1School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box: 19395-5746, Tehran, Islamic Republic of Iran
2Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, P. O. Box: 35195-363, Semnan, Islamic Republic of Iran.
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 as a solution is regarded. This paper develops approximate solutions for boundary value problems of differential equations with non-integer order by using the Shannon wavelet bases. Wavelet bases have different resolution capability for approximating of different functions. Since for Shannon-type wavelets, the scaling function and the mother wavelet are not necessarily absolutely integrable, the partial sums of the wavelet series behave differently and a more stringent condition, such as bounded variation, is needed for convergence of Shannon wavelet series. With nominate Shannon wavelet operational matrices of integration, the solutions are approximated in the form of convergent series with easily computable terms. Also, by applying collocation points the exact solutions of fractional differential equations can be achieved by well-known series solutions. Illustrative examples are presented to demonstrate the applicability and validity of the wavelet base technique. To highlight the convergence, the numerical experiments are solved for different values of bounded series approximation.
 J. J. Benedetto, P. J. S. G. Ferreira, Modern Sampling Theory, Springer Science and Business Media, New York, 2001.  Y. M. Chen, Y. B. Wu, Wavelet method for a class of fractional convection-diffusion equation with variable coefficients, J. Comput. Sci., 1 (2010), 146-149.  K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, 2010.  V. J. Ervin, J.P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Dierential Equations, 22 (2006), 558-576.  V. D. Gejji, H. Jafari, Solving a multi-order fractional differential equation, Appl. Math. Comput., 189 (2007) 541-548.  I. Hashim, O. Abdulaziz, S. Momani, Homotopy analysis method for fractional IVPs, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 674-684.  J. He, Nonlinear oscillation with fractional derivative and its applications, International Conference on Vibrating Engineering, Dalian, China, (1998), 288-291.  J. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci. Technol., 15 (1999), 86-90.  S. Hosseinnia, A. Ranjbar, S. Momani, Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part, Comput. Math. Appl., 56 (2008), 3138-3149.  M. Inc, The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method, J. Math. Anal. Appl., 345 (2008), 476-484.  A. Kilicman, Z.A.A. Al Zhour, Kronecker operational matrices for fractional calculus and some applications, Appl. Math. Comput., 187 (2007), 250-265.  K. Moaddy, S. Momani, I. Hashim, The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics, Comput. Math. Appl., 61 (2011), 1209-1216.  S. Momani, Z. Odibat, Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. Lett. A, 355 (2006), 271-279.  Y. Nawaz, Variational iteration method and homotopy perturbation method for fourth-order fractional integrodifferential equations, Comput. Math. Appl., 61 (2011), 2330-2341.  Z. Odibat, S. Momani, The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics, Comput. Math. Appl., 58 (2009), 2199-2208.  I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.  S. S. Ray, K. S. Chaudhuri, R. K. Bera, Analytical approximate solution of nonlinear dynamic system containing fractional derivative by modified decomposition method, Appl. Math. Comput., 182 (2006), 544-552.  M. U. Rehman, R. A. Khan, A numerical method for solving boundary value problems for fractional differential equations, Applied Mathematical Modelling, 36 (2012), 894-907.  A. Saadatmandi, M. Dehghan, A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl., 59 (2010), 1326-1336.  H. Saeedi, M. Mohseni Moghadam, Numerical solution of nonlinear Volterra integro-differential equations of arbitrary order by CAS wavelets, Appl. Math. Comput., 16 (2011), 1216-1226.  H. Saeedi, M. Mohseni Moghadam, N. Mollahasani, G. N. Chuev, A CAS wavelet method for solving nonlinear Fredholm integro-differential equations of fractional order, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 1154-1163.  N. H. Sweilam, M. M. Khader, R. F. Al-Bar, Numerical studies for a multi-order fractional differential equation, Phys. Lett. A, 371 (2007), 26-33.  Q. Wang, Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method, Appl. Math. Comput., 182 (2006), 1048-1055.  M. Zurigat, S. Momani, A. Alawneh, Analytical approximate solutions of systems of fractional algebraicdifferential equations by homotopy analysis method, Comput. Math. Appl., 59 (2010), 1227-1235.