Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations

In this paper, a new numerical method for solving fractional differential equations is presented. The fractional derivative is described in the Caputo sense. The method is based upon Bernoulli wavelet approximations. The Bernoulli wavelet is first presented. An operational matrix of fractional order integration is derived and is utilized to reduce the initial and boundary value problems to system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

[1]  A. E. M. El-Mesiry,et al.  Numerical solution for multi-term fractional (arbitrary) orders differential equations , 2004 .

[2]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[3]  M. Meerschaert,et al.  Finite difference approximations for two-sided space-fractional partial differential equations , 2006 .

[4]  T. S. Chow,et al.  Fractional dynamics of interfaces between soft-nanoparticles and rough substrates , 2005 .

[5]  L. Gaul,et al.  Damping description involving fractional operators , 1991 .

[6]  M. Razzaghi,et al.  SEMIORTHOGONAL SPLINE WAVELETS APPROXIMATION FOR FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS , 2006 .

[7]  Dan Li,et al.  Solving two-point boundary value problems using combined homotopy perturbation method and Green's function method , 2009, Appl. Math. Comput..

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

[9]  Khosrow Maleknejad,et al.  Computational method based on Bernstein operational matrices for nonlinear Volterra–Fredholm–Hammerstein integral equations , 2012 .

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

[11]  Zaid M. Odibat,et al.  Generalized Taylor's formula , 2007, Appl. Math. Comput..

[12]  Rahmat Ali Khan,et al.  The Legendre wavelet method for solving fractional differential equations , 2011 .

[13]  Francesco Dell'Accio,et al.  A new approach to Bernoulli polynomials , 2006 .

[14]  Richard T. Baillie,et al.  Long memory processes and fractional integration in econometrics , 1996 .

[15]  Peter J. Torvik,et al.  Fractional calculus in the transient analysis of viscoelastically damped structures , 1983 .

[16]  E. Kreyszig.  Introductory Functional Analysis With Applications , 1978 .

[17]  Yanxin Wang,et al.  The second kind Chebyshev wavelet method for solving fractional differential equations , 2012, Appl. Math. Comput..

[18]  M. Razzaghi,et al.  Hybrid functions approach for optimal control of systems described by integro-differential equations , 2013 .

[19]  Mohsen Razzaghi,et al.  Linear quadratic optimal control problems via shifted Legendre state parametrization , 1994 .

[20]  Mohsen Razzaghi,et al.  Solution of Hallen's integral equation using multiwavelets , 2005, Comput. Phys. Commun..

[21]  I. Ozkol,et al.  Solution of fractional integro-differential equations by using fractional differential transform method , 2009 .

[22]  Kamel Al-khaled,et al.  Numerical solutions for systems of fractional differential equations by the decomposition method , 2005, Appl. Math. Comput..

[23]  Mohsen Razzaghi,et al.  HYBRID FUNCTIONS APPROACH FOR LINEARLY CONSTRAINED QUADRATIC OPTIMAL CONTROL PROBLEMS , 2003 .

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

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

[26]  M. Rehman,et al.  A numerical method for solving boundary value problems for fractional differential equations , 2012 .

[27]  Mohsen Razzaghi,et al.  Instabilities in the solution of a heat conduction problem using taylor series and alternative approaches , 1989 .

[28]  Wei-Sun Jiang,et al.  The Haar wavelets operational matrix of integration , 1996, Int. J. Syst. Sci..

[29]  Hossein Jafari,et al.  Application of Legendre wavelets for solving fractional differential equations , 2011, Comput. Math. Appl..

[30]  G. Arfken.  Mathematical Methods for Physicists , 1967 .

[31]  C. Chui.  Wavelets: A Mathematical Tool for Signal Analysis , 1997 .

[32]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[33]  Luis E. Suarez,et al.  An Eigenvector Expansion Method for the Solution of Motion Containing Fractional Derivatives , 1997 .

[34]  M. Razzaghi,et al.  A hybrid functions approach for the Duffing equation , 2013 .

[35]  M. Shitikova,et al.  Applications of Fractional Calculus to Dynamic Problems of Linear and Nonlinear Hereditary Mechanics of Solids , 1997 .

[36]  Yuanlu Li,et al.  Solving a nonlinear fractional differential equation using Chebyshev wavelets , 2010 .

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