Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems

We present a direct solution technique for approximating linear multiterm fractional differential equations (FDEs) on semi-infinite interval, using generalized Laguerre polynomials. We derive the operational matrix of Caputo fractional derivative of the generalized Laguerre polynomials which is applied together with generalized Laguerre tau approximation for implementing a spectral solution of linear multiterm FDEs on semi-infinite interval subject to initial conditions. The generalized Laguerre pseudo-spectral approximation based on the generalized Laguerre operational matrix is investigated to reduce the nonlinear multiterm FDEs and its initial conditions to nonlinear algebraic system, thus greatly simplifying the problem. Through several numerical examples, we confirm the accuracy and performance of the proposed spectral algorithms. Indeed, the methods yield accurate results, and the exact solutions are achieved for some tested problems.

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

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

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

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

[5]  E. A. Rawashdeh,et al.  Numerical solution of fractional integro-differential equations by collocation method , 2006, Appl. Math. Comput..

[6]  A. Kılıçman,et al.  Robustness of Operational Matrices of Differentiation for Solving State-Space Analysis and Optimal Control Problems , 2013 .

[7]  M. El-Kady,et al.  FRACTIONAL DIFFERENTIATION MATRICES FOR SOLVING FRACTIONAL ORDERS DIFFERENTIAL EQUATIONS , 2013 .

[8]  S. Momani,et al.  Solving Linear and Nonlinear Fractional Differential Equations Using Spline Functions , 2012 .

[9]  Sohrab Effati,et al.  A New Piecewise-Spectral Homotopy Analysis Method for Solving Chaotic Systems of Initial Value Problems , 2013 .

[10]  Dumitru Baleanu,et al.  Variational iteration method for fractional calculus - a universal approach by Laplace transform , 2013 .

[11]  Dumitru Baleanu,et al.  Positive Solutions of an Initial Value Problem for Nonlinear Fractional Differential Equations , 2012 .

[12]  Saeed Kazem,et al.  A modification of the homotopy analysis method based on Chebyshev operational matrices , 2013, Math. Comput. Model..

[13]  J. Machado,et al.  On a Generalized Laguerre Operational Matrix of Fractional Integration , 2013 .

[14]  A. A. Hemeda.  New Iterative Method: An Application for Solving Fractional Physical Differential Equations , 2013 .

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

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

[17]  Saeid Abbasbandy,et al.  An Adaptive Pseudospectral Method for Fractional Order Boundary Value Problems , 2012 .

[18]  S. Das,et al.  Functional Fractional Calculus for System Identification and Controls , 2007 .

[19]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

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

[21]  Mustafa Gülsu,et al.  Numerical approach for solving fractional Fredholm integro-differential equation , 2013, Int. J. Comput. Math..

[22]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[23]  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..

[24]  D. Baleanu,et al.  A modified variational iteration method for solving fractional Riccati differential equation by Adomian polynomials , 2012 .

[25]  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 .

[26]  K. Parand,et al.  Application of the Exact Operational Matrices Based on the Bernstein Polynomials , 2013 .

[27]  Francisco Marcellán,et al.  Journal of Mathematical Analysis and Applications Monotonicity of Zeros of Laguerre–sobolev-type Orthogonal Polynomials , 2022 .

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