A Modified Generalized Laguerre Spectral Method for Fractional Differential Equations on the Half Line

This paper deals with modified generalized Laguerre spectral tau and collocation methods for solving linear and nonlinear multiterm fractional differential equations (FDEs) on the half line. A new formula expressing the Caputo fractional derivatives of modified generalized Laguerre polynomials of any degree and for any fractional order in terms of the modified generalized Laguerre polynomials themselves is derived. An efficient direct solver technique is proposed for solving the linear multiterm FDEs with constant coefficients on the half line using a modified generalized Laguerre tau method. The spatial approximation with its Caputo fractional derivatives is based on modified generalized Laguerre polynomials with , , and , and is the polynomial degree. We implement and develop the modified generalized Laguerre collocation method based on the modified generalized Laguerre-Gauss points which is used as collocation nodes for solving nonlinear multiterm FDEs on the half line.

[1]  Ahmed M. A. El-Sayed,et al.  Bell polynomials of arbitrary (fractional) orders+ , 1999, Appl. Math. Comput..

[2]  Fanhai Zeng,et al.  Spectral approximations to the fractional integral and derivative , 2012 .

[3]  B. Guo,et al.  A new generalized Laguerre spectral approximation and its applications , 2005 .

[4]  Chao Zhang,et al.  The spectral method for high order problems with proper simulations of asymptotic behaviors at infinity , 2013, J. Comput. Appl. Math..

[5]  Aydin Kurnaz,et al.  The solution of the Bagley-Torvik equation with the generalized Taylor collocation method , 2010, J. Frankl. Inst..

[6]  Yonghong Wu,et al.  The iterative solutions of nonlinear fractional differential equations , 2013, Appl. Math. Comput..

[7]  T. A. Zang,et al.  Spectral Methods: Fundamentals in Single Domains , 2010 .

[8]  Şuayip Yüzbaşı,et al.  Numerical solution of the Bagley–Torvik equation by the Bessel collocation method , 2013 .

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

[10]  Ali Ahmadian,et al.  An Operational Matrix Based on Legendre Polynomials for Solving Fuzzy Fractional-Order Differential Equations , 2013 .

[11]  Mehmet Sezer,et al.  Laguerre polynomial approach for solving linear delay difference equations , 2011, Appl. Math. Comput..

[12]  Robert A. Van Gorder,et al.  Computation of certain infinite series of the form Sigma f(n)nk for arbitrary real-valued k , 2009, Appl. Math. Comput..

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

[14]  E. H. Doha,et al.  EFFICIENT CHEBYSHEV SPECTRAL METHODS FOR SOLVING MULTI-TERM FRACTIONAL ORDERS DIFFERENTIAL EQUATIONS , 2011 .

[15]  G. Ben-yu,et al.  Generalized Laguerre Interpolation and Pseudospectral Method for Unbounded Domains , 2006 .

[16]  K. Diethelm,et al.  Fractional Calculus: Models and Numerical Methods , 2012 .

[17]  Ali H. Bhrawy,et al.  A quadrature tau method for fractional differential equations with variable coefficients , 2011, Appl. Math. Lett..

[18]  G. H. Erjaee,et al.  Chebyshev operational matrix method for solving multi-order fractional ordinary differential equations , 2013 .

[19]  Ali Ahmadian,et al.  An Eigenvalue-Eigenvector Method for Solving a System of Fractional Differential Equations with Uncertainty , 2013 .

[20]  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..

[21]  Saad Zagloul Rida,et al.  Noncommutative Bell polynomials , 1996, Int. J. Algebra Comput..

[22]  B. Guo,et al.  A new generalized Laguerre spectral approximation and its applications , 2005 .

[23]  Ali H. Bhrawy,et al.  A new modified generalized Laguerre operational matrix of fractional integration for solving fractional differential equations on the half line , 2012 .

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

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

[26]  Ali H. Bhrawy,et al.  A Jacobi–Gauss collocation method for solving nonlinear Lane–Emden type equations , 2012 .

[27]  Eid H. Doha,et al.  An efficient direct solver for multidimensional elliptic Robin boundary value problems using a Legendre spectral-Galerkin method , 2012, Comput. Math. Appl..

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

[29]  Dumitru Baleanu,et al.  On shifted Jacobi spectral approximations for solving fractional differential equations , 2013, Appl. Math. Comput..

[30]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[31]  Dumitru Baleanu,et al.  A Jacobi operational matrix for solving a fuzzy linear fractional differential equation , 2013 .

[32]  Santanu Saha Ray,et al.  Analytical solution of the Bagley Torvik equation by Adomian decomposition method , 2005, Appl. Math. Comput..

[33]  Yao-Lin Jiang,et al.  Waveform relaxation methods for fractional differential equations with the Caputo derivatives , 2013, J. Comput. Appl. Math..

[34]  A. Bhrawy,et al.  A new formula for fractional integrals of Chebyshev polynomials: Application for solving multi-term fractional differential equations , 2013 .

[35]  N. Ford,et al.  Numerical Solution of the Bagley-Torvik Equation , 2002, BIT Numerical Mathematics.

[36]  Junaid Ali Khan,et al.  Solution of Fractional Order System of Bagley-Torvik Equation Using Evolutionary Computational Intelligence , 2011 .

[37]  Fengshun Lu,et al.  Fractional Variational Iteration Method versus Adomian's Decomposition Method in Some Fractional Partial Differential Equations , 2013, J. Appl. Math..

[38]  B. Guo,et al.  A Collocation Method for Initial Value Problems of Second-Order ODEs by Using Laguerre Functions , 2011 .

[39]  Santanu Saha Ray,et al.  On Haar wavelet operational matrix of general order and its application for the numerical solution of fractional Bagley Torvik equation , 2012, Appl. Math. Comput..

[40]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies) , 2006 .

[41]  Ben-yu Guo,et al.  Generalized Laguerre Interpolation and Pseudospectral Method for Unbounded Domains , 2006, SIAM J. Numer. Anal..

[42]  Manuel Duarte Ortigueira,et al.  Introduction to fractional linear systems. Part 1. Continuous-time case , 2000 .