Quartic and pantic B-spline operational matrix of fractional integration

Document Type: Research Paper


1 Depatrment of Mathematics Graduate University of Advanced Technology

2 Shahid Bahonar University of Kerman

3 Shahid Bahonar University of Kerman, Kerman, Iran


In this work, we proposed an effective method based on cubic and pantic B-spline scaling functions to solve partial differential equations of fractional order. Our method is based on dual functions of B-spline scaling functions. We derived the operational matrix of fractional integration of cubic and pantic B-spline scaling functions and used them to transform the mentioned equations to a system of algebraic equations. Some examples are presented to show the applicability and effectivity of the technique.


[1] M. Dehghan, J. Manafian, A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Methods Partial Differ. Equations, 26(2)(2010), 448-479.
[2] J. Goswami, A. Chan, Fundamentals of wavelets theory, algorithms and applications, John Wiley and Sons, Inc., 1999.
[3] T. Ismaeelpour, A. Askari Hemmat and H. Saeedi, B-spline Operational Matrix of Fractional Integration, Optik- International Journal for Light and Electron Optics, 130(2017), 291-305.
[4] K. AL-Khaled, Numerical solution of time-fractional partial differential equations Using Sumudu decomposition method, Rom. J. Phys., 60(1-2)(2015),
[5] A. A. Kilbas, H. M. Srivastava and J.J. Trujillo, Theory and application of fractional differential equations.
North-Holland Mathematics studies, Vol.204, Elsevier, 2006.
[6] M. Lakestani, M. Dehghan, S. Irandoust-pakchin. The construction of operational matrix of fractional derivatives using B-spline functions, Commun. Nonlinear Sci. Numer. Simul., 17(3)(2012), 1149 - 1162.
[7] Y. Li, Solving a nonlinear fractional differential equations using chebyshev wavelets, Commun. Nonlinear. Sci. Numer. Simul., 15(9)(2010), 2284 - 2292.
[8] K.S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley,
New York, 1993.
[9] D. Sh. Mohammed, Numerical solution of fractional integro-differential equations by least squares method and shifted Chebyshev polynomia, Math. Probl. Eng., 2014.
[10] I. Podlubny, Fractional differential equations. Academic Press, New York, 1999.
[11] H. Saeedi, Applicaion of Haar wavelets in solving nonlinear fractional Fredholm integro-differential equations, J. Mahani Math. Res. Cent., 2 (1) (2013), 15 - 28.
[12] 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.
[13] M. Unser, Approximation power of biorthogonal wavelet expansions, IEEE Trans. Signal Process., 44 (39)
(1996), 519-527.
[14] J.L. Wu, A wavelet operational method for solving fractional partial differential equations numerically, Appl. Math. Comput., 214 (1) (2009), 31 - 40.