010, Studies in Fuzziness and Soft Computing.

[50]  Y Chalco Cano,et al.  ON NEW SOLUTIONS OF FUZZY DIFFERENTIAL EQUATIONS , 2008 .

[51]  Ali Vahidian Kamyad,et al.  Modified fractional Euler method for solving Fuzzy Fractional Initial Value Problem , 2013, Commun. Nonlinear Sci. Numer. Simul..

[52]  Hossein Jafari,et al.  A mathematical model for simulation of a water table profile between two parallel subsurface drains using fractional derivatives , 2013, Comput. Math. Appl..

[53]  Kai Diethelm,et al.  Multi-order fractional differential equations and their numerical solution , 2004, Appl. Math. Comput..

[54]  Sorin G. Gal,et al.  Almost periodic fuzzy-number-valued functions , 2004, Fuzzy Sets Syst..

[55]  Hans-Jürgen Zimmermann,et al.  Fuzzy Set Theory - and Its Applications , 1985 .

[56]  N. Ford,et al.  A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations , 2013 .

[57]  O. P. Singh,et al.  An analytic algorithm for the space-time fractional advection-dispersion equation , 2011, Comput. Phys. Commun..

[58]  Xiaoning Zheng.  High-order numerical methods and algorithms: Eigen-based spectral element approach , 2012 .

[59]  C. Canuto.  Spectral methods in fluid dynamics , 1991 .

[60]  M. Shitikova,et al.  Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results , 2010 .

[61]  Luis Vázquez Martínez,et al.  Fractional dynamics of populations , 2011, Appl. Math. Comput..

[62]  S. Momani,et al.  Numerical methods for nonlinear partial differential equations of fractional order , 2008 .