The use of Jacobi polynomials in the numerical solution of coupled system of fractional differential equations

References

• E.H. Doha, A.H. Bahraway, and S.S. Ezz-Eliden, A Chebeshev spectral method based on operational matrix for initial and boundary value problems of fractional order, Comput. Math. Appl. 62 (2011), pp. 2364–2373. doi: 10.1016/j.camwa.2011.07.024
   Web of Science ®Google Scholar
• E.H. Doha, A.H. Bahraway, and S.S. Ezz-Eliden, Efficient Chebeshev spectral method for solving multi term fraction order differential equations, Appl. Math. Model. 35 (2011), pp. 5662–5672. doi: 10.1016/j.apm.2011.05.011
   Web of Science ®Google Scholar
• E.H. Doha, A.H. Bhrawy, and S.S. Ezz-Eldien, A new Jacobi operational matrix: An application for solving fractional differential equations, Appl. Math. Model. 36 (2012), pp. 4931–4943. doi: 10.1016/j.apm.2011.12.031
   Web of Science ®Google Scholar
• S. Esmaeili and M. Shamsi, A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), pp. 3646–3654. doi: 10.1016/j.cnsns.2010.12.008
   Web of Science ®Google Scholar
• M. Gasea and T. Sauer, On the history of multivariate polynomial interpolation, J. Comput. Appl. Math. 122 (2000), pp. 23–35. doi: 10.1016/S0377-0427(00)00353-8
   Web of Science ®Google Scholar
• J.S. Hesthaven, S. Gottlieb, and D. Gottlieb, Spectral Methods for Time-Dependent Problems, 1st ed., Cambridge University, Cambridge, 2007.
   Google Scholar
• Z. Kalateh Bojdi, S. Ahmadi-Asl, and A. Aminataei, The general two dimensional shifted Jacobi matrix method for solving the second order linear partial difference-differential equations with variable coefficients, Univ. J. Appl. Math. 1(2) (2013), pp. 142–155.
   Google Scholar
• H. Khalil and R.A. Khan, New operational matrix of integration and coupled system of Fredholm integral equations, Chin. J. Math. 2014 (2014). Article ID 146013.
   Google Scholar
• H. Khalil and R.A. Khan, A new method based on Legendre polynomials for solutions of the fractional two-dimensional heat conduction equation, Comput. Math. Appl. 67 (2014), pp. 1938–1953. doi: 10.1016/j.camwa.2014.03.008
   Web of Science ®Google Scholar
• H. Khalil and R.A. khan, A new method based on Legender polynomials for solution of system of fractional order partial differential equation, Int. J. Comput. Math. doi:10.1080/00207160.2014.880781 (ID: 880781).
   Google Scholar
• A.A. Kilbas, H.M. Srivastava, and J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, 2006.
   Google Scholar
• C. Li, F. Zeng, and F. Liu, Spectral approximation of the fractional integral and derivative, Frac. Calc. Appl. Anal. 15 (2014), pp. 241–258.
   Web of Science ®Google Scholar
• S. Nemati and Y. Ordokhani, Legender expansion methods for the numerical solution of nonlinear 2D Fredholm integral equations of the second kind, J. Appl. Math. Inform. 31 (2013), pp. 609–621. doi: 10.14317/jami.2013.609
   Google Scholar
• K.B. Oldhalm and J. Spainer, The Fractional Calculus, Academic Press, New York, 1974.
   Google Scholar
• P.N. Paraskevopolus, P.D. Saparis, and S.G. Mouroutsos, The Fourier series operational matrix of integration, Int. J. Syst. Sci. 16 (1985), pp. 171–176. doi: 10.1080/00207728508926663
   Web of Science ®Google Scholar
• I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
   Google Scholar
• I. Podlubny, Fractional Differential Equations, Academic Press, San Diedo, CA, 1999.
   Google Scholar
• M. Razzaghi and S. Yousefi, The Legender wavelets operational matrix of integration, Int. J. Syst. Sci. 32 (2001), pp. 495–502. doi: 10.1080/00207720120227
   Web of Science ®Google Scholar
• M. Razzaghi and S. Yousefi, Sine–cosine wavelets operational matrix of integration and its application in the calculus of variation, Int. J. Syst. Sci. 33 (2002), pp. 805–810. doi: 10.1080/00207720210161768
   Web of Science ®Google Scholar
• M. ur Rehman and R.A. Khan, The Legender wavelet method for solving fractional differential equation, Commun. Nonlinear Sci. Numer. Simul. (2011), pp. 10. doi:(1016)/j.cnsns2011.01.014
   Google Scholar
• B. Ross (Ed.), Fractional Calculus and Its Applications, Lecture Notes in Mathematics, Vol. 457, Springer, Berlin-Heidelberg, NY, 1975.
   Google Scholar
• A. Saadatmandi and M. Deghan, A new operational matrix for solving fractional-order differential equation, Comput. Math. Appl. 59 (2010), pp. 1326–1336. doi: 10.1016/j.camwa.2009.07.006
   Web of Science ®Google Scholar
• E. Scalas, M. Raberto, and F. Mainardi, Fractional calculus and continuous-time finance, Phys. A: Stat. Mech. Appl. 284 (2000), pp. 376–384. doi: 10.1016/S0378-4371(00)00255-7
   Web of Science ®Google Scholar
• W.Y. Tian, W.H. Deng, and Y.J. Wu, Polynomial spectral collection method for the space fractional advection–diffusion equation, Numer. Methods Partial Differ. Equ. 30 (2014), pp. 514–535. doi: 10.1002/num.21822
   Web of Science ®Google Scholar