Application of Shannon wavelet for solving boundary value problems of fractional differential equations I

Document Type : Research Paper

Authors

1 School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box: 19395-5746, Tehran, Islamic Republic of Iran

2 Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, P. O. Box: 35195-363, Semnan, Islamic Republic of Iran.

Abstract

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.

Keywords

References

[1] J. J. Benedetto, P. J. S. G. Ferreira, Modern Sampling Theory, Springer Science and Business Media, New York,
2001.
[2] 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.
[3] K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, 2010.
[4] V. J. Ervin, J.P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Dierential Equations, 22 (2006), 558-576.
[5] V. D. Gejji, H. Jafari, Solving a multi-order fractional differential equation, Appl. Math. Comput., 189 (2007)
541-548.
[6] I. Hashim, O. Abdulaziz, S. Momani, Homotopy analysis method for fractional IVPs, Commun. Nonlinear Sci.
Numer. Simul., 14 (2009), 674-684.
[7] J. He, Nonlinear oscillation with fractional derivative and its applications, International Conference on Vibrating Engineering, Dalian, China, (1998), 288-291.
[8] J. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci.
Technol., 15 (1999), 86-90.
[9] 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.
[10] 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.
[11] A. Kilicman, Z.A.A. Al Zhour, Kronecker operational matrices for fractional calculus and some applications,
Appl. Math. Comput., 187 (2007), 250-265.
[12] 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.
[13] S. Momani, Z. Odibat, Analytical approach to linear fractional partial differential equations arising in fluid
mechanics, Phys. Lett. A, 355 (2006), 271-279.
[14] Y. Nawaz, Variational iteration method and homotopy perturbation method for fourth-order fractional integrodifferential equations, Comput. Math. Appl., 61 (2011), 2330-2341.
[15] 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.
[16] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[17] 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.
[18] 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.
[19] A. Saadatmandi, M. Dehghan, A new operational matrix for solving fractional-order differential equations,
Comput. Math. Appl., 59 (2010), 1326-1336.
[20] 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.
[21] 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.
[22] 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.
[23] Q. Wang, Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method, Appl.
Math. Comput., 182 (2006), 1048-1055.
[24] 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.

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