An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations

Abstract This paper aims to construct a general formulation for the Jacobi operational matrix of fractional integral operator. Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, a reliable and efficient technique for the solution of them is too important. For the concept of fractional derivative we will adopt Caputo’s definition by using Riemann–Liouville fractional integral operator. Our main aim is to generalize the Jacobi integral operational matrix to the fractional calculus. These matrices together with the Tau method are then utilized to reduce the solution of this problem to the solution of a system of algebraic equations. The method is applied to solve linear and nonlinear fractional differential equations. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.

[1]  Elena Grigorenko,et al.  Erratum: Chaotic Dynamics of the Fractional Lorenz System [Phys. Rev. Lett.91, 034101 (2003)] , 2006 .

[2]  Hongguang Sun,et al.  Fractional diffusion equations by the Kansa method , 2010, Comput. Math. Appl..

[3]  Ahmed Alawneh,et al.  Analytical approximate solutions of systems of fractional algebraic-differential equations by homotopy analysis method , 2010, Comput. Math. Appl..

[4]  S. Momani,et al.  Analytical approach to linear fractional partial differential equations arising in fluid mechanics , 2006 .

[5]  Qi Wang,et al.  Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method , 2006, Appl. Math. Comput..

[6]  Igor Podlubny,et al.  Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation , 2001, math/0110241.

[7]  V. Ervin,et al.  Variational formulation for the stationary fractional advection dispersion equation , 2006 .

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

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

[10]  K. S. Chaudhuri,et al.  Analytical approximate solution of nonlinear dynamic system containing fractional derivative by modified decomposition method , 2006, Appl. Math. Comput..

[11]  Francesco Mainardi,et al.  Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics , 2012, 1201.0863.

[12]  Elena Grigorenko,et al.  Chaotic dynamics of the fractional Lorenz system. , 2003, Physical review letters.

[13]  Yasir Nawaz,et al.  Variational iteration method and homotopy perturbation method for fourth-order fractional integro-differential equations , 2011, Comput. Math. Appl..

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

[15]  Ben-yu Guo,et al.  Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces , 2004, J. Approx. Theory.

[16]  S. Hassan HosseinNia,et al.  Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part , 2008, Comput. Math. Appl..

[17]  I. Hashim,et al.  HOMOTOPY ANALYSIS METHOD FOR FRACTIONAL IVPS , 2009 .

[18]  Shaher Momani,et al.  The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics , 2011, Comput. Math. Appl..

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

[20]  S. Momani,et al.  Application of Variational Iteration Method to Nonlinear Differential Equations of Fractional Order , 2006 .

[21]  S. Momani,et al.  AN ALGORITHM FOR THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER , 2008 .

[22]  Shaher Momani,et al.  Homotopy perturbation method for nonlinear partial differential equations of fractional order , 2007 .

[23]  Lingpeng Yang,et al.  Document reranking by term distribution and maximal marginal relevance for chinese information retrieval , 2007, Information Processing & Management.

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

[25]  Santos B. Yuste,et al.  Weighted average finite difference methods for fractional diffusion equations , 2004, J. Comput. Phys..

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

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

[28]  Hossein Jafari,et al.  Homotopy analysis method for solving multi-term linear and nonlinear diffusion-wave equations of fractional order , 2010, Comput. Math. Appl..

[29]  Ji-Huan He Approximate analytical solution for seepage flow with fractional derivatives in porous media , 1998 .

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

[31]  K. Diethelm AN ALGORITHM FOR THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER , 1997 .

[32]  S. Momani,et al.  Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order , 2008 .

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

[34]  Pankaj Kumar,et al.  An approximate method for numerical solution of fractional differential equations , 2006, Signal Process..

[35]  Shaher Momani,et al.  The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics , 2009, Comput. Math. Appl..

[36]  Shaher Momani,et al.  Decomposition method for solving fractional Riccati differential equations , 2006, Appl. Math. Comput..

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

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

[39]  Fawang Liu,et al.  Numerical solution of the space fractional Fokker-Planck equation , 2004 .